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| author | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:08:04 -0700 |
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| committer | Roger Frank <rfrank@pglaf.org> | 2025-10-14 20:08:04 -0700 |
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You may copy it, give it away or +re-use it under the terms of the Project Gutenberg License included +with this eBook or online at www.gutenberg.org + + +Title: Encyclopaedia Britannica, 11th Edition, Volume 11, Slice 6 + "Geodesy" to "Geometry" + +Author: Various + +Release Date: September 17, 2011 [EBook #37461] + +Language: English + +Character set encoding: ISO-8859-1 + +*** START OF THIS PROJECT GUTENBERG EBOOK ENCYCLOPAEDIA BRITANNICA *** + + + + +Produced by Marius Masi, Don Kretz and the Online +Distributed Proofreading Team at http://www.pgdp.net + + + + + + +</pre> + + + +<table border="0" cellpadding="10" style="background-color: #dcdcdc; color: #696969; " summary="Transcriber's note"> +<tr> +<td style="width:25%; vertical-align:top"> +Transcriber’s note: +</td> +<td class="norm"> +A few typographical errors have been corrected. They +appear in the text <span class="correction" title="explanation will pop up">like this</span>, and the +explanation will appear when the mouse pointer is moved over the marked +passage. Sections in Greek will yield a transliteration +when the pointer is moved over them, and words using diacritic characters in the +Latin Extended Additional block, which may not display in some fonts or browsers, will +display an unaccented version. <br /><br /> +<a name="artlinks">Links to other EB articles:</a> Links to articles residing in other EB volumes will +be made available when the respective volumes are introduced online. +</td> +</tr> +</table> +<div style="padding-top: 3em; "> </div> + +<h2>THE ENCYCLOPÆDIA BRITANNICA</h2> + +<h2>A DICTIONARY OF ARTS, SCIENCES, LITERATURE AND GENERAL INFORMATION</h2> + +<h3>ELEVENTH EDITION</h3> +<div style="padding-top: 3em; "> </div> + +<hr class="full" /> +<h3>VOLUME XI SLICE VI<br /><br /> +Geodesy to Geometry</h3> +<hr class="full" /> +<div style="padding-top: 3em; "> </div> + +<p class="center1" style="font-size: 150%; font-family: 'verdana';">Articles in This Slice</p> +<table class="reg" style="width: 90%; font-size: 90%; border: gray 2px solid;" cellspacing="8" summary="Contents"> + +<tr><td class="tcl"><a href="#ar1">GEODESY</a></td> <td class="tcl"><a href="#ar11">GEOFFROY, ÉTIENNE FRANÇOIS</a></td></tr> +<tr><td class="tcl"><a href="#ar2">GEOFFREY</a> (Martel)</td> <td class="tcl"><a href="#ar12">GEOFFROY, JULIEN LOUIS</a></td></tr> +<tr><td class="tcl"><a href="#ar3">GEOFFREY</a> (Plantagenet)</td> <td class="tcl"><a href="#ar13">GEOFFROY SAINT-HILAIRE, ÉTIENNE</a></td></tr> +<tr><td class="tcl"><a href="#ar4">GEOFFREY</a> (duke of Brittany)</td> <td class="tcl"><a href="#ar14">GEOFFROY SAINT-HILAIRE, ISIDORE</a></td></tr> +<tr><td class="tcl"><a href="#ar5">GEOFFREY</a> (archbishop of York)</td> <td class="tcl"><a href="#ar15">GEOGRAPHY</a></td></tr> +<tr><td class="tcl"><a href="#ar6">GEOFFREY DE MONTBRAY</a></td> <td class="tcl"><a href="#ar16">GEOID</a></td></tr> +<tr><td class="tcl"><a href="#ar7">GEOFFREY OF MONMOUTH</a></td> <td class="tcl"><a href="#ar17">GEOK-TEPE</a></td></tr> +<tr><td class="tcl"><a href="#ar8">GEOFFREY OF PARIS</a></td> <td class="tcl"><a href="#ar18">GEOLOGY</a></td></tr> +<tr><td class="tcl"><a href="#ar9">GEOFFREY THE BAKER</a></td> <td class="tcl"><a href="#ar19">GEOMETRICAL CONTINUITY</a></td></tr> +<tr><td class="tcl"><a href="#ar10">GEOFFRIN, MARIE THÉRÈSE RODET</a></td> <td class="tcl"><a href="#ar20">GEOMETRY</a></td></tr> +</table> + +<hr class="art" /> +<p><span class="pagenum"><a name="page607" id="page607"></a>607</span></p> +<p><span class="bold">GEODESY<a name="ar1" id="ar1"></a></span> (from the Gr. <span class="grk" title="gê">γῆ</span>, the earth, and <span class="grk" title="daiein">δαίειν</span>, to divide), +the science of surveying (<i>q.v.</i>) extended to large tracts of country, +having in view not only the production of a system of maps of +very great accuracy, but the determination of the curvature of +the surface of the earth, and eventually of the figure and +dimensions of the earth. This last, indeed, may be the sole +object in view, as was the case in the operations conducted in +Peru and in Lapland by the celebrated French astronomers +P. Bouguer, C.M. de la Condamine, P.L.M. de Maupertuis, +A.C. Clairault and others; and the measurement of the meridian +<span class="pagenum"><a name="page608" id="page608"></a>608</span> +arc of France by P.F.A. Méchain and J.B.J. Delambre had +for its end the determination of the true length of the “metre” +which was to be the legal standard of length of France (see +<span class="sc"><a href="#artlinks">Earth, Figure of the</a></span>).</p> + +<p>The basis of every extensive survey is an accurate triangulation, +and the operations of geodesy consist in the measurement, by +theodolites, of the angles of the triangles; the measurement of +one or more sides of these triangles on the ground; the determination +by astronomical observations of the azimuth of the whole +network of triangles; the determination of the actual position +of the same on the surface of the earth by observations, first for +latitude at some of the stations, and secondly for longitude; the +determination of altitude for all stations.</p> + +<p>For the computation, the points of the actual surface of the +earth are imagined as projected along their plumb lines on the +mathematical figure, which is given by the stationary sea-level, +and the extension of the sea through the continents by a system +of imaginary canals. For many purposes the mathematical +surface is assumed to be a plane; in other cases a sphere of +radius 6371 kilometres (20,900,000 ft.). In the case of extensive +operations the surface must be considered as a compressed +ellipsoid of rotation, whose minor axis coincides with the earth’s +axis, and whose compression, flattening, or ellipticity is about +1/298.</p> + +<p class="pt2 center"><i>Measurement of Base Lines.</i></p> + +<div class="condensed"> +<p>To determine by actual measurement on the ground the length of a +side of one of the triangles (“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.</p> + +<p>In order to avoid the difficulty in exactly determining the temperature +of a bar by the mercury thermometer, F.W. Bessel introduced +in 1834 near Königsberg a compound bar which constituted a +metallic thermometer.<a name="fa1a" id="fa1a" href="#ft1a"><span class="sp">1</span></a> A zinc bar is laid on an iron bar two toises +long, both bars being perfectly planed and in free contact, the zinc +bar being slightly shorter and the two bars rigidly united at one end. +As the temperature varies, the difference of the lengths of the bars, +as perceived by the other end, also varies, and affords a quantitative +correction for temperature variations, which is applied to reduce the +length to standard temperature. During the measurement of the +base line the bars were not allowed to come into contact, the interval +being measured by the insertion of glass wedges. The results of the +comparisons of four measuring rods with one another and with the +standards were elaborately computed by the method of least-squares. +The probable error of the measured length of 935 toises (about +6000 ft.) has been estimated as 1/863500 or 1.2 μ (μ denoting a +millionth). With this apparatus fourteen base lines were measured +in Prussia and some neighbouring states; in these cases a somewhat +higher degree of accuracy was obtained.</p> + +<p>The principal triangulation of Great Britain and Ireland has seven +base lines: five have been measured by steel chains, and two, +more exactly, by the compensation bars of General T.F. Colby, an +apparatus introduced in 1827-1828 at Lough Foyle in Ireland. Ten +base lines were measured in India in 1831-1869 by the same apparatus. +This is a system of six compound-bars self-correcting for temperature. +The bars may be thus described: Two bars, one of brass and the +other of iron, are laid in parallelism side by side, firmly united at +their centres, from which they may freely expand or contract; at +the standard temperature they are of the same length. Let AB be +one bar, A′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.</p> + +<p>The last base line measured in India had a length of 8913 ft. In +consequence of some suspicion as to the accuracy of the compensation +apparatus, the measurement was repeated four times, the operations +being conducted so as to determine the actual values of the probable +errors of the apparatus. The direction of the line (which is at Cape +Comorin) is north and south. In two of the measurements the brass +component was to the west, in the others to the east; the differences +between the individual measurements and the mean of the four were ++0.0017, −0.0049, −0.0015, +0.0045 ft. These differences are +very small; an elaborate investigation of all sources of error shows +that the probable error of a base line in India is on the average +±2.8 μ. These compensation bars were also used by Sir Thomas +Maclear in the measurement of the base line in his extension of +Lacaille’s arc at the Cape. The account of this operation will be +found in a volume entitled <i>Verification and Extension of Lacaille’s +Arc of Meridian at the Cape of Good Hope</i>, by Sir Thomas Maclear, +published in 1866. A rediscussion has been given by Sir David +Gill in his <i>Report on the Geodetic Survey of South Africa, &c., 1896</i>.</p> + +<p>A very simple base apparatus was employed by W. Struve in his +triangulations in Russia from 1817 to 1855. This consisted of four +wrought-iron bars, each two toises (rather more than 13 ft.) long; +one end of each bar is terminated in a small steel cylinder presenting +a slightly convex surface for contact, the other end carries a contact +lever rigidly connected with the bar. The shorter arm of the lever +terminates below in a polished hemisphere, the upper and longer +arm traversing a vertical divided arc. In measuring, the plane end +of one bar is brought into contact with the short arm of the contact +lever (pushed forward by a weak spring) of the next bar. Each bar +has two thermometers, and a level for determining the inclination +of the bar in measuring. The manner of transferring the end of a +bar to the ground is simply this: under the end of the bar a stake +is driven very firmly into the ground, carrying on its upper surface +a disk, capable of movement in the direction of the measured line +by means of slow-motion screws. A fine mark on this disk is +brought vertically under the end of the bar by means of a theodolite +which is planted at a distance of 25 ft. from the stake in a direction +perpendicular to the base. Struve investigated for each base the +probable errors of the measurement arising from each of these seven +causes: Alignment, inclination, comparisons with standards, readings +of index, personal errors, uncertainties of temperature, and the +probable errors of adopted rates of expansion. He found that +±0.8 μ was the mean of the probable errors of the seven bases +measured by him. The Austro-Hungarian apparatus is similar; +the distance of the rods is measured by a slider, which rests on one +of the ends of each rod. Twenty-two base lines were measured in +1840-1899.</p> + +<p>General Carlos Ibañez employed in 1858-1879, for the measurement +of nine base lines in Spain, two apparatus similar to the +apparatus previously employed by Porro in Italy; one is complicated, +the other simplified. The first, an apparatus of the brothers Brunner +of Paris, was a thermometric combination of two bars, one of platinum +and one of brass, in length 4 metres, furnished with three levels and +four thermometers. Suppose A, B, C three micrometer microscopes +very firmly supported at intervals of 4 metres with their axes vertical, +and aligned in the plane of the base line by means of a transit +instrument, their micrometer screws being in the line of measurement. +The measuring bar is brought under say A and B, and those micrometers +read; the bar is then shifted and brought under B and C. By +repetition of this process, the reading of a micrometer indicating the +end of each position of the bar, the measurement is made.</p> + +<p>Quite similar apparatus (among others) has been employed by the +French and Germans. Since, however, it only permitted a distance +of about 300 m. to be measured daily, Ibañez introduced a simplification; +the measuring rod being made simply of steel, and provided +with inlaid mercury thermometers. This apparatus was used in +Switzerland for the measurement of three base lines. The accuracy +is shown by the estimated probable errors: ±0.2 μ to ±0.8 μ. +The distance measured daily amounts at least to 800 m.</p> + +<p>A greater daily distance can be measured with the same accuracy +by means of Bessel’s apparatus; this permits the ready measurement +of 2000 m. daily. For this, however, it is important to notice +that a large staff and favourable ground are necessary. An important +improvement was introduced by Edward Jäderin of Stockholm, +who measures with stretched wires of about 24 metres long; +these wires are about 1.65 mm. in diameter, and when in use are +stretched by an accurate spring balance with a tension of 10 kg.<a name="fa2a" id="fa2a" href="#ft2a"><span class="sp">2</span></a> +The nature of the ground has a very trifling effect on this method. +The difficulty of temperature determinations is removed by employing +wires made of invar, an alloy of steel (64%) and nickel (36%) +which has practically no linear expansion for small thermal changes +<span class="pagenum"><a name="page609" id="page609"></a>609</span> +at ordinary temperatures; this alloy was discovered in 1896 by +Benôit and Guillaume of the International Bureau of Weights and +Measures at Breteuil. Apparently the future of base-line measurements +rests with the invar wires of the Jäderin apparatus; next +comes Porro’s apparatus with invar bars 4 to 5 metres long.</p> + +<p>Results have been obtained in the United States, of great importance +in view of their accuracy, rapidity of determination and +economy. For the measurement of the arc of meridian in longitude +98° E., in 1900, nine base lines of a total length of 69.2 km. were +measured in six months. The total cost of one base was $1231. +At the beginning and at the end of the field-season a distance of +exactly 100 m. was measured with R.S. Woodward’s “5-m. ice-bar” +(invented in 1891); by means of the remeasurement of this +length the standardization of the apparatus was done under the same +conditions as existed in the case of the base measurements. For +the measurements there were employed two steel tapes of 100 m. +long, provided with supports at distances of 25 m., two of 50 m., +and the duplex apparatus of Eimbeck, consisting of four 5-m. rods. +Each base was divided into sections of about 1000 m.; one of these, +the “test kilometre,” was measured with all the five apparatus, +the others only with two apparatus, mostly tapes. The probable +error was about ±0.8 μ, and the day’s work a distance of about +2000 m. Each of the four rods of the duplex apparatus consists of +two bars of brass and steel. Mercury thermometers are inserted +in both bars; these serve for the measurement of the length of the +base lines by each of the bars, as they are brought into their consecutive +positions, the contact being made by an elastic-sliding +contact. The length of the base lines may be calculated for each +bar only, and also by the supposition that both bars have the same +temperature. The apparatus thus affords three sets of results, +which mutually control themselves, and the contact adjustments +permit rapid work. The same device has been applied to the older +bimetallic-compensating apparatus of Bache-Würdemann (six +bases, 1847-1857) and of Schott. There was also employed a single +rod bimetallic apparatus on F. Porro’s principle, constructed by the +brothers Repsold for some base lines. Excellent results have been +more recently obtained with invar tapes.</p> + +<p>The following results show the lengths of the same German base +lines as measured by different apparatus:</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl" colspan="4"> </td> <td class="tcc">metres.</td></tr> +<tr><td class="tcl">Base at Berlin</td> <td class="tcc">1864</td> <td class="tcc">Apparatus of</td> <td class="tcl">Bessel</td> <td class="tcr">2336·3920</td></tr> +<tr><td class="tcl">    ”    ”</td> <td class="tcc">1880</td> <td class="tcc">”</td> <td class="tcl">Brunner</td> <td class="tcr">·3924</td></tr> +<tr><td class="tcl">Base at Strehlen</td> <td class="tcc">1854</td> <td class="tcc">”</td> <td class="tcl">Bessel</td> <td class="tcr">2762·5824</td></tr> +<tr><td class="tcl">    ”    ”</td> <td class="tcc">1879</td> <td class="tcc">”</td> <td class="tcl">Brunner</td> <td class="tcr">·5852</td></tr> +<tr><td class="tcl">Old base at Bonn</td> <td class="tcc">1847</td> <td class="tcc">”</td> <td class="tcl">Bessel</td> <td class="tcr">2133·9095</td></tr> +<tr><td class="tcl">    ”    ”</td> <td class="tcc">1892</td> <td class="tcc">”</td> <td class="tcc">”</td> <td class="tcr">·9097</td></tr> +<tr><td class="tcl">New base at Bonn</td> <td class="tcc">1892</td> <td class="tcc">”</td> <td class="tcc">”</td> <td class="tcr">2512·9612</td></tr> +<tr><td class="tcl">    ”    ”</td> <td class="tcc">1892</td> <td class="tcc">”</td> <td class="tcl">Brunner</td> <td class="tcr">·9696</td></tr> +</table> + +<p>It is necessary that the altitude above the level of the sea of every +part of a base line be ascertained by spirit levelling, in order that +the measured length may be reduced to what it would have been +had the measurement been made on the surface of the sea, produced +in imagination. Thus if l be the length of a measuring bar, h its +height at any given position in the measurement, r the radius of +the earth, then the length radially projected on to the level of the +sea is l(1 − h/r). In the Salisbury Plain base line the reduction to +the level of the sea is −0.6294 ft.</p> + +<table class="flt" style="float: right; width: 250px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:203px; height:347px" src="images/img609.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 1.</span></td></tr></table> + +<p>The total number of base lines measured in Europe up to the +present time is about one hundred and ten, nineteen of which do +not exceed in length 2500 metres, or about 1½ miles, and three—one +in France, the others in Bavaria—exceed +19,000 metres. The question +has been frequently discussed whether +or not the advantage of a long base is +sufficiently great to warrant the expenditure +of time that it requires, or +whether as much precision is not obtainable +in the end by careful triangulation +from a short base. But the answer +cannot be given generally; it must +depend on the circumstances of each +particular case. With Jäderin’s apparatus, +provided with invar wires, bases +of 20 to 30 km. long are obtained without +difficulty.</p> + +<p>In working away from a base line ab, +stations c, d, e, f are carefully selected so +as to obtain from well-shaped triangles +gradually increasing sides. Before, however, +finally leaving the base line, it is +usual to verify it by triangulation thus: +during the measurement two or more +points, as p, q (fig. 1), are marked in the +base in positions such that the lengths of +the different segments of the line are +known; then, taking suitable external stations, as h, k, the angles of +the triangles bhp, phq, hqk, kqa are measured. From these angles +can be computed the ratios of the segments, which must agree, if all +operations are correctly performed, with the ratios resulting from +the measures. Leaving the base line, the sides increase up to 10, +30 or 50 miles occasionally, but seldom reaching 100 miles. The +triangulation points may either be natural objects presenting themselves +in suitable positions, such as church towers; or they may be +objects specially constructed in stone or wood on mountain tops +or other prominent ground. In every case it is necessary that the +precise centre of the station be marked by some permanent mark. +In India no expense is spared in making permanent the principal +trigonometrical stations—costly towers in masonry being erected. +It is essential that every trigonometrical station shall present a fine +object for observation from surrounding stations.</p> + +<p class="pt2 center"><i>Horizontal Angles.</i></p> + +<p>In placing the theodolite over a station to be observed from, the +first point to be attended to is that it shall rest upon a perfectly +solid foundation. The method of obtaining this desideratum must +depend entirely on the nature of the ground; the instrument must +if possible be supported on rock, or if that be impossible a solid +foundation must be obtained by digging. When the theodolite is +required to be raised above the surface of the ground in order to +command particular points, it is necessary to build two scaffolds,—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.</p> + +<p>At every station it is necessary to examine and correct the adjustments +of the theodolite, which are these: the line of collimation +of the telescope must be perpendicular to its axis of rotation; this +axis perpendicular to the vertical axis of the instrument; and the +latter perpendicular to the plane of the horizon. The micrometer +microscopes must also measure correct quantities on the divided +circle or circles. The method of observing is this. Let A, B, C ... +be the stations to be observed taken in order of azimuth; the +telescope is first directed to A and the cross-hairs of the telescope +made to bisect the object presented by A, then the microscopes or +verniers of the horizontal circle (also of the vertical circle if necessary) +are read and recorded. The telescope is then turned to B, which +is observed in the same manner; then C and the other stations. +Coming round by continuous motion to A, it is again observed, and +the agreement of this second reading with the first is some test of +the stability of the instrument. In taking this round of angles—or +“arc,” as it is called on the Ordnance Survey—it is desirable +that the interval of time between the first and second observations +of A should be as small as may be consistent with due care. Before +taking the next arc the horizontal circle is moved through 20° or +30°; thus a different set of divisions of the circle is used in each +arc, which tends to eliminate the errors of division.</p> + +<p>It is very desirable that all arcs at a station should contain one +point in common, to which all angular measurements are thus +referred,—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.</p> + +<p>This method seems at first sight very advantageous; but if, +however, it be desired to attain the highest accuracy, it is better, +as shown by General Schreiber of Berlin in 1878, to measure only +single angles, and as many of these as possible between the directions +to be determined. Division-errors are thus more perfectly eliminated, +and errors due to the variation in the stability, &c., of the instruments +are diminished. This method is rapidly gaining precedence.</p> + +<p>The theodolites used in geodesy vary in pattern and in size—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 +<span class="pagenum"><a name="page610" id="page610"></a>610</span> +number of measures of an angle is never less than 20. An examination +of 1407 angles showed that the probable error of an observed +angle is on the average ±0″.28.</p> + +<p>For the observations of very distant stations it is usual to employ +a heliotrope (from the Gr. <span class="grk" title="hêlios">ἥλιος</span>, sun; <span class="grk" title="tropos">τρόπος</span>, a turn), invented by +Gauss at Göttingen in 1821. In its simplest form this is a plane +mirror, 4, 6, or 8 in. in diameter, capable of rotation round a horizontal +and a vertical axis. This mirror is placed at the station to be observed, +and in fine weather it is kept so directed that the rays of the +sun reflected by it strike the distant observing telescope. To the +observer the heliotrope presents the appearance of a star of the +first or second magnitude, and is generally a pleasant object for +observing.</p> + +<p>Observations at night, with the aid of light-signals, have been +repeatedly made, and with good results, particularly in France +by General François 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 +Ibañez and Perrier (over a distance of 270 km.), of Sicily and Malta +in 1900, and of the islands of Elba and Sardinia in 1902 by Dr +Guarducci (over distances up to 230 km.); in these cases artificial +light was employed: in the first case electric light and in the two +others acetylene lamps.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:431px; height:692px" src="images/img610a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 2.</span>—Altazimuth Theodolite.</td></tr></table> + +<p class="pt2 center"><i>Astronomical Observations.</i></p> + +<p>The direction of the meridian is determined either by a theodolite +or a portable transit instrument. In the former case the operation +consists in observing the angle between a terrestrial object—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 2δ/sin z, in seconds of angle, omitting +smaller terms, δ being the star’s declination and z its zenith distance. +The collimation and level errors are very carefully determined +before and after these observations, and it is usual to arrange the +observations by the reversal of the telescope so that collimation +error shall disappear. If b, c be the level and collimation errors, +the correction to the circle reading is b cot z ± c cosec z, b being +positive when the west end of the axis is high. It is clear that any +uncertainty as to the real state of the level will produce a corresponding +uncertainty in the resulting value of the azimuth,—an +uncertainty which increases with the latitude and is very large +in high latitudes. This may be partly remedied by observing in +connexion with the star its reflection in mercury. In determining +the value of “one division” of a level tube, it is necessary to bear +in mind that in some the value varies considerably with the temperature. +By experiments on the level of Ramsden’s 3-foot theodolite, +it was found that though at the ordinary temperature of 66° the +value of a division was about one second, yet at 32° it was about +five seconds.</p> + +<p>In a very excellent portable transit used on the Ordnance Survey, +the uprights carrying the telescope are constructed of mahogany, +each upright being built of several pieces glued and screwed together; +the base, which is a solid and heavy plate of iron, carries a reversing +apparatus for lifting the telescope out of its bearings, reversing it +and letting it down again. Thus is avoided the change of temperature +which the telescope would incur by being lifted by the hands +of the observer. Another form of transit is the German diagonal +form, in which the rays of light after passing through the object-glass +are turned by a total reflection prism through one of the transverse +arms of the telescope, at the extremity of which arm is the +eye-piece. The unused half of the ordinary telescope being cut away +is replaced by a counterpoise. In this instrument there is the +advantage that the observer without moving the position of his eye +commands the whole meridian, and that the level may remain on +the pivots whatever be the elevation of the telescope. But there is +the disadvantage that the flexure of the transverse axis causes a +variable collimation error depending on the zenith distance of the +star to which it is directed; and moreover it has been found that in +some cases the personal error of an observer is not the same in the +two positions of the telescope.</p> + +<p>To determine the direction of the meridian, it is well to erect two +marks at nearly equal angular distances on either side of the north +meridian line, so that the pole star crosses the vertical of each mark +a short time before and after attaining its greatest eastern and +western azimuths.</p> + +<p>If now the instrument, perfectly levelled, is adjusted to have its +centre wire on one of the marks, then when elevated to the star, +the star will traverse the wire, and its exact position in the field at +any moment can be measured by the micrometer wire. Alternate +observations of the star and the terrestrial mark, combined with +careful level readings and reversals of the instrument, will enable +one, even with only one mark, to determine the direction of the +meridian in the course of an hour with a probable error of less than +a second. The second mark enables one to complete the station +more rapidly and gives a check upon the work. As an instance, +at Findlay Seat, in latitude 57° 35′, the resulting azimuths of the +two marks were 177° 45′ 37″.29 ± 0″.20 and 182° 17′ 15″.61 ± 0″.13, +while the angle between the two marks directly measured by a +theodolite was found to be 4° 31′ 37″.43 ± 0″.23.</p> + +<table class="flt" style="float: right; width: 260px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:208px; height:207px" src="images/img610b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 3.</span></td></tr></table> + +<p>We now come to the consideration of the determination of time +with the transit instrument. Let fig. 3 represent the sphere stereographically +projected on the plane of +the horizon,—ns being the meridian, +we the prime vertical, Z, P the zenith +and the pole. Let p be the point in +which the production of the axis of +the instrument meets the celestial +sphere, S the position of a star when +observed on a wire whose distance +from the collimation centre is c. Let +a be the azimuthal deviation, namely, +the angle wZp, b the level error so +that Zp = 90° − b. Let also the hour +angle corresponding to p be 90° − n, +and the declination of the same = m, +the star’s declination being δ, and the +latitude φ. Then to find the hour +angle ZPS = τ of the star when observed, in the triangles pPS, pPZ +we have, since pPS = 90 + τ − n,</p> + +<table class="reg" style="clear: both;" summary="poem"><tr><td> <div class="poemr"> +<p>    − Sin c = sin m sin δ + cos m cos δ sin (n − τ),</p> +<p>    Sin m = sin b sin φ − cos b cos φ sin a,</p> +<p>Cos m sin n = sin b cos φ + cos b sin φ sin a.</p> +</div> </td></tr></table> + +<p class="noind">And these equations solve the problem, however large be the errors +of the instrument. Supposing, as usual, a, b, m, n to be small, +we have at once τ = n + c sec δ + m tan δ, 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 δ. Another very convenient form +for stars near the zenith is τ = b sec φ + c sec δ + m (tan δ − tan φ).</p> + +<p>Suppose that in commencing to observe at a station the error of the +chronometer is not known; then having secured for the instrument +a very solid foundation, removed as far as possible level and collimation +errors, and placed it by estimation nearly in the meridian, +let two stars differing considerably in declination be observed—the +instrument not being reversed between them. From these two +stars, neither of which should be a close circumpolar star, a good +approximation to the chronometer error can be obtained; thus +<span class="pagenum"><a name="page611" id="page611"></a>611</span> +let ε<span class="su">1</span>, ε<span class="su">2</span>, be the apparent clock errors given by these stars if δ<span class="su">1</span>, δ<span class="su">2</span> +be their declinations the real error is</p> + +<p class="center">ε = ε<span class="su">1</span> + (ε<span class="su">1</span> − ε<span class="su">2</span>) (tan φ − tan δ<span class="su">1</span>) / (tan δ<span class="su">1</span> − tan δ<span class="su">2</span>).</p> + +<p class="noind">Of course this is still only approximate, but it will enable the observer +(who by the help of a table of natural tangents can compute ε 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.</p> + +<p>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.</p> + +<p>The transit instrument is also used in the prime vertical for the +determination of latitudes. In the preceding figure let q be the point +in which the northern extremity of the axis of the instrument +produced meets the celestial sphere. Let nZq be the azimuthal +deviation = a, and b being the level error, Zq = 90° − b; let also +nPq = τ and Pq = ψ. 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</p> + +<table class="reg" style="clear: both;" summary="poem"><tr><td> <div class="poemr"> +<p>    −Sin c = sin δ cos ψ − cos δ sin ψ cos (h + τ),</p> +<p>    Cos ψ = sin b sin φ + cos b cos φ cos a,</p> +<p>Sin ψ sin τ = cos b sin a.</p> +</div> </td></tr></table> + +<p>Now when a and b are very small, we see from the last two equations +that ψ = φ − b, a = τ sin ψ, and if we calculate φ′ by the formula +cot φ′ = cot δ cos h, the first equation leads us to this result—</p> + +<p class="center">φ = φ′ + (a sin z + b cos z + c) / cos z,</p> + +<p class="noind">the correction for instrumental error being very similar to that +applied to the observed time of transit in the case of meridian +observations. When a is not very small and z is small, the formulae +required are more complicated.</p> + +<table class="flt" style="float: right; width: 360px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:312px; height:644px" src="images/img611.jpg" alt="" /></td></tr> +<tr><td class="caption1"><span class="sc">Fig. 4.</span>—Zenith Telescope constructed +for the International Stations at Mizusawa, Carloforte, Gaithersburg and +Ukiah, by Hermann Wanschaff, Berlin.</td></tr></table> + +<p>The method of determining latitude by transits in the prime +vertical has the disadvantage of being a somewhat slow process, +and of requiring a very precise knowledge of the time, a disadvantage +from which the zenith telescope is free. In principle this instrument +is based on the proposition +that when the meridian +zenith distances of +two stars at their upper +culminations—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 δ, and +that of another culminating +to the north with +zenith distance Z′ and +declination δ′, then clearly +the latitude is ½(δ + δ′) + +½(Z − Z′). Now the zenith +telescope does away with +the divided circle, and +substitutes the measurement +micrometrically of +the quantity Z′ − Z.</p> + +<p>In fig. 4 is shown a +zenith telescope by H. +Wanschaff of Berlin, +which is the type used +(according to the Central +Bureau at Potsdam) since +about 1890 for the determination +of the variations +of latitude due to different, +but as yet imperfectly +understood, influences. +The instrument is supported +on a strong tripod, +fitted with levelling +screws; to this tripod is +fixed the azimuth circle +and a long vertical steel +axis. Fitting on this axis +is a hollow axis which +carries on its upper end a +short transverse horizontal +axis with a level. This +latter carries the telescope, which, supported at the centre of its +length, is free to rotate in a vertical plane. The telescope is thus +mounted eccentrically with respect to the vertical axis around +which it revolves. Two extremely sensitive levels are attached to +the telescope, which latter carries a micrometer in its eye-piece, +with a screw of long range for measuring differences of zenith distance. +Two levels are employed for controlling and increasing the +accuracy. For this instrument stars are selected in pairs, passing +north and south of the zenith, culminating within a few minutes +of time and within about twenty minutes (angular) of zenith distance +of each other. When a pair of stars is to be observed, the +telescope is set to the mean of the zenith distances and in the plane +of the meridian. The first star on passing the central meridional +wire is bisected by the micrometer; then the telescope is rotated +very carefully through 180° round the vertical axis, and the second +star on passing through the field is bisected by the micrometer on +the centre wire. The micrometer has thus measured the difference +of the zenith distances, and the calculation to get the latitude is +most simple. Of course it is necessary to read the level, and the +observations are not necessarily confined to the centre wire. In +fact if n, s be the north and south readings of the level for the south +star, n′, s′ the same for the north star, l the value of one division +of the level, m the value of one division of the micrometer, r, r′ the +refraction corrections, μ, μ′ the micrometer readings of the south +and north star, the micrometer being supposed to read from the +zenith, then, supposing the observation made on the centre wire,—</p> + +<p class="center">φ = ½ (δ + δ′) + ½ (μ − mu′)m + ¼ (n + n′ − s − s′)l + ½ (r − r′).</p> + +<p>It is of course of the highest importance that the value m of the +screw be well determined. This is done most effectually by observing +the vertical movement of a close circumpolar star when at its greatest +azimuth.</p> + +<p>In a single night with this instrument a very accurate result, +say with a probable error of about 0″.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.</p> + +<p>The zenith telescope is a particularly pleasant instrument to +work with, and an observer has been known (a sergeant of Royal +Engineers, on one occasion) to take every star in his list during +eleven hours on a stretch, namely, from 6 o’clock <span class="scs">P.M.</span> until 5 <span class="scs">A.M.</span>, +and this on a very cold November night on one of the highest points +of the Grampians. Observers accustomed to geodetic operations +attain considerable powers of endurance. Shortly after the commencement +of the observations on one of the hills in the Isle of Skye +a storm carried away the wooden houses of the men and left the +observatory roofless. Three observatory roofs were subsequently +demolished, and for some time the observatory was used without a +roof, being filled with snow every night and emptied every morning. +Quite different, however, was the experience of the same party when +on the top of Ben Nevis, 4406 ft. high. For about a fortnight the +state of the atmosphere was unusually calm, so much so, that a +lighted candle could often be carried between the tents of the men +and the observatory, whilst at the foot of the hill the weather was +wild and stormy.</p> + +<p>The determination of the difference of longitude between two +stations A and B resolves itself into the determination of the local +time at each of the stations, and the comparison by signals of the +clocks at A and B. Whenever telegraphic lines are available these +comparisons are made by telegraphy. A small and delicately-made +apparatus introduced into the mechanism of an astronomical clock +or chronometer breaks or closes by the action of the clock an electric +circuit every second. In order to record the minutes as well as +seconds, one second in each minute, namely that numbered 0 or 60, +is omitted. The seconds are recorded on a chronograph, which +consists of a cylinder revolving uniformly at the rate of one revolution +per minute covered with white paper, on which a pen having a slow +movement in the direction of the axis of the cylinder describes a +continuous spiral. This pen is deflected through the agency of an +electromagnet every second, and thus the seconds of the clock are +recorded on the chronograph by offsets from the spiral curve. An +observer having his hand on a contact key in the same circuit can +record in the same manner his observed times of transits of stars. +The method of determination of difference of longitude is, therefore, +virtually as follows. After the necessary observations for instrumental +corrections, which are recorded only at the station of observation, +the clock at A is put in connexion with the circuit so as to +write on both chronographs, namely, that at A and that at B. +Then the clock at B is made to write on both chronographs. It is +clear that by this double operation one can eliminate the effect of the +small interval of time consumed in the transmission of signals, for +the difference of longitude obtained from the one chronograph +will be in excess by as much as that obtained from the other will be +in defect. The determination of the personal errors of the observers +in this delicate operation is a matter of the greatest importance, +as therein lies probably the chief source of residual error.</p> + +<p><span class="pagenum"><a name="page612" id="page612"></a>612</span></p> + +<p>These errors can nevertheless be almost entirely avoided by using +the impersonal micrometer of Dr Repsold (Hamburg, 1889). In +this device there is a movable micrometer wire which is brought by +hand into coincidence with the star and moved along with it; at +fixed points there are electrical contacts, which replace the fixed +wires. Experiments at the Geodetic Institute and Central Bureau +at Potsdam in 1891 gave the following personal equations in the case +of four observers:—</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc"> </td> <td class="tcc">Older Procedure.</td> <td class="tcc">New Procedure.</td></tr> + +<tr><td class="tcc">A − B</td> <td class="tcc">−0<span class="sp">s</span>.108</td> <td class="tcc">−0<span class="sp">s</span>.004</td></tr> +<tr><td class="tcc">A − G</td> <td class="tcc">−0<span class="sp">s</span>.314</td> <td class="tcc">−0<span class="sp">s</span>.035</td></tr> +<tr><td class="tcc">A − S</td> <td class="tcc">−0<span class="sp">s</span>.184</td> <td class="tcc">−0<span class="sp">s</span>.027</td></tr> +<tr><td class="tcc">B − G</td> <td class="tcc">−0<span class="sp">s</span>.225</td> <td class="tcc">+0<span class="sp">s</span>.013</td></tr> +<tr><td class="tcc">B − S</td> <td class="tcc">−0<span class="sp">s</span>.086</td> <td class="tcc">−0<span class="sp">s</span>.023</td></tr> +<tr><td class="tcc">G − S</td> <td class="tcc">+0<span class="sp">s</span>.109</td> <td class="tcc">−0<span class="sp">s</span>.006</td></tr> +</table> + +<p>These results show that in the later method the personal equation +is small and not so variable; and consequently the repetition of +longitude determinations with exchanged observers and apparatus +entirely eliminates the constant errors, the probable error of such +determinations on ten nights being scarcely ±0<span class="sp">s</span>.01.</p> + +<p class="pt2 center"><i>Calculation of Triangulation.</i></p> + +<p>The surface of Great Britain and Ireland is uniformly covered by +triangulation, of which the sides are of various lengths from 10 to +111 miles. The largest triangle has one angle at Snowdon in Wales, +another on Slieve Donard in Ireland, and a third at Scaw Fell in +Cumberland; each side is over a hundred miles and the spherical +excess is 64″. 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.</p> + +<p>The inevitable errors of observation, which are inseparable from +all angular as well as other measurements, introduce a great difficulty +into the calculation of the sides of a triangulation. Starting from a +given base in order to get a required distance, it may generally be +obtained in several different ways—that is, by using different sets +of triangles. The results will certainly differ one from another, +and probably no two will agree. The experience of the computer +will then come to his aid, and enable him to say which is the most +trustworthy result; but no experience or ability will carry him +through a large network of triangles with anything like assurance. +The only way to obtain trustworthy results is to employ the method +of least squares. We cannot here give any illustration of this method +as applied to general triangulation, for it is most laborious, even for +the simplest cases.</p> + +<p>Three stations, projected on the surface of the sea, give a spherical +or spheroidal triangle according to the adoption of the sphere or +the ellipsoid as the form of the surface. A spheroidal triangle differs +from a spherical triangle, not only in that the curvatures of the sides +are different one from another, but more especially in this that, +while in the spherical triangle the normals to the surface at the angular +points meet at the centre of the sphere, in the spheroidal triangle +the normals at the angles A, B, C meet the axis of revolution of the +spheroid in three different points, which we may designate α, β, γ +respectively. Now the angle A of the triangle as measured by a +theodolite is the inclination of the planes BAα and CAα, and the angle +at B is that contained by the planes ABβ and CBβ. But the planes +ABα and ABβ containing the line AB in common cut the surface in +two distinct plane curves. In order, therefore, that a spheroidal +triangle may be exactly defined, it is necessary that the nature of the +lines joining the three vertices be stated. In a mathematical point +of view the most natural definition is that the sides be geodetic or +shortest lines. C.C.G. Andrae, of Copenhagen, has also shown +that other lines give a less convenient computation.</p> + +<p>K.F. Gauss, in his treatise, <i>Disquisitiones generales circa superficies +curvas</i>, entered fully into the subject of geodetic (or geodesic) +triangles, and investigated expressions for the angles of a geodetic +triangle whose sides are given, not certainly finite expressions, but +approximations inclusive of small quantities of the fourth order, the +side of the triangle or its ratio to the radius of the nearly spherical +surface being a small quantity of the first order. The terms of the +fourth order, as given by Gauss for any surface in general, are very +complicated even when the surface is a spheroid. If we retain small +quantities of the second order only, and put <span class="got">A</span>, <span class="got">B</span>, <span class="got">C</span> for the angles +of the geodetic triangle, while A, B, C are those of a plane triangle +having sides equal respectively to those of the geodetic triangle, +then, σ being the area of the plane triangle and <span class="got">a</span>, <span class="got">b</span>, <span class="got">c</span> the measures +of curvature at the angular points,</p> + +<table class="reg" summary="poem"><tr><td> <div class="poemr"> +<p><span class="got">A</span> = A + σ(2<span class="got">a</span> + <span class="got">b</span> + <span class="got">c</span>) / 12,</p> +<p><span class="got">B</span> = B + σ(<span class="got">a</span> + 2<span class="got">b</span> + <span class="got">c</span>) / 12,</p> +<p><span class="got">C</span> = C + σ(<span class="got">a</span> + <span class="got">b</span> + 2<span class="got">c</span>) / 12.</p> +</div> </td></tr></table> + +<p class="noind">For the sphere <span class="got">a</span> = <span class="got">b</span> = <span class="got">c</span>, and making this simplification, we obtain the +theorem previously given by A.M. Legendre. With the terms of the +fourth order, we have (after Andrae):</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="got">A</span> − A =</td> <td>ε</td> +<td rowspan="2">+</td> <td>σ</td> +<td rowspan="2">k <span class="f150">(</span></td> <td>m² − a²</td> +<td rowspan="2">k +</td> <td><span class="got">a</span> − k</td> +<td rowspan="2"><span class="f150">)</span>,</td></tr> +<tr><td class="denom">3</td> <td class="denom">3</td> +<td class="denom">20</td> <td class="denom">4k</td></tr></table> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="got">B</span> − B =</td> <td>ε</td> +<td rowspan="2">+</td> <td>σ</td> +<td rowspan="2">k <span class="f150">(</span></td> <td>m² − b²</td> +<td rowspan="2">k +</td> <td><span class="got">b</span> − k</td> +<td rowspan="2"><span class="f150">)</span>,</td></tr> +<tr><td class="denom">3</td> <td class="denom">3</td> +<td class="denom">20</td> <td class="denom">4k</td></tr></table> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="got">C</span> − C =</td> <td>ε</td> +<td rowspan="2">+</td> <td>σ</td> +<td rowspan="2">k <span class="f150">(</span></td> <td>m² − c²</td> +<td rowspan="2">k +</td> <td><span class="got">c</span> − k</td> +<td rowspan="2"><span class="f150">)</span>,</td></tr> +<tr><td class="denom">3</td> <td class="denom">3</td> +<td class="denom">20</td> <td class="denom">4k</td></tr></table> + +<p class="noind">in which ε = σk {1 + (m²k / 8)}, 3m² = a² + b² + c², 3k = <span class="got">a</span> + <span class="got">b</span> + <span class="got">c</span>. For the +ellipsoid of rotation the measure of curvature is equal to 1/ρn, +ρ and n being the radii of curvature of the meridian and perpendicular.</p> + +<p>It is rarely that the terms of the fourth order are required. As a +rule spheroidal triangles are calculated as spherical (after Legendre), +<i>i.e.</i> like plane triangles with a decrease of each angle of about ε/3; +ε must, however, be calculated for each triangle separately with its +mean measure of curvature k.</p> + +<p>The geodetic line being the shortest that can be drawn on any +surface between two given points, we may be conducted to its most +important characteristics by the following considerations: let p, q +be adjacent points on a curved surface; through s the middle point +of the chord pq imagine a plane drawn perpendicular to pq, and let +S be any point in the intersection of this plane with the surface; +then pS + Sq is evidently least when sS is a minimum, which is +when sS is a normal to the surface; hence it follows that of all +plane curves on the surface joining p, q, when those points are indefinitely +near to one another, that is the shortest which is made +by the normal plane. That is to say, the osculating plane at any +point of a geodetic line contains the normal to the surface at that +point. Imagine now three points in space, A, B, C, such that AB = +BC = c; let the direction cosines of AB be l, m, n, those of BC l’, +m′, n′, then x, y, z being the co-ordinates of B, those of A and C will +be respectively—</p> + +<table class="reg" summary="poem"><tr><td> <div class="poemr"> +<p>x − cl : y − cm : z − cn</p> +<p>x + cl′ : y + cm′ : z + cn′.</p> +</div> </td></tr></table> + +<p class="noind">Hence the co-ordinates of the middle point M of AC are x + ½c(l′ − l), +y + ½c(m′ − m), z + ½c(n′ − n), and the direction cosines of BM are +therefore proportional to l′ − l: m′ − m: n′ − n. If the angle made +by BC with AB be indefinitely small, the direction cosines of BM +are as δl : δm : δn. Now if AB, BC be two contiguous elements of +a geodetic, then BM must be a normal to the surface, and since δl, +δm, δn are in this case represented by δ(dx/ds), δ(dy/ds), δ(dz/ds), +and if the equation of the surface be u = 0, we have</p> + +<table class="math0" summary="math"> +<tr><td>d²x</td> +<td rowspan="2"><span class="f200">/</span></td> <td>du</td> +<td rowspan="2">=</td> <td>d²y</td> +<td rowspan="2"><span class="f200">/</span></td> <td>du</td> +<td rowspan="2">=</td> <td>d²z</td> +<td rowspan="2"><span class="f200">/</span></td> <td>du</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">ds²</td> <td class="denom">dx</td> +<td class="denom">ds²</td> <td class="denom">dy</td> +<td class="denom">ds²</td> <td class="denom">dz</td></tr></table> + +<p class="noind">which, however, are equivalent to only one equation. In the case +of the spheroid this equation becomes</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">y</td> <td>d²x</td> +<td rowspan="2">−</td> <td>d²y</td> +<td rowspan="2">= 0,</td></tr> +<tr><td class="denom">ds²</td> <td class="denom">ds²</td></tr></table> + +<p class="noind">which integrated gives ydx − 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.</p> + +<p>From this it may be shown that the azimuth at A of the geodetic +joining AB is not the same as the astronomical azimuth at A of B +or that determined by the vertical plane AαB. Generally speaking, +the geodetic lies between the two plane section curves joining A and +B which are formed by the two vertical planes, supposing these points +not far apart. If, however, A and B are nearly in the same latitude, +the geodetic may cross (between A and B) that plane curve which +lies nearest the adjacent pole of the spheroid. The condition of +crossing is this. Suppose that for a moment we drop the consideration +of the earth’s non-sphericity, and draw a perpendicular from +the pole C on AB, meeting it in S between A and B. Then A being +that point which is nearest the pole, the geodetic will cross the plane +curve if AS be between ¼AB and <span class="spp">3</span>⁄<span class="suu">8</span>AB. If AS lie between this last +value and ½AB, the geodetic will lie wholly to the north of both +plane curves, that is, supposing both points to be in the northern +hemisphere.</p> + +<p>The difference of the azimuths of the vertical section AB and of +the geodetic AB, <i>i.e.</i> the astronomical and geodetic azimuths, is +very small for all observable distances, being approximately:—</p> + +<p>Geod. azimuth = Astr. azimuth −1/12 [e²/(1 − e²)] [(s²/ρn (cos²φ sin 2α + (s/4a) | sin 2φ sin α)], +in which: e and a are the numerical eccentricity +and semi-major axis respectively of the meridian ellipse, φ and α are +the latitude and azimuth at A, s = AB, and ρ and n are the radii of +curvature of the meridian and perpendicular at A. For s = 100 +kilometres, only the first term is of moment; its value is 0″.028 +cos² φ sin 2α, and it lies well within the errors of observation. If we +imagine the geodetic AB, it will generally trisect the angles between +the vertical sections at A and B, so that the geodetic at A is near +<span class="pagenum"><a name="page613" id="page613"></a>613</span> +the vertical section AB, and at B near the section BA.<a name="fa3a" id="fa3a" href="#ft3a"><span class="sp">3</span></a> The +greatest distance of the vertical sections one from another is +e²s³ cos² φ<span class="su">0</span> sin 2α<span class="su">0</span>/16a², in which φ<span class="su">0</span> and α<span class="su">0</span> are the mean latitude +and azimuth respectively of the middle point of AB. For the value +s = 64 kilometres, the maximum distance is 3 mm.</p> + +<p>An idea of the course of a longer geodetic line may be gathered +from the following example. Let the line be that joining Cadiz and +St Petersburg, whose approximate positions are—</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc">Cadiz.</td> <td class="tcc">St Petersburg.</td></tr> +<tr><td class="tcl">Lat.   36° 22′ N.</td> <td class="tcc">59° 56′ N.</td></tr> +<tr><td class="tcl">Long. 6°   18′ W.</td> <td class="tcc">30° 17′ E.</td></tr> +</table> + +<p class="noind">If G be the point on the geodetic corresponding to F on that one +of the plane curves which contains the normal at Cadiz (by “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.</p> + +<p>The azimuth of a geodetic line cannot be observed, so that the +line does not enter of necessity into practical geodesy, although +many formulae connected with its use are of great simplicity and +elegance. The geodetic line has always held a more important place +in the science of geodesy among the mathematicians of France, +Germany and Russia than has been assigned to it in the operations +of the English and Indian triangulations. Although the observed +angles of a triangulation are not geodetic angles, yet in the calculation +of the distance and reciprocal bearings of two points which +are far apart, and are connected by a long chain of triangles, we may +fall upon the geodetic line in this manner:—</p> + +<p>If A, Z be the points, then to start the calculation from A, we +obtain by some preliminary calculation the approximate azimuth +of Z, or the angle made by the direction of Z with the side AB or +AC of the first triangle. Let P<span class="su">1</span> be the point where this line intersects +BC; then, to find P<span class="su">2</span>, where the line cuts the next triangle +side CD, we make the angle BP<span class="su">1</span>P<span class="su">2</span> such that BP<span class="su">1</span>P<span class="su">2</span> + BP<span class="su">1</span>A = 180°. +This fixes P<span class="su">2</span>, and P<span class="su">3</span> is fixed by a repetition of the same process; +so for P<span class="su">4</span>, P<span class="su">5</span> .... Now it is clear that the points P<span class="su">1</span>, P<span class="su">2</span>, P<span class="su">3</span> so computed +are those which would be actually fixed by an observer with +a theodolite, proceeding in the following manner. Having set the +instrument up at A, and turned the telescope in the direction of +the computed bearing, an assistant places a mark P<span class="su">1</span> on the line +BC, adjusting it till bisected by the cross-hairs of the telescope at +A. The theodolite is then placed over P<span class="su">1</span>, and the telescope turned +to A; the horizontal circle is then moved through 180°. The +assistant then places a mark P<span class="su">2</span> on the line CD, so as to be bisected +by the telescope, which is then moved to P<span class="su">2</span>, and in the same manner +P<span class="su">3</span> is fixed. Now it is clear that the series of points P<span class="su">1</span>, P<span class="su">2</span>, P<span class="su">3</span> +approaches to the geodetic line, for the plane of any two consecutive +elements P<span class="su">n−1</span> P<span class="su">n</span>, P<span class="su">n</span> P<span class="su">n+1</span> contains the normal at P<span class="su">n</span>.</p> + +<p>If the objection be raised that not the geodetic azimuths but the +astronomical azimuths are observed, it is necessary to consider that +the observed vertical sections do not correspond to points on the +sea-level but to elevated points. Since the normals of the ellipsoid +of rotation do not in general intersect, there consequently arises an +influence of the height on the azimuth. In the case of the measurement +of the azimuth from A to B, the instrument is set to a point A′ +over the surface of the ellipsoid (the sea-level), and it is then adjusted +to a point B′, also over the surface, say at a height h′. The vertical +plane containing A′ and B′ also contains A but not B: it must +therefore be rotated through a small azimuth in order to contain B. +The correction amounts approximately to −e²h′ cos²φ sin 2α/2a; +in the case of h′ = 1000 m., its value is 0″.108 cos²φ sin 2α.</p> + +<p>This correction is therefore of greater importance in the case of +observed azimuths and horizontal angles than in the previously +considered case of the astronomical and the geodetic azimuths. The +observed azimuths and horizontal angles must therefore also be +corrected in the case, where it is required to dispense with geodetic +lines.</p> + +<p>When the angles of a triangulation have been adjusted by the +method of least squares, and the sides are calculated, the next +process is to calculate the latitudes and longitudes of all the stations +starting from one given point. The calculated latitudes, longitudes +and azimuths, which are designated geodetic latitudes, longitudes +and azimuths, are not to be confounded with the observed latitudes, +longitudes and azimuths, for these last are subject to somewhat +large errors. Supposing the latitudes of a number of stations in the +triangulation to be observed, practically the mean of these determines +the position in latitude of the network, taken as a whole. So the +orientation or general azimuth of the whole is inferred from all the +azimuth observations. The triangulation is then supposed to be +projected on a spheroid of given elements, representing as nearly as +one knows the real figure of the earth. Then, taking the latitude +of one point and the direction of the meridian there as given—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 φ, φ′ be the latitudes of two stations +A and B; α, α<span class="sp">*</span> their mutual azimuths counted from north by east +continuously from 0° to 360°; ω their difference of longitude +measured from west to east; and s the distance AB.</p> + +<p>First compute a latitude φ<span class="su">1</span> by means of the formula φ<span class="su">1</span> = φ ++ (s cos α)/ρ, where ρ is the radius of curvature of the meridian at the +latitude φ; this will require but four places of logarithms. Then, +in the first two of the following, five places are sufficient—</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ε =</td> <td>s²</td> +<td rowspan="2">sin α cos a,   η =</td> <td>s²</td> +<td rowspan="2">sin² α tan φ<span class="su">1</span>,</td></tr> +<tr><td class="denom">2ρn</td> <td class="denom">2ρn</td></tr></table> + +<table class="math0" summary="math"> +<tr><td rowspan="2">φ′ − φ =</td> <td>s</td> +<td rowspan="2">cos (α − <span class="spp">2</span>⁄<span class="suu">3</span>ε) − η,</td></tr> +<tr><td class="denom">rho<span class="su">0</span></td></tr></table> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ω =</td> <td>s sin (alpha − <span class="spp">1</span>⁄<span class="suu">3</span>ε)</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">n cos (φ′ + <span class="spp">1</span>⁄<span class="suu">3</span>η)</td></tr></table> + +<p class="center">α<span class="sp">*</span> − α = ω sin (φ′ + <span class="spp">2</span>⁄<span class="suu">3</span>η) − ε + 180°.</p> + +<p class="noind">Here n is the normal or radius of curvature perpendicular to the +meridian; both n and ρ correspond to latitude φ<span class="su">1</span>, and ρ<span class="su">0</span> to latitude +½(φ + φ′). For calculations of latitude and longitude, tables of the +logarithmic values of ρ sin 1″, n sin 1″, and 2 n ρ sin 1″ are necessary. +The following table contains these logarithms for every ten minutes +of latitude from 52° to 53° computed with the elements a = 20926060 +and a : b = 295 : 294 :—</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc allb">Lat.</td> <td class="tcc allb">Log. 1/ρ sin 1″.</td> <td class="tcc allb">Log. 1/n sin 1″.</td> <td class="tcc allb">Log. 1/2ρn sin 1″.</td></tr> + +<tr><td class="tcr lb rb">°   ′</td> <td class="rb"> </td> <td class="rb"> </td> <td class="rb"> </td></tr> + +<tr><td class="tcr lb rb">52 0</td> <td class="tcr rb">7.9939434</td> <td class="tcr rb">7.9928231</td> <td class="tcr rb">0.37131</td></tr> +<tr><td class="tcr lb rb">10</td> <td class="tcr rb">9309</td> <td class="tcr rb">8190</td> <td class="tcr rb">29</td></tr> +<tr><td class="tcr lb rb">20</td> <td class="tcr rb">9185</td> <td class="tcr rb">8148</td> <td class="tcr rb">28</td></tr> +<tr><td class="tcr lb rb">30</td> <td class="tcr rb">9060</td> <td class="tcr rb">8107</td> <td class="tcr rb">26</td></tr> +<tr><td class="tcr lb rb">40</td> <td class="tcr rb">8936</td> <td class="tcr rb">8065</td> <td class="tcr rb">24</td></tr> +<tr><td class="tcr lb rb">50</td> <td class="tcr rb">8812</td> <td class="tcr rb">8024</td> <td class="tcr rb">23</td></tr> +<tr><td class="tcr lb rb bb">53 0</td> <td class="tcr rb bb">8688</td> <td class="tcr rb bb">7982</td> <td class="tcr rb bb">22</td></tr> +</table> + +<p>The logarithm in the last column is that required also for the +calculation of spherical excesses, the spherical excess of a triangle +being expressed by a b sin C/(2ρn) sin 1″.</p> + +<p>It is frequently necessary to obtain the co-ordinates of one point +with reference to another point; that is, let a perpendicular arc be +drawn from B to the meridian of A meeting it in P, then, α being +the azimuth of B at A, the co-ordinates of B with reference to A are</p> + +<p class="center">AP = s cos (α − <span class="spp">2</span>⁄<span class="suu">3</span>ε), BP = s sin (α − <span class="spp">1</span>⁄<span class="suu">3</span>ε),</p> + +<p class="noind">where ε is the spherical excess of APB, viz. s² sin α cos α multiplied +by the quantity whose logarithm is in the fourth column of the above +table.</p> + +<p>If it be necessary to determine the geographical latitude and +longitude as well as the azimuths to a greater degree of accuracy +than is given by the above formulae, we make use of the following +formula: given the latitude φ of A, and the azimuth α and the +distance s of B, to determine the latitude φ′ and longitude ω of B, +and the back azimuth α′. Here it is understood that α′ is symmetrical +to α, so that α<span class="sp">*</span> + α′ = 360°.</p> + +<p class="noind">Let</p> + +<p class="center">θ = sΔ / a, where Δ = (1 − e² sin² φ)<span class="sp">1/2</span></p> + +<p class="noind">and</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ξ =</td> <td>e² θ²</td> +<td rowspan="2">cos² φ sin 2α,   ξ′ =</td> <td>e² θ³</td> +<td rowspan="2">cos² φ cos² α;</td></tr> +<tr><td class="denom">4 (1 − e²)</td> <td class="denom">6 (1 − e²)</td></tr></table> + +<p>ξ, ξ′ are always very minute quantities even for the longest distances; +then, putting κ = 90° − φ,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">tan</td> <td>α′ + ξ − ω</td> +<td rowspan="2">=</td> <td>sin ½(κ − θ − ξ′)</td> +<td rowspan="2">cot</td> <td>α</td></tr> +<tr><td class="denom">2</td> <td class="denom">sin ½(κ + θ + ξ′)</td> +<td class="denom">2</td></tr></table> + +<table class="math0" summary="math"> +<tr><td rowspan="2">tan</td> <td>α′ + ξ − ω</td> +<td rowspan="2">=</td> <td>cos ½(κ − θ − ξ′)</td> +<td rowspan="2">cot</td> <td>α</td></tr> +<tr><td class="denom">2</td> <td class="denom">cos ½(κ + θ + ξ′)</td> +<td class="denom">2</td></tr></table> + +<table class="math0" summary="math"> +<tr><td rowspan="2">φ′ − φ =</td> <td>s sin ½(α′ + ξ − α)</td> +<td rowspan="2"><span class="f150">(</span> 1 +</td> <td>θ²</td> +<td rowspan="2">cos²</td> <td>α′ − α</td> +<td rowspan="2"><span class="f150">)</span>;</td></tr> +<tr><td class="denom">ρ<span class="su">0</span> sin ½(α′ + ξ + α)</td> <td class="denom">12</td> +<td class="denom">2</td></tr></table> + +<p class="noind">here ρ<span class="su">0</span> is the radius of curvature of the meridian for the mean +latitude ½(φ + φ′). These formulae are approximate only, but they +are sufficiently precise even for very long distances.</p> + +<p>For lines of any length the formulae of F.W. Bessel (<i>Astr. Nach.</i>, +1823, iv. 241) are suitable.</p> + +<p>If the two points A and B be defined by their geographical +<span class="pagenum"><a name="page614" id="page614"></a>614</span> +co-ordinates, we can accurately calculate the corresponding astronomical +azimuths, <i>i.e.</i> those of the vertical section, and then proceed, +in the case of not too great distances, to determine the length and +the azimuth of the shortest lines. For <i>any</i> distances recourse must +again be made to Bessel’s formula.<a name="fa4a" id="fa4a" href="#ft4a"><span class="sp">4</span></a></p> + +<p>Let α, α′ be the mutual azimuths of two points A, B on a spheroid, +k the chord line joining them, μ, μ′ the angles made by the chord +with the normals at A and B, φ, φ′, ω their latitudes and difference of +longitude, and (x² + y²)/a² + z² b² = 1 the equation of the surface; +then if the plane xz passes through A the co-ordinates of A and B +will be</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">x = (a/Δ) cos φ,</td> <td class="tcl">x′ = (a/Δ’) cos φ′ cos ω,</td></tr> + +<tr><td class="tcl">y = 0</td> <td class="tcl">y′ = (a/Δ’) cos φ′ sin ω,</td></tr> + +<tr><td class="tcl">z = (a/Δ) (1 − e²) sin φ,</td> <td class="tcl">z′ = (a/Δ′) (1 − e²) sin φ′,</td></tr> +</table> + +<p class="noind">where Δ = (1 − e² sin² φ)<span class="sp">1/2</span>, Δ′ = (1 − e² sin² φ′)<span class="sp">1/2</span>, and e is the eccentricity. +Let f, g, h be the direction cosines of the normal to that +plane which contains the normal at A and the point B, and whose +inclinations to the meridian plane of A is = α; 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</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcr">f (x′ − x) + gy′ + h (z′ − z) = 0</td></tr> + +<tr><td class="tcr">fl + gm + hn = 0</td></tr> + +<tr><td class="tcr">fl′ + gm′ + hn′ = 0.</td></tr> +</table> + +<p>Eliminate f, g, h from these equations, and substitute</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">l = cos φ</td> <td class="tcl">l′ = − sin φ cos α</td></tr> + +<tr><td class="tcl">m = 0</td> <td class="tcl">m′ = sin α</td></tr> + +<tr><td class="tcl">n = sin φ</td> <td class="tcl">n′ = cos φ cos α,</td></tr> +</table> + +<p class="noind">and we get</p> + +<p class="center">(x′ − x) sin φ + y′ cot α − (z′ − z) cos φ = 0.</p> + +<p class="noind">The substitution of the values of x, z, x′, y′, z′ in this equation will +give immediately the value of cot α; and if we put ζ, ζ’ for the +corresponding azimuths on a sphere, or on the supposition e = 0, +the following relations exist</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">cot α − cot ζ = e²</td> <td>cos φ Q</td></tr> +<tr><td class="denom">cos φ′ Δ</td></tr></table> + +<table class="math0" summary="math"> +<tr><td rowspan="2">cot α′ − cot ζ′ = −e²</td> <td>cos φ′ Q</td> +<td rowspan="2"></td></tr> +<tr><td class="denom">cos φ Δ′</td></tr></table> + +<p class="center">Δ′ sin φ − Δ sin φ′ = Q sin ω.</p> + +<p>If from B we let fall a perpendicular on the meridian plane of A, +and from A let fall a perpendicular on the meridian plane of B, +then the following equations become geometrically evident:</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">k sin μ sin α = (a/Δ′) cos φ′ sin ω</td></tr> +<tr><td class="tcl">k sin μ′ sin α′ = (a/Δ) cos φ sin ω.</td></tr> +</table> + +<p>Now in any surface u = 0 we have</p> + +<p class="center">k² = (x′ − x)² + (y′ − y)² + (z′ − z)²</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">−cos μ = <span class="f150">[</span> (x′ − x)</td> <td>du</td> +<td rowspan="2">+ (y′ − y)</td> <td>du</td> +<td rowspan="2">+ (z′ − z)</td> <td>du</td> +<td rowspan="2"><span class="f150">] /</span> k <span class="f150">(</span></td> <td>du²</td> +<td rowspan="2">+</td> <td>du²</td> +<td rowspan="2">+</td> <td>du²</td> +<td rowspan="2"><span class="f150">)</span></td> <td><span class="sp">1/2</span></td></tr> +<tr><td class="denom">dx</td> <td class="denom">dy</td> +<td class="denom">dz</td> <td class="denom">dx²</td> +<td class="denom">dy²</td> <td class="denom">dz²</td></tr></table> + +<table class="math0" summary="math"> +<tr><td rowspan="2">cos μ′ = <span class="f150">[</span> (x′ − x)</td> <td>du</td> +<td rowspan="2">+ (y′ − y)</td> <td>du</td> +<td rowspan="2">+ (z′ − z)</td> <td>du</td> +<td rowspan="2"><span class="f150">] /</span> k <span class="f150">(</span></td> <td>du²</td> +<td rowspan="2">+</td> <td>du²</td> +<td rowspan="2">+</td> <td>du²</td> +<td rowspan="2"><span class="f150">)</span></td> <td><span class="sp">1/2</span></td> <td rowspan="2">.</td></tr> +<tr><td class="denom">dx′</td> <td class="denom">dy′</td> +<td class="denom">dz′</td> <td class="denom">dx′²</td> +<td class="denom">dy′²</td> <td class="denom">dz′²</td></tr></table> + +<p class="noind">In the present case, if we put</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">1 −</td> <td>xx′</td> +<td rowspan="2">−</td> <td>zz′</td> +<td rowspan="2">= U,</td></tr> +<tr><td class="denom">a²</td> <td class="denom">b²</td></tr></table> + +<p class="noind">then</p> + +<table class="math0" summary="math"> +<tr><td>k²</td> +<td rowspan="2">= 2U − e² <span class="f150">(</span></td> <td>z′ − z</td> +<td rowspan="2"><span class="f150">)</span></td> <td>²</td></tr> +<tr><td class="denom">a²</td> <td class="denom">b</td></tr></table> + +<p class="center">cos μ = (a/k) ΔU; cos μ′ = (a/k) Δ′U.</p> + +<p class="noind">Let u be such an angle that</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcr">(1 − e²)<span class="sp">½</span> sin φ = Δ sin u</td></tr> + +<tr><td class="tcr">cos φ = Δ cos u,</td></tr> +</table> + +<p class="noind">then on expressing x, x′, z, z′ in terms of u and u′,</p> + +<p class="center">U = 1 − cos u cos u′ cos ω − sin u sin u′;</p> + +<p class="noind">also, if v be the third side of a spherical triangle, of which two +sides are ½π − u and ½π − u′ and the included angle ω, using a subsidiary +angle ψ such that</p> + +<p class="center">sin ψ sin ½v = e sin ½ (u′ − u) cos ½ (u′ + u),</p> + +<p class="noind">we obtain finally the following equations:—</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcr">k</td> <td class="tcl">= 2a cos ψ sin ½v</td></tr> +<tr><td class="tcr">cos μ</td> <td class="tcl">= Δ sec ψ sin ½v</td></tr> +<tr><td class="tcr">cos μ′</td> <td class="tcl">= Δ′ sec ψ sin ½v</td></tr> +<tr><td class="tcr">sin μ sin α</td> <td class="tcl">= (a/k) cos u′ sin ω</td></tr> +<tr><td class="tcr">sin μ′ sin α′</td> <td class="tcl">= (a/k) cos u sin ω.</td></tr> +</table> + +<p>These determine rigorously the distance, and the mutual zenith +distances and azimuths, of any two points on a spheroid whose +latitudes and difference of longitude are given.</p> + +<p>By a series of reductions from the equations containing ζ, ζ′ it +may be shown that</p> + +<p class="center">α + α′ = ζ + ζ′ + ¼e<span class="sp">4</span>ω (φ′ − φ)² cos<span class="sp">4</span> φ<span class="su">0</span> sin φ<span class="su">0</span> + ...,</p> + +<p class="noind">where φ<span class="su">0</span> is the mean of φ and φ′, 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 α + α′ on the +spheroid is equal to the sum of the spherical azimuths, whence +follows this very important theorem (known as Dalby′s theorem). +If φ, φ′ be the latitudes of two points on the surface of a spheroid, ω +their difference of longitude, α, α′ their reciprocal azimuths,</p> + +<p class="center">tan ½ω = cot ½ (α + α′) {cos ½ (φ′ − φ) / sin ½ (φ′ + φ)}.</p> + +<p>The computation of the geodetic from the astronomical azimuths +has been given above. From k we can now compute the length s +of the vertical section, and from this the shortest length. The +difference of length of the geodetic line and either of the plane +curves is</p> + +<p class="center">e<span class="sp">4</span>s<span class="sp">5</span> cos<span class="sp">4</span> φ<span class="su">0</span> sin² 2α<span class="su">0</span>/360 a<span class="sp">4</span>.</p> + +<p class="noind">At least this is an approximate expression. Supposing s = 0.1a, +this quantity would be less than one-hundredth of a millimetre. +The line s is now to be calculated as a circular arc with a mean radius r +along AB. If φ<span class="su">0</span> = ½ (φ + φ′), α<span class="su">0</span> = ½ (180° + α − α′), Δ<span class="su">0</span> = (1 − e² sin² φ<span class="su">0</span>)<span class="sp">1/2</span>, +then 1/r = Δ<span class="su">0</span>/a [1 + (e²/(1 − e²) cos² φ<span class="su">0</span> cos² α<span class="su">0</span>], and approximately sin (s/2r) = +k/2r. These formulae give, in the case of k = 0.1a, values certain to +eight logarithmic decimal places. An excellent series of formulae +for the solution of the problem, to determine the azimuths, chord +and distance along the surface from the geographical co-ordinates, +was given in 1882 by Ch. M. Schols (<i>Archives Néerlandaises</i>, vol. xvii.).</p> + +<p class="pt2 center"><i>Irregularities of the Earth’s Surface.</i></p> + +<p>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 ρ 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 ρ, a function of r only, but is +expressed by ρ + ρ′, where ρ′ is a function of three co-ordinates θ, φ, r. +Then ρ′ 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 ρ′ 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 ρ′, M the +mass of the earth (the attraction-constant is assumed equal to unity)</p> + +<table class="math0" summary="math"> +<tr><td>M</td> +<td rowspan="2">+ V = C =</td> <td>M</td> +<td rowspan="2">−</td> <td>M</td> +<td rowspan="2">N + V.</td></tr> +<tr><td class="denom">a + N</td> <td class="denom">a</td> +<td class="denom">a²</td></tr></table> + +<p class="noind">As far as we know, N is always a very small quantity, and we have +with sufficient approximation N = 3V/4πδa, where δ 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 <i>Astronomische Nachrichten</i>, Nos. 329, 330, 331, in a +paper entitled “Ueber den Einfluss der Unregelmässigkeiten der +Figur der Erde auf geodätische Arbeiten, &c.” But without entering +into further details it is not difficult to see how local attraction at +any station affects the determinations of latitude, longitude and +azimuth there.</p> + +<p>Let there be at the station an attraction to the north-east throwing +the zenith to the south-west, so that it takes in the celestial sphere a +position Z′, its undisturbed position being Z. Let the rectangular +components of the displacement ZZ′ be ξ measured southwards +<span class="pagenum"><a name="page615" id="page615"></a>615</span> +and η measured westwards. Now the great circle joining Z′ with +the pole of the heavens P makes there an angle with the meridian +PZ = η cosec PZ′ = η sec φ, where φ is the latitude of the station. +Also this great circle meets the horizon in a point whose distance +from the great circle PZ is η sec φ sin φ = η tan φ. 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 ξ; the observed longitude a correction η sec φ; and +any observed azimuth a correction η tan φ. Here it is supposed +that azimuths are measured from north by east, and longitudes +eastwards. The horizontal angles are also influenced by the deflections +of the plumb-line, in fact, just as if the direction of the vertical +axis of the theodolite varied by the same amount. This influence, +however, is slight, so long as the sights point almost horizontally +at the objects, which is always the case in the observation of distant +points.</p> + +<p>The expression given for N enables one to form an approximate +estimate of the effect of a compact mountain in raising the sea-level. +Take, for instance, Ben Nevis, which contains about a couple of +cubic miles; a simple calculation shows that the elevation produced +would only amount to about 3 in. In the case of a mountain mass +like the Himalayas, stretching over some 1500 miles of country with +a breadth of 300 and an average height of 3 miles, although it is difficult +or impossible to find an expression for V, yet we may ascertain +that an elevation amounting to several hundred feet may exist +near their base. The geodetical operations, however, rather negative +this idea, for it was shown by Colonel Clarke (<i>Phil. Mag.</i>, 1878) +that the form of the sea-level along the Indian arc departs but slightly +from that of the mean figure of the earth. If this be so, the action +of the Himalayas must be counteracted by subterranean tenuity.</p> + +<p>Suppose now that A, B, C, ... are the stations of a network of +triangulation projected on or lying on a spheroid of semiaxis major +and eccentricity a, e, this spheroid having its axis parallel to the axis +of rotation of the earth, and its surface coinciding with the mathematical +surface of the earth at A. Then basing the calculations +on the observed elements at A, the calculated latitudes, longitudes +and directions of the meridian at the other points will be the true +latitudes, &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 ξ and to the east by the quantity η. +Then in starting the calculation of geodetic latitudes, longitudes and +azimuths from A, we must take, not the observed elements φ, α, +but for φ, φ + ξ, and for α, α + η tan φ, and zero longitude must be +replaced by η sec φ. 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 (φ′), (α′), (ω) be the +numerical elements at B as obtained in the first calculation, viz. +before the shifting and alteration of the spheroid; they will now +take the form</p> + +<table class="reg" summary="poem"><tr><td> <div class="poemr"> +<p>(φ′) + fξ + gη + hda + kde,</p> +<p>(α′) + f′ξ + g′η + h′da + k′de,</p> +<p>ω + f″ξ + g″η + h″da + k″de,</p> +</div> </td></tr></table> + +<p class="noind">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 ξ′, η′ be the components of the inclination at that +point, then we have</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcr">ξ′</td> <td class="tcl">= (φ′) − φ′ + fξ + gη + hda + kde,</td></tr> + +<tr><td class="tcr">η′ tan φ′</td> <td class="tcl">= (α′) − α′ + f′ξ + g′η + h′da + k′de,</td></tr> + +<tr><td class="tcr">η′ sec φ′</td> <td class="tcl">= (ω) − ω + f″ξ + g″η + h″da + k″de,</td></tr> +</table> + +<p class="noind">where φ′, α′, ω are the observed elements at B. Here it appears +that the observation of longitude gives no additional information, +but is available as a check upon the azimuthal observations.</p> + +<p>If now there be a number of astronomical stations in the triangulation, +and we form equations such as the above for each point, +then we can from them determine those values of ξ, η, da, de, which +make the quantity ξ² + η² + ξ′² + η′² + ... a minimum. Thus we +obtain that spheroid which best represents the surface covered by the +triangulation.</p> + +<p>In the <i>Account of the Principal Triangulation of Great Britain and +Ireland</i> will be found the determination, from 75 equations, of the +spheroid best representing the surface of the British Isles. Its +elements are a = 20927005 ± 295 ft., b : a − b = 280 ± 8; and it is so +placed that at Greenwich Observatory ξ = 1″.864, η = −0″.546.</p> + +<p>Taking Durham Observatory as the origin, and the tangent plane +to the surface (determined by ξ = −0″.664, η = −4″.117) as the plane +of x and y, the former measured northwards, and z measured vertically +downwards, the equation to the surface is</p> + +<p class="center">.99524953 x² + .99288005 y² + .99763052 z² − 0.00671003xz − 41655070z = 0.</p> + +<p class="pt2 center"><i>Altitudes.</i></p> + +<p>The precise determination of the altitude of his station is a matter +of secondary importance to the geodesist; nevertheless it is usual +to observe the zenith distances of all trigonometrical points. Of +great importance is a knowledge of the height of the base for its reduction +to the sea-level. Again the height of a station does influence +a little the observation of terrestrial angles, for a vertical line at B +does not lie generally in the vertical plane of A (see above). The +height above the sea-level also influences the geographical latitude, +inasmuch as the centrifugal force is increased and the magnitude and +direction of the attraction of the earth are altered, and the effect +upon the latitude is a very small term expressed by the formula +h (g′ − g) sin 2 φ/ag, where g, g′ are the values of gravity at the equator +and at the pole. This is h sin 2 φ/5820 seconds, h being in metres, +a quantity which may be neglected, since for ordinary mountain +heights it amounts to only a few hundredths of a second. We +can assume this amount as joined with the northern component of +the plumb-line perturbations.</p> + +<p>The uncertainties of terrestrial refraction render it impossible to +determine accurately by vertical angles the heights of distant points. +Generally speaking, refraction is greatest at about daybreak; from +that time it diminishes, being at a minimum for a couple of hours +before and after mid-day; later in the afternoon it again increases. +This at least is the general march of the phenomenon, but it is by +no means regular. The vertical angles measured at the station on +Hart Fell showed on one occasion in the month of September a +refraction of double the average amount, lasting from 1 <span class="scs">P.M.</span> to 5 <span class="scs">P.M.</span> +The mean value of the coefficient of refraction k determined from a +very large number of observations of terrestrial zenith distances in +Great Britain is .0792 ± .0047; and if we separate those rays which +for a considerable portion of their length cross the sea from those +which do not, the former give k = .0813 and the latter k = .0753. +These values are determined from high stations and long distances; +when the distance is short, and the rays graze the ground, the amount +of refraction is extremely uncertain and variable. A case is noted +in the Indian survey where the zenith distance of a station 10.5 miles +off varied from a depression of 4′ 52″.6 at 4.30 <span class="scs">P.M.</span> to an elevation +of 2′ 24″.0 at 10.50 <span class="scs">P.M.</span></p> + +<p>If h, h′ be the heights above the level of the sea of two stations, +90° + δ, 90° + δ′ their mutual zenith distances (δ being that observed +at h), s their distance apart, the earth being regarded as a sphere of +radius = a, then, with sufficient precision,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">h′ − h = s tan <span class="f150">(</span> s</td> <td>1 − 2k</td> +<td rowspan="2">− δ<span class="f150">)</span>,   h − h′ = s tan <span class="f150">(</span> s</td> <td>1 − 2k</td> +<td rowspan="2">− δ′<span class="f150">)</span>.</td></tr> +<tr><td class="denom">2a</td> <td class="denom">2a</td></tr></table> + +<p class="noind">If from a station whose height is h the horizon of the sea be observed +to have a zenith distance 90° + δ, then the above formula gives for h +the value</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">h =</td> <td>a</td> +<td rowspan="2"> </td> <td>tan² δ</td> +<td rowspan="2"></td></tr> +<tr><td class="denom">2</td> <td class="denom">1 − 2k</td></tr></table> + +<p>Suppose the depression δ to be n minutes, then h = 1.054n² if +the ray be for the greater part of its length crossing the sea; if +otherwise, h = 1.040n². To take an example: the mean of eight +observations of the zenith distance of the sea horizon at the top of +Ben Nevis is 91° 4′ 48″, or δ = 64.8; the ray is pretty equally disposed +over land and water, and hence h = 1.047n² = 4396 ft. The +actual height of the hill by spirit-levelling is 4406 ft., so that the error +of the height thus obtained is only 10 ft.</p> + +<p>The determination of altitudes by means of spirit-levelling is +undoubtedly the most exact method, particularly in its present +development as precise-levelling, by which there have been determined +in all civilized countries close-meshed nets of elevated points +covering the entire land.</p> +</div> +<div class="author">(A. R. C; F. R. H.)</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1a" id="ft1a" href="#fa1a"><span class="fn">1</span></a> An arrangement acting similarly had been previously introduced +by Borda.</p> + +<p><a name="ft2a" id="ft2a" href="#fa2a"><span class="fn">2</span></a> <i>Geodetic Survey of South Africa</i>, vol. iii. (1905), p. viii; <i>Les Nouveaux +Appareils pour la mesure rapide des bases géod.</i>, par J. René Benoît +et Ch. Éd. Guillaume (1906).</p> + +<p><a name="ft3a" id="ft3a" href="#fa3a"><span class="fn">3</span></a> See a paper “On the Course of Geodetic Lines on the Earth’s +Surface” in the <i>Phil. Mag.</i> 1870; Helmert, <i>Theorien der höheren +Geodäsie</i>, 1. 321.</p> + +<p><a name="ft4a" id="ft4a" href="#fa4a"><span class="fn">4</span></a> Helmert, Theorien der höheren Geodäsie, 1. 232, 247.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">GEOFFREY<a name="ar2" id="ar2"></a></span>, surnamed <span class="sc">Martel</span> (1006-1060), count of Anjou, +son of the count Fulk Nerra (<i>q.v.</i>) and of the countess Hildegarde +or Audegarde, was born on the 14th of October 1006. During his +father’s lifetime he was recognized as suzerain by Fulk l’Oison +(“the Gosling”), count of Vendôme, 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-Couër near Saint-Jouin-de-Marnes on the 20th of September +1033. He then tried to win recognition as dukes of Aquitaine for +the sons of his wife Agnes by William the Great, who were still +minors, but Fulk Nerra promptly took up arms to defend his +suzerain William the Fat, from whom he held the Loudunois and +<span class="pagenum"><a name="page616" id="page616"></a>616</span> +Saintonge in fief against his son. In 1036 Geoffrey Martel had to +liberate William the Fat, on payment of a heavy ransom, but the +latter having died in 1038, and the second son of William the +Great, Odo, duke of Gascony, having fallen in his turn at the +siege of Mauzé (10th of March 1039) Geoffrey made peace with his +father in the autumn of 1039, and had his wife’s two sons recognized +as dukes. About this time, also, he had interfered in the +affairs of Maine, though without much result, for having sided +against Gervais, bishop of Le Mans, who was trying to make +himself guardian of the young count of Maine, Hugh, he had been +beaten and forced to make terms with Gervais in 1038. In 1040 +he succeeded his father in Anjou and was able to conquer Touraine +(1044) and assert his authority over Maine (see <span class="sc"><a href="#artlinks">Anjou</a></span>). About +1050 he repudiated Agnes, his first wife, and married Grécie, the +widow of Bellay, lord of Montreuil-Bellay (before August 1052), +whom he subsequently left in order to marry Adela, daughter of a +certain Count Odo. Later he returned to Grécie, but again left +her to marry Adelaide the German. When, however, he died on +the 14th of November 1060, at the monastery of St Nicholas at +Angers, he left no children, and transmitted the countship to +Geoffrey the Bearded, the eldest of his nephews (see ANJOU).</p> + +<div class="condensed"> +<p>See Louis Halphen, <i>Le Comté d’Anjou au XI<span class="sp">e</span> siècle</i> (Paris, 1906). +A summary biography is given by Célestin Port, <i>Dictionnaire +historique, géographique et biographique de Maine-et-Loire</i> (3 vols., +Paris-Angers, 1874-1878), vol. ii. pp. 252-253, and a sketch of the +wars by Kate Norgate, <i>England under the Angevin Kings</i> (2 vols., +London, 1887), vol. i. chs. iii. iv.</p> +</div> +<div class="author">(L. H.*)</div> + + +<hr class="art" /> +<p><span class="bold">GEOFFREY,<a name="ar3" id="ar3"></a></span> surnamed <span class="sc">Plantagenet</span> [or <span class="sc">Plantegenet</span>] +(1113-1151), count of Anjou, was the son of Count Fulk the Young +and of Eremburge (or Arembourg of La Flèche); 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 (<i>genêt</i>). In 1127 he was made a knight, and on the 2nd of +June 1129 married Matilda, daughter of Henry I. of England, and +widow of the emperor Henry V. Some months afterwards he +succeeded to his father, who gave up the countship when he +definitively went to the kingdom of Jerusalem. The years of his +government were spent in subduing the Angevin barons and in +conquering Normandy (see <span class="sc"><a href="#artlinks">Anjou</a></span>). In 1151, while returning +from the siege of Montreuil-Bellay, he took cold, in consequence of +bathing in the Loir at Château-du-Loir, and died on the 7th of +September. He was buried in the cathedral of Le Mans. By his +wife Matilda he had three sons: Henry Plantagenet, born at Le +Mans on Sunday, the 5th of March 1133; Geoffrey, born at +Argentan on the 1st of June 1134; and William Long-Sword, born +on the 22nd of July 1136.</p> + +<div class="condensed"> +<p>See Kate Norgate, <i>England under the Angevin Kings</i> (2 vols., +London, 1887), vol. i. chs. v.-viii.; Célestin Port, <i>Dictionnaire +historique, géographique et biographique de Maine-et-Loire</i> (3 vols., +Paris-Angers, 1874-1878), vol. ii. pp. 254-256. A history of +Geoffrey le Bel has yet to be written; there is a biography of him +written in the 12th century by Jean, a monk of Marmoutier, <i>Historia +Gaufredi, ducis Normannorum et comitis Andegavorum</i>, published by +Marchegay et Salmon; “Chroniques des comtes d’Anjou” (<i>Société +de l’histoire de France</i>, Paris, 1856), pp. 229-310.</p> +</div> +<div class="author">(L. H.*)</div> + + +<hr class="art" /> +<p><span class="bold">GEOFFREY<a name="ar4" id="ar4"></a></span> (1158-1186), duke of Brittany, fourth son of the +English king Henry II. and his wife Eleanor of Aquitaine, was +born on the 23rd of September 1158. In 1167 Henry suggested a +marriage between Geoffrey and Constance (d. 1201), daughter and +heiress of Conan IV., duke of Brittany (d. 1171); and Conan not +only assented, perhaps under compulsion, to this proposal, but +surrendered the greater part of his unruly duchy to the English +king. Having received the homage of the Breton nobles, +Geoffrey joined his brothers, Henry and Richard, who, in alliance +with Louis VII. of France, were in revolt against their father; but +he made his peace in 1174, afterwards helping to restore order in +Brittany and Normandy, and aiding the new French king, Philip +Augustus, to crush some rebellious vassals. In July 1181 his +marriage with Constance was celebrated, and practically the +whole of his subsequent life was spent in warfare with his brother +Richard. In 1183 he made peace with his father, who had come +to Richard’s assistance; but a fresh struggle soon broke out for +the possession of Anjou, and Geoffrey was in Paris treating for +aid with Philip Augustus, when he died on the 19th of August +1186. He left a daughter, Eleanor, and his wife bore a +posthumous son, the unfortunate Arthur.</p> + + +<hr class="art" /> +<p><span class="bold">GEOFFREY<a name="ar5" id="ar5"></a></span> (<i>c.</i> 1152-1212), archbishop of York, was a bastard +son of Henry II., king of England. He was distinguished from +his legitimate half-brothers by his consistent attachment and +fidelity to his father. He was made bishop of Lincoln at the age +of twenty-one (1173); but though he enjoyed the temporalities +he was never consecrated and resigned the see in 1183. He then +became his father’s chancellor, holding a large number of lucrative +benefices in plurality. Richard nominated him archbishop of +York in 1189, but he was not consecrated till 1191, or enthroned +till 1194. Geoffrey, though of high character, was a man of +uneven temper; his history in chiefly one of quarrels, with the +see of Canterbury, with the chancellor <span class="correction" title="amended from Willian">William</span> Longchamp, with +his half-brothers Richard and John, and especially with his +canons at York. This last dispute kept him in litigation before +Richard and the pope for many years. He led the clergy in their +refusal to be taxed by John and was forced to fly the kingdom in +1207. He died in Normandy on the 12th of December 1212.</p> + +<div class="condensed"> +<p>See Giraldus Cambrensis, <i>Vita Galfridi</i>; Stubbs’s prefaces to +<i>Roger de Hoveden</i>, vols. iii. and iv. (Rolls Series).</p> +</div> +<div class="author">(H. W. C. D.)</div> + + +<hr class="art" /> +<p><span class="bold">GEOFFREY DE MONTBRAY<a name="ar6" id="ar6"></a></span> (d. 1093), bishop of Coutances +(<i>Constantiensis</i>), a right-hand man of William the Conqueror, was +a type of the great feudal prelate, warrior and administrator at +need. He knew, says Orderic, more about marshalling mailed +knights than edifying psalm-singing clerks. Obtaining, as a young +man, in 1048, the see of Coutances, by his brother’s influence +(see <span class="sc"><a href="#artlinks">Mowbray</a></span>), he raised from his fellow nobles and from their +Sicilian spoils funds for completing his cathedral, which was +consecrated in 1056. With bishop Odo, a warrior like himself, +he was on the battle-field of Hastings, exhorting the Normans to +victory; and at William’s coronation it was he who called on +them to acclaim their duke as king. His reward in England was a +mighty fief scattered over twelve counties. He accompanied +William on his visit to Normandy (1067), but, returning, led a +royal force to the relief of Montacute in September 1069. In 1075 +he again took the field, leading with Bishop Odo a vast host +against the rebel earl of Norfolk, whose stronghold at Norwich +they besieged and captured.</p> + +<p>Meanwhile the Conqueror had invested him with important +judicial functions. In 1072 he had presided over the great +Kentish suit between the primate and Bishop Odo, and about the +same time over those between the abbot of Ely and his despoilers, +and between the bishop of Worcester and the abbot of Ely, and +there is some reason to think that he acted as a Domesday +commissioner (1086), and was placed about the same time in +charge of Northumberland. The bishop, who attended the +Conqueror’s funeral, joined in the great rising against William +Rufus next year (1088), making Bristol, with which (as +Domesday shows) he was closely connected and where he had +built a strong castle, his base of operations. He burned Bath and +ravaged Somerset, but had submitted to the king before the end +of the year. He appears to have been at Dover with William in +January 1090, but, withdrawing to Normandy, died at Coutances +three years later. In his fidelity to Duke Robert he seems to +have there held out for him against his brother Henry, when the +latter obtained the Cotentin.</p> + +<div class="condensed"> +<p>See E.A. Freeman, <i>Norman Conquest</i> and <i>William Rufus</i>; J.H. +Round, <i>Feudal England</i>; and, for original authorities, the works of +Orderic Vitalis and William of Poitiers, and of Florence of Worcester; +the Anglo-Saxon Chronicle; William of Malmesbury’s <i>Gesta pontificum</i>, +and Lanfranc’s works, ed. Giles; Domesday Book.</p> +</div> +<div class="author">(J. H. R.)</div> + + +<hr class="art" /> +<p><span class="bold">GEOFFREY OF MONMOUTH<a name="ar7" id="ar7"></a></span> (d. 1154), bishop of St Asaph +and writer on early British history, was born about the year 1100. +Of his early life little is known, except that he received a liberal +education under the eye of his paternal uncle, Uchtryd, who was +at that time archdeacon, and subsequently bishop, of Llandaff. +In 1129 Geoffrey appears at Oxford among the witnesses of an +Oseney charter. He subscribes himself Geoffrey Arturus; +from this we may perhaps infer that he had already begun his +experiments in the manufacture of Celtic mythology. A first +edition of his <i>Historia Britonum</i> was in circulation by the year +<span class="pagenum"><a name="page617" id="page617"></a>617</span> +1139, although the text which we possess appears to date from +1147. This famous work, which the author has the audacity +to place on the same level with the histories of William of +Malmesbury and Henry of Huntingdon, professes to be a translation +from a Celtic source; “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 <i>Historia</i> to +preclude us from supposing that Geoffrey drew partly upon +confused traditions, partly on his own powers of invention, and +to a very slight degree upon the accepted authorities for early +British history. His chronology is fantastic and incredible; +William of Newburgh justly remarks that, if we accepted the +events which Geoffrey relates, we should have to suppose that +they had happened in another world. William of Newburgh +wrote, however, in the reign of Richard I. when the reputation +of Geoffrey’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.</p> + +<p>Before his death the <i>Historia Britonum</i> had already become a +model and a quarry for poets and chroniclers. The list of +imitators begins with Geoffrey Gaimar, the author of the <i>Estorie +des Engles</i> (<i>c.</i> 1147), and Wace, whose <i>Roman de Brut</i> (1155) is +partly a translation and partly a free paraphrase of the <i>Historia</i>. +In the next century the influence of Geoffrey is unmistakably +attested by the <i>Brut</i> of Layamon, and the rhyming English +chronicle of Robert of Gloucester. Among later historians who +were deceived by the <i>Historia Britonum</i> it is only needful to +mention Higdon, Hardyng, Fabyan (1512), Holinshed (1580) +and John Milton. Still greater was the influence of Geoffrey +upon those writers who, like Warner in <i>Albion’s England</i> (1586), +and Drayton in <i>Polyolbion</i> (1613), deliberately made their +accounts of English history as poetical as possible. The stories +which Geoffrey preserved or invented were not infrequently +a source of inspiration to literary artists. The earliest English +tragedy, <i>Gorboduc</i> (1565), the <i>Mirror for Magistrates</i> (1587), and +Shakespeare’s Lear, are instances in point. It was, however, +the Arthurian legend which of all his fabrications attained the +greatest vogue. In the work of expanding and elaborating this +theme the successors of Geoffrey went as far beyond him as he +had gone beyond Nennius; but he retains the credit due to the +founder of a great school. Marie de France, who wrote at the +court of Henry II., and Chrétien de Troyes, her French contemporary, +were the earliest of the avowed romancers to take +up the theme. The succeeding age saw the Arthurian story +popularized, through translations of the French romances, as +far afield as Germany and Scandinavia. It produced in England +the <i>Roman du Saint Graal</i> and the <i>Roman de Merlin</i>, both from +the pen of Robert de Borron; the <i>Roman de Lancelot</i>; the <i>Roman +de Tristan</i>, which is attributed to a fictitious Lucas de Gast. In +the reign of Edward IV. Sir Thomas Malory paraphrased and +arranged the best episodes of these romances in English prose. +His <i>Morte d’Arthur</i>, 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 <i>Historia Britonum</i> may be +illustrated in another way, by enumerating the more familiar +of the legends to which it first gave popularity. Of the twelve +books into which it is divided only three (Bks. IX., X., XI.) are +concerned with Arthur. Earlier in the work, however, we have +the adventures of Brutus; of his follower Corineus, the vanquisher +of the Cornish giant Goemagol (Gogmagog); of Locrinus and +his daughter Sabre (immortalized in Milton’s <i>Comus</i>); of Bladud +the builder of Bath; of Lear and his daughters; of the three +pairs of brothers, Ferrex and Porrex, Brennius and Belinus, +Elidure and Peridure. The story of Vortigern and Rowena +takes its final form in the <i>Historia Britonum</i>; and Merlin makes +his first appearance in the prelude to the Arthur legend. Besides +the <i>Historia Britonum</i> Geoffrey is also credited with a <i>Life of +Merlin</i> composed in Latin verse. The authorship of this work +has, however, been disputed, on the ground that the style is distinctly +superior to that of the <i>Historia</i>. A minor composition, the +<i>Prophecies of Merlin</i>, was written before 1136, and afterwards incorporated +with the <i>Historia</i>, of which it forms the seventh book.</p> + +<div class="condensed"> +<p>For a discussion of the manuscripts of Geoffrey’s work, see Sir +T.D. Hardy’s <i>Descriptive Catalogue</i> (Rolls Series), i. pp. 341 ff. The +<i>Historia Britonum</i> has been critically edited by San Marte (Halle, +1854). There is an English translation by J.A. Giles (London, 1842). +The <i>Vita Merlini</i> has been edited by F. Michel and T. Wright (Paris, +1837). See also the <i>Dublin Univ. Magazine</i> for April 1876, for an +article by T. Gilray on the literary influence of Geoffrey; G. Heeger’s +<i>Trojanersage der Britten</i> (1889); and La Borderie’s <i>Études historiques +bretonnes</i> (1883).</p> +</div> +<div class="author">(H. W. C. D.)</div> + + +<hr class="art" /> +<p><span class="bold">GEOFFREY OF PARIS<a name="ar8" id="ar8"></a></span> (d. <i>c.</i> 1320), French chronicler, was +probably the author of the <i>Chronique métrique de Philippe le +Bel, or Chronique rimée de Geoffroi de Paris</i>. This work, which +deals with the history of France from 1300 to 1316, contains +7918 verses, and is valuable as that of a writer who had a personal +knowledge of many of the events which he relates. Various short +historical poems have also been attributed to Geoffrey, but there +is no certain information about either his life or his writings.</p> + +<div class="condensed"> +<p>The <i>Chronique</i> was published by J.A. Buchon in his <i>Collection des +chroniques</i>, tome ix. (Paris, 1827), and it has also been printed in +tome xxii. of the <i>Recueil des historiens des Gaules et de la France</i> +(Paris, 1865). See G. Paris, <i>Histoire de la littérature française au +moyen âge</i> (Paris, 1890); and A. Molinier, <i>Les Sources de l’histoire de +France</i>, tome iii. (Paris, 1903).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">GEOFFREY THE BAKER<a name="ar9" id="ar9"></a></span> (d. <i>c.</i> 1360), English chronicler, +is also called Walter of Swinbroke, and was probably a secular +clerk at Swinbrook in Oxfordshire. He wrote a <i>Chronicon +Angliae temporibus Edwardi II. et Edwardi III.</i>, which deals +with the history of England from 1303 to 1356. From the beginning +until about 1324 this work is based upon Adam Murimuth’s +<i>Continuatio chronicarum</i>, but after this date it is valuable and +interesting, containing information not found elsewhere, and +closing with a good account of the battle of Poitiers. The author +obtained his knowledge about the last days of Edward II. from +William Bisschop, a companion of the king’s murderers, Thomas +Gurney and John Maltravers. Geoffrey also wrote a <i>Chroniculum</i> +from the creation of the world until 1336, the value of +which is very slight. His writings have been edited with notes +by Sir E.M. Thompson as the <i>Chronicon Galfridi le Baker de +Swynebroke</i> (Oxford, 1889). Some doubt exists concerning +Geoffrey’s share in the compilation of the <i>Vita et mors Edwardi +II.</i>, usually attributed to Sir Thomas de la More, or Moor, and +printed by Camden in his <i>Anglica scripta</i>. It has been maintained +by Camden and others that More wrote an account of Edward’s +reign in French, and that this was translated into Latin by +Geoffrey and used by him in compiling his <i>Chronicon</i>. Recent +scholarship, however, asserts that More was no writer, and that +the <i>Vita et mors</i> is an extract from Geoffrey’s <i>Chronicon</i>, 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 <i>Vita et mors</i> in his <i>Chronicles of the reigns of +Edward I. and Edward II.</i> (London, 1883). The manuscripts +of Geoffrey’s works are in the Bodleian library at Oxford.</p> + + +<hr class="art" /> +<p><span class="bold">GEOFFRIN, MARIE THÉRÈSE RODET<a name="ar10" id="ar10"></a></span> (1699-1777), a +Frenchwoman who played an interesting part in French literary +and artistic life, was born in Paris in 1699. She married, on the +19th of July 1713, Pierre François Geoffrin, a rich manufacturer +and lieutenant-colonel of the National Guard, who died in 1750. +It was not till Mme Geoffrin was nearly fifty years of age that we +begin to hear of her as a power in Parisian society. She had +learned much from Mme de Tencin, and about 1748 began to +gather round her a literary and artistic circle. She had every +week two dinners, on Monday for artists, and on Wednesday for +her friends the Encyclopaedists and other men of letters. She +received many foreigners of distinction, Hume and Horace +Walpole among others. Walpole spent much time in her society +before he was finally attached to Mme du Deffand, and speaks of +her in his letters as a model of common sense. She was indeed +somewhat of a small tyrant in her circle. She had adopted the +pose of an old woman earlier than necessary, and her coquetry, if +<span class="pagenum"><a name="page618" id="page618"></a>618</span> +such it can be called, took the form of being mother and mentor to +her guests, many of whom were indebted to her generosity for +substantial help. Although her aim appears to have been to +have the <i>Encyclopédie</i> in conversation and action around her, she +was extremely displeased with any of her friends who were so +rash as to incur open disgrace. Marmontel lost her favour after +the official censure of <i>Bélisaire</i>, and her advanced views did not +prevent her from observing the forms of religion. A devoted +Parisian, Mme Geoffrin rarely left the city, so that her journey to +Poland in 1766 to visit the king, Stanislas Poniatowski, whom she +had known in his early days in Paris, was a great event in her life. +Her experiences induced a sensible gratitude that she had been +born “<i>Française</i>” and “<i>particulière</i>.” In her last illness her +daughter, Thérèse, marquise de la Ferté Imbault, excluded her +mother’s old friends so that she might die as a good Christian, a +proceeding wittily described by the old lady: “My daughter is +like Godfrey de Bouillon, she wished to defend my tomb from +the infidels.” Mme Geoffrin died in Paris on the 6th of October +1777.</p> + +<div class="condensed"> +<p>See <i>Correspondance inédite du roi Stanislas Auguste Poniatowski et +de Madame Geoffrin</i>, edited by the comte de Mouÿ (1875); P. de +Ségur, <i>Le Royaume de la rue Saint-Honoré, Madame Geoffrin et sa +fille</i> (1897); A. Tornezy, <i>Un Bureau d’esprit au XVIII<span class="sp">e</span> siècle: le +salon de Madame Geoffrin</i> (1895); and Janet Aldis, <i>Madame Geoffrin, +her Salon and her Times, 1750-1777</i> (1905).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">GEOFFROY, ÉTIENNE FRANÇOIS<a name="ar11" id="ar11"></a></span> (1672-1731), French +chemist, born in Paris on the 13th of February 1672, was first +an apothecary and then practised medicine. After studying at +Montpellier he accompanied Marshal Tallard on his embassy to +London in 1698 and thence travelled to Holland and Italy. +Returning to Paris he became professor of chemistry at the +Jardin du Roi and of pharmacy and medicine at the Collège de +France, and dean of the faculty of medicine. He died in Paris on +the 6th of January 1731. His name is best known in connexion +with his tables of affinities (<i>tables des rapports</i>), which he presented +to the French Academy in 1718 and 1720. These were lists, +prepared by collating observations on the actions of substances +one upon another, showing the varying degrees of affinity exhibited +by analogous bodies for different reagents, and they retained +their vogue for the rest of the century, until displaced by the +profounder conceptions introduced by C.L. Berthollet. Another +of his papers dealt with the delusions of the philosopher’s stone, +but nevertheless he believed that iron could be artificially formed +in the combustion of vegetable matter. His <i>Tractatus de materia +medica</i>, published posthumously in 1741, was long celebrated.</p> + +<p>His brother <span class="sc">Claude Joseph</span>, known as Geoffroy the younger +(1685-1752), was also an apothecary and chemist who, having a +considerable knowledge of botany, devoted himself especially to +the study of the essential oils in plants.</p> + + +<hr class="art" /> +<p><span class="bold">GEOFFROY, JULIEN LOUIS<a name="ar12" id="ar12"></a></span> (1743-1814), French critic, was +born at Rennes in 1743. He studied in the school of his native +town and at the Collège Louis le Grand in Paris. He took orders +and fulfilled for some time the humble functions of an usher, +eventually becoming professor of rhetoric at the <i>Collège Mazarin</i>. +A bad tragedy, Caton, was accepted at the <i>Théâtre Français</i>, but +was never acted. On the death of Élie Fréron in 1776 the other +collaborators in the <i>Année littéraire</i> asked Geoffroy to succeed him, +and he conducted the journal until in 1792 it ceased to appear. +Geoffroy was a bitter critic of Voltaire and his followers, and +made for himself many enemies. An enthusiastic royalist, +he published with Fréron’s brother-in-law, the abbé Thomas +Royou (1741-1792), a journal, <i>L’Ami du roi</i> (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 <i>Année littéraire</i> failed, and Geoffroy undertook the +dramatic feuilleton of the <i>Journal des débats</i>. His scathing +criticisms had a success of notoriety, but their popularity was +ephemeral, and the publication of them (5 vols., 1819-1820) as +<i>Cours de littérature dramatique</i> proved a failure. He was also the +author of a perfunctory <i>Commentaire</i> on the works of Racine +prefixed to Lenormant’s edition (1808). He died in Paris on the +27th of February 1814.</p> + + +<hr class="art" /> +<p><span class="bold">GEOFFROY SAINT-HILAIRE, ÉTIENNE<a name="ar13" id="ar13"></a></span> (1772-1844), French +naturalist, was the son of Jean Gèrard Geoffroy, procurator and +magistrate of Étampes, Seine-et-Oise, where he was born on the +15th of April 1772. Destined for the church he entered the +college of Navarre, in Paris, where he studied natural philosophy +under M.J. Brisson; and in 1788 he obtained one of the canonicates +of the chapter of Sainte Croix at Étampes, and also a +benefice. Science, however, offered him a more congenial career, +and he gained from his father permission to remain in Paris, and +to attend the lectures at the Collège de France and the Jardin des +Plantes, on the condition that he should also read law. He +accordingly took up his residence at Cardinal Lemoine’s college, +and there became the pupil and soon the esteemed associate of +Brisson’s friend, the abbé Haüy, the mineralogist. Having, +before the close of the year 1790, taken the degree of bachelor in +law, he became a student of medicine, and attended the lectures of +A.F. de Fourcroy at the Jardin des Plantes, and of L.J.M. +Daubenton at the Collège de France. His studies at Paris were at +length suddenly interrupted, for, in August 1792, Haüy and the +other professors of Lemoine’s college, as also those of the college +of Navarre, were arrested by the revolutionists as priests, and +confined in the prison of St Firmin. Through the influence of +Daubenton and others Geoffroy on the 14th of August obtained +an order for the release of Haüy in the name of the Academy; +still the other professors of the two colleges, save C.F. Lhomond, +who had been rescued by his pupil J.L. Tallien, remained in +confinement. Geoffroy, foreseeing their certain destruction if +they remained in the hands of the revolutionists, determined if +possible to secure their liberty by stratagem. By bribing one of +the officials at St Firmin, and disguising himself as a commissioner +of prisons, he gained admission to his friends, and entreated them +to effect their escape by following him. All, however, dreading +lest their deliverance should render the doom of their fellow-captives +the more certain, refused the offer, and one priest only, +who was unknown to Geoffroy, left the prison. Already on the +night of the 2nd of September the massacre of the proscribed had +begun, when Geoffroy, yet intent on saving the life of his friends +and teachers, repaired to St Firmin. At 4 o’clock on the morning +of the 3rd of September, after eight hours’ waiting, he by means +of a ladder assisted the escape of twelve ecclesiastics, not of the +number of his acquaintance, and then the approach of dawn and +the discharge of a gun directed at him warned him, his chief +purpose unaccomplished, to return to his lodgings. Leaving Paris +he retired to Étampes, where, in consequence of the anxieties of +which he had lately been the prey, and the horrors which he had +witnessed, he was for some time seriously ill. At the beginning +of the winter of 1792 he returned to his studies in Paris, and in +March of the following year Daubenton, through the interest of +Bernardin de Saint Pierre, procured him the office of sub-keeper +and assistant demonstrator of the cabinet of natural history, +vacant by the resignation of B.G.E. Lacépède. By a law +passed in June 1793, Geoffroy was appointed one of the twelve +professors of the newly constituted museum of natural history, +being assigned the chair of zoology. In the same year he +busied himself with the formation of a menagerie at that +institution.</p> + +<p>In 1794 through the introduction of A.H. Tessier he entered +into correspondence with Georges Cuvier, to whom, after the +perusal of some of his manuscripts, he wrote: “Venez jouer +parmi nous le rôle de Linné, d’un autre législateur de l’histoire +naturelle.” Shortly after the appointment of Cuvier as assistant +at the Muséum d’Histoire Naturelle, Geoffroy received him into +his house. The two friends wrote together five memoirs on +natural history, one of which, on the classification of mammals, +puts forward the idea of the subordination of characters upon +which Cuvier based his zoological system. It was in a paper +entitled “Histoire des Makis, ou singes de Madagascar,” written +in 1795, that Geoffroy first gave expression to his views on “the +unity of organic composition,” the influence of which is perceptible +in all his subsequent writings; nature, he observes, +presents us with only one plan of construction, the same in +principle, but varied in its accessory parts.</p> + +<p><span class="pagenum"><a name="page619" id="page619"></a>619</span></p> + +<p>In 1798 Geoffroy was chosen a member of the great scientific +expedition to Egypt, and on the capitulation of Alexandria in +August 1801, he took part in resisting the claim made by the +British general to the collections of the expedition, declaring that, +were that demand persisted in, history would have to record +that he also had burnt a library in Alexandria. Early in January +1802 Geoffroy returned to his accustomed labours in Paris. He +was elected a member of the academy of sciences of that city +in September 1807. In March of the following year the emperor, +who had already recognized his national services by the award +of the cross of the legion of honour, selected him to visit the +museums of Portugal, for the purpose of procuring collections +from them, and in the face of considerable opposition from the +British he eventually was successful in retaining them as a +permanent possession for his country. In 1809, the year after +his return to France, he was made professor of zoology at the +faculty of sciences at Paris, and from that period he devoted +himself more exclusively than before to anatomical study. In +1818 he gave to the world the first part of his celebrated <i>Philosophie +anatomique</i>, the second volume of which, published in +1822, and subsequent memoirs account for the formation of +monstrosities on the principle of arrest of development, and of +the attraction of similar parts. When, in 1830, Geoffroy proceeded +to apply to the invertebrata his views as to the unity of +animal composition, he found a vigorous opponent in Georges +Cuvier, and the discussion between them, continued up to the +time of the death of the latter, soon attracted the attention of +the scientific throughout Europe. Geoffroy, a synthesist, contended, +in accordance with his theory of unity of plan in organic +composition, that all animals are formed of the same elements, +in the same number, and with the same connexions: homologous +parts, however they differ in form and size, must remain associated +in the same invariable order. With Goethe he held that there +is in nature a law of compensation or balancing of growth, so +that if one organ take on an excess of development, it is at the +expense of some other part; and he maintained that, since +nature takes no sudden leaps, even organs which are superfluous +in any given species, if they have played an important part in +other species of the same family, are retained as rudiments, +which testify to the permanence of the general plan of creation. +It was his conviction that, owing to the conditions of life, the +same forms had not been perpetuated since the origin of all +things, although it was not his belief that existing species are +becoming modified. Cuvier, who was an analytical observer of +facts, admitted only the prevalence of “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.</p> + +<p>In July 1840 Geoffroy became blind, and some months later +he had a paralytic attack. From that time his strength gradually +failed him. He resigned his chair at the museum in 1841, and +died at Paris on the 19th of June 1844.</p> + +<div class="condensed"> +<p>Geoffroy wrote: <i>Catalogue des mammifères du Muséum National +d’Histoire Naturelle</i> (1813), not quite completed; <i>Philosophie anatomique</i>—t. +i., <i>Des organes respiratoires</i> (1818), and t. ii., <i>Des monstruosités +humaines</i> (1822); <i>Système dentaire des mammifères et des +oiseaux</i> (1st pt., 1824); <i>Sur le principe de l’unité de composition +organique</i> (1828); <i>Cours de l’histoire naturelle des mammifères</i> +(1829); <i>Principes de philosophie zoologique</i> (1830); <i>Études progressives +d’un naturaliste</i> (1835); <i>Fragments biographiques</i> (1832); +<i>Notions synthétiques, historiques et physiologiques de philosophie +naturelle</i> (1838), and other works; also part of the <i>Description de +l’Égypte par la commission des sciences</i> (1821-1830); and, with +Frédéric Cuvier (1773-1838), a younger brother of G. Cuvier, <i>Histoire +naturelle des mammifères</i> (4 vols., 1820-1842); besides numerous +papers on such subjects as the anatomy of marsupials, ruminants +and electrical fishes, the vertebrate theory of the skull, the opercula +of fishes, teratology, palaeontology and the influence of surrounding +conditions in modifying animal forms.</p> + +<p>See <i>Vie, travaux, et doctrine scientifique d’Étienne Geoffroy Saint-Hilaire, +par son fils M. Isidore Geoffroy Saint-Hilaire</i> (Paris and +Strasburg, 1847), to which is appended a list of Geoffroy’s works; +and Joly, in <i>Biog. universelle</i>, t. xvi. (1856).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">GEOFFROY SAINT-HILAIRE, ISIDORE<a name="ar14" id="ar14"></a></span> (1805-1861), French +zoologist, son of the preceding, was born at Paris on the 16th of +December 1805. In his earlier years he showed an aptitude for +mathematics, but eventually he devoted himself to the study +of natural history and of medicine, and in 1824 he was appointed +assistant naturalist to his father. On the occasion of his taking +the degree of doctor of medicine in September 1829, he read a +thesis entitled <i>Propositions sur la monstruosité, considérée chez +l’homme et les animaux</i>; and in 1832-1837 was published his +great teratological work, <i>Histoire générale et particulière des +anomalies de l’organisation chez l’homme et les animaux</i>, 3 vols. +8vo. with 20 plates. In 1829 he delivered for his father the second +part of a course of lectures on ornithology, and during the three +following years he taught zoology at the Athénée, and teratology +at the École pratique. He was elected a member of the academy +of sciences at Paris in 1833, was in 1837 appointed to act as +deputy for his father at the faculty of sciences in Paris, and in +the following year was sent to Bordeaux to organize a similar +faculty there. He became successively inspector of the academy +of Paris (1840), professor of the museum on the retirement of +his father (1841), inspector-general of the university (1844), +a member of the royal council for public instruction (1845), and +on the death of H.M.D. de Blainville, professor of zoology +at the faculty of sciences (1850). In 1854 he founded the +Acclimatization Society of Paris, of which he was president. +He died at Paris on the 10th of November 1861.</p> + +<div class="condensed"> +<p>Besides the above-mentioned works, he wrote: <i>Essais de zoologie +générale</i> (1841); <i>Vie ... d’Étienne Geoffroy Saint-Hilaire</i> (1847); +<i>Acclimatation et domestication des animaux utiles</i> (1849; 4th ed., +1861); <i>Lettres sur les substances alimentaires et particulièrement sur +la viande de cheval</i> (1856); and <i>Histoire naturelle générale des règnes +organiques</i> (3 vols., 1854-1862), which was not quite completed. +He was the author also of various papers on zoology, comparative +anatomy and palaeontology.</p> +</div> + + +<hr class="art" /> +<p><span class="bold">GEOGRAPHY<a name="ar15" id="ar15"></a></span> (Gr. <span class="grk" title="gê">γῆ</span>, earth, and <span class="grk" title="graphein">γράφειν</span>, to write), the +exact and organized knowledge of the distribution of phenomena +on the surface of the earth. The fundamental basis of geography +is the vertical relief of the earth’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.</p> + +<p>The fundamental conception of geography is form, including +<span class="pagenum"><a name="page620" id="page620"></a>620</span> +the figure of the earth and the varieties of crustal relief. Hence +mathematical geography (see <span class="sc"><a href="#artlinks">Map</a></span>), including cartography as +a practical application, comes first. It merges into physical +geography, which takes account of the forms of the lithosphere +(geomorphology), and also of the distribution of the hydrosphere +and the rearrangements resulting from the workings of solar +energy throughout the hydrosphere and atmosphere (oceanography +and climatology). Next follows the distribution of plants +and animals (biogeography), and finally the distribution of +mankind and the various artificial boundaries and redistributions +(anthropogeography). The applications of anthropogeography +to human uses give rise to political and commercial geography, +in the elucidation of which all the earlier departments or stages +have to be considered, together with historical and other purely +human conditions. The evolutionary idea has revolutionized +and unified geography as it did biology, breaking down the old +hard-and-fast partitions between the various departments, and +substituting the study of the nature and influence of actual +terrestrial environments for the earlier motive, the discovery +and exploration of new lands.</p> + +<p class="pt2 center sc">History of Geographical Theory</p> + +<div class="condensed"> +<p>The earliest conceptions of the earth, like those held by the primitive +peoples of the present day, are difficult to discover and almost +impossible fully to grasp. Early generalizations, as far as they were +made from known facts, were usually expressed in symbolic language, +and for our present purpose it is not profitable to speculate on the +underlying truths which may sometimes be suspected in the old +mythological cosmogonies.</p> + +<p>The first definite geographical theories to affect the western world +were those evolved, or at least first expressed, by the Greeks.<a name="fa1b" id="fa1b" href="#ft1b"><span class="sp">1</span></a> +The earliest theoretical problem of geography was the +<span class="sidenote">Early Greek ideas.<br />Flat earth of Homer.</span> +form of the earth. The natural supposition that the earth +is a flat disk, circular or elliptical in outline, had in the +time of Homer acquired a special definiteness by the +introduction of the idea of the ocean river bounding the whole, an +application of imperfectly understood observations. Thales of +Miletus is claimed as the first exponent of the idea of a +spherical earth; but, although this does not appear to be +warranted, his disciple Anaximander (<i>c.</i> 580 <span class="scs">B.C.</span>) put +forward the theory that the earth had the figure of a solid body +hanging freely in the centre of the hollow sphere of the starry heavens. +The Pythagorean school of philosophers adopted the theory of a +spherical earth, but from metaphysical rather than scientific reasons; +their convincing argument was that a sphere being the most perfect +solid figure was the only one worthy to circumscribe the dwelling-place +of man. The division of the sphere into parallel zones and +some of the consequences of this generalization seem to have presented +themselves to Parmenides (<i>c.</i> 450 <span class="scs">B.C.</span>); but these ideas did +not influence the Ionian school of philosophers, who in their treatment +of geography preferred to deal with facts demonstrable by +<span class="sidenote">Hecataeus.<br /><br />Herodotus.</span> +travel rather than with speculations. Thus Hecataeus, +claimed by H.F. Tozer<a name="fa2b" id="fa2b" href="#ft2b"><span class="sp">2</span></a> as the father of geography on +account of his <i>Periodos</i>, or general treatise on the earth, did not +advance beyond the primitive conception of a circular disk. He +systematized the form of the land within the ring of ocean—the +<span class="grk" title="oikoumenê">οἰκουμένη</span>, 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 <i>oekumene</i>, which he +knew to be longer from east to west than it was broad from north to +south. He also pointed out reasons for accepting a division of the +land into three continents—Europe, Asia and Africa. Beyond the +limits of his personal travels Herodotus applied the characteristically +Greek theory of symmetry to complete, in the unknown, outlines +<span class="sidenote">The idea of symmetry.</span> +of lands and rivers analogous to those which had been +explored. Symmetry was in fact the first geographical +theory, and the effect of Herodotus’s hypothesis that the +Nile must flow from west to east before turning north in +order to balance the Danube running from west to east before turning +south lingered in the maps of Africa down to the time of Mungo +Park.<a name="fa3b" id="fa3b" href="#ft3b"><span class="sp">3</span></a></p> + +<p>To Aristotle (384-322 <span class="scs">B.C.</span>) must be given the distinction of founding +scientific geography. He demonstrated the sphericity of the +earth by three arguments, two of which could be tested by observation. +These were: (1) that the earth must be spherical, because +<span class="sidenote">Aristotle and the sphere.</span> +of the tendency of matter to fall together towards a common +centre; (2) that only a sphere could always throw a +circular shadow on the moon during an eclipse; and (3) +that the shifting of the horizon and the appearance of +new constellations, or the disappearance of familiar stars, as one +travelled from north to south, could only be explained on the hypothesis +that the earth was a sphere. Aristotle, too, gave greater +definiteness to the idea of zones conceived by Parmenides, who had +pictured a torrid zone uninhabitable by reason of heat, two frigid +zones uninhabitable by reason of cold, and two intermediate temperate +zones fit for human occupation. Aristotle defined the temperate +zone as extending from the tropic to the arctic circle, but there is +some uncertainty as to the precise meaning he gave to the term +“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.</p> + +<p>While the theory of the sphere was being elaborated the efforts of +practical geographers were steadily directed towards ascertaining +the outline and configuration of the <i>oekumene</i>, or habitable +world, the only portion of the terrestrial surface known +<span class="sidenote">Fitting the oekumene to the sphere.</span> +to the ancients and to the medieval peoples, and still +retaining a shadow of its old monopoly of geographical +attention in its modern name of the “Old World.” The +fitting of the <i>oekumene</i> to the sphere was the second theoretical +problem. The circular outline had given way in geographical +opinion to the elliptical with the long axis lying east and west, and +Aristotle was inclined to view it as a very long and relatively narrow +band almost encircling the globe in the temperate zone. His argument +as to the narrowness of the sea between West Africa and East +Asia, from the occurrence of elephants at both extremities, is difficult +to understand, although it shows that he looked on the distribution +of animals as a problem of geography.</p> + +<p>Pythagoras had speculated as to the existence of antipodes, but +it was not until the first approximately accurate measurements of +the globe and estimates of the length and breadth of the +<i>oekumene</i> were made by Eratosthenes (<i>c.</i> 250 <span class="scs">B.C.</span>) that +<span class="sidenote">Problem of the Antipodes.</span> +the fact that, as then known, it occupied less than a quarter +of the surface of the sphere was clearly recognized. It was +natural, if not strictly logical, that the ocean river should be extended +from a narrow stream to a world-embracing sea, and here again +Greek theory, or rather fancy, gave its modern name to the greatest +feature of the globe. The old instinctive idea of symmetry must +often have suggested other <i>oekumene</i> balancing the known world +in the other quarters of the globe. The Stoic philosophers, especially +Crates of Mallus, arguing from the love of nature for life, placed an +<i>oekumene</i> in each quarter of the sphere, the three unknown world-islands +being those of the Antoeci, Perioeci and Antipodes. This +was a theory not only attractive to the philosophical mind, but +eminently adapted to promote exploration. It had its opponents, +however, for Herodotus showed that sea-basins existed cut off from +the ocean, and it is still a matter of controversy how far the pre-Ptolemaic +geographers believed in a water-connexion between the +Atlantic and Indian oceans. It is quite clear that Pomponius Mela +(<i>c.</i> <span class="scs">A.D.</span> 40), following Strabo, held that the southern temperate zone +contained a habitable land, which he designated by the name +<i>Antichthones</i>.</p> + +<p>Aristotle left no work on geography, so that it is impossible to +know what facts he associated with the science of the earth’s surface. +The word geography did not appear before Aristotle, +the first use of it being in the <span class="grk" title="Peri kosmôn">Περὶ κόσμων</span>, which is one +<span class="sidenote">Aristotle’s geographical views.</span> +of the writings doubtfully ascribed to him, and H. Berger +considers that the expression was introduced by Eratosthenes.<a name="fa4b" id="fa4b" href="#ft4b"><span class="sp">4</span></a> +Aristotle was certainly conversant with many +facts, such as the formation of deltas, coast-erosion, and to a certain +extent the dependence of plants and animals on their physical +surroundings. He formed a comprehensive theory of the variations +of climate with latitude and season, and was convinced of the necessity +of a circulation of water between the sea and rivers, though, +like Plato, he held that this took place by water rising from the sea +through crevices in the rocks, losing its dissolved salts in the process. +He speculated on the differences in the character of races of mankind +living in different climates, and correlated the political forms of +communities with their situation on a seashore, or in the neighbourhood +of natural strongholds.</p> + +<p>Strabo (<i>c.</i> 50 <span class="scs">B.C.</span>-<span class="scs">A.D.</span> 24) followed Eratosthenes rather than +Aristotle, but with sympathies which went out more to the human +interests than the mathematical basis of geography. He +<span class="sidenote">Strabo.</span> +compiled a very remarkable work dealing, in large measure +from personal travel, with the countries surrounding the Mediterranean. +He may be said to have set the pattern which was followed +in succeeding ages by the compilers of “political geographies” +<span class="pagenum"><a name="page621" id="page621"></a>621</span> +dealing less with theories than with facts, and illustrating rather than +formulating the principles of the science.</p> + +<p>Claudius Ptolemaeus (<i>c.</i> <span class="scs">A.D.</span> 150) concentrated in his writings the +final outcome of all Greek geographical learning, and passed it across +the gulf of the middle ages by the hands of the Arabs, +to form the starting-point of the science in modern times. +<span class="sidenote">Ptolemy.</span> +His geography was based more immediately on the work of his +predecessor, Marinus of Tyre, and on that of Hipparchus, the +follower and critic of Eratosthenes. It was the ambition of Ptolemy +to describe and represent accurately the surface of the <i>oekumene</i>, +for which purpose he took immense trouble to collect all existing +determinations of the latitude of places, all estimates of longitude, +and to make every possible rectification in the estimates of distances +by land or sea. His work was mainly cartographical in its aim, +and theory was as far as possible excluded. The symmetrically +placed hypothetical islands in the great continuous ocean disappeared, +and the <i>oekumene</i> acquired a new form by the representation of the +Indian Ocean as a larger Mediterranean completely cut off by land +from the Atlantic. The <i>terra incognita</i> uniting Africa and Farther +Asia was an unfortunate hypothesis which helped to retard exploration. +Ptolemy used the word <i>geography</i> to signify the description +of the whole <i>oekumene</i> on mathematical principles, while <i>chorography</i> +signified the fuller description of a particular region, and +<i>topography</i> the very detailed description of a smaller locality. He +introduced the simile that geography represented an artist’s sketch +of a whole portrait, while chorography corresponded to the careful +and detailed drawing of an eye or an ear.<a name="fa5b" id="fa5b" href="#ft5b"><span class="sp">5</span></a></p> + +<p>The Caliph al-Mamūn (<i>c.</i> <span class="scs">A.D.</span> 815), the son and successor of +Hārūn al-Rashīd, caused an Arabic version of Ptolemy’s great +astronomical work (<span class="grk" title="Suntaxis megistê">Σύνταξις μεγίστη</span>) to be made, which is known +as the <i>Almagest</i>, the word being nothing more than the Gr. <span class="grk" title="megistê">μεγίστη</span> +with the Arabic article <i>al</i> prefixed. The geography of Ptolemy was +also known and is constantly referred to by Arab writers. The +Arab astronomers measured a degree on the plains of Mesopotamia, +thereby deducing a fair approximation to the size of the earth. +The caliph’s librarian, Abu Jafar Muhammad Ben Musa, wrote a +geographical work, now unfortunately lost, entitled <i>Rasm el Arsi</i> (“A +Description of the World”), which is often referred to by subsequent +writers as having been composed on the model of that of Ptolemy.</p> + +<p>The middle ages saw geographical knowledge die out in Christendom, +although it retained, through the Arabic translations of +Ptolemy, a certain vitality in Islam. The verbal interpretation +of Scripture led Lactantius (<i>c.</i> <span class="scs">A.D.</span> 320) and +<span class="sidenote">Geography in the middle ages.</span> +other ecclesiastics to denounce the spherical theory of the +earth as heretical. The wretched subterfuge of Cosmas +(<i>c.</i> <span class="scs">A.D.</span> 550) to explain the phenomena of the apparent +movements of the sun by means of an earth modelled on the plan +of the Jewish Tabernacle gave place ultimately to the wheel-maps—the +T in an O—which reverted to the primitive ignorance of the +times of Homer and Hecataeus.<a name="fa6b" id="fa6b" href="#ft6b"><span class="sp">6</span></a></p> + +<p>The journey of Marco Polo, the increasing trade to the East and +the voyages of the Arabs in the Indian Ocean prepared the way for +the reacceptance of Ptolemy’s ideas when the sealed books of the +Greek original were translated into Latin by Angelus in 1410.</p> + +<p>The old arguments of Aristotle and the old measurements of +Ptolemy were used by Toscanelli and Columbus in urging a westward +voyage to India; and mainly on this account did the +<span class="sidenote">Revival of geography.</span> +crossing of the Atlantic rank higher in the history of +scientific geography than the laborious feeling out of the +coast-line of Africa. But not until the voyage of Magellan shook +the scales from the eyes of Europe did modern geography begin to +advance. Discovery had outrun theory; the rush of new facts +made Ptolemy practically obsolete in a generation, after having been +the fount and origin of all geography for a millennium.</p> + +<p>The earliest evidence of the reincarnation of a sound theoretical +geography is to be found in the text-books by Peter Apian and +Sebastian Münster. Apian in his <i>Cosmographicus liber</i>, +published in 1524, and subsequently edited and added to +<span class="sidenote">Apianus.</span> +by Gemma Frisius under the title of <i>Cosmographia</i>, based the whole +science on mathematics and measurement. He followed Ptolemy +closely, enlarging on his distinction between geography and chorography, +and expressing the artistic analogy in a rough diagram. +This slender distinction was made much of by most subsequent +writers until Nathanael Carpenter in 1625 pointed out that the +difference between geography and chorography was simply one of +degree, not of kind.</p> + +<p>Sebastian Münster, on the other hand, in his <i>Cosmographia +universalis</i> of 1544, paid no regard to the mathematical basis of +geography, but, following the model of Strabo, described +<span class="sidenote">Münster.</span> +the world according to its different political divisions, +and entered with great zest into the question of the productions +of countries, and into the manners and costumes of the various +peoples. Thus early commenced the separation between what were +long called mathematical and political geography, the one subject +appealing mainly to mathematicians, the other to historians.</p> + +<p>Throughout the 16th and 17th centuries the rapidly accumulating +store of facts as to the extent, outline and mountain and river +systems of the lands of the earth were put in order by the generation +of cartographers of which Mercator was the chief; but the writings +of Apian and Münster held the field for a hundred years without a +serious rival, unless the many annotated editions of Ptolemy might +be so considered. Meanwhile the new facts were the subject of +original study by philosophers and by practical men without reference +to classical traditions. Bacon argued keenly on geographical +matters and was a lover of maps, in which he observed and reasoned +upon such resemblances as that between the outlines of South +America and Africa.</p> + +<p>Philip Cluver’s <i>Introductio in geographiam universam tam veterem +quam novam</i> 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 +<span class="sidenote">Cluverius.</span> +with the earth alone, not with the universe, and from chorography +and topography by dealing with the whole earth, not with a country +or a place. The first book, of fourteen short chapters, is concerned +with the general properties of the globe; the remaining six books +treat in considerable detail of the countries of Europe and of the +other continents. Each country is described with particular regard +to its people as well as to its surface, and the prominence given to +the human element is of special interest.</p> + +<p>A little-known book which appears to have escaped the attention +of most writers on the history of modern geography was published +at Oxford in 1625 by Nathanael Carpenter, fellow of +<span class="sidenote">Carpenter.</span> +Exeter College, with the title <i>Geographie delineated forth +in Two Bookes, containing the Sphericall and Topicall parts thereof</i>. +It is discursive in its style and verbose; but, considering the period +at which it appeared, it is remarkable for the strong common sense +displayed by the author, his comparative freedom from prejudice, +and his firm application of the methods of scientific reasoning to +the interpretation of phenomena. Basing his work on the principles +of Ptolemy, he brings together illustrations from the most recent +travellers, and does not hesitate to take as illustrative examples +the familiar city of Oxford and his native county of Devon. He +divides geography into <i>The Spherical Part</i>, or that for the study of +which mathematics alone is required, and <i>The Topical Part</i>, or the +description of the physical relations of parts of the earth’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.</p> + +<p>A much more important work in the history of geographical +method is the <i>Geographia generalis</i> of Bernhard Varenius, a German +medical doctor of Leiden, who died at the age of twenty-eight +in 1650, the year of the publication of his book. +<span class="sidenote">Varenius.</span> +Although for a time it was lost sight of on the continent, Sir Isaac +Newton thought so highly of this book that he prepared an annotated +edition which was published in Cambridge in 1672, with the addition +of the plates which had been planned by Varenius, but not produced +by the original publishers. “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.”<a name="fa7b" id="fa7b" href="#ft7b"><span class="sp">7</span></a> The treatise of Varenius is a +model of logical arrangement and terse expression; it is a work of +science and of genius; one of the few of that age which can still be +studied with profit. The English translation renders the definition +thus: “Geography is that part of <i>mixed mathematics</i> which explains +the state of the earth and of its parts, depending on quantity, viz. +its figure, place, magnitude and motion, with the celestial appearances, +&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.”</p> + +<p>Varenius was reluctant to include the human side of geography in +his system, and only allowed it as a concession to custom, and in +order to attract readers by imparting interest to the sterner details +of the science. His division of geography was into two parts—(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 <i>Absolute</i> part, dealing with the form, dimensions, +position and substance of the earth, the distribution of land and +water, mountains, woods and deserts, hydrography (including all +the waters of the earth) and the atmosphere; (2) the <i>Relative</i> part, +including the celestial properties, <i>i.e.</i> latitude, climate zones, longitude, +&c.; and (3) the <i>Comparative</i> part, which “considers the +<span class="pagenum"><a name="page622" id="page622"></a>622</span> +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) +<i>Terrestrial</i>, including position, outline, boundaries, mountains, +mines, woods and deserts, waters, fertility and fruits, and living +creatures; (2) <i>Celestial</i>, including appearance of the heavens and +the climate; (3) <i>Human</i>, but this was added out of deference to +popular usage.</p> + +<p>This system of geography founded a new epoch, and the book—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.</p> + +<p>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.</p> + +<p>The next marked advance in the theory of geography may be +taken as the nearly simultaneous studies of the physical earth +carried out by the Swedish chemist, Torbern Bergman, +acting under the impulse of Linnaeus, and by the German +<span class="sidenote">Bergman.</span> +philosopher, Immanuel Kant. Bergman’s <i>Physical Description of +the Earth</i> was published in Swedish in 1766, and translated into +English in 1772 and into German in 1774. It is a plain, straightforward +description of the globe, and of the various phenomena +of the surface, dealing only with definitely ascertained facts in the +natural order of their relationships, but avoiding any systematic +classification or even definitions of terms.</p> + +<p>The problems of geography had been lightened by the destructive +criticism of the French cartographer D’Anville (who had purged +the map of the world of the last remnants of traditional +fact unverified by modern observations) and rendered +<span class="sidenote">Kant.</span> +richer by the dawn of the new era of scientific travel, when Kant +brought his logical powers to bear upon them. Kant’s lectures on +physical geography were delivered in the university of Königsberg +from 1765 onwards.<a name="fa8b" id="fa8b" href="#ft8b"><span class="sp">8</span></a> Geography appealed to him as a valuable +educational discipline, the joint foundation with anthropology of +that “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.</p> + +<p>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) <i>Mathematical geography</i>, which +deals with the form, size and movements of the earth and its place +in the solar system; (2) <i>Moral geography</i>, or an account of the +different customs and characters of mankind according to the region +they inhabit; (3) <i>Political geography</i>, the divisions according to +their organized governments; (4) <i>Mercantile geography</i>, dealing +with the trade in the surplus products of countries; (5) <i>Theological +geography</i>, or the distribution of religions. Here there is a clear and +formal statement of the interaction and causal relation of all the +phenomena of distribution on the earth’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.</p> + +<p>Alexander von Humboldt (1769-1859) was the first modern geographer +to become a great traveller, and thus to acquire an extensive +stock of first-hand information on which an improved +system of geography might be founded. The impulse +<span class="sidenote">Humboldt.</span> +given to the study of natural history by the example of Linnaeus; +the results brought back by Sir Joseph Banks, Dr Solander and the +two Forsters, who accompanied Cook in his voyages of discovery; +the studies of De Saussure in the Alps, and the lists of desiderata +in physical geography drawn up by that investigator, combined to +prepare the way for Humboldt. The theory of geography was +advanced by Humboldt mainly by his insistence on the great +principle of the unity of nature. He brought all the “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 <i>Encyclopaedia Britannica</i>) showed the +effect produced in Great Britain by the stimulus of Humboldt’s work.</p> + +<p>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 +<span class="sidenote">Ritter.</span> +itself, but from the comparison of the phenomena of many places. +Impressed by the influence of terrestrial relief and climate on human +movements, Ritter was led deeper and deeper into the study of history +and archaeology. His monumental <i>Vergleichende Geographie</i>, which +was to have made the whole world its theme, died out in a wilderness +of detail in twenty-one volumes before it had covered more of the +earth’s surface than Asia and a portion of Africa. Some of his +followers showed a tendency to look on geography rather as an +auxiliary to history than as a study of intrinsic worth.</p> + +<p>During the rapid development of physical geography many +branches of the study of nature, which had been included in the +cosmography of the early writers, the physiography of +Linnaeus and even the <i>Erdkunde</i> of Ritter, had been +<span class="sidenote">Geography as a natural science.</span> +so much advanced by the labours of specialists that +their connexion was apt to be forgotten. Thus geology, +meteorology, oceanography and anthropology developed +into distinct sciences. The absurd attempt was, and sometimes +is still, made by geographers to include all natural science in geography; +but it is more common for specialists in the various detailed +sciences to think, and sometimes to assert, that the ground of +physical geography is now fully occupied by these sciences. Political +geography has been too often looked on from both sides as a mere +summary of guide-book knowledge, useful in the schoolroom, a poor +relation of physical geography that it was rarely necessary to +recognize.</p> + +<p>The science of geography, passed on from antiquity by Ptolemy, +re-established by Varenius and Newton, and systematized by Kant, +included within itself definite aspects of all those terrestrial phenomena +which are now treated exhaustively under the heads of geology, +meteorology, oceanography and anthropology; and the inclusion +of the requisite portions of the perfected results of these sciences in +geography is simply the gathering in of fruit matured from the seed +scattered by geography itself.</p> + +<p>The study of geography was advanced by improvements in cartography +(see <span class="sc"><a href="#artlinks">Map</a></span>), not only in the methods of survey and projection, +but in the representation of the third dimension by means +of contour lines introduced by Philippe Buache in 1737, and the +more remarkable because less obvious invention of isotherms +introduced by Humboldt in 1817.</p> + +<p>The “argument from design” had been a favourite form of +reasoning amongst Christian theologians, and, as worked out by +Paley in his <i>Natural Theology</i>, it served the useful purpose +of emphasizing the fitness which exists between all the +<span class="sidenote">The teleological argument in geography.</span> +inhabitants of the earth and their physical environment. +It was held that the earth had been created so as to fit +the wants of man in every particular. This argument was +tacitly accepted or explicitly avowed by almost every writer on the +theory of geography, and Carl Ritter distinctly recognized and +adopted it as the unifying principle of his system. As a student of +nature, however, he did not fail to see, and as professor of geography +he always taught, that man was in very large measure conditioned +by his physical environment. The apparent opposition of the +observed fact to the assigned theory he overcame by looking upon +the forms of the land and the arrangement of land and sea as instruments +of Divine Providence for guiding the destiny as well as for +supplying the requirements of man. This was the central theme of +Ritter’s philosophy; his religion and his geography were one, and +the consequent fervour with which he pursued his mission goes far +to account for the immense influence he acquired in Germany.</p> + +<p>The evolutionary theory, more than hinted at in Kant’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 +<span class="sidenote">The theory of evolution in geography.</span> +cooling of the surface of the planet throughout the long +geological periods, the guiding power of environment on +the circulation of water and of air, on the distribution +of plants and animals, and finally on the movements of man, give +to geography a philosophical dignity and a scientific completeness +<span class="pagenum"><a name="page623" id="page623"></a>623</span> +which it never previously possessed. The influence of environment +on the organism may not be quite so potent as it was once believed +to be, in the writings of Buckle, for instance,<a name="fa9b" id="fa9b" href="#ft9b"><span class="sp">9</span></a> and certainly man, +the ultimate term in the series, reacts upon and greatly modifies his +environment; yet the fact that environment does influence all +distributions is established beyond the possibility of doubt. In +this way also the position of geography, at the point where physical +science meets and mingles with mental science, is explained and +justified. The change which took place during the 19th century +in the substance and style of geography may be well seen by comparing +the eight volumes of Malte-Brun’s <i>Géographie universelle</i> +(Paris, 1812-1829) with the twenty-one volumes of Reclus’s <i>Géographie +universelle</i> (Paris, 1876-1895).</p> + +<p>In estimating the influence of recent writers on geography it is +usual to assign to Oscar Peschel (1826-1875) the credit of having +corrected the preponderance which Ritter gave to the historical +element, and of restoring physical geography to its old pre-eminence.<a name="fa10b" id="fa10b" href="#ft10b"><span class="sp">10</span></a> +As a matter of fact, each of the leading modern exponents of theoretical +geography—such as Ferdinand von Richthofen, Hermann +Wagner, Friedrich Ratzel, William M. Davis, A. Penck, A. de +Lapparent and Elisée Reclus—has his individual point of view, +one devoting more attention to the results of geological processes, +another to anthropological conditions, and the rest viewing the +subject in various blendings of the extreme lights.</p> + +<p>The two conceptions which may now be said to animate the theory +of geography are the genetic, which depends upon processes of +origin, and the morphological, which depends on facts of form and +distribution.</p> + +<p class="pt2 center sc">Progress of Geographical Discovery</p> + +<p>Exploration and geographical discovery must have started from +more than one centre, and to deal justly with the matter one ought +to treat of these separately in the early ages before the whole civilized +world was bound together by the bonds of modern intercommunication. +At the least there should be some consideration of four +separate systems of discovery—the Eastern, in which Chinese and +Japanese explorers acquired knowledge of the geography of Asia, +and felt their way towards Europe and America; the Western, in +which the dominant races of the Mexican and South American +plateaus extended their knowledge of the American continent +before Columbus; the Polynesian, in which the conquering races +of the Pacific Islands found their way from group to group; and +the Mediterranean. For some of these we have no certain information, +and regarding others the tales narrated in the early records +are so hard to reconcile with present knowledge that they are better +fitted to be the battle-ground of scholars championing rival theories +than the basis of definite history. So it has come about that the +only practicable history of geographical exploration starts from the +Mediterranean centre, the first home of that civilization which has +come to be known as European, though its field of activity has long +since overspread the habitable land of both temperate zones, eastern +Asia alone in part excepted.</p> + +<p>From all centres the leading motives of exploration were probably +the same—commercial intercourse, warlike operations, whether +resulting in conquest or in flight, religious zeal expressed in pilgrimages +or missionary journeys, or, from the other side, the avoidance +of persecution, and, more particularly in later years, the +advancement of knowledge for its own sake. At different times one +or the other motive predominated.</p> + +<p>Before the 14th century <span class="scs">B.C.</span> the warrior kings of Egypt had carried +the power of their arms southward from the delta of the Nile well-nigh +to its source, and eastward to the confines of Assyria. The +hieroglyphic inscriptions of Egypt and the cuneiform inscriptions of +Assyria are rich in records of the movements and achievements of +armies, the conquest of towns and the subjugation of peoples; but +though many of the recorded sites have been identified, their discovery +by wandering armies was isolated from their subsequent +history and need not concern us here.</p> + +<p>The Phoenicians are the earliest Mediterranean people in the +consecutive chain of geographical discovery which joins pre-historic +time with the present. From Sidon, and later from its +more famous rival Tyre, the merchant adventurers of +<span class="sidenote">The Phoenicians.</span> +Phoenicia explored and colonized the coasts of the Mediterranean +and fared forth into the ocean beyond. They traded also +on the Red sea, and opened up regular traffic with India as well +as with the ports of the south and west, so that it was natural for +Solomon to employ the merchant navies of Tyre in his oversea trade. +The western emporium known in the scriptures as Tarshish was +probably situated in the south of Spain, possibly at Cadiz, although +some writers contend that it was Carthage in North Africa. Still +more diversity of opinion prevails as to the southern gold-exporting +port of Ophir, which some scholars place in Arabia, others at one or +another point on the east coast of Africa. Whether associated +with the exploitation of Ophir (<i>q.v.</i>) or not the first great voyage of +African discovery appears to have been accomplished by the Phoenicians +sailing the Red Sea. Herodotus (himself a notable traveller +in the 5th century <span class="scs">B.C.</span>) relates that the Egyptian king Necho of +the XXVIth Dynasty (<i>c.</i> 600 <span class="scs">B.C.</span>) built a fleet on the Red Sea, +and confided it to Phoenician sailors with the orders to sail southward +and return to Egypt by the Pillars of Hercules and the Mediterranean +sea. According to the tradition, which Herodotus quotes +sceptically, this was accomplished; but the story is too vague to +be accepted as more than a possibility.</p> + +<p>The great Phoenician colony of Carthage, founded before 800 <span class="scs">B.C.</span>, +perpetuated the commercial enterprise of the parent state, and extended +the sphere of practical trade to the ocean shores of Africa +and Europe. The most celebrated voyage of antiquity undertaken +for the express purpose of discovery was that fitted out by the +senate of Carthage under the command of Hanno, with the intention +of founding new colonies along the west coast of Africa. According +to Pliny, the only authority on this point, the period of the voyage +was that of the greatest prosperity of Carthage, which may be taken +as somewhere between 570 and 480 <span class="scs">B.C.</span> The extent of this voyage +is doubtful, but it seems probable that the farthest point reached +was on the east-running coast which bounds the Gulf of Guinea +on the north. Himilco, a contemporary of Hanno, was charged +with an expedition along the west coast of Iberia northward, and +as far as the uncertain references to this voyage can be understood, +he seems to have passed the Bay of Biscay and possibly sighted the +coast of England.</p> + +<p>The sea power of the Greek communities on the coast of Asia +Minor and in the Archipelago began to be a formidable rival to the +Phoenician soon after the time of Hanno and Himilco, +and peculiar interest attaches to the first recorded Greek +<span class="sidenote">The Greeks.</span> +voyage beyond the Pillars of Hercules. Pytheas, a +navigator of the Phocean colony of Massilia (Marseilles), determined +the latitude of that port with considerable precision by the somewhat +clumsy method of ascertaining the length of the longest day, and +when, about 330 <span class="scs">B.C.</span>, he set out on exploration to the northward +in search of the lands whence came gold, tin and amber, he followed +this system of ascertaining his position from time to time. If on +each occasion he himself made the observations his voyage must +have extended over six years; but it is not impossible that he +ascertained the approximate length of the longest day in some cases +by questioning the natives. Pytheas, whose own narrative is not +preserved, coasted the Bay of Biscay, sailed up the English Channel +and followed the coast of Britain to its most northerly point. Beyond +this he spoke of a land called <i>Thule</i>, which, if his estimate of the +length of the longest day is correct, may have been Shetland, but +was possibly Iceland; and from some confused statements as to a +sea which could not be sailed through, it has been assumed that +Pytheas was the first of the Greeks to obtain direct knowledge of +the Arctic regions. During this or a second voyage Pytheas entered +the Baltic, discovered the coasts where amber is obtained and returned +to the Mediterranean. It does not seem that any maritime +trade followed these discoveries, and indeed it is doubtful whether +his contemporaries accepted the truth of Pytheas’s narrative; +Strabo four hundred years later certainly did not, but the critical +studies of modern scholars have rehabilitated the Massilian explorer.</p> + +<p>The Greco-Persian wars had made the remoter parts of Asia +Minor more than a name to the Greek geographers before the time +of Alexander the Great, but the campaigns of that conqueror +<span class="sidenote">Alexander the Great.</span> +from 329 to 325 <span class="scs">B.C.</span> opened up the greater Asia +to the knowledge of Europe. His armies crossed the plains +beyond the Caspian, penetrated the wild mountain passes north-west +of India, and did not turn back until they had entered on the +Indo-Gangetic plain. This was one of the few great epochs of +geographical discovery.</p> + +<p>The world was henceforth viewed as a very large place stretching +far on every side beyond the Midland or Mediterranean Sea, and the +land journey of Alexander resulted in a voyage of discovery in the +outer ocean from the mouth of the Indus to that of the Tigris, +thus opening direct intercourse between Grecian and Hindu civilization. +The Greeks who accompanied Alexander described with care +the towns and villages, the products and the aspect of the country. +The conqueror also intended to open up trade by sea between Europe +and India, and the narrative of his general Nearchus records this +famous voyage of discovery, the detailed accounts of the chief +pilot Onesicritus being lost. At the beginning of October 326 <span class="scs">B.C.</span> +Nearchus left the Indus with his fleet, and the anchorages sought for +each night are carefully recorded. He entered the Persian Gulf, +and rejoined Alexander at Susa, when he was ordered to prepare +another expedition for the circumnavigation of Arabia. Alexander +died at Babylon in 323 <span class="scs">B.C.</span>, and the fleet was dispersed without +making the voyage.</p> + +<p>The dynasties founded by Alexander’s generals, Seleucus, Antiochus +and Ptolemy, encouraged the same spirit of enterprise which +their master had fostered, and extended geographical knowledge +in several directions. Seleucus Nicator established the Greco-Bactrian +empire and continued the intercourse with India. Authentic +information respecting the great valley of the Ganges was supplied +by Megasthenes, an ambassador sent by Seleucus, who reached the +remote city of Patali-putra, the modern Patna.</p> + +<p>The Ptolemies in Egypt showed equal anxiety to extend the +bounds of geographical knowledge. Ptolemy Euergetes (247-222 <span class="scs">B.C.</span>) +<span class="pagenum"><a name="page624" id="page624"></a>624</span> +rendered the greatest service to geography by the protection and +<span class="sidenote">The Ptolemies.</span> +encouragement of Eratosthenes, whose labours gave the first approximate +knowledge of the true size of the spherical +earth. The second Euergetes and his successor Ptolemy +Lathyrus (118-115 <span class="scs">B.C.</span>) furnished Eudoxus with a fleet +to explore the Arabian sea. After two successful voyages, Eudoxus, +impressed with the idea that Africa was surrounded by ocean on the +south, left the Egyptian service, and proceeded to Cadiz and other +Mediterranean centres of trade seeking a patron who would finance +an expedition for the purpose of African discovery; and we learn +from Strabo that the veteran explorer made at least two voyages +southward along the coast of Africa. The Ptolemies continued to +send fleets annually from their Red Sea ports of Berenice and Myos +Hormus to Arabia, as well as to ports on the coasts of Africa and +India.</p> + +<p>The Romans did not encourage navigation and commerce with +the same ardour as their predecessors; still the luxury of Rome, +which gave rise to demands for the varied products +of all the countries of the known world, led to an active +<span class="sidenote">The Romans.</span> +trade both by ships and caravans. But it was the military +genius of Rome, and the ambition for universal empire, which led, +not only to the discovery, but also to the survey of nearly all Europe, +and of large tracts in Asia and Africa. Every new war produced +a new survey and itinerary of the countries which were conquered, +and added one more to the imperishable roads that led from every +quarter of the known world to Rome. In the height of their power +the Romans had surveyed and explored all the coasts of the Mediterranean, +Italy, Greece, the Balkan Peninsula, Spain, Gaul, western +Germany and southern Britain. In Africa their empire included +Egypt, Carthage, Numidia and Mauritania. In Asia they held +Asia Minor and Syria, had sent expeditions into Arabia, and were +acquainted with the more distant countries formerly invaded by +Alexander, including Persia, Scythia, Bactria and India. Roman +intercourse with India especially led to the extension of geographical +knowledge.</p> + +<p>Before the Roman legions were sent into a new region to extend +the limits of the empire, it was usual to send out exploring expeditions +to report as to the nature of the country. It is narrated by Pliny +and Seneca that the emperor Nero sent out two centurions on such +a mission towards the source of the Nile (probably about <span class="scs">A.D.</span> 60), +and that the travellers pushed southwards until they reached vast +marshes through which they could not make their way either on +foot or in boats. This seems to indicate that they had penetrated +to about 9° N. Shortly before <span class="scs">A.D.</span> 79 Hippalus took advantage of +the regular alternation of the monsoons to make the voyage from +the Red Sea to India across the open ocean out of sight of land. +Even though this sea-route was known, the author of the <i>Periplus +of the Erythraean Sea</i>, published after the time of Pliny, recites the +old itinerary around the coast of the Arabian Gulf. It was, however, +in the reigns of Severus and his immediate successors that Roman +intercourse with India was at its height, and from the writings of +Pausanias (<i>c.</i> 174) it appears that direct communication between +Rome and China had already taken place.</p> + +<p>After the division of the Roman empire, Constantinople became +the last refuge of learning, arts and taste; while Alexandria continued +to be the emporium whence were imported the commodities +of the East. The emperor Justinian (483-565), in whose reign the +greatness of the Eastern empire culminated, sent two Nestorian +monks to China, who returned with eggs of the silkworm concealed +in a hollow cane, and thus silk manufactures were established in +the Peloponnesus and the Greek islands. It was also in the reign +of Justinian that Cosmas Indicopleustes, an Egyptian merchant, +made several voyages, and afterwards composed his <span class="grk" title="Christianikê +topographia">Χριστιανικὴ τοπογραφία</span> (Christian Topography), containing, in addition to his +absurd cosmogony, a tolerable description of India.</p> + +<p>The great outburst of Mahommedan conquest in the 7th century +was followed by the Arab civilization, having its centres at Bagdad +and Cordova, in connexion with which geography again +received a share of attention. The works of the ancient +<span class="sidenote">The Arabs.</span> +Greek geographers were translated into Arabic, and starting with a +sound basis of theoretical knowledge, exploration once more made +progress. From the 9th to the 13th century intelligent Arab +travellers wrote accounts of what they had seen and heard in distant +lands. The earliest Arabian traveller whose observations have come +down to us is the merchant Sulaiman, who embarked in the Persian +Gulf and made several voyages to India and China, in the middle of +the 9th century. Abu Zaid also wrote on India, and his work is the +most important that we possess before the epoch-making discoveries +of Marco Polo. Masudi, a great traveller who knew from personal +experience all the countries between Spain and China, described the +plains, mountains and seas, the dynasties and peoples, in his <i>Meadows +of Gold</i>, an abstract made by himself of his larger work <i>News of the +Time</i>. He died in 956, and was known, from the comprehensiveness +of his survey, as the Pliny of the East. Amongst his contemporaries +were Istakhri, who travelled through all the Mahommedan +countries and wrote his <i>Book of Climates</i> in 950, and Ibn Haukal, +whose <i>Book of Roads and Kingdoms</i>, based on the work of Istakhri, +was written in 976. Idrisi, the best known of the Arabian geographical +authors, after travelling far and wide in the first half of +the 12th century, settled in Sicily, where he wrote a treatise descriptive +of an armillary sphere which he had constructed for Roger II., +the Norman king, and in this work he incorporated all accessible +results of contemporary travel.</p> + +<p>The Northmen of Denmark and Norway, whose piratical adventures +were the terror of all the coasts of Europe, and who established +themselves in Great Britain and Ireland, in France and +Sicily, were also geographical explorers in their rough but +<span class="sidenote">The Northmen.</span> +practical way during the darkest period of the middle ages. +All Northmen were not bent on rapine and plunder; many were +peaceful merchants. Alfred the Great, king of the Saxons in +England, not only educated his people in the learning of the past +ages; he inserted in the geographical works he translated many +narratives of the travel of his own time. Thus he placed on record +the voyages of the merchant Ulfsten in the Baltic, including particulars +of the geography of Germany. And in particular he told of +the remarkable voyage of Other, a Norwegian of Helgeland, who +was the first authentic Arctic explorer, the first to tell of the rounding +of the North Cape and the sight of the midnight sun. This voyage +of the middle of the 9th century deserves to be held in happy memory, +for it unites the first Norwegian polar explorer with the first English +collector of travels. Scandinavian merchants brought the products +of India to England and Ireland. From the 8th to the 11th century +a commercial route from India passed through Novgorod to the +Baltic, and Arabian coins found in Sweden, and particularly in +the island of Gotland, prove how closely the enterprise of the Northmen +and of the Arabs intertwined. Five-sixths of these coins +preserved at Stockholm were from the mints of the Samanian +dynasty, which reigned in Khorasan and Transoxiana from about +<span class="scs">A.D.</span> 900 to 1000. It was the trade with the East that originally gave +importance to the city of Visby in Gotland.</p> + +<p>In the end of the 9th century Iceland was colonized from Norway; +and about 985 the intrepid viking, Eric the Red, discovered Greenland, +and induced some of his Icelandic countrymen to settle on its inhospitable +shores. His son, Leif Ericsson, and others of his followers +were concerned in the discovery of the North American coast (see +<span class="sc"><a href="#artlinks">Vinland</a></span>), which, but for the isolation of Iceland from the centres +of European awakening, would have had momentous consequences. +As things were, the importance of this discovery passed unrecognized. +The story of two Venetians, Nicolo and Antonio Zeno, who gave a +vague account of voyages in the northern seas in the end of the 13th +century, is no longer to be accepted as history.</p> + +<p>At length the long period of barbarism which accompanied and +followed the fall of the Roman empire drew to a close in Europe. +The Crusades had a favourable influence on the intellectual +state of the Western nations. Interesting regions, +<span class="sidenote">Close of the dark ages.</span> +known only by the scant reports of pilgrims, were made +the objects of attention and study; while religious zeal, +and the hope of gain, combined with motives of mere curiosity, +induced several persons to travel by land into remote regions of the +East, far beyond the countries to which the operations of the crusaders +extended. Among these was Benjamin of Tudela, who set out from +Spain in 1160, travelled by land to Constantinople, and having +visited India and some of the eastern islands, returned to Europe +by way of Egypt after an absence of thirteen years.</p> + +<p>Joannes de Plano Carpini, a Franciscan monk, was the head of +one of the missions despatched by Pope Innocent to call the chief +and people of the Tatars to a better mind. He reached +the headquarters of Batu, on the Volga, in February +<span class="sidenote">Asiatic journeys.</span> +1246; and, after some stay, went on to the camp of the +great khan near Karakorum in central Asia, and returned safely +in the autumn of 1247. A few years afterwards, a Fleming named +Rubruquis was sent on a similar mission, and had the merit of being +the first traveller of this era who gave a correct account of the Caspian +Sea. He ascertained that it had no outlet. At nearly the same +time Hayton, king of Armenia, made a journey to Karakorum in +1254, by a route far to the north of that followed by Carpini and +Rubruquis. He was treated with honour and hospitality, and +returned by way of Samarkand and Tabriz, to his own territory. +The curious narrative of King Hayton was translated by Klaproth.</p> + +<p>While the republics of Italy, and above all the state of Venice, +were engaged in distributing the rich products of India and the Far +East over the Western world, it was impossible that motives of +curiosity, as well as a desire of commercial advantage, should not be +awakened to such a degree as to impel some of the merchants to +visit those remote lands. Among these were the brothers Polo, who +traded with the East and themselves visited Tatary. The recital +of their travels fired the youthful imagination of young Marco Polo, +son of Nicolo, and he set out for the court of Kublai Khan, with his +father and uncle, in 1265. Marco remained for seventeen years +in the service of the Great Khan, and was employed on many +important missions. Besides what he learnt from his own observation, +he collected much information from others concerning +countries which he did not visit. He returned to Europe possessed +of a vast store of knowledge respecting the eastern parts of the +world, and, being afterwards made a prisoner by the Genoese, he +dictated the narrative of his travels during his captivity. The +work of Marco Polo is the most valuable narrative of travels that +appeared during the middle ages, and despite a cold reception and +many denials of the accuracy of the record, its substantial truthfulness +has been abundantly proved.</p> + +<p><span class="pagenum"><a name="page625" id="page625"></a>625</span></p> + +<p>Missionaries continued to do useful geographical work. Among +them were John of Monte Corvino, a Franciscan monk, Andrew of +Perugia, John Marignioli and Friar Jordanus, who visited the west +coast of India, and above all Friar Odoric of Pordenone. Odoric +set out on his travels about 1318, and his journeys embraced parts +of India, the Malay Archipelago, China and even Tibet, where he +was the first European to enter Lhasa, not yet a forbidden city.</p> + +<p>Ibn Batuta, the great Arab traveller, is separated by a wide space +of time from his countrymen already mentioned, and he finds his +proper place in a chronological notice after the days of Marco Polo, +for he did not begin his wanderings until 1325, his career thus coinciding +in time with the fabled journeyings of Sir John Mandeville. +While Arab learning flourished during the darkest ages of European +ignorance, the last of the Arab geographers lived to see the dawn of +the great period of the European awakening. Ibn Batuta went by +land from Tangier to Cairo, then visited Syria, and performed the +pilgrimages to Medina and Mecca. After exploring Persia, and again +residing for some time at Mecca, he made a voyage down the Red +sea to Yemen, and travelled through that country to Aden. Thence +he visited the African coast, touching at Mombasa and Quiloa, and +then sailed across to Ormuz and the Persian Gulf. He crossed +Arabia from Bahrein to Jidda, traversed the Red sea and the desert +to Syene, and descended the Nile to Cairo. After this he revisited +Syria and Asia Minor, and crossed the Black sea, the desert from +Astrakhan to Bokhara, and the Hindu Kush. He was in the service +of Muhammad Tughluk, ruler of Delhi, about eight years, and was +sent on an embassy to China, in the course of which the ambassadors +sailed down the west coast of India to Calicut, and then visited the +Maldive Islands and Ceylon. Ibn Batuta made the voyage through +the Malay Archipelago to China, and on his return he proceeded +from Malabar to Bagdad and Damascus, ultimately reaching Fez, +the capital of his native country, in November 1349. After a journey +into Spain he set out once more for Central Africa in 1352, and +reached Timbuktu and the Niger, returning to Fez in 1353. His +narrative was committed to writing from his dictation.</p> + +<p>The European country which had come the most completely +under the influence of Arab culture now began to send forth explorers +to distant lands, though the impulse came not from the +Moors but from Italian merchant navigators in Spanish +<span class="sidenote">Spanish exploration.</span> +service. The peaceful reign of Henry III. of Castile is +famous for the attempts of that prince to extend the +diplomatic relations of Spain to the remotest parts of the earth. +He sent embassies to all the princes of Christendom and to the +Moors. In 1403 the Spanish king sent a knight of Madrid, Ruy +Gonzalez de Clavijo, to the distant court of Timur, at Samarkand. +He returned in 1406, and wrote a valuable narrative of his travels.</p> + +<p>Italians continued to make important journeys in the East +during the 15th century. Among them was Nicolo Conti, who +passed through Persia, sailed along the coast of Malabar, visited +Sumatra, Java and the south of China, returned by the Red sea, +and got home to Venice in 1444 after an absence of twenty-five years. +He related his adventures to Poggio Bracciolini, secretary to Pope +Eugenius IV.; and the narrative contains much interesting information. +One of the most remarkable of the Italian travellers was +Ludovico di Varthema, who left his native land in 1502. He went +to Egypt and Syria, and for the sake of visiting the holy cities became +a Mahommedan. He was the first European who gave an account +of the interior of Yemen. He afterwards visited and described +many places in Persia, India and the Malay Archipelago, returning +to Europe in a Portuguese ship after an absence of five years.</p> + +<p>In the 15th century the time was approaching when the discovery +of the Cape of Good Hope was to widen the scope of geographical +enterprise. This great event was preceded by the general +utilization in Europe of the polarity of the magnetic +<span class="sidenote">Portuguese exploration—Prince Henry the Navigator.</span> +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 Cão 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 Covilhão 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 João 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 Covilhão, +having heard that a Christian ruler reigned in the mountains of +Ethiopia, penetrated into Abyssinia in 1490. He delivered the letter +which João II. had addressed to Prester John to the Negus Alexander +of Abyssinia, but he was detained by that prince and never allowed +to leave the country.</p> + +<p>The Portuguese, following the lead of Prince Henry, continued to +look for the road to India by the Cape of Good Hope. The same +end was sought by Christopher Columbus, following the +suggestion of Toscanelli, and under-estimating the diameter +<span class="sidenote">Columbus.</span> +of the globe, by sailing due west. The voyages of Columbus +(1492-1498) resulted in the discovery of the West Indies and North +America which barred the way to the Far East. In 1493 the pope, +Alexander VI., issued a bull instituting the famous “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 <span class="sc"><a href="#artlinks">Columbus, Christopher</a></span>; <span class="sc"><a href="#artlinks">Gama, Vasco da</a></span>.</p> + +<p>The three voyages of Vasco da Gama (who died on the scene of his +labours, at Cochin, in 1524) revolutionized the commerce of the +East. Until then the Venetians held the carrying trade +<span class="sidenote">Vasco da Gama.</span> +of India, which was brought by the Persian Gulf and Red +sea into Syria and Egypt, the Venetians receiving the +products of the East at Alexandria and Beirut and distributing +them over Europe. This commerce was a great source of wealth +to Venice; but after the discovery of the new passage round the +Cape, and the conquests of the Portuguese, the trade of the East +passed into other hands.</p> + +<p>The discoveries of Columbus awakened a spirit of enterprise in +Spain which continued in full force for a century; adventurers +flocked eagerly across the Atlantic, and discovery followed +discovery in rapid succession. Many of the companions +<span class="sidenote">Spaniards in America.</span> +of Columbus continued his work. Vicente Yañez Pinzon +in 1500 reached the mouth of the Amazon. In the same +year Alonso de Ojeda, accompanied by Juan de la Cosa, from whose +maps we learn much of the discoveries of the 16th century navigators, +and by a Florentine named Amerigo Vespucci, touched the +coast of South America somewhere near Surinam, following the shore +as far as the Gulf of Maracaibo. Vespucci afterwards made three +voyages to the Brazilian coast; and in 1504 he wrote an account +of his four voyages, which was widely circulated, and became the +means of procuring for its author at the hands of the cartographer +Waldseemüller in 1507 the disproportionate distinction of giving his +name to the whole continent. In 1508 Alonso de Ojeda obtained the +government of the coast of South America from Cabo de la Vela +to the Gulf of Darien; Ojeda landed at Cartagena in 1510, and +sustained a defeat from the natives, in which his lieutenant, Juan +de la Cosa, was killed. After another reverse on the east side of the +Gulf of Darien Ojeda returned to Hispaniola and died there. The +Spaniards in the Gulf of Darien were left by Ojeda under the command +of Francisco Pizarro, the future conqueror of Peru. After +suffering much from famine and disease, Pizarro resolved to leave, +and embarked the survivors in small vessels, but outside the harbour +they met a ship which proved to be that of Martin Fernandez Enciso, +Ojeda’s partner, coming with provisions and reinforcements. One +of the crew of Enciso’s ship, Vasco Nuñez de Balboa, the future discoverer +of the Pacific Ocean, induced his commander to form a +settlement on the other side of the Gulf of Darien. The soldiers +became discontented and deposed Enciso, who was a man of learning +and an accomplished cosmographer. His work <i>Suma de Geografia</i>, +which was printed in 1519, is the first Spanish book which gives an +account of America. Vasco Nuñez, 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 Nuñez was beheaded in 1517 by Pedrarias de Avila, +who was sent out to supersede him. This was one of the greatest +calamities that could have happened to South America; for the +discoverer of the South sea was on the point of sailing with a little +fleet into his unknown ocean, and a humane and judicious man would +probably have been the conqueror of Peru, instead of the cruel and +ignorant Pizarro. In the year 1519 Panama was founded by +Pedrarias; and the conquest of Peru by Pizarro followed a few years +afterwards. Hernan Cortes overran and conquered Mexico from +1518 to 1521, and the discovery and conquest of Guatemala by +Alvarado, the invasion of Florida by De Soto, and of Nueva Granada +by Quesada, followed in rapid succession. The first detailed account +of the west coast of South America was written by a keenly observant +old soldier, Pedro de Cieza de Leon, who was travelling in South +America from 1533 to 1550, and published his story at Seville +in 1553.</p> + +<p>The great desire of the Spanish government at that time was +to find a westward route to the Moluccas. For this purpose Juan +Diaz de Solis was despatched in October 1515, and in +January 1516 he discovered the mouth of the Rio de la +<span class="sidenote">Pacific Ocean.</span> +Plata. He was, however, killed by the natives, and his +ships returned. In the following year the Portuguese Ferdinando +Magalhães, familiarly known as Magellan, laid before Charles V., +at Valladolid, a scheme for reaching the Spice Islands by sailing +westward. He started on the 21st of September 1519, entered the +strait which now bears his name in October 1520, worked his way +through between Patagonia and Tierra del Fuego, and entered on +<span class="pagenum"><a name="page626" id="page626"></a>626</span> +the vast Pacific which he crossed without sighting any of its innumerable +island groups. This was unquestionably the greatest of +the voyages which followed from the impulse of Prince Henry, and it +was rendered possible only by the magnificent courage of the commander +in spite of rebellion, mutiny and starvation. It was the +6th of March 1521 when he reached the Ladrone Islands. Thence +Magellan proceeded to the Philippines, and there his career ended +in an unimportant encounter with hostile natives. Eventually a +Biscayan named Sebastian del Cano, sailing home by way of the +Cape of Good Hope, reached San Lucar in command of the “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 <i>Primus circumdedisti me</i>.</p> + +<p>While the Spaniards were circumnavigating the +world and completing their knowledge of the coasts of +Central and South America, the Portuguese were actively +<span class="sidenote">Portuguese in Africa and the East.</span> +engaged on similar work as regards Africa and the East +Indies.</p> + +<p>With Abyssinia the mission of Covilhão led to further intercourse. +In April 1520 Vasco da Gama, as viceroy of the Indies, took a fleet +into the Red sea, and landed an embassy consisting of Dom Rodriguez +de Lima and Father Francisco Alvarez, a priest whose detailed narrative +is the earliest and not the least interesting account we possess +of Abyssinia. It was not until 1526 that the embassy was dismissed; +and not many years afterwards the negus entreated the help of the +Portuguese against Mahommedan invaders, and the viceroy sent an +expeditionary force, commanded by his brother Cristoforo da Gama, +with 450 musketeers. Da Gama was taken prisoner and killed, but +his followers enabled the Christians of Abyssinia to regain their +power, and a Jesuit mission remained in the country. The Portuguese +also established a close connexion with the kingdom of Congo +on the west side of Africa, and obtained much information respecting +the interior of the continent. Duarte Lopez, a Portuguese settled +in the country, was sent on a mission to Rome by the king of Congo, +and Pope Sixtus V. caused him to recount to his chamberlain, +Felipe Pigafetta, all he had learned during the nine years he had been +in Africa, from 1578 to 1587. This narrative, under the title of +<i>Description of the Kingdom of Congo</i>, was published at Rome by +Pigafetta in 1591. A map was attached on which several great +equatorial lakes are shown, and the empire of Monomwezi or Unyamwezi +is laid down. The most valuable work on Africa about +this time is, however, that written by the Moor Leo Africanus in +the early part of the 16th century. Leo travelled extensively in +the north and west of Africa, and was eventually taken by pirates +and sold to a master who presented him to Pope Leo X. At the +pope’s desire he translated his work on Africa into Italian.</p> + +<p>In Further India and the Malay Archipelago the Portuguese +acquired predominating influence at sea, establishing factories on +the Malabar coast, in the Persian Gulf, at Malacca, and in the Spice +Islands, and extending their commercial enterprises from the Red +sea to China. Their missionaries were received at the court of +Akbar, and Benedict Goes, a native of the Azores, was despatched +on a journey overland from Agra to China. He started in 1603, +and, after traversing the least-known parts of Central Asia, he +reached the confines of China. He appears to have ascended from +Kabul to the plateau of the Pamir, and thence onwards by Yarkand, +Khotan and Aksu. He died on the journey in March 1607; and +thus, as one of the brethren pronounced his epitaph, “seeking +Cathay he found heaven.”</p> + +<p>The activity and love of adventure, which became a passion for +two or three generations in Spain and Portugal, spread to other +countries. It was the spirit of the age; and England, +Holland and France were fired by it. English enterprise +<span class="sidenote">English, Dutch and French.</span> +was first aroused by John and Sebastian Cabot, father +and son, who came from Venice and settled at Bristol +in the time of Henry VII. The Cabots received a patent in 1496, +empowering them to seek unknown lands; and John Cabot discovered +Newfoundland and part of the coast of America. Sebastian +afterwards made a voyage to Rio de la Plata in the service of Spain, +but he returned to England in 1548 and received a pension from +Edward VI. At his suggestion a voyage was undertaken for the discovery +of a north-east passage to Cathay, with Sir Hugh Willoughby +as captain-general of the fleet and Richard Chancellor as pilot-major. +They sailed in May 1553, but Willoughby and all his crew +perished on the Lapland coast. Chancellor, however, was more +fortunate. He reached the White Sea, performed the journey +overland to Moscow, where he was well received, and may be said +to have been the founder of the trade between Russia and England. +He returned to Archangel and brought his ship back in safety to +England. On a second voyage, in 1556, Chancellor was drowned; +and three subsequent voyages, led by Stephen Burrough, Arthur +Pet and Charles Jackman, in small craft of 50 tons and under, +carried on an examination of the straits which lead into the Kara +sea.</p> + +<p>The French followed closely on the track of John Cabot, and +Norman and Breton fishermen frequented the banks of Newfoundland +at the beginning of the 16th century. In 1524 Francis I. sent +Giovanni da Verazzano of Florence on an expedition of discovery +to the coast of North America; and the details of his voyage were +embodied in a letter addressed by him to the king of France from +Dieppe, in July 1524. In 1534 Jacques Cartier set out to continue +the discoveries of Verazzano, and visited Newfoundland and the +Gulf of St Lawrence. In the following year he made another +voyage, discovered the island of Anticosti, and ascended the St +Lawrence to Hochelaga, now Montreal. He returned, after passing +two winters in Canada; and on another occasion he also failed to +establish a colony. Admiral de Coligny made several unsuccessful +endeavours to form a colony in Florida under Jean Ribault +of Dieppe, René de Laudonnière and others, but the settlers +were furiously assailed by the Spaniards and the attempt was +abandoned.</p> + +<p>The reign of Elizabeth is famous for the gallant enterprises that +were undertaken by sea and land to discover and bring to light the +unknown parts of the earth. The great promoter of +geographical discovery in the Elizabethan period was +<span class="sidenote">The Elizabethan era.</span> +Richard Hakluyt (1553-1616), who was active in the formation +of the two companies for colonizing Virginia in +1606; and devoted his life to encouraging and recording similar +undertakings. He published much, and left many valuable papers +at his death, most of which, together with many other narratives, +were published in 1622 in the great work of the Rev. Samuel Purchas, +entitled <i>Hakluytus Posthumus, or Purchas his Pilgrimes</i>.</p> + +<p>It is from these works that our knowledge of the gallant deeds of +the English and other explorers of the Elizabethan age is mainly +derived. The great and splendidly illustrated collections of voyages +and travels of Theodorus de Bry and Hulsius served a similar useful +purpose on the continent of Europe. One important object of +English maritime adventurers of those days was to discover a route +to Cathay by the north-west, a second was to settle Virginia, and a +third was to raid the Spanish settlements in the West Indies. Nor +was the trade to Muscovy and Turkey neglected; while latterly +a resolute and successful attempt was made to establish direct +commercial relations with India.</p> + +<p>The conception of the north-western route to Cathay now leads +the story of exploration, for the first time as far as important and +sustained efforts are concerned, towards the Arctic seas. This part +of the story is fully told under the heading of <span class="sc"><a href="#artlinks">Polar Regions</a></span>, and +only the names of Martin Frobisher (1576), John Davis (1585), +Henry Hudson (1607) and William Baffin (1616) need be mentioned +here in order to preserve the complete conspectus of the history of +discovery. The Dutch emulated the British in the Arctic seas during +this period, directing their efforts mainly towards the discovery of +a north-east passage round the northern end of Novaya Zemlya; +and William Barents or Barendsz (1594-1597) is the most famous +name in this connexion, his boat voyage along the coast of Novaya +Zemlya after losing his ship and wintering in a high latitude, being +one of the most remarkable achievements in polar annals.</p> + +<p>Many English voyages were also made to Guinea and the West +Indies, and twice English vessels followed in the track of Magellan, +and circumnavigated the globe. In 1577 Francis Drake, who had +previously served with Hawkins in the West Indies, undertook his +celebrated voyage round the world. Reaching the Pacific through +the Strait of Magellan, Drake proceeded northward along the west +coast of America, resolved to attempt the discovery of a northern +passage from the Pacific to the Atlantic. The coast from the +southern extremity of the Californian peninsula to Cape Mendocino +had been discovered by Juan Rodriguez Cabrillo and Francisco de +Ulloa in 1539. Drake’s discoveries extended from Cape Mendocino +to 48° N., in which latitude he gave up his quest, sailed across the +Pacific and reached the Philippine Islands, returning home round +the Cape of Good Hope in 1580.</p> + +<p>Thomas Cavendish, emulous of Drake’s example, fitted out three +vessels for an expedition to the South sea in 1586. He took the +same route as Drake along the west coast of America. From Cape +San Lucas Cavendish steered across the Pacific, seeing no land until +he reached the Ladrone Islands. He returned to England in 1588. +The third English voyage into the Pacific was not so fortunate. +Sir Richard Hawkins (1593) on reaching the bay of Atacames, in 1°N. +in 1594, was attacked by a Spanish fleet, and, after a desperate +naval engagement, was forced to surrender. Hawkins declared +his object to be discovery and the survey of unknown lands, and +his voyage, though terminating in disaster, bore good fruit. <i>The +Observations of Sir Richard Hawkins in his Voyage into the South Sea</i>, +published in 1622, are very valuable. It was long before another +British ship entered the Pacific Ocean. Sir John Narborough took +two ships through the Strait of Magellan in 1670 and touched on +the coast of Chile, but it was not until 1685 that Dampier sailed over +the part of the Pacific where Hawkins met his defeat.</p> + +<p>The exploring enterprise of the Spanish nation did not wane +after the conquest of Peru and Mexico, and the acquisition of the +vast empire of the Indies. It was spurred into renewed activity +by the audacity of Sir John Hawkins in the West Indies, and by +the appearance of Drake, Cavendish and Richard Hawkins in the +Pacific.</p> + +<p>In the interior of South America the Spanish conquerors had +explored the region of the Andes from the isthmus of Panama to +Chile. Pedro de Valdivia in 1540 made an expedition into the +country of the Araucanian Indians of Chile, and was the first to +<span class="pagenum"><a name="page627" id="page627"></a>627</span> +explore the eastern base of the Andes in what is now Argentine +Patagonia. In 1541 Francisco de Orellana discovered the whole +course of the Amazon from its source in the Andes to the Atlantic. +A second voyage on the Amazon was made in 1561 by the mad pirate +Lope de Aguirre; but it was not until 1639 that a full account was +written of the great river by Father Cristoval de Acuña, who ascended +it from its mouth and reached the city of Quito.</p> + +<p>The voyage of Drake across the Pacific was preceded by that of +Alvaro de Mendaña, who was despatched from Peru in 1567 to +discover the great Antarctic continent which was believed +to extend far northward into the South sea, the search +<span class="sidenote">Spaniards in the Pacific.</span> +for which now became one of the leading motives of +exploration. After a voyage of eighty days across the +Pacific, Mendaña 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 Mendaña 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 Mendaña 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.</p> + +<p>The Portuguese, in the early part of the 17th century (1578-1640), +were under the dominion of Spain, and their enterprise was +to some extent damped; but their missionaries extended geographical +knowledge in Africa. Father Francisco Paez acquired great influence +in Abyssinia, and explored its highlands from 1600 to 1622. Fathers +Mendez and Lobo traversed the deserts between the coast of the +Red sea and the mountains, became acquainted with Lake Tsana, +and discovered the sources of the Blue Nile in 1624-1633.</p> + +<p>But the attention of the Portuguese was mainly devoted to vain +attempts to maintain their monopoly of the trade of India against +the powerful rivalry of the English and Dutch. The +English enterprises were persevering, continuous and +<span class="sidenote">Rivalry in the East.</span> +successful. James Lancaster made a voyage to the Indian +Ocean from 1591 to 1594; and in 1599 the merchants and adventurers +of London resolved to form a company, with the object of +establishing a trade with the East Indies. On the 31st of December +1599 Queen Elizabeth granted the charter of incorporation to the +East India Company, and Sir James Lancaster, one of the directors, +was appointed general of their first fleet. He was accompanied +by John Davis, the great Arctic navigator, as pilot-major. This +voyage was eminently successful. The ships touched at Achin in +Sumatra and at Java, returning with full ladings of pepper in 1603. +The second voyage was commanded by Sir Henry Middleton; but +it was in the third voyage, under Keelinge and Hawkins, that the +mainland of India was first reached in 1607. Captain Hawkins +landed at Surat and travelled overland to Agra, passing some time +at the court of the Great Mogul. In the voyage of Sir Edward +Michelborne in 1605, John Davis lost his life in a fight with a Japanese +junk. The eighth voyage, led by Captain Saris, extended the +operations of the company to Japan; and in 1613 the Japanese +government granted privileges to the company; but the British +retired in 1623, giving up their factory. The chief result of this +early intercourse between Great Britain and Japan was the interesting +series of letters written by William Adams from 1611 to 1617. From +the tenth voyage of the East India Company, commanded by +Captain Best, who left England in 1612, dates the establishment of +permanent British factories on the coast of India. It was Captain +Best who secured a regular <i>firman</i> for trade from the Great Mogul. +From that time a fleet was despatched every year, and the company’s +operations greatly increased geographical knowledge of India +and the Eastern Archipelago. British visits to Eastern countries, +at this time, were not confined to the voyages of the company. +Journeys were also made by land, and, among others, the entertaining +author of the <i>Crudities</i>, Thomas Coryate, of Odcombe in +Somersetshire, wandered on foot from France to India, and died +(1617) in the company’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.</p> + +<p>The Dutch nation, as soon as it was emancipated from Spanish +tyranny, displayed an amount of enterprise, which, for a long time, +was fully equal to that of the British. The Arctic voyages +of Barents were quickly followed by the establishment of +<span class="sidenote">Dutch exploration, 16th-17th centuries.</span> +a Dutch East India Company; and the Dutch, ousting +the Portuguese, not only established factories on the +mainland of India and in Japan, but acquired a preponderating +influence throughout the Malay Archipelago. In 1583 Jan +Hugen van Linschoten made a voyage to India with a Portuguese +fleet, and his full and graphic descriptions of India, Africa, China +and the Malay Archipelago must have been of no small use to his +countrymen in their distant voyages. The first of the Dutch Indian +voyages was performed by ships which sailed in April 1595, and +rounded the Cape of Good Hope. A second large Dutch fleet sailed +in 1598; and, so eager was the republic to extend her commerce +over the world that another fleet, consisting of five ships of Rotterdam, +was sent in the same year by way of Magellan’s Strait, under +Jacob Mahu as admiral, with William Adams as pilot. Mahu died +on the passage out, and was succeeded by Simon de Cordes, who +was killed on the coast of Chile. In September 1599 the fleet had +entered the Pacific. The ships were then steered direct for Japan, +and anchored off Bungo in April 1600. In the same year, 1598, a +third expedition was despatched under Oliver van Noort, a native +of Utrecht, but the voyage contributed nothing to geography. The +Dutch Company in 1614 again resolved to send a fleet to the Moluccas +by the westward route, and Joris Spilbergen was appointed to the +command as admiral, with a commission from the States-General. +He was furnished with four ships of Amsterdam, two of Rotterdam +and one from Zeeland. On the 6th of May 1615 Spilbergen entered +the Pacific Ocean, and touched at several places on the coast of Chile +and Peru, defeating the Spanish fleet in a naval engagement off +Chilca. After plundering Payta and making requisitions at Acapulco, +the Dutch fleet crossed the Pacific and reached the Moluccas in +March 1616.</p> + +<p>The Dutch now resolved to discover a passage into the Pacific +to the south of Tierra del Fuego, the insular nature of which had +been ascertained by Sir Francis Drake. The vessels fitted out for +this purpose were the “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.</p> + +<p>There were several early indications of the existence of the great +Australian continent, and the Dutch endeavoured to obtain further +knowledge concerning the country and its extent; but only its +northern and western coasts had been visited before the time of +Governor van Diemen. Dirk Hartog had been on the west coast +in latitude 26° 30′ S. in 1616. Pelsert struck on a reef called “Houtman’s +Abrolhos” on the 4th of June 1629. In 1697 the Dutch +captain Vlamingh landed on the west coast of Australia, then called +New Holland, in 31° 43′ S., and named the Swan river from the black +swans he discovered there. In 1642 the governor and council of +Batavia fitted out two ships to prosecute the discovery of the south +land, then believed to be part of a vast Antarctic continent, and +entrusted the command to Captain Abel Jansen Tasman. This +voyage proved to be the most important to geography that had been +undertaken since the first circumnavigation of the globe. Tasman +sailed from Batavia in 1642, and on the 24th of November sighted +high land in 42° 30′ S., which was named van Diemen’s Land, and +after landing there proceeded to the discovery of the western coast +of New Zealand; at first called Staten Land, and supposed to be connected +with the Antarctic continent from which this voyage proved +New Holland to be separated. He then reached Tongatabu, one +of the Friendly Islands of Cook; and returned by the north coast +of New Guinea to Batavia. In 1644 Tasman made a second voyage +to effect a fuller discovery of New Guinea.</p> + +<p><span class="pagenum"><a name="page628" id="page628"></a>628</span></p> + +<p>The French directed their enterprise more in the direction of +North America than of the Indies. One of their most distinguished +explorers was Samuel Champlain, a captain in the navy, +<span class="sidenote">French in North America.</span> +who, after a remarkable journey through Mexico and the +West Indies from 1599 to 1602, established his historic +connexion with Canada, to the geographical knowledge +of which he made a very large addition.</p> + +<p>The principles and methods of surveying and position finding +had by this time become well advanced, and the most remarkable +example of the early application of these improvements +is to be found in the survey of China by Jesuit missionaries. +<span class="sidenote">Missionaries in the East.</span> +They first prepared a map of the country round Peking, +which was submitted to the emperor Kang-hi, and, +being satisfied with the accuracy of the European method of surveying, +he resolved to have a survey made of the whole empire on the +same principles. This great work was begun in July 1708, and the +completed maps were presented to the emperor in 1718. The +records preserved in each city were examined, topographical information +was diligently collected, and the Jesuit fathers checked their +triangulation by meridian altitudes of the sun and pole star and by a +system of remeasurements. The result was a more accurate map of +China than existed, at that time, of any country in Europe. Kang-hi +next ordered a similar map to be made of Tibet, the survey being +executed by two lamas who were carefully trained as surveyors +by the Jesuits at Peking. From these surveys were constructed +the well-known maps which were forwarded to Duhalde, and which +D’Anville utilized for his atlas.</p> + +<p>Several European missionaries had previously found their way +from India to Tibet. Antonio Andrada, in 1624, was the first +European to enter Tibet since the visit of Friar Odoric +<span class="sidenote">The 18th century.</span> +in 1325. The next journey was that of Fathers Grueber +and Dorville about 1660, who succeeded in passing from +China, through Tibet, into India. In 1715 Fathers Desideri and +Freyre made their way from Agra, across the Himalayas, to Lhasa, +and the Capuchin Friar Orazio della Penna resided in that city +from 1735 until 1747. But the most remarkable journey in this +direction was performed by a Dutch traveller named Samuel van de +Putte. He left Holland in 1718, went by land through Persia to +India, and eventually made his way to Lhasa, where he resided for a +long time. He went thence to China, returned to Lhasa, and was +in India in time to be an eye-witness of the sack of Delhi by Nadir +<span class="sidenote">Asia.</span> +Shah in 1737. In 1743 he left India and died at Batavia +on the 27th of September 1745. The premature death +of this illustrious traveller is the more to be lamented because his +vast knowledge died with him. Two English missions sent by +Warren Hastings to Tibet, one led by George Bogle in 1774, and the +other by Captain Turner in 1783, complete Tibetan exploration in +the 18th century.</p> + +<p>From Persia much new information was supplied by Jean Chardin, +Jean Tavernier, Charles Hamilton, Jean de Thévenot and Father +Jude Krusinski, and by English traders on the Caspian. In 1738 +John Elton traded between Astrakhan and the Persian port of +Enzelî 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.</p> + +<p>Arabia received very careful attention, in the 18th century, +from the Danish scientific mission, which included Carsten Niebuhr +among its members. Niebuhr landed at Loheia, on the coast of +Yemen, in December 1762, and went by land to Sana. All the other +members of the mission died, but he proceeded from Mokha to +Bombay. He then made a journey through Persia and Syria to +Constantinople, returning to Copenhagen in 1767. His valuable +work, the <i>Description of Arabia</i>, was published in 1772, and was +followed in 1774-1778 by two volumes of travels in Asia. The great +traveller survived until 1815, when he died at the age of eighty-two.</p> + +<p>James Bruce of Kinnaird, the contemporary of Niebuhr, was +equally devoted to Eastern travel; and his principal geographical +work was the tracing of the Blue Nile from its source to +its junction with the White Nile. Before the death of +<span class="sidenote">Africa.</span> +Bruce an African Association was formed, in 1788, for collecting +information respecting the interior of that continent, with Major +Rennell and Sir Joseph Banks as leading members. The association +first employed John Ledyard (who had previously made an extraordinary +journey into Siberia) to cross Africa from east to west +on the parallel of the Niger, and William Lucas to cross the Sahara +to Fezzan. Lucas went from Tripoli to Mesurata, obtained some +information respecting Fezzan and returned in 1789. One of the +chief problems the association wished to solve was that of the existence +and course of the river Niger, which was believed by some +authorities to be identical with the Congo. Mungo Park, then an +assistant surgeon of an Indiaman, volunteered his services, which +were accepted by the association, and in 1795 he succeeded in +reaching the town of Segu on the Niger, but was prevented from +continuing his journey to Timbuktu. Five years later he accepted +an offer from the government to command an expedition into the +interior of Africa, the plan being to cross from the Gambia to the +Niger and descend the latter river to the sea. After losing most of +his companions he himself and the rest perished in a rapid on the +Niger at Busa, having been attacked from the shore by order of a +chief who thought he had not received suitable presents. His work, +however, had established the fact that the Niger was not identical +with the Congo.</p> + +<p>While the British were at work in the direction of the Niger, the +Portuguese were not unmindful of their old exploring fame. In +1798 Dr F.J.M. de Lacerda, an accomplished astronomer, was +appointed to command a scientific expedition of discovery to the +north of the Zambesi. He started in July, crossed the Muchenja +Mountains, and reached the capital of the Cazembe, where he died +of fever. Lacerda left a valuable record of his adventurous journey; +but with Mungo Park and Lacerda the history of African exploration +in the 18th century closes.</p> + +<p>In South America scientific exploration was active during this +period. The great geographical event of the century, as regards +that continent, was the measurement of an arc of the +meridian. The undertaking was proposed by the French +<span class="sidenote">South America.</span> +Academy as part of an investigation with the object +of ascertaining the length of the degree near the equator and near the +pole respectively so as to determine the figure of the earth. A +commission left Paris in 1735, consisting of Charles Marie de la +Condamine, Pierre Bouguer, Louis Godin and Joseph de Jussieu +the naturalist. Spain appointed two accomplished naval officers, +the brothers Ulloa, as coadjutors. The operations were carried on +during eight years on a plain to the south of Quito; and, in addition +to his memoir on this memorable measurement, La Condamine +collected much valuable geographical information during a voyage +down the Amazon. The arc measured was 3° 7′ 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.</p> + +<p>The British and French governments despatched several expeditions +of discovery into the Pacific and round the world during the +18th century. They were preceded by the wonderful +and romantic voyages of the buccaneers. The narratives +<span class="sidenote">The Pacific Ocean.</span> +of such men as Woodes Rogers, Edward Davis, George +Shelvocke, Clipperton and William Dampier, can never +fail to interest, while they are not without geographical value. +The works of Dampier are especially valuable, and the narratives +of William Funnell and Lionel Wafer furnished the best accounts +then extant of the Isthmus of Darien. Dampier’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.</p> + +<p>The voyage of Lord Anson to the Pacific in 1740-1744 was of a +predatory character, and he lost more than half his men from scurvy; +while it is not pleasant to reflect that at the very time when the +French and Spaniards were measuring an arc of the meridian at +Quito, the British under Anson were pillaging along the coast of the +Pacific and burning the town of Payta. But a romantic interest +attaches to the wreck of the “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.</p> + +<p>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 Mendaña +and Quiros; and discovered the strait separating New Britain from +New Ireland. He reached Spithead again in February 1769. Wallis +and Carteret were followed very closely by the French expedition +of Bougainville, which sailed from Nantes in November 1766. +Bougainville had first to perform the unpleasant task of delivering +up the Falkland Islands, where he had encouraged the formation +of a French settlement, to the Spaniards. He then entered the +Pacific, and reached Tahiti in April 1768. Passing through the New +<span class="pagenum"><a name="page629" id="page629"></a>629</span> +Hebrides group he touched at Batavia, and arrived at St Malo after +an absence of two years and four months.</p> + +<p>The three voyages of Captain James Cook form an era in the history +of geographical discovery. In 1767 he sailed for Tahiti, with the +object of observing the transit of Venus, accompanied +by two naturalists, Sir Joseph Banks and Dr Solander, +<span class="sidenote">Captain Cook.</span> +a pupil of Linnaeus, as well as by two astronomers. The +transit was observed on the 3rd of June 1769. After exploring +Tahiti and the Society group, Cook spent six months surveying New +Zealand, which he discovered to be an island, and the coast of New +South Wales from latitude 38° S. to the northern extremity. The +belief in a vast Antarctic continent stretching far into the temperate +zone had never been abandoned, and was vehemently asserted by +Charles Dalrymple, a disappointed candidate nominated by the +Royal Society for the command of the Transit expedition of 1769. +In 1772 the French explorer Yves Kerguelen de Tremarec had discovered +the land that bears his name in the South Indian Ocean +without recognizing it to be an island, and naturally believed it +to be part of the southern continent.</p> + +<p>Cook’s second voyage was mainly intended to settle the question +of the existence of such a continent once for all, and to define the +limits of any land that might exist in navigable seas towards the +Antarctic circle. James Cook at his first attempt reached a south +latitude of 57° 15′. On a second cruise from the Society Islands, +in 1773, he, first of all men, crossed the Antarctic circle, and was +stopped by ice in 71° 10′ S. During the second voyage Cook visited +Easter Island, discovered several islands of the New Hebrides and +New Caledonia; and on his way home by Cape Horn, in March 1774, +he discovered the Sandwich Island group and described South +Georgia. He proved conclusively that any southern continent +that might exist lay under the polar ice. The third voyage was +intended to attempt the passage from the Pacific to the Atlantic by +the north-east. The “Resolution” and “Discovery” sailed in +1776, and Cook again took the route by the Cape of Good Hope. +On reaching the North American coast, he proceeded northward, +fixed the position of the western extremity of America and surveyed +Bering Strait. He was stopped by the ice in 70° 41′ N., and named +the farthest visible point on the American shore Icy Cape. He then +visited the Asiatic shore and discovered Cape North. Returning to +Hawaii, Cook was murdered by the natives. On the 14th of February +1779, his second, Captain Edward Clerke, took command, and +proceeding to Petropavlovsk in the following summer, he again +examined the edge of the ice, but only got as far as 70° 33′ N. The +ships returned to England in October 1780.</p> + +<p>In 1785 the French government carefully fitted out an expedition +of discovery at Brest, which was placed under the command of +François La Pérouse, an accomplished and experienced officer. +After touching at Concepcion in Chile and at Easter Island, La +Pérouse proceeded to Hawaii and thence to the coast of California, +of which he has given a very interesting account. He then crossed +the Pacific to Macao, and in July 1787 he proceeded to explore the +Gulf of Tartary and the shores of Sakhalin, remaining some time at +Castries Bay, so named after the French minister of marine. Thence +he went to the Kurile Islands and Kamchatka, and sailed from the +far north down the meridian to the Navigator and Friendly Islands. +He was in Botany Bay in January 1788; and sailing thence, the +explorer, his ship and crew were never seen again. Their fate was +long uncertain. In September 1791 Captain Antoine d’Entrecasteaux +sailed from Brest with two vessels to seek for tidings. +He visited the New Hebrides, Santa Cruz, New Caledonia and Solomon +Islands, and made careful though rough surveys of the Louisiade +Archipelago, islands north of New Britain and part of New Guinea. +D’Entrecasteaux died on board his ship on the 20th of July 1793, +without ascertaining the fate of La Pérouse. Captain Peter Dillon +at length ascertained, in 1828, that the ships of La Pérouse had been +wrecked on the island of Vanikoro during a hurricane.</p> + +<p>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.</p> + +<p>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.</p> + +<p>The 18th century saw the Arctic coast of North America reached +at two points, as well as the first scientific attempt to reach the +North Pole. The Hudson Bay Company had been incorporated +in 1670, and its servants soon extended their +<span class="sidenote">Arctic regions.</span> +operations over a wide area to the north and west of +Canada. In 1741 Captain Christopher Middleton was ordered to +solve the question of a passage from Hudson Bay to the westward. +Leaving Fort Churchill in July 1742, he discovered the Wager river +and Repulse Bay. He was followed by Captain W. Moor in 1746, +and Captain Coats in 1751, who examined the Wager Inlet up to the +end. In November 1769 Samuel Hearne was sent by the Hudson +Bay Company to discover the sea on the north side of America, +but was obliged to return. In February 1770 he set out again from +Fort Prince of Wales; but, after great hardships, he was again +forced to return to the fort. He started once more in December +1771, and at length reached the Coppermine river, which he surveyed +to its mouth, but his observations are unreliable. With the same +object Alexander Mackenzie, with a party of Canadians, set out from +Fort Chippewyan on the 3rd of June 1789, and descending the great +river which now bears the explorer’s name reached the Arctic sea.</p> + +<p>In February 1773 the Royal Society submitted a proposal to the +king for an expedition towards the North Pole. The expedition was +fitted out under Captains Constantine Phipps and Skeffington +Lutwidge, and the highest latitude reached was 80° 48′ N., but no +opening was discovered in the heavy Polar pack. The most important +Arctic work in the 18th century was performed by the +Russians, for they succeeded in delineating the whole of the northern +coast of Siberia. Some of this work was possibly done at a still +earlier date. The Cossack Simon Dezhneff is thought to have made a +voyage, in the summer of 1648, from the river Kolyma, through +Bering Strait (which was rediscovered by Vitus Bering in 1728) to +Anadyr. Between 1738 and 1750 Manin and Sterlegoff made their +way in small sloops from the mouth of the Yenesei as far north as +75° 15′ N. The land from Taimyr to Cape Chelyuskin, the most +northern extremity of Siberia, was mapped in many years of patient +exploration by Chelyuskin, who reached the extreme point +(77° 34′ N.) in May 1742. To the east of Cape Chelyuskin the +Russians encountered greater difficulties. They built small vessels +at Yakutsk on the Lena, 900 m. from its mouth, whence the first +expedition was despatched under Lieut. Prontschichev in 1735. He +sailed from the mouth of the Lena to the mouth of the Olonek, +where he wintered, and on the 1st of September 1736 he got as far +as 77° 29′ N., within 5 m. of Cape Chelyuskin. Both he and his +young wife died of scurvy, and the vessel returned. A second +expedition, under Lieut. Laptyev, started from the Lena in 1739, +but encountered masses of drift ice in Chatanga bay, and with this +ended the voyages to the westward of the Lena. Several attempts +were also made to navigate the sea from the Lena to the Kolyma. +In 1736 Lieut. Laptyev sailed, but was stopped by the drift ice in +August, and in 1739, during another trial, he reached the mouth +of the Indigirka, where he wintered. In the season of 1740 he +continued his voyage to beyond the Kolyma, wintering at Nizhni +Kolymsk. In September 1740 Vitus Bering sailed from Okhotsk +on a second Arctic voyage with George William Steller on board +as naturalist. In June 1741 he named the magnificent peak on the +coast of North America Mount St Elias and explored the Aleutian +Islands. In November the ship was wrecked on Bering Island; +and the gallant Dane, worn out with scurvy, died there on the +8th of December 1741. In March 1770 a merchant named Liakhov +saw a large herd of reindeer coming from the north to the Siberian +coast, which induced him to start in a sledge in the direction whence +they came. Thus he reached the New Siberian or Liakhov Islands, +and for years afterwards the seekers for fossil ivory resorted to them. +The Russian Captain Vassili Chitschakov in 1765 and 1766 made two +persevering attempts to penetrate the ice north of Spitsbergen, +and reached 80° 30′ N., while Russian parties twice wintered at Bell +Sound.</p> + +<p>In reviewing the progress of geographical discovery thus far, it +has been possible to keep fairly closely to a chronological order. +But in the 19th century and after exploring work was so +generally and steadily maintained in all directions, and +<span class="sidenote">Geographical societies.</span> +was in so many cases narrowed down from long journeys +to detailed surveys within relatively small areas, that it +becomes desirable to cover the whole period at one view for certain +great divisions of the world. (See <span class="sc"><a href="#artlinks">Africa</a></span>; <span class="sc"><a href="#artlinks">Asia</a></span>; <span class="sc"><a href="#artlinks">Australia</a></span>; <span class="sc"><a href="#artlinks">Polar +Regions</a></span>; &c.) Here, however, may be noticed the development +of geographical societies devoted to the encouragement of exploration +and research. The first of the existing geographical societies was +<span class="pagenum"><a name="page630" id="page630"></a>630</span> +that of Paris, founded in 1825 under the title of La Société de +Géographie. The Berlin Geographical Society (Gesellschaft für +Erdkunde) is second in order of seniority, having been founded in +1827. The Royal Geographical Society, which was founded in +London in 1830, comes third on the list; but it may be viewed as a +direct result of the earlier African Association founded in 1788. +Sir John Barrow, Sir John Cam Hobhouse (Lord Broughton), Sir +Roderick Murchison, Mr Robert Brown and Mr Bartle Frere formed +the foundation committee of the Royal Geographical Society, and +the first president was Lord Goderich. The action of the society in +supplying practical instruction to intending travellers, in astronomy, +surveying and the various branches of science useful to collectors, +has had much to do with advancement of discovery. Since the war +of 1870 many geographical societies have been established on the +continent of Europe. At the close of the 19th century there were +upwards of 100 such societies in the world, with more than 50,000 +members, and over 150 journals were devoted entirely to geographical +subjects.<a name="fa11b" id="fa11b" href="#ft11b"><span class="sp">11</span></a> The great development of photography has been a notable +aid to explorers, not only by placing at their disposal a faithful and +ready means of recording the features of a country and the types +of inhabitants, but by supplying a method of quick and accurate +topographical surveying.</p> + +<p class="pt2 center sc">The Principles of Geography</p> + +<p>As regards the scope of geography, the order of the various +departments and their inter-relation, there is little difference of +opinion, and the principles of geography<a name="fa12b" id="fa12b" href="#ft12b"><span class="sp">12</span></a> are now generally accepted +by modern geographers. The order in which the various subjects +are treated in the following sketch is the natural succession from +fundamental to dependent facts, which corresponds also to the +evolution of the diversities of the earth’s crust and of its inhabitants.</p> + +<p>The fundamental geographical conceptions are mathematical, the +relations of space and form. The figure and dimensions of the +earth are the first of these. They are ascertained by a +combination of actual measurement of the highest +<span class="sidenote">Mathematical geography.</span> +precision on the surface and angular observations of the +positions of the heavenly bodies. The science of geodesy +is part of mathematical geography, of which the arts of surveying +and cartography are applications. The motions of the earth +as a planet must be taken into account, as they render possible +the determination of position and direction by observations of the +heavenly bodies. The diurnal rotation of the earth furnishes two +fixed points or poles, the axis joining which is fixed or nearly so in its +direction in space. The rotation of the earth thus fixes the directions +of north and south and defines those of east and west. The angle +which the earth’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.<a name="fa13b" id="fa13b" href="#ft13b"><span class="sp">13</span></a></p> + +<p>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 +<span class="sidenote">Physical geography.</span> +term which appears to have been introduced by Linnaeus, +and was reinvented as a substitute for the cosmography of the middle +ages by Professor Huxley. Although the term has since been limited +by some writers to one particular part of the subject, it seems best +to maintain the original and literal meaning. In the stricter sense, +physical geography is that part of geography which involves the +processes of contemporary change in the crust and the circulation +of the fluid envelopes. It thus draws upon physics for the explanation +of the phenomena with the space-relations of which it is specially +concerned. Physical geography naturally falls into three divisions, +dealing respectively with the surface of the lithosphere—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.</p> + +<p>Geomorphology is the part of geography which deals with terrestrial +relief, including the submarine as well as the subaërial portions +of the crust. The history of the origin of the various forms belongs +to geology, and can be completely studied only by geological +<span class="sidenote">Geomorphology.</span> +methods. But the relief of the crust is not a finished piece of sculpture; +the forms are for the most part transitional, owing +their characteristic outlines to the process by which they +are produced; therefore the geographer must, for strictly +geographical purposes, take some account of the processes which are +now in action modifying the forms of the crust. Opinion still differs +as to the extent to which the geographer’s work should overlap that +of the geologist.</p> + +<p>The primary distinction of the forms of the crust is that between +elevations and depressions. Granting that the geoid or mean +surface of the ocean is a uniform spheroid, the distribution of land +and water approximately indicates a division of the surface of the +globe into two areas, one of elevation and one of depression. The +increasing number of measurements of the height of land in all +continents and islands, and the very detailed levellings in those +countries which have been thoroughly surveyed, enable the average +elevation of the land above sea-level to be fairly estimated, although +many vast gaps in accurate knowledge remain, and the estimate +is not an exact one. The only part of the sea-bed the configuration +of which is at all well known is the zone bordering the coasts where +the depth is less than about 100 fathoms or 200 metres, <i>i.e.</i> those +parts which sailors speak of as “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.</p> + +<p>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 +<span class="sidenote">Crustal relief.</span> +unveiled of water would exhibit a great region of elevation +arranged with a certain rough radiate symmetry round the north +pole, and extending southwards in three unequal arms which taper +to points in the south. A depression surrounds the little-known +south polar region in a continuous ring and extends northwards in +three vast hollows lying between the arms of the elevated area. So +far only is it possible to speak with certainty, but it is permissible +to take a few steps into the twilight of dawning knowledge and +indicate the chief subdivisions which are likely to be established +in the great crust-hollow and the great crust-heap. The boundary +between these should obviously be the mean surface of the +sphere.</p> + +<p>Sir John Murray deduced the mean height of the land of the globe +as about 2250 ft. above sea-level, and the mean depth of the oceans +as 2080 fathoms or 12,480 ft. below sea-level.<a name="fa14b" id="fa14b" href="#ft14b"><span class="sp">14</span></a> Calculating the area +of the land at 55,000,000 sq. m. (or 28.6% of the surface), and that +of the oceans as 137,200,000 sq. m. (or 71.4% of the surface), he +found that the volume of the land above sea-level was 23,450,000 +cub. m., the volume of water below sea-level 323,800,000, and the +total volume of the water equal to about <span class="spp">1</span>⁄<span class="suu">666</span>th of the volume of the +whole globe. From these data, as revised by A. Supan,<a name="fa15b" id="fa15b" href="#ft15b"><span class="sp">15</span></a> H.R. Mill +calculated the position of mean sphere-level at about 10,000 ft. or +1700 fathoms below sea-level. He showed that an imaginary +spheroidal shell, concentric with the earth and cutting the slope +between the elevated and depressed areas at the contour-line of 1700 +fathoms, would not only leave above it a volume of the crust equal +to the volume of the hollow left below it, but would also divide the +surface of the earth so that the area of the elevated region was +equal to that of the depressed region.<a name="fa16b" id="fa16b" href="#ft16b"><span class="sp">16</span></a></p> + +<p>A similar observation was made almost simultaneously by +Romieux,<a name="fa17b" id="fa17b" href="#ft17b"><span class="sp">17</span></a> who further speculated on the equilibrium between the +weight of the elevated land mass and that of the total +waters of the ocean, and deduced some interesting relations +<span class="sidenote">Areas of the crust according to Murray.</span> +between them. Murray, as the result of his study, +divided the earth’s surface into three zones—the <i>continental +area</i> containing all dry land, the <i>transitional area</i> including +the submarine slopes down to 1000 fathoms, and the <i>abysmal area</i> +consisting of the floor of the ocean beyond that depth; and Mill +proposed to take the line of mean-sphere level, instead of the empirical +depth of 1000 fathoms, as the boundary between the transitional +and abysmal areas.</p> + +<p>An elaborate criticism of all the existing data regarding the +volume relations of the vertical relief of the globe was made in +1894 by Professor Hermann Wagner, whose recalculations of volumes +<span class="pagenum"><a name="page631" id="page631"></a>631</span> +and mean heights—the best results which have yet been obtained—led +to the following conclusions.<a name="fa18b" id="fa18b" href="#ft18b"><span class="sp">18</span></a></p> + +<p>The area of the dry land was taken as 28.3% of the surface of the +globe, and that of the oceans as 71.7%. The mean height deduced +for the land was 2300 ft. above sea-level, the mean depth +of the sea 11,500 ft. below, while the position of mean-sphere +<span class="sidenote">Areas of the crust according to Wagner.</span> +level comes out as 7500 ft. (1250 fathoms) below +sea-level. From this it would appear that 43% of the +earth’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.</p> + +<p>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,<a name="fa19b" id="fa19b" href="#ft19b"><span class="sp">19</span></a> Wagner +showed his results graphically.</p> + +<p>This curve with the values reduced from metres to feet is reproduced +below.</p> + +<p>Wagner subdivides the earth’s surface, according to elevation, +into the following five regions:</p> + +<p class="pt2 center"><i>Wagner’s Divisions of the Earth’s Crust:</i></p> + +<table class="ws" summary="Contents"> +<tr><td class="tccm allb">Name.</td> <td class="tccm allb">Per cent of<br />Surface.</td> <td class="tccm allb">From</td> <td class="tccm allb">To</td></tr> + +<tr><td class="tcl lb rb">Depressed area</td> <td class="tcr rb">3</td> <td class="tcc rb">Deepest.</td> <td class="tcr rb">−16,400 feet.</td></tr> +<tr><td class="tcl lb rb">Oceanic plateau</td> <td class="tcr rb">54</td> <td class="tcr rb">−16,400 feet.</td> <td class="tcr rb">− 7,400 feet.</td></tr> +<tr><td class="tcl lb rb">Continental slope</td> <td class="tcr rb">9</td> <td class="tcr rb">− 7,400 feet.</td> <td class="tcr rb">−   660 feet.</td></tr> +<tr><td class="tcl lb rb">Continental plateau</td> <td class="tcr rb">28</td> <td class="tcr rb">−   660 feet.</td> <td class="tcr rb">+ 3,000 feet.</td></tr> +<tr><td class="tcl lb rb bb">Culminating area</td> <td class="tcr rb bb">6</td> <td class="tcr rb bb">+ 3,300 feet.</td> <td class="tcc rb bb">Highest.</td></tr> +</table> + +<p>The continental plateau might for purposes of detailed study be +divided into the <i>continental shelf</i> from -660 ft. to sea-level, and +<i>lowlands</i> from sea-level to +660 ft. (corresponding to +the mean level of the whole globe).<a name="fa20b" id="fa20b" href="#ft20b"><span class="sp">20</span></a> <i>Uplands</i> reaching +from 660 ft. to 2300 (the approximate mean level of +the land), and <i>highlands</i>, from 2300 upwards, might +also be distinguished.</p> + +<div class="center ptb2"><img style="width:600px; height:364px; vertical-align: middle;" src="images/img631.jpg" alt="" /></div> + +<p>A striking fact in the configuration of the crust is +that each continent, or elevated mass of the crust, is +diametrically opposite to an ocean basin or great depression; +the only partial exception being in the case of southern +<span class="sidenote">Arrangement of world-ridges and hollows.</span> +South America, which is antipodal to eastern Asia. +Professor C. Lapworth has generalized the grand features +of crustal relief in a scheme of attractive simplicity. He +sees throughout all the chaos of irregular crust-forms the +recurrence of a certain harmony, a succession of folds or +waves which build up all the minor features.<a name="fa21b" id="fa21b" href="#ft21b"><span class="sp">21</span></a> One +great series of crust waves from east to west is crossed by a +second great series of crust waves from north to south, giving rise +by their interference to six great elevated masses (the continents), +arranged in three groups, each consisting of a northern and a +southern member separated by a minor depression. These elevated +masses are divided from one another by similar great depressions.</p> + +<p>He says: “The surface of each of our great continental masses of +land resembles that of a long and broad arch-like form, of +which we see the simplest type in the New World. The +surface of the North American arch is sagged downwards +<span class="sidenote">Lapworth’s fold-theory.</span> +in the middle into a central depression which +lies between two long marginal plateaus, and these +plateaus are finally crowned by the wrinkled crests which form its +two modern mountain systems. The surface of each of our ocean +floors exactly resembles that of a continent turned upside down. +Taking the Atlantic as our simplest type, we may say that the +surface of an ocean basin resembles that of a mighty trough or +syncline, buckled up more or less centrally in a medial ridge, which +is bounded by two long and deep marginal hollows, in the cores +of which still deeper grooves sink to the profoundest depths. This +complementary relationship descends even to the minor features +of the two. Where the great continental sag sinks below the ocean +level, we have our gulfs and our Mediterraneans, seen in our type +continent, as the Mexican Gulf and Hudson Bay. Where the +central oceanic buckle attains the water-line we have our oceanic +islands, seen in our type ocean, as St Helena and the Azores. Although +the apparent crust-waves are neither equal in size nor +symmetrical in form, this complementary relationship between +them is always discernible. The broad Pacific depression seems to +answer to the broad elevation of the Old World—the narrow trough +of the Atlantic to the narrow continent of America.”</p> + +<p>The most thorough discussion of the great features of terrestrial +relief in the light of their origin is that by Professor E. Suess,<a name="fa22b" id="fa22b" href="#ft22b"><span class="sp">22</span></a> who +points out that the plan of the earth is the result of +two movements of the crust—one, subsidence over +<span class="sidenote">Suess’s theory.</span> +wide areas, giving rise to oceanic depressions and leaving +the continents protuberant; the other, folding along comparatively +narrow belts, giving rise to mountain ranges. This theory of crust +blocks dropped by subsidence is opposed to Lapworth’s theory of +vast crust-folds, but geology is the science which has to decide +between them.</p> + +<p>Geomorphology is concerned, however, in the suggestions which +have been made as to the cause of the distribution of heap and +hollow in the larger features of the crust. Élie de Beaumont, in +his speculations on the relation between the direction of mountain +ranges and their geological age and character, was feeling towards a +comprehensive theory of the forms of crustal relief; but his ideas +were too geometrical, and his theory that the earth is a spheroid +built up on a rhombic dodecahedron, the pentagonal faces of which +determined the direction of mountain ranges, could not be proved.<a name="fa23b" id="fa23b" href="#ft23b"><span class="sp">23</span></a> +The “tetrahedral theory” brought forward by Lowthian Green,<a name="fa24b" id="fa24b" href="#ft24b"><span class="sp">24</span></a> +that the form of the earth is a spheroid based on a regular tetrahedron, +is more serviceable, because it accounts for three very +interesting facts of the terrestrial plan—(1) the antipodal +position of continents and ocean basins; (2) the triangular +outline of the continents; and (3) the excess of +sea in the southern hemisphere. Recent investigations +have recalled attention to the work of Lowthian Green, +but the question is still in the controversial stage.<a name="fa25b" id="fa25b" href="#ft25b"><span class="sp">25</span></a> The +study of tidal strain in the earth’s crust by Sir George +Darwin has led that physicist to indicate the possibility +of the triangular form and southerly direction of the +continents being a result of the differential or tidal +attraction of the sun and moon. More recently Professor +A.E.H. Love has shown that the great features of the +relief of the lithosphere may be expressed by spherical +harmonics of the first, second and third degrees, and their +formation related to gravitational action in a sphere of +unequal density.<a name="fa26b" id="fa26b" href="#ft26b"><span class="sp">26</span></a></p> + +<p>In any case it is fully recognized that the plan of the earth is so +clear as to leave no doubt as to its being due to some general cause +which should be capable of detection.</p> + +<p>If the level of the sea were to become coincident with the mean +level of the lithosphere, there would result one tri-radiate land-mass +of nearly uniform outline and one continuous sheet of water +<span class="pagenum"><a name="page632" id="page632"></a>632</span> +broken by few islands. The actual position of sea-level lies so near +<span class="sidenote">The continents.</span> +the summit of the crust-heap that the varied relief of the upper +portion leads to the formation of a complicated coast-line +and a great number of detached portions of land. +The hydrosphere is, in fact, continuous, and the land is +all in insular masses: the largest is the Old World of Europe, +Asia and Africa; the next in size, America; the third, possibly, +Antarctica; the fourth, Australia; the fifth, Greenland. After +this there is a considerable gap before New Guinea, Borneo, Madagascar, +Sumatra and the vast multitude of smaller islands descending +in size by regular gradations to mere rocks. The contrast between +island and mainland was natural enough in the days before the +discovery of Australia, and the mainland of the Old World was +traditionally divided into three continents. These “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.</p> + +<p>The typical continental form is triangular as regards its sea-level +outline. The relief of the surface typically includes a central plain, +sometimes dipping below sea-level, bounded by lateral +highlands or mountain ranges, loftier on one side than +<span class="sidenote">Homology of continents.</span> +on the other, the higher enclosing a plateau shut in by +mountains. South America and North America follow +this type most closely; Eurasia (the land mass of Europe and Asia) +comes next, while Africa and Australia are farther removed from +the type, and the structure of Antarctica and Greenland is unknown.</p> + +<p>If the continuous, unbroken, horizontal extent of land in a continent +is termed its <i>trunk</i>,<a name="fa27b" id="fa27b" href="#ft27b"><span class="sp">27</span></a> and the portions cut up by inlets or +channels of the sea into islands and peninsulas the <i>limbs</i>, it is possible +to compare the continents in an instructive manner.</p> + +<p>The following table is from the statistics of Professor H. Wagner,<a name="fa28b" id="fa28b" href="#ft28b"><span class="sp">28</span></a> +his metric measurements being transposed into British units:</p> + +<p class="pt2 center"><i>Comparison of the Continents.</i></p> + +<table class="ws" summary="Contents"> +<tr><td class="tccm allb"> </td> <td class="tccm allb">Area<br />total<br />mil.<br />sq. m.</td> <td class="tccm allb">Mean<br />height,<br />feet.</td> <td class="tccm allb">Area<br />trunk,<br />mil.<br />sq. m.</td> <td class="tccm allb">Area<br />penin-<br />sulas,<br />mil.<br />sq. m.</td> <td class="tccm allb">Area<br />islands,<br />mil.<br />sq. m.</td> <td class="tccm allb">Area<br />limbs,<br />mil.<br />sq. m.</td> <td class="tccm allb">Area<br />limbs,<br />per<br />cent.</td></tr> + +<tr><td class="tcl lb rb">Old World</td> <td class="tcr rb">35.8 </td> <td class="tcr rb">2360</td> <td class="tcr rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcr rb"> </td> <td class="tcc rb"> </td></tr> +<tr><td class="tcl lb rb">New World</td> <td class="tcr rb">16.2 </td> <td class="tcr rb">2230</td> <td class="tcr rb"> </td> <td class="tcc rb"> </td> <td class="tcc rb"> </td> <td class="tcr rb"> </td> <td class="tcc rb"> </td></tr> +<tr><td class="tcl lb rb">Eurasia</td> <td class="tcr rb">20.85</td> <td class="tcr rb">2620</td> <td class="tcr rb">15.42</td> <td class="tcc rb">4.09</td> <td class="tcc rb">1.34</td> <td class="tcc rb">5.43</td> <td class="tcr rb">26 </td></tr> +<tr><td class="tcl lb rb">Africa</td> <td class="tcr rb">11.46</td> <td class="tcr rb">2130</td> <td class="tcr rb">11.22</td> <td class="tcc rb">..</td> <td class="tcc rb">0.24</td> <td class="tcc rb">0.24</td> <td class="tcr rb">2.1</td></tr> +<tr><td class="tcl lb rb">North America</td> <td class="tcr rb">9.26</td> <td class="tcr rb">2300</td> <td class="tcr rb">6.92</td> <td class="tcc rb">0.78</td> <td class="tcc rb">1.56</td> <td class="tcc rb">2.34</td> <td class="tcr rb">25 </td></tr> +<tr><td class="tcl lb rb">South America</td> <td class="tcr rb">6.84</td> <td class="tcr rb">1970</td> <td class="tcr rb">6.76</td> <td class="tcc rb">0.02</td> <td class="tcc rb">0.06</td> <td class="tcc rb">0.08</td> <td class="tcr rb">1.1</td></tr> +<tr><td class="tcl lb rb">Australia</td> <td class="tcr rb">3.43</td> <td class="tcr rb">1310</td> <td class="tcr rb">2.77</td> <td class="tcc rb">0.16</td> <td class="tcc rb">0.50</td> <td class="tcc rb">0.66</td> <td class="tcr rb">19 </td></tr> +<tr><td class="tcl lb rb">Asia</td> <td class="tcr rb">17.02</td> <td class="tcr rb">3120</td> <td class="tcr rb">12.93</td> <td class="tcc rb">3.05</td> <td class="tcc rb">1.04</td> <td class="tcc rb">4.09</td> <td class="tcr rb">24 </td></tr> +<tr><td class="tcl lb rb bb">Europe</td> <td class="tcr rb bb">3.83</td> <td class="tcr rb bb">980</td> <td class="tcr rb bb">2.49</td> <td class="tcc rb bb">1.04</td> <td class="tcc rb bb">0.30</td> <td class="tcc rb bb">1.34</td> <td class="tcr rb bb">35 </td></tr> +</table> + +<p>The usual classification of islands is into continental and oceanic. +The former class includes all those which rise from the continental +shelf, or show evidence in the character of their rocks of +having at one time been continuous with a neighbouring +<span class="sidenote">Islands.</span> +continent. The latter rise abruptly from the oceanic abysses. +Oceanic islands are divided according to their geological character +into volcanic islands and those of organic origin, including coral +islands. More elaborate subdivisions according to structure, origin and +position have been proposed.<a name="fa29b" id="fa29b" href="#ft29b"><span class="sp">29</span></a> In some cases a piece of land is only +an island at high water, and by imperceptible gradation the form +passes into a peninsula. The typical peninsula is connected with the +mainland by a relatively narrow isthmus; the name is, however, extended +to any limb projecting from the trunk of the mainland, even +when, as in the Indian peninsula, it is connected by its widest part.</p> + +<p>Small peninsulas are known as promontories or headlands, and +the extremity as a cape. The opposite form, an inlet of the sea, is +known when wide as a gulf, bay or bight, according +to size and degree of inflection, or as a fjord or ria when +<span class="sidenote">Coasts.</span> +long and narrow. It is convenient to employ a specific name for a +projection of a coast-line less pronounced than a peninsula, and for +an inlet less pronounced than a bay or bight; outcurve and incurve +may serve the turn. The varieties of coast-lines were reduced to an +exact classification by Richthofen, who grouped them according to +the height and slope of the land into cliff-coasts (<i>Steilküsten</i>)—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.</p> + +<p>On large-scale maps it is necessary to show two coast-lines, one +for the highest, the other for the lowest tide; but in small-scale +maps a single line is usually wider than is required to +represent the whole breadth of the inter-tidal zone. +<span class="sidenote">Coast-lines.</span> +The measurement of a coast-line is difficult, because +the length will necessarily be greater when measured on a large-scale +map where minute irregularities can be taken into account. +It is usual to distinguish between the general coast-line measured +from point to point of the headlands disregarding the smaller bays, +and the detailed coast-line which takes account of every inflection +shown by the map employed, and follows up river entrances to the +point where tidal action ceases. The ratio between these two +coast-lines represents the “coastal development” of any region.</p> + +<p>While the forms of the sea-bed are not yet sufficiently well known +to admit of exact classification, they are recognized to be as a rule +distinct from the forms of the land, and the importance +of using a distinctive terminology is felt. Efforts have +<span class="sidenote">Submarine forms.</span> +been made to arrive at a definite international agreement +on this subject, and certain terms suggested by a committee were +adopted by the Eighth International Geographical Congress at New +York in 1904.<a name="fa30b" id="fa30b" href="#ft30b"><span class="sp">30</span></a> The forms of the ocean floor include the “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.</p> + +<p>The forms of the dry land are of infinite variety, and have been +studied in great detail.<a name="fa31b" id="fa31b" href="#ft31b"><span class="sp">31</span></a> From the descriptive or topographical +point of view, geometrical form alone should be considered; +<span class="sidenote">Land forms.</span> +but the origin and geological structure of +land forms must in many cases be taken into account +when dealing with the function they exercise in the control of +mobile distributions. The geographers who have hitherto given +most attention to the forms of the land have been trained as geologists, +and consequently there is a general tendency to make origin +or structure the basis of classification rather than form alone.</p> + +<p>The fundamental form-elements may be reduced to the six +proposed by Professor Penck as the basis of his double system of +classification by form and origin.<a name="fa32b" id="fa32b" href="#ft32b"><span class="sp">32</span></a> These may be looked +<span class="sidenote">The six elementary land forms.</span> +upon as being all derived by various modifications or +arrangements of the single form-unit, the <i>slope</i> or inclined +plane surface. No one form occurs alone, but always +grouped together with others in various ways to make up districts, +regions and lands of distinctive characters. The form-elements are:</p> + +<p><span class="pagenum"><a name="page633" id="page633"></a>633</span></p> + +<p>1. The <i>plain</i> or gently inclined uniform surface.</p> + +<p>2. The <i>scarp</i> or steeply inclined slope; this is necessarily of +small extent except in the direction of its length.</p> + +<p>3. The <i>valley</i>, composed of two lateral parallel slopes inclined +towards a narrow strip of plain at a lower level which itself slopes +downwards in the direction of its length. Many varieties of this +fundamental form may be distinguished.</p> + +<p>4. The <i>mount</i>, composed of a surface falling away on every side +from a particular place. This place may either be a point, as +in a volcanic cone, or a line, as in a mountain range or ridge of +hills.</p> + +<p>5. The <i>hollow</i> or form produced by a land surface sloping inwards +from all sides to a particular lowest place, the converse of a mount.</p> + +<p>6. The <i>cavern</i> or space entirely surrounded by a land surface.</p> + +<p>These forms never occur scattered haphazard over a region, +but always in an orderly subordination depending on their mode +of origin. The dominant forms result from crustal +movements, the subsidiary from secondary reactions +<span class="sidenote">Geology and land forms.</span> +during the action of the primitive forms on mobile distributions. +The geological structure and the mineral composition +of the rocks are often the chief causes determining the +character of the land forms of a region. Thus the scenery of a limestone +country depends on the solubility and permeability of the +rocks, leading to the typical Karst-formations of caverns, swallow-holes +and underground stream courses, with the contingent phenomena +of dry valleys and natural bridges. A sandy beach or desert +owes its character to the mobility of its constituent sand-grains, +which are readily drifted and piled up in the form of dunes. A +region where volcanic activity has led to the embedding of dykes or +bosses of hard rock amongst softer strata produces a plain broken by +abrupt and isolated eminences.<a name="fa33b" id="fa33b" href="#ft33b"><span class="sp">33</span></a></p> + +<p>It would be impracticable to go fully into the varieties of each +specific form; but, partly as an example of modern geographical +classification, partly because of the exceptional importance +of mountains amongst the features of the land, one +<span class="sidenote">Classification of mountains.</span> +exception may be made. The classification of mountains +into types has usually had regard rather to geological +structure than to external form, so that some geologists would even +apply the name of a mountain range to a region not distinguished +by relief from the rest of the country if it bear geological evidence +of having once been a true range. A mountain may be described +(it cannot be defined) as an elevated region of irregular surface +rising comparatively abruptly from lower ground. The actual +elevation of a summit above sea-level does not necessarily affect its +mountainous character; a gentle eminence, for instance, rising a +few hundred feet above a tableland, even if at an elevation of say +15,000 ft., could only be called a hill.<a name="fa34b" id="fa34b" href="#ft34b"><span class="sp">34</span></a> But it may be said that +any abrupt slope of 2000 ft. or more in vertical height may justly +be called a mountain, while abrupt slopes of lesser height may +be called hills. Existing classifications, however, do not take +account of any difference in kind between mountain and hills, +although it is common in the German language to speak of <i>Hügelland</i>, +<i>Mittelgebirge</i> and <i>Hochgebirge</i> with a definite significance.</p> + +<p>The simple classification employed by Professor James Geikie<a name="fa35b" id="fa35b" href="#ft35b"><span class="sp">35</span></a> +into mountains of accumulation, mountains of elevation and mountains +of circumdenudation, is not considered sufficiently thorough +by German geographers, who, following Richthofen, generally +adopt a classification dependent on six primary divisions, each of +which is subdivided. The terms employed, especially for the subdivisions, +cannot be easily translated into other languages, and the +English equivalents in the following table are only put forward +tentatively:—</p> + +<p class="pt2 center sc">Richthofen’s Classification of Mountains<a name="fa36b" id="fa36b" href="#ft36b"><span class="sp">36</span></a></p> + +<table class="reg" summary="poem"><tr><td> <div class="poemr"> +<p>I. <i>Tektonische Gebirge</i>—Tectonic mountains.</p> + <p class="i2">(<i>a</i>) <i>Bruchgebirge oder Schollengebirge</i>—Block mountains.</p> + <p class="i3">1. <i>Einseitige Schollengebirge oder Schollenrandgebirge</i>—Scarp or tilted block mountains.</p> + <p class="i5">(i.) <i>Tafelscholle</i>—Table blocks.</p> + <p class="i5">(ii.) <i>Abrasionsscholle</i>—Abraded blocks.</p> + <p class="i5">(iii.) <i>Transgressionsscholle</i>—Blocks of unconformable strata.</p> + <p class="i3">2. <i>Flexurgebirge</i>—Flexure mountains.</p> + <p class="i3">3. <i>Horstgebirge</i>—Symmetrical block mountains.</p> + <p class="i2">(<i>b</i>) <i>Faltungsgebirge</i>—Fold mountains.</p> + <p class="i3">1. <i>Homöomorphe Faltungsgebirge</i>—Homomorphic fold mountains.</p> + <p class="i3">2. <i>Heteromorphe Faltungsgebirge</i>—Heteromorphic fold mountains.</p> + +<p class="s">II. <i>Rumpfgebirge oder Abrasionsgebirge</i>—Trunk or abraded mountains.</p> +<p class="s">III. <i>Ausbruchsgebirge</i>—Eruptive mountains.</p> +<p class="s">IV. <i>Aufschüttungsgebirge</i>—Mountains of accumulation.</p> +<p class="s">V. <i>Flachböden</i>—Plateaux.</p> + <p class="i2">(<i>a</i>) <i>Abrasionsplatten</i>—Abraded plateaux.</p> + <p class="i2">(<i>b</i>) <i>Marines Flachland</i>—Plain of marine erosion.</p> + <p class="i2">(<i>c</i>) <i>Schichtungstafelland</i>—Horizontally stratified tableland.</p> + <p class="i2">(<i>d</i>) <i>Übergusstafelland</i>—Lava plain.</p> + <p class="i2">(<i>e</i>) <i>Stromflachland</i>—River plain.</p> + <p class="i2">(<i>f</i>) <i>Flachböden der atmosphärischen Aufschüttung</i>—Plains of aeolian formation.</p> +<p class="s">VI. <i>Erosionsgebirge</i>—Mountains of erosion.</p> +</div> </td></tr></table> + +<p>From the morphological point of view it is more important to +distinguish the associations of forms, such as the <i>mountain mass</i> +or group of mountains radiating from a centre, with the +valleys furrowing their flanks spreading towards every +<span class="sidenote">Mountain forms.</span> +direction; the <i>mountain chain</i> or line of heights, forming a +long narrow ridge or series of ridges separated by parallel valleys; +the <i>dissected plateau</i> or highland, divided into mountains of circumdenudation +by a system of deeply-cut valleys; and the <i>isolated +peak</i>, usually a volcanic cone or a hard rock mass left projecting after +the softer strata which embedded it have been worn away (Monadnock +of Professor Davis).</p> + +<p>The geographical distribution of mountains is intimately associated +with the great structural lines of the continents of which they form +the culminating region. Lofty lines of fold mountains +form the “backbones” of North America in the Rocky +<span class="sidenote">Distribution of mountains.</span> +Mountains and the west coast systems, of South America +in the Cordillera of the Andes, of Europe in the Pyrenees, +Alps, Carpathians and Caucasus, and of Asia in the mountains of +Asia Minor, converging on the Pamirs and diverging thence in the +Himalaya and the vast mountain systems of central and eastern +Asia. The remarkable line of volcanoes around the whole coast +of the Pacific and along the margin of the Caribbean and Mediterranean +seas is one of the most conspicuous features of the globe.</p> + +<p>If land forms may be compared to organs, the part they serve in +the economy of the earth may, without straining the term, be +characterized as functions. The first and simplest +<span class="sidenote">Functions of land forms.<br /><br />Land waste.</span> +function of the land surface is that of guiding loose +material to a lower level. The downward pull of gravity +suffices to bring about the fall of such material, but the +path it will follow and the distance it will travel before coming to +rest depend upon the land form. The loose material may, and in +an arid region does, consist only of portions of the higher +parts of the surface detached by the expansion and +contraction produced by heating and cooling due to +radiation. Such broken material rolling down a uniform scarp +would tend to reduce its steepness by the loss of material in the +upper part and by the accumulation of a mound or scree against +the lower part of the slope. But where the side is not a uniform +scarp, but made up of a series of ridges and valleys, the tendency +will be to distribute the detritus in an irregular manner, directing +it away from one place and collecting it in great masses in another, +so that in time the land form assumes a new appearance. Snow +accumulating on the higher portions of the land, when compacted +into ice and caused to flow downwards by gravity, gives rise, on +<span class="sidenote">Glaciers.</span> +account of its more coherent character, to continuous +glaciers, which mould themselves to the slopes down +which they are guided, different ice-streams converging to send +forward a greater volume. Gradually coming to occupy definite +beds, which are deepened and polished by the friction, they impress +a characteristic appearance on the land, which guides them as they +traverse it, and, although the ice melts at lower levels, vast quantities +of clay and broken stones are brought down and deposited in terminal +moraines where the glacier ends.</p> + +<p>Rain is by far the most important of the inorganic mobile distributions +upon which land forms exercise their function of guidance +and control. The precipitation of rain from the aqueous +<span class="sidenote">Rain.</span> +vapour of the atmosphere is caused in part by vertical +movements of the atmosphere involving heat changes and apparently +independent of the surface upon which precipitation occurs; but in +greater part it is dictated by the form and altitude of the land surface +and the direction of the prevailing winds, which itself is largely +influenced by the land. It is on the windward faces of the highest +ground, or just beyond the summit of less dominant heights upon the +leeward side, that most rain falls, and all that does not evaporate +or percolate into the ground is conducted back to the sea by a route +which depends only on the form of the land. More mobile and more +searching than ice or rock rubbish, the trickling drops are guided by +the deepest lines of the hillside in their incipient flow, and as these +<span class="sidenote">River systems.</span> +lines converge, the stream, gaining strength, proceeds in +its torrential course to carve its channel deeper and entrench +itself in permanent occupation. Thus the stream-bed, +from which at first the water might be blown away into a new +channel by a gale of wind, ultimately grows to be the strongest line +of the landscape. As the main valley deepens, the tributary stream-beds +are deepened also, and gradually cut their way headwards, +enlarging the area whence they draw their supplies. Thus new +land forms are created—valleys of curious complexity, for example—by +<span class="pagenum"><a name="page634" id="page634"></a>634</span> +the “capture” and diversion of the water of one river by another, +leading to a change of watershed.<a name="fa37b" id="fa37b" href="#ft37b"><span class="sp">37</span></a> The minor tributaries become +more numerous and more constant, until the system of torrents +has impressed its own individuality on the mountain side. As +the river leaves the mountain, ever growing by the accession of +tributaries, it ceases, save in flood time, to be a formidable instrument +of destruction; the gentler slope of the land surface gives to +it only power sufficient to transport small stones, gravel, sand and +ultimately mud. Its valley banks are cut back by the erosion of +minor tributaries, or by rain-wash if the climate be moist, or left +steep and sharp while the river deepens its bed if the climate be +arid. The outline of the curve of a valley’s sides ultimately depends +on the angle of repose of the detritus which covers them, if there +has been no subsequent change, such as the passage of a glacier +along the valley, which tends to destroy the regularity of the cross-section. +The slope of the river bed diminishes until the plain compels +the river to move slowly, swinging in <i>meanders</i> proportioned to its +size, and gradually, controlled by the flattening land, ceasing to +transport material, but raising its banks and silting up its bed by +the dropped sediment, until, split up and shoaled, its distributaries +struggle across its delta to the sea. This is the typical river of which +there are infinite varieties, yet every variety would, if time were +given, and the land remained unchanged in level relatively to the sea, +ultimately approach to the type. Movements of the land +<span class="sidenote">Adjustment of rivers to land.</span> +either of subsidence or elevation, changes in the land by +the action of erosion in cutting back an escarpment or +cutting through a col, changes in climate by affecting the +rainfall and the volume of water, all tend to throw the +river valley out of harmony with the actual condition of +its stream. There is nothing more striking in geography than the +perfection of the adjustment of a great river system to its valleys +when the land has remained stable for a very lengthened period. +Before full adjustment has been attained the river bed may be +broken in places by waterfalls or interrupted by lakes; after adjustment +the bed assumes a permanent outline, the slope diminishing +more and more gradually, without a break in its symmetrical descent. +Excellent examples of the indecisive drainage of a new land surface, +on which the river system has not had time to impress itself, are to be +seen in northern Canada and in Finland, where rivers are separated +by scarcely perceptible divides, and the numerous lakes frequently +belong to more than one river system.</p> + +<p>The action of rivers on the land is so important that it has been +made the basis of a system of physical geography by Professor +W.M. Davis, who classifies land surfaces in terms of +the three factors—structure, process and time.<a name="fa38b" id="fa38b" href="#ft38b"><span class="sp">38</span></a> Of +<span class="sidenote">The geographical cycle.</span> +these time, during which the process is acting on the +structure, is the most important. A land may thus be +characterized by its position in the “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.</p> + +<p>The whole question of the régime of rivers and lakes is sometimes +treated under the name hydrography, a name used by some writers +in the sense of marine surveying, and by others as synonymous with +oceanography. For the study of rivers alone the name potamology<a name="fa39b" id="fa39b" href="#ft39b"><span class="sp">39</span></a> +has been suggested by Penck, and the subject being of much practical +importance has received a good deal of attention.<a name="fa40b" id="fa40b" href="#ft40b"><span class="sp">40</span></a></p> + +<p>The study of lakes has also been specialized under the name of +limnology (see <span class="sc"><a href="#artlinks">Lake</a></span>).<a name="fa41b" id="fa41b" href="#ft41b"><span class="sp">41</span></a> The existence of lakes in hollows of the land +depends upon the balance between precipitation and evaporation. +A stream flowing into a hollow will tend to fill it up, and +<span class="sidenote">Lakes and internal drainage.</span> +the water will begin to escape as soon as its level rises high +enough to reach the lowest part of the rim. In the case +of a large hollow in a very dry climate the rate of +evaporation may be sufficient to prevent the water from ever rising +to the lip, so that there is no outflow to the sea, and a basin of internal +drainage is the result. This is the case, for instance, in the Caspian +sea, the Aral and Balkhash lakes, the Tarim basin, the Sahara, inner +Australia, the great basin of the United States and the Titicaca +basin. These basins of internal drainage are calculated to amount +to 22% of the land surface. The percentages of the land surface +draining to the different oceans are approximately—Atlantic, 34.3%; +Arctic sea, 16.5%; Pacific, 14.4%; Indian Ocean, 12.8%.<a name="fa42b" id="fa42b" href="#ft42b"><span class="sp">42</span></a></p> + +<p>The parts of a river system have not been so clearly defined as is +desirable, hence the exaggerated importance popularly attached to +“the source” of a river. A well-developed river system +has in fact many equally important and widely-separated +<span class="sidenote">Terminology of river systems.</span> +sources, the most distant from the mouth, the highest, +or even that of largest initial volume not being necessarily +of greater geographical interest than the rest. +The whole of the land which directs drainage towards one river is +known as its basin, catchment area or drainage area—sometimes, +by an incorrect expression, as its valley or even its watershed. +The boundary line between one drainage area and others is rightly +termed the watershed, but on account of the ambiguity which has +been tolerated it is better to call it water-parting or, as in America, +divide. The only other important term which requires to be noted +here is <i>talweg</i>, a word introduced from the German into French +and English, and meaning the deepest line along the valley, which +is necessarily occupied by a stream unless the valley is dry.</p> + +<p>The functions of land forms extend beyond the control of the +circulation of the atmosphere, the hydrosphere and the water which +is continually being interchanged between them; they are exercised +with increased effect in the higher departments of biogeography and +anthropogeography.</p> + +<p>The sum of the organic life on the globe is termed by some geographers +the biosphere, and it has been estimated that the whole +mass of living substance in existence at one time would +cover the surface of the earth to a depth of one-fifth of +<span class="sidenote">Biogeography.</span> +an inch.<a name="fa43b" id="fa43b" href="#ft43b"><span class="sp">43</span></a> The distribution of living organisms is a +complex problem, a function of many factors, several of which +are yet but little known. They include the biological nature of +the organism and its physical environment, the latter involving +conditions in which geographical elements, direct or indirect, preponderate. +The direct geographical elements are the arrangement +of land and sea (continents and islands standing in sharp contrast) +and the vertical relief of the globe, which interposes barriers of a +less absolute kind between portions of the same land area or oceanic +depression. The indirect geographical elements, which, as a rule, +act with and intensify the direct, are mainly climatic; the prevailing +winds, rainfall, mean and extreme temperatures of every +locality depending on the arrangement of land and sea and of land +forms. Climate thus guided affects the weathering of rocks, and +so determines the kind and arrangement of soil. Different species +of organisms come to perfection in different climates; and it may +be stated as a general rule that a species, whether of plant or animal, +once established at one point, would spread over the whole zone +of the climate congenial to it unless some barrier were interposed +to its progress. In the case of land and fresh-water organisms +the sea is the chief barrier; in the case of marine organisms, the +land. Differences in land forms do not exert great influence on the +distribution of living creatures directly, but indirectly such land +forms as mountain ranges and internal drainage basins are very +potent through their action on soil and climate. A snow-capped +mountain ridge or an arid desert forms a barrier between different +forms of life which is often more effective than an equal breadth of +sea. In this way the surface of the land is divided into numerous +natural regions, the flora and fauna of each of which include some +distinctive species not shared by the others. The distribution of +life is discussed in the various articles in this <i>Encyclopaedia</i> dealing +with biological, botanical and zoological subjects.<a name="fa44b" id="fa44b" href="#ft44b"><span class="sp">44</span></a></p> + +<p><span class="pagenum"><a name="page635" id="page635"></a>635</span></p> + +<p>The classification of the land surface into areas inhabited by +distinctive groups of plants has been attempted by many phyto-geographers, +but without resulting in any scheme of +general acceptance. The simplest classification is perhaps +<span class="sidenote">Floral zones.</span> +that of Drude according to climatic zones, subdivided +according to continents. This takes account of—(1) the <i>Arctic-Alpine</i> +zone, including all the vegetation of the region bordering +on perpetual snow; (2) the <i>Boreal</i> zone, including the temperate +lands of North America, Europe and Asia, all of which are substantially +alike in botanical character; (3) the <i>Tropical</i> zone, divided +sharply into (<i>a</i>) the tropical zone of the New World, and (<i>b</i>) the +tropical zone of the Old World, the forms of which differ in a significant +degree; (4) the <i>Austral</i> zone, comprising all continental +land south of the equator, and sharply divided into three regions +the floras of which are strikingly distinct—(<i>a</i>) South American, +(<i>b</i>) South African and (<i>c</i>) Australian; (5) the <i>Oceanic</i>, comprising +all oceanic islands, the flora of which consists exclusively of forms +whose seeds could be drifted undestroyed by ocean currents or +carried by birds. To these might be added the antarctic, which is +still very imperfectly known. Many subdivisions and transitional +zones have been suggested by different authors.</p> + +<p>From the point of view of the economy of the globe this classification +by species is perhaps less important than that by mode +of life and physiological character in accordance with +environment. The following are the chief areas of +<span class="sidenote">Vegetation areas.</span> +vegetational activity usually recognized: (1) The ice-deserts +of the arctic and antarctic and the highest mountain regions, +where there is no vegetation except the lowest forms, like that +which causes “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.</p> + +<p>The divisions of the earth into faunal regions by Dr P.L. Sclater +have been found to hold good for a large number of groups of animals +as different in their mode of life as birds and mammals, +and they may thus be accepted as based on nature. +<span class="sidenote">Faunal realms.</span> +They are six in number: (1) <i>Palaearctic</i>, including +Europe, Asia north of the Himalaya, and Africa north of the Sahara; +(2) <i>Ethiopian</i>, consisting of Africa south of the Atlas range, and +Madagascar; (3) <i>Oriental</i>, including India, Indo-China and the +Malay Archipelago north of Wallace’s line, which runs between +Bali and Lombok; (4) <i>Australian</i>, including Australia, New Zealand, +New Guinea and Polynesia; (5) <i>Nearctic</i> or North America, north +of Mexico; and (6) <i>Neotropical</i> or South America. Each of these +divisions is the home of a special fauna, many species of which +are confined to it alone; in the Australian region, indeed, practically +the whole fauna is peculiar and distinctive, suggesting a prolonged +period of complete biological isolation. In some cases, such as the +Ethiopian and Neotropical and the Palaearctic and Nearctic regions, +the faunas, although distinct, are related, several forms on opposite +sides of the Atlantic being analogous, <i>e.g.</i> the lion and puma, ostrich +and rhea. Where two of the faunal realms meet there is usually, +though not always, a mixing of faunas. These facts have led some +naturalists to include the Palaearctic and Nearctic regions in one, +termed <i>Holarctic</i>, and to suggest transitional regions, such as the +<i>Sonoran</i>, between North and South America, and the <i>Mediterranean</i>, +between Europe and Africa, or to create sub-regions, such as Madagascar +and New Zealand. Oceanic islands have, as a rule, distinctive +faunas and floras which resemble, but are not identical with, those of +other islands in similar positions.</p> + +<p>The study of the evolution of faunas and the comparison of the +faunas of distant regions have furnished a trustworthy +instrument of pre-historic geographical research, which +<span class="sidenote">Biological distribution as a means of geographical research.</span> +enables earlier geographical relations of land and sea to +be traced out, and the approximate period, or at least the +chronological order of the larger changes, to be estimated. +In this way, for example, it has been suggested that a +land, “Lemuria,” once connected Madagascar with the +Malay Archipelago, and that a northern extension of +the antarctic land once united the three southern continents.</p> + +<p>The distribution of fossils frequently makes it possible to map out +approximately the general features of land and sea in long-past +geological periods, and so to enable the history of crustal relief to be +traced.<a name="fa45b" id="fa45b" href="#ft45b"><span class="sp">45</span></a></p> + +<p>While the tendency is for the living forms to come into harmony +with their environment and to approach the state of equilibrium +by successive adjustments if the environment should +happen to change, it is to be observed that the action +<span class="sidenote">Reaction of organisms on environment.</span> +of organisms themselves often tends to change their +environment. Corals and other quick-growing calcareous +marine organisms are the most powerful in this +respect by creating new land in the ocean. Vegetation of all sorts +acts in a similar way, either in forming soil and assisting in breaking +up rocks, in filling up shallow lakes, and even, like the mangrove, +in reclaiming wide stretches of land from the sea. Plant life, +utilizing solar light to combine the inorganic elements of water, +soil and air into living substance, is the basis of all animal life. +This is not by the supply of food alone, but also by the withdrawal +of carbonic acid from the atmosphere, by which vegetation maintains +the composition of the air in a state fit for the support of animal +life. Man in the primitive stages of culture is scarcely to be distinguished +from other animals as regards his subjection to environment, +but in the higher grades of culture the conditions of control +and reaction become much more complicated, and the department +of anthropogeography is devoted to their consideration.</p> + +<p>The first requisites of all human beings are food and protection, +in their search for which men are brought into intimate relations +with the forms and productions of the earth’s surface. +The degree of dependence of any people upon environment +<span class="sidenote">Anthropogeography.</span> +varies inversely as the degree of culture or civilization, +which for this purpose may perhaps be defined as the power +of an individual to exercise control over the individual and over +the environment for the benefit of the community. The development +of culture is to a certain extent a question of race, and although +forming one species, the varieties of man differ in almost imperceptible +gradations with a complexity defying classification (see <span class="sc"><a href="#artlinks">Anthropology</a></span>). +Professor Keane groups man round four leading types, +which may be named the black, yellow, red and white, or the Ethiopic, +Mongolic, American and Caucasic. Each may be subdivided, +though not with great exactness, into smaller groups, either according +to physical characteristics, of which the form of the head is most +important, or according to language.</p> + +<p>The black type is found only in tropical or sub-tropical countries, +and is usually in a primitive condition of culture, unless educated +by contact with people of the white type. They follow +the most primitive forms of religion (mainly fetishism), +<span class="sidenote">Types of man.</span> +live on products of the woods or of the chase, with the +minimum of work, and have only a loose political organization. +The red type is peculiar to America, inhabiting every climate from +polar to equatorial, and containing representatives of many stages +of culture which had apparently developed without the aid or +interference of people of any other race until the close of the 15th +century. The yellow type is capable of a higher culture, cherishes +higher religious beliefs, and inhabits as a rule the temperate zone, +although extending to the tropics on one side and to the arctic +regions on the other. The white type, originating in the north +temperate zone, has spread over the whole world. They have +attained the highest culture, profess the purest forms of monotheistic +religion, and have brought all the people of the black type +and many of those of the yellow under their domination.</p> + +<p>The contrast between the yellow and white types has been softened +by the remarkable development of the Japanese following the +assimilation of western methods.</p> + +<p>The actual number of human inhabitants in the world has been +calculated as follows:</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl"> </td> <td class="tcc f80">By Continents.<a name="fa46b" id="fa46b" href="#ft46b"><span class="sp">46</span></a></td> <td class="tcc f80"> </td> <td class="tcc f80">By Race.<a name="fa47b" id="fa47b" href="#ft47b"><span class="sp">47</span></a></td></tr> + +<tr><td class="tcl">Asia</td> <td class="tcr rb">875,000,000</td> <td class="tcl">White (Caucasic)</td> <td class="tcr">770,000,000</td></tr> +<tr><td class="tcl">Europe</td> <td class="tcr rb">392,000,000</td> <td class="tcl">Yellow (Mong.)</td> <td class="tcr">540,000,000</td></tr> +<tr><td class="tcl">Africa</td> <td class="tcr rb">170,000,000</td> <td class="tcl">Black (Ethiopic)</td> <td class="tcr">175,000,000</td></tr> +<tr><td class="tcl">America</td> <td class="tcr rb">143,000,000</td> <td class="tcl">Red (American)</td> <td class="tcr">22,000,000</td></tr> +<tr><td class="tcl">Australia and Polynesia</td> <td class="tcr rb">7,000,000</td> <td class="tcl"> </td> <td class="tcr">—————</td></tr> +<tr><td class="tcl"> </td> <td class="tcr rb">—————</td> <td class="tcc">Total</td> <td class="tcr">1,507,000,000</td></tr> +<tr><td class="tcc">Total</td> <td class="tcr rb">1,587,000,000</td> <td class="tcl"> </td> <td class="tcl"> </td></tr> +</table> + +<p>In round numbers the population of the world is about +1,600,000,000, and, according to an estimate by Ravenstein,<a name="fa48b" id="fa48b" href="#ft48b"><span class="sp">48</span></a> the +maximum population which it will be possible for the earth to +maintain is 6000 millions, a number which, if the average rate of +increase in 1891 continued, would be reached within 200 years.</p> + +<p>While highly civilized communities are able to evade many of +the restrictions of environment, to overcome the barriers to intercommunication +interposed by land or sea, to counteract the adverse +<span class="pagenum"><a name="page636" id="page636"></a>636</span> +influence of climate, and by the development of trade even to +inhabit countries which cannot yield a food-supply, the mass of +mankind is still completely under the control of those conditions +which in the past determined the distribution and the mode of life +of the whole human race.</p> + +<p>In tropical forests primitive tribes depend on the collection of +wild fruits, and in a minor degree on the chase of wild animals, for +their food. Clothing is unnecessary; hence there is +little occasion for exercising the mental faculties beyond +<span class="sidenote">Influence of environment on man.</span> +the sense of perception to avoid enemies, or the inventive +arts beyond what is required for the simplest +weapons and the most primitive fortifications. When +the pursuit of game becomes the chief occupation of a people there +is of necessity a higher development of courage, skill, powers of +observation and invention; and these qualities are still further +enhanced in predatory tribes who take by force the food, clothing +and other property prepared or collected by a feebler people. The +fruit-eating savage cannot stray beyond his woods which bound +his life as the water bounds that of a fish; the hunter is free to +live on the margin of forests or in open country, while the robber +or warrior from some natural stronghold of the mountains sweeps +over the adjacent plains and carries his raids into distant lands. +Wide grassy steppes lead to the organization of the people as nomads +whose wealth consists in flocks and herds, and their dwellings +are tents. The nomad not only domesticates and turns to his +own use the gentler and more powerful animals, such as sheep, +cattle, horses, camels, but even turns some predatory creatures, +like the dog, into a means of defending their natural prey. They +hunt the beasts of prey destructive to their flocks, and form armed +bands for protection against marauders or for purposes of aggression +on weaker sedentary neighbours. On the fertile low grounds along +the margins of rivers or in clearings of forests, agricultural communities +naturally take their rise, dwelling in villages and cultivating +the wild grains, which by careful nurture and selection have been +turned into rich cereals. The agriculturist as a rule is rooted to +the soil. The land he tills he holds, and acquires a closer connexion +with a particular patch of ground than either the hunter or the herdsman. +In the temperate zone, where the seasons are sharply contrasted, +but follow each other with regularity, foresight and self-denial +were fostered, because if men did not exercise these qualities seed-time +or harvest might pass into lost opportunities and the tribes would +suffer. The more extreme climates of arid regions on the margins of +the tropics, by the unpredictable succession of droughts and floods, +confound the prevision of uninstructed people, and make prudence +and industry qualities too uncertain in their results to be worth +cultivating. Thus the civilization of agricultural peoples of the +temperate zone grew rapidly, yet in each community a special type +arose adapted to the soil, the crop and the climate. On the seashore +fishing naturally became a means of livelihood, and dwellers +by the sea, in virtue of the dangers to which they are exposed from +storm and unseaworthy craft, are stimulated to a higher degree of +foresight, quicker observation, prompter decision and more energetic +action in emergencies than those who live inland. The building +and handling of vessels also, and the utilization of such uncontrollable +powers of nature as wind and tide, helped forward mechanical +invention. To every type of coast there may be related a special +type of occupation and even of character; the deep and gloomy +fjord, backed by almost impassable mountains, bred bold mariners +whose only outlet for enterprise was seawards towards other lands—the +<i>viks</i> created the vikings. On the gently sloping margin of the +estuary of a great river a view of tranquil inland life was equally +presented to the shore-dweller, and the ocean did not present the +only prospect of a career. Finally the mountain valley, with its +patches of cultivable soil on the alluvial fans of tributary torrents, +its narrow pastures on the uplands only left clear of snow in summer, +its intensified extremes of climates and its isolation, almost equal to +that of an island, has in all countries produced a special type of +brave and hardy people, whose utmost effort may bring them comfort, +but not wealth, by honest toil, who know little of the outer +world, and to whom the natural outlet for ambition is marauding +on the fertile plains. The highlander and viking, products of the +valleys raised high amid the mountains or half-drowned in the sea, +are everywhere of kindred spirit.</p> + +<p>It is in some such manner as these that the natural conditions +of regions, which must be conformed to by prudence and utilized +by labour to yield shelter and food, have led to the growth of peoples +differing in their ways of life, thought and speech. The initial +differences so produced are confirmed and perpetuated by the +same barriers which divide the faunal or floral regions, the sea, +mountains, deserts and the like, and much of the course of past +history and present politics becomes clear when the combined +results of differing race and differing environment are taken into +account.<a name="fa49b" id="fa49b" href="#ft49b"><span class="sp">49</span></a></p> + +<p>The specialization which accompanies the division of labour has +important geographical consequences, for it necessitates communication +between communities and the interchange of their products. +<span class="sidenote">Density of population.</span> +Trade makes it possible to work mineral resources +in localities where food can only be grown with great +difficulty and expense, or which are even totally barren +and waterless, entirely dependent on supplies from distant sources.</p> + +<p>The population which can be permanently supported by a given +area of land differs greatly according to the nature of the resources +and the requirements of the people. Pastoral communities are +always scattered very thinly over large areas; agricultural populations +may be almost equally sparse where advanced methods of +agriculture and labour-saving machinery are employed; but where +a frugal people are situated on a fertile and inexhaustible soil, such +as the deltas and river plains of Egypt, India and China, an enormous +population may be supported on a small area. In most cases, +however, a very dense population can only be maintained in regions +where mineral resources have fixed the site of great manufacturing +industries. The maximum density of population which a given +region can support is very difficult to determine; it depends partly +on the race and standard of culture of the people, partly on the +nature and origin of the resources on which they depend, partly +on the artificial burdens imposed and very largely on the climate. +Density of population is measured by the average number of people +residing on a unit of area; but in order to compare one part of the +world with another the average should, strictly speaking, be taken +for regions of equal size or of equal population; and the portions +of the country which are permanently uninhabitable ought to be +excluded from the calculation.<a name="fa50b" id="fa50b" href="#ft50b"><span class="sp">50</span></a> Considering the average density +of population within the political limits of countries, the following +list is of some value; the figures for a few smaller divisions of +large countries are added (in brackets) for comparison:</p> + +<p class="pt2 center"><i>Average Population on 1 sq. m.</i> (<i>For 1900 or 1901.</i>)</p> + +<table class="ws" summary="Contents"> +<tr><td class="tccm allb">Country.</td> <td class="tccm allb">Density<br />of pop.</td> <td class="tccm allb">Country.</td> <td class="tccm allb">Density<br />of pop.</td></tr> + +<tr><td class="tcl lb rb">(Saxony)</td> <td class="tcl rb">743*</td> <td class="tcl rb">Ceylon</td> <td class="tcl rb">141**</td></tr> +<tr><td class="tcl lb rb">Belgium</td> <td class="tcl rb">589*</td> <td class="tcl rb">Greece</td> <td class="tcl rb"> 97</td></tr> +<tr><td class="tcl lb rb">Java</td> <td class="tcl rb">568**</td> <td class="tcl rb">European Turkey</td> <td class="tcl rb"> 90</td></tr> +<tr><td class="tcl lb rb">(England and Wales)</td> <td class="tcl rb">558</td> <td class="tcl rb">Spain</td> <td class="tcl rb"> 97</td></tr> +<tr><td class="tcl lb rb">(Bengal)</td> <td class="tcl rb">495**</td> <td class="tcl rb">European Russia</td> <td class="tcl rb"> 55**</td></tr> +<tr><td class="tcl lb rb">Holland</td> <td class="tcl rb">436</td> <td class="tcl rb">Sweden</td> <td class="tcl rb"> 30</td></tr> +<tr><td class="tcl lb rb">United Kingdom</td> <td class="tcl rb">344</td> <td class="tcl rb">United States</td> <td class="tcl rb"> 25</td></tr> +<tr><td class="tcl lb rb">Japan</td> <td class="tcl rb">317</td> <td class="tcl rb">Mexico</td> <td class="tcl rb"> 18</td></tr> +<tr><td class="tcl lb rb">Italy</td> <td class="tcl rb">293</td> <td class="tcl rb">Norway</td> <td class="tcl rb"> 18</td></tr> +<tr><td class="tcl lb rb">China proper</td> <td class="tcl rb">270**</td> <td class="tcl rb">Persia</td> <td class="tcl rb"> 15</td></tr> +<tr><td class="tcl lb rb">German Empire</td> <td class="tcl rb">270</td> <td class="tcl rb">New Zealand</td> <td class="tcl rb"> 7</td></tr> +<tr><td class="tcl lb rb">Austria</td> <td class="tcl rb">226</td> <td class="tcl rb">Argentina</td> <td class="tcl rb"> 5</td></tr> +<tr><td class="tcl lb rb">Switzerland</td> <td class="tcl rb">207</td> <td class="tcl rb">Brazil</td> <td class="tcl rb"> 4.5</td></tr> +<tr><td class="tcl lb rb">France</td> <td class="tcl rb">188</td> <td class="tcl rb">Eastern States of</td> <td class="tcl rb"> </td></tr> +<tr><td class="tcl lb rb">Indian Empire</td> <td class="tcl rb">167**</td> <td class="tcl rb"> Australia</td> <td class="tcl rb"> 3</td></tr> +<tr><td class="tcl lb rb">Denmark</td> <td class="tcl rb">160**</td> <td class="tcl rb">Dominion of Canada</td> <td class="tcl rb"> 1.5</td></tr> +<tr><td class="tcl lb rb">Hungary</td> <td class="tcl rb">154**</td> <td class="tcl rb">Siberia</td> <td class="tcl rb"> 1</td></tr> +<tr><td class="tcl lb rb bb">Portugal</td> <td class="tcl rb bb">146</td> <td class="tcl rb bb">West Australia</td> <td class="tcl rb bb"> 0.2</td></tr> + +<tr><td class="tcl" colspan="4"> * Almost exclusively industrial.</td></tr> + +<tr><td class="tcl" colspan="4"> ** Almost exclusively agricultural.</td></tr> +</table> + +<p>The movement of people from one place to another without the +immediate intention of returning is known as migration, and according +to its origin it may be classed as centrifugal (directed +<i>from</i> a particular area) and centripetal (directed <i>towards</i> +<span class="sidenote">Migration.</span> +a particular area). Centrifugal migration is usually a matter of +compulsion; it may be necessitated by natural causes, such as a +change of climate leading to the withering of pastures or destruction +of agricultural land, to inundation, earthquake, pestilence or to an +excess of population over means of support; or to artificial causes, +such as the wholesale deportation of a conquered people; or to +political or religious persecution. In any case the people are driven +out by some adverse change; and when the urgency is great they +may require to drive out in turn weaker people who occupy a desirable +territory, thus propagating the wave of migration, the direction of +which is guided by the forms of the land into inevitable channels. +Many of the great historic movements of peoples were doubtless due +to the gradual change of geographical or climatic conditions; and the +slow desiccation of Central Asia has been plausibly suggested as the +real cause of the peopling of modern Europe and of the medieval +wars of the Old World, the theatres of which were critical points on +the great natural lines of communication between east and west.</p> + +<p>In the case of centripetal migrations people flock to some particular +place where exceptionally favourable conditions have been found to +exist. The rushes to gold-fields and diamond-fields are typical instances; +the growth of towns on coal-fields and near other sources +of power, and the rapid settlement of such rich agricultural districts +as the wheat-lands of the American prairies and great plains are +other examples.</p> + +<p>There is, however, a tendency for people to remain rooted to the +<span class="pagenum"><a name="page637" id="page637"></a>637</span> +land of their birth, when not compelled or induced by powerful +external causes to seek a new home.</p> + +<p>Thus arises the spirit of patriotism, a product of purely geographical +conditions, thereby differing from the sentiment of loyalty, +which is of racial origin. Where race and soil conspire to +evoke both loyalty and patriotism in a people, the moral +<span class="sidenote">Political geography.</span> +qualities of a great and permanent nation are secured. +It is noticeable that the patriotic spirit is strongest in those places +where people are brought most intimately into relation with the land; +dwellers in the mountain or by the sea, and, above all, the people of +rugged coasts and mountainous archipelagoes, have always been +renowned for love of country, while the inhabitants of fertile plains +and trading communities are frequently less strongly attached to +their own land.</p> + +<p>Amongst nomads the tribe is the unit of government, the political +bond is personal, and there is no definite territorial association +of the people, who may be loyal but cannot be patriotic. The idea +of a country arises only when a nation, either homogeneous or +composed of several races, establishes itself in a region the boundaries +of which may be defined and defended against aggression from +without. Political geography takes account of the partition of the +earth amongst organized communities, dealing with the relation of +races to regions, and of nations to countries, and considering the +conditions of territorial equilibrium and instability.</p> + +<p>The definition of boundaries and their delimitation is one of the +most important parts of political geography. Natural boundaries +are always the most definite and the strongest, lending +themselves most readily to defence against aggression. +<span class="sidenote">Boundaries.</span> +The sea is the most effective of all, and an island state is +recognized as the most stable. Next in importance comes a mountain +range, but here there is often difficulty as to the definition of +the actual crest-line, and mountain ranges being broad regions, it +may happen that a small independent state, like Switzerland or +Andorra, occupies the mountain valleys between two or more great +countries. Rivers do not form effective international boundaries, +although between dependent self-governing communities they are +convenient lines of demarcation. A desert, or a belt of country +left purposely without inhabitants, like the mark, marches or +debatable lands of the middle ages, was once a common means +of separating nations which nourished hereditary grievances. The +“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.<a name="fa51b" id="fa51b" href="#ft51b"><span class="sp">51</span></a></p> + +<p>The modes of government amongst civilized peoples have little +influence on political geography; some republics are as arbitrary +and exacting in their frontier regulations as some absolute +monarchies. It is, however, to be noticed that absolute +<span class="sidenote">Forms of government.</span> +monarchies are confined to the east of Europe and to +Asia, Japan being the only established constitutional +monarchy east of the Carpathians. Limited monarchies are (with +the exception of Japan) peculiar to Europe, and in these the degree +of democratic control may be said to diminish as one passes eastwards +from the United Kingdom. Republics, although represented +in Europe, are the peculiar form of government of America and +are unknown in Asia.</p> + +<p>The forms of government of colonies present a series of transitional +types from the autocratic administration of a governor +appointed by the home government to complete democratic +self-government. The latter occurs only in the temperate possessions +of the British empire, in which there is no great preponderance +of a coloured native population. New colonial forms have been +developed during the partition of Africa amongst European powers, +the sphere of influence being especially worthy of notice. This +is a vaguer form of control than a protectorate, and frequently +amounts merely to an agreement amongst civilized powers to respect +the right of one of their number to exercise government within +a certain area, if it should decide to do so at any future time.</p> + +<p>The central governments of all civilized countries concerned with +external relations are closely similar in their modes of action, but +the internal administration may be very varied. In this respect a +country is either centralized, like the United Kingdom or France, +or federated of distinct self-governing units like Germany (where +the units include kingdoms, at least three minor types of monarchies, +municipalities and a crown land under a nominated governor), or the +United States, where the units are democratic republics. The ultimate +cause of the predominant form of federal government may be +the geographical diversity of the country, as in the cantons occupying +the once isolated mountain valleys of Switzerland, the racial diversity +of the people, as in Austria-Hungary, or merely political expediency, +as in republics of the American type.</p> + +<p>The minor subdivisions into provinces, counties and parishes, or +analogous areas, may also be related in many cases to natural +features or racial differences perpetuated by historical causes. The +territorial divisions and subdivisions often survive the conditions +which led to their origin; hence the study of political geography is +allied to history as closely as the study of physical geography is allied +to geology, and for the same reason.</p> + +<p>The aggregation of population in towns was at one time mainly +brought about by the necessity for defence, a fact indicated by the +defensive sites of many old towns. In later times, +towns have been more often founded in proximity to +<span class="sidenote">Towns.</span> +valuable mineral resources, and at critical points or nodes on lines +of communication. These are places where the mode of travelling +or of transport is changed, such as seaports, river ports and railway +termini, or natural resting-places, such as a ford, the foot of a +steep ascent on a road, the entrance of a valley leading up from a +plain into the mountains, or a crossing-place of roads or railways.<a name="fa52b" id="fa52b" href="#ft52b"><span class="sp">52</span></a> +The existence of a good natural harbour is often sufficient to +give origin to a town and to fix one end of a line of land communication.</p> + +<p>In countries of uniform surface or faint relief, roads and railways +may be constructed in any direction without regard to the configuration. +In places where the low ground is marshy, +roads and railways often follow the ridge-lines of hills, +<span class="sidenote">Lines of communication.</span> +or, as in Finland, the old glacial eskers, which run parallel +to the shore. Wherever the relief of the land is pronounced, +roads and railways are obliged to occupy the lowest ground +winding along the valleys of rivers and through passes in the mountains. +In exceptional cases obstructions which it would be impossible +or too costly to turn are overcome by a bridge or tunnel, the magnitude +of such works increasing with the growth of engineering skill +and financial enterprise. Similarly the obstructions offered to +water communication by interruption through land or shallows are +overcome by cutting canals or dredging out channels. The economy +and success of most lines of communication depend on following +as far as possible existing natural lines and utilizing existing natural +sources of power.<a name="fa53b" id="fa53b" href="#ft53b"><span class="sp">53</span></a></p> + +<p>Commercial geography may be defined as the description of the +earth’s surface with special reference to the discovery, production, +transport and exchange of commodities. The transport +concerns land routes and sea routes, the latter being +<span class="sidenote">Commercial geography.</span> +the more important. While steam has been said to +make a ship independent of wind and tide, it is still +true that a long voyage even by steam must be planned so as to +encounter the least resistance possible from prevailing winds and +permanent currents, and this involves the application of oceanographical +and meteorological knowledge. The older navigation by +utilizing the power of the wind demands a very intimate knowledge +of these conditions, and it is probable that a revival of sailing +ships may in the present century vastly increase the importance of +the study of maritime meteorology.</p> + +<p>The discovery and production of commodities require a knowledge +of the distribution of geological formations for mineral products, +of the natural distribution, life-conditions and cultivation +or breeding of plants and animals and of the labour market. Attention +must also be paid to the artificial restrictions of political geography, +to the legislative restrictions bearing on labour and trade +as imposed in different countries, and, above all, to the incessant +fluctuations of the economic conditions of supply and demand and +the combinations of capitalists or workers which affect the market.<a name="fa54b" id="fa54b" href="#ft54b"><span class="sp">54</span></a> +The term “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.</p> + +<p>It will be seen that as each successive aspect of geographical +science is considered in its natural sequence the conditions become +<span class="pagenum"><a name="page638" id="page638"></a>638</span> +<span class="sidenote">Conclusion.</span> +more numerous, complex, variable and practically important. +From the underlying abstract mathematical considerations all +through the superimposed physical, biological, anthropological, +political and commercial development of the +subject runs the determining control exercised by crust-forms +acting directly or indirectly on mobile distributions; and this +is the essential principle of geography.</p> +</div> +<div class="author">(H. R. M.)</div> + +<hr class="foot" style="clear: both;" /> <div class="note"> + +<p><a name="ft1b" id="ft1b" href="#fa1b"><span class="fn">1</span></a> A concise sketch of the whole history of geographical method or +theory as distinguished from the history of geographical discovery +(see later section of this article) is only to be found in the introduction +to H. Wagner’s <i>Lehrbuch der Geographie</i>, vol. i. (Leipzig, 1900), +which is in every way the most complete treatise on the principles of +geography.</p> + +<p><a name="ft2b" id="ft2b" href="#fa2b"><span class="fn">2</span></a> <i>History of Ancient Geography</i> (Cambridge, 1897), p. 70.</p> + +<p><a name="ft3b" id="ft3b" href="#fa3b"><span class="fn">3</span></a> See J.L. Myres, “An Attempt to reconstruct the Maps used by +Herodotus,” <i>Geographical Journal</i>, viii. (1896), p. 605.</p> + +<p><a name="ft4b" id="ft4b" href="#fa4b"><span class="fn">4</span></a> <i>Geschichte der wissenschaftlichen Erdkunde der Griechen</i> (Leipzig, +1891), Abt. 3, p. 60.</p> + +<p><a name="ft5b" id="ft5b" href="#fa5b"><span class="fn">5</span></a> Bunbury’s <i>History of Ancient Geography</i> (2 vols., London, 1879), +Müller’s <i>Geographi Graeci minores</i> (2 vols., Paris, 1855, 1861) and +Berger’s <i>Geschichte der wissenschaftlichen Erdkunde der Griechen</i> +(4 vols., Leipzig, 1887-1893) are standard authorities on the Greek +geographers.</p> + +<p><a name="ft6b" id="ft6b" href="#fa6b"><span class="fn">6</span></a> The period of the early middle ages is dealt with in Beazley’s +<i>Dawn of Modern Geography</i> (London; part i., 1897; part ii., 1901; +part iii., 1906); see also Winstedt, <i>Cosmos Indicopleustes</i> (1910).</p> + +<p><a name="ft7b" id="ft7b" href="#fa7b"><span class="fn">7</span></a> From translator’s preface to the English version by Mr Dugdale +(1733), entitled <i>A Complete System of General Geography</i>, revised +by Dr Peter Shaw (London, 1756).</p> + +<p><a name="ft8b" id="ft8b" href="#fa8b"><span class="fn">8</span></a> Printed in <i>Schriften zur physischen Geographie</i>, vol. vi. of +Schubert’s edition of the collected works of Kant (Leipzig, 1839). +First published with notes by Rink in 1802.</p> + +<p><a name="ft9b" id="ft9b" href="#fa9b"><span class="fn">9</span></a> <i>History of Civilization</i>, vol. i. (1857).</p> + +<p><a name="ft10b" id="ft10b" href="#fa10b"><span class="fn">10</span></a> See H.J. Mackinder in <i>British Association Report</i> (Ipswich), +1895, p. 738, for a summary of German opinion, which has been +expressed by many writers in a somewhat voluminous literature.</p> + +<p><a name="ft11b" id="ft11b" href="#fa11b"><span class="fn">11</span></a> H. Wagner’s year-book, <i>Geographische Jahrbuch</i>, published at +Gotha, is the best systematic record of the progress of geography +in all departments; and Haack’s <i>Geographen Kalender</i>, also published +annually at Gotha, gives complete lists of the geographical societies +and geographers of the world.</p> + +<p><a name="ft12b" id="ft12b" href="#fa12b"><span class="fn">12</span></a> This phrase is old, appearing in one of the earliest English works +on geography, William Cuningham’s <i>Cosmographical Glasse conteinyng +the pleasant Principles of Cosmographie, Geographie, Hydrographie +or Navigation</i> (London, 1559).</p> + +<p><a name="ft13b" id="ft13b" href="#fa13b"><span class="fn">13</span></a> See also S. Günther, <i>Handbuch der mathematischen Geographie</i> +(Stuttgart, 1890).</p> + +<p><a name="ft14b" id="ft14b" href="#fa14b"><span class="fn">14</span></a> “On the Height of the Land and the Depth of the Ocean,” <i>Scot. +Geog. Mag.</i> iv. (1888), p. 1. Estimates had been made previously by +Humboldt, De Lapparent, H. Wagner, and subsequently by Penck +and Heiderich, and for the oceans by Karstens.</p> + +<p><a name="ft15b" id="ft15b" href="#fa15b"><span class="fn">15</span></a> <i>Petermanns Mitteilungen</i>, xxv. (1889), p. 17.</p> + +<p><a name="ft16b" id="ft16b" href="#fa16b"><span class="fn">16</span></a> <i>Proc. Roy. Soc. Edin.</i> xvii. (1890) p. 185.</p> + +<p><a name="ft17b" id="ft17b" href="#fa17b"><span class="fn">17</span></a> <i>Comptes rendus Acad. Sci.</i> (Paris, 1890), vol. iii. p. 994.</p> + +<p><a name="ft18b" id="ft18b" href="#fa18b"><span class="fn">18</span></a> “Areal und mittlere Erhebung der Landflächen sowie der Erdkruste” +in Gerland’s <i>Beiträge zur Geophysik</i>, ii. (1895) p. 667. See +also <i>Nature</i>, 54 (1896), p. 112.</p> + +<p><a name="ft19b" id="ft19b" href="#fa19b"><span class="fn">19</span></a> <i>Petermanns Mitteilungen</i>, xxxv. (1889) p. 19.</p> + +<p><a name="ft20b" id="ft20b" href="#fa20b"><span class="fn">20</span></a> The areas of the continental shelf and lowlands are approximately +equal, and it is an interesting circumstance that, taken as a +whole, the actual coast-line comes just midway on the most nearly +level belt of the earth’s surface, excepting the ocean floor. The configuration +of the continental slope has been treated in detail by +Nansen in <i>Scientific Results of Norwegian North Polar Expedition</i>, +vol. iv. (1904), where full references to the literature of the subject +will be found.</p> + +<p><a name="ft21b" id="ft21b" href="#fa21b"><span class="fn">21</span></a> <i>British Association Report</i> (Edinburgh, 1892), p. 699.</p> + +<p><a name="ft22b" id="ft22b" href="#fa22b"><span class="fn">22</span></a> <i>Das Antlitz der Erde</i> (4 vols., Leipzig, 1885, 1888, 1901). Translated +under the editorship of E. de Margerie, with much additional +matter, as <i>La Face de la terre</i>, vols. i. and ii. (Paris, 1897, 1900), and +into English by Dr Hertha Sollas as <i>The Face of the Earth</i>, vols. i. +and ii. (Oxford, 1904, 1906).</p> + +<p><a name="ft23b" id="ft23b" href="#fa23b"><span class="fn">23</span></a> Élie de Beaumont, <i>Notice sur les systèmes de montagnes</i> (3 vols., +Paris, 1852).</p> + +<p><a name="ft24b" id="ft24b" href="#fa24b"><span class="fn">24</span></a> <i>Vestiges of the Molten Globe</i> (London, 1875).</p> + +<p><a name="ft25b" id="ft25b" href="#fa25b"><span class="fn">25</span></a> See J.W. Gregory, “The Plan of the Earth and its Causes,” +<i>Geog. Journal</i>, xiii. (1899) p. 225; Lord Avebury, <i>ibid.</i> xv. (1900) +p. 46; Marcel Bertrand, “Déformation tétraédrique de la terre et +déplacement du pôle,” <i>Comptes rendus Acad. Sci.</i> (Paris, 1900), +vol. cxxx. p. 449; and A. de Lapparent, <i>ibid.</i> p. 614.</p> + +<p><a name="ft26b" id="ft26b" href="#fa26b"><span class="fn">26</span></a> See A.E.H. Love, “Gravitational Stability of the Earth,” <i>Phil. +Trans.</i> ser. A. vol. ccvii. (1907) p. 171.</p> + +<p><a name="ft27b" id="ft27b" href="#fa27b"><span class="fn">27</span></a> <i>Rumpf</i>, in German, the language in which this distinction was +first made.</p> + +<p><a name="ft28b" id="ft28b" href="#fa28b"><span class="fn">28</span></a> <i>Lehrbuch der Geographie</i> (Hanover and Leipzig, 1900), Bd. i. S. +245, 249.</p> + +<p><a name="ft29b" id="ft29b" href="#fa29b"><span class="fn">29</span></a> See, for example, F.G. Hahn’s <i>Insel-Studien</i> (Leipzig, 1883).</p> + +<p><a name="ft30b" id="ft30b" href="#fa30b"><span class="fn">30</span></a> See <i>Geographical Journal</i>, xxii. (1903) pp. 191-194.</p> + +<p><a name="ft31b" id="ft31b" href="#fa31b"><span class="fn">31</span></a> The most important works on the classification of land forms are +F. von Richthofen, <i>Führer für Forschungsreisende</i> (Berlin, 1886); +G. de la Noë and E. de Margerie, <i>Les Formes du terrain</i> (Paris, 1888); +and above all A. Penck, <i>Morphologie der Erdoberfläche</i> (2 vols., +Stuttgart, 1894). Compare also A. de Lapparent, <i>Leçons de géographie +physique</i> (2nd ed., Paris, 1898), and W.M. Davis, <i>Physical +Geography</i> (Boston, 1899).</p> + +<p><a name="ft32b" id="ft32b" href="#fa32b"><span class="fn">32</span></a> “Geomorphologie als genetische Wissenschaft,” in <i>Report of +Sixth International Geog. Congress</i> (London, 1895), p. 735 (English +Abstract, p. 748).</p> + +<p><a name="ft33b" id="ft33b" href="#fa33b"><span class="fn">33</span></a> On this subject see J. Geikie, <i>Earth Sculpture</i> (London, 1898); +J.E. Marr, <i>The Scientific Study of Scenery</i> (London, 1900); Sir A. +Geikie, <i>The Scenery and Geology of Scotland</i> (London, 2nd ed., 1887); +Lord Avebury (Sir J. Lubbock), <i>The Scenery of Switzerland</i> (London, +1896) and <i>The Scenery of England</i> (London, 1902).</p> + +<p><a name="ft34b" id="ft34b" href="#fa34b"><span class="fn">34</span></a> Some geographers distinguish a mountain from a hill by origin; +thus Professor Seeley says “a mountain implies elevation and a hill +implies denudation, but the external forms of both are often identical.” +<i>Report VI. Int. Geog. Congress</i> (London, 1895), p. 751.</p> + +<p><a name="ft35b" id="ft35b" href="#fa35b"><span class="fn">35</span></a> “Mountains,” in <i>Scot. Geog. Mag.</i> ii. (1896) p. 145.</p> + +<p><a name="ft36b" id="ft36b" href="#fa36b"><span class="fn">36</span></a> <i>Führer für Forschungsreisende</i>, pp. 652-685.</p> + +<p><a name="ft37b" id="ft37b" href="#fa37b"><span class="fn">37</span></a> See, for a summary of river-action, A. Phillipson, <i>Studien über +Wasserscheiden</i> (Leipzig, 1886); also I.C. Russell, <i>River Development</i>, +(London, 1898) (published as <i>The Rivers of North America</i>, New York, +1898).</p> + +<p><a name="ft38b" id="ft38b" href="#fa38b"><span class="fn">38</span></a> W.M. Davis, “The Geographical Cycle,” <i>Geog. Journ.</i> xiv. +(1899) p. 484.</p> + +<p><a name="ft39b" id="ft39b" href="#fa39b"><span class="fn">39</span></a> A. Penck, “Potamology as a Branch of Physical Geography,” +<i>Geog. Journ.</i> x. (1897) p. 619.</p> + +<p><a name="ft40b" id="ft40b" href="#fa40b"><span class="fn">40</span></a> See, for instance, E. Wisotzki, <i>Hauptfluss und Nebenfluss</i> +(Stettin, 1889). For practical studies see official reports on the +Mississippi, Rhine, Seine, Elbe and other great rivers.</p> + +<p><a name="ft41b" id="ft41b" href="#fa41b"><span class="fn">41</span></a> F.A. Forel, <i>Handbuch der Seenkunde: allgemeine Limnologie</i> +(Stuttgart, 1901); F.A. Forel, “La Limnologie, branche de la géographie,” +<i>Report VI. Int. Geog. Congress</i> (London, 1895), p. 593; +also <i>Le Léman</i> (2 vols., Lausanne, 1892, 1894); H. Lullies, “Studien +über Seen,” <i>Jubiläumsschrift der Albertus-Universität</i> (Königsberg, +1894); and G.R. Credner, “Die Reliktenseen,” <i>Petermanns Mitteilungen</i>, +Ergänzungshefte 86 and 89 (Gotha., 1887, 1888).</p> + +<p><a name="ft42b" id="ft42b" href="#fa42b"><span class="fn">42</span></a> J. Murray, “Drainage Areas of the Continents,” <i>Scot. Geog. Mag.</i> +ii. (1886) p. 548.</p> + +<p><a name="ft43b" id="ft43b" href="#fa43b"><span class="fn">43</span></a> Wagner, <i>Lehrbuch der Geographie</i> (1900), i. 586.</p> + +<p><a name="ft44b" id="ft44b" href="#fa44b"><span class="fn">44</span></a> For details, see A.R. Wallace, <i>Geographical Distribution of +Animals and Island Life</i>; A. Heilprin, <i>Geographical and Geological +Distribution of Animals</i> (1887); O. Drude, <i>Handbuch der Pflanzengeographie</i>; +A. Engler, <i>Entwickelungsgeschichte der Pflanzenwelt</i>; +also Beddard, <i>Zoogeography</i> (Cambridge, 1895); and Sclater, <i>The +Geography of Mammals</i> (London, 1899).</p> + +<p><a name="ft45b" id="ft45b" href="#fa45b"><span class="fn">45</span></a> See particularly A. de Lapparent, <i>Traité de géologie</i> (4th ed., +Paris, 1900).</p> + +<p><a name="ft46b" id="ft46b" href="#fa46b"><span class="fn">46</span></a> Estimate for 1900. H. Wagner, <i>Lehrbuch der Geographie</i>, i. +P. 658.</p> + +<p><a name="ft47b" id="ft47b" href="#fa47b"><span class="fn">47</span></a> Estimate for year not stated. A.H. Keane in <i>International +Geography</i>, p. 108.</p> + +<p><a name="ft48b" id="ft48b" href="#fa48b"><span class="fn">48</span></a> In <i>Proc. R. G. S.</i> xiii. (1891) p. 27.</p> + +<p><a name="ft49b" id="ft49b" href="#fa49b"><span class="fn">49</span></a> On the influence of land on people see Shaler, <i>Nature and +Man in America</i> (New York and London, 1892); and Ellen C. +Semple’s <i>American History and its Geographic Conditions</i> (Boston, +1903).</p> + +<p><a name="ft50b" id="ft50b" href="#fa50b"><span class="fn">50</span></a> See maps of density of population in Bartholomew’s great large-scale +atlases, <i>Atlas of Scotland</i> and <i>Atlas of England</i>.</p> + +<p><a name="ft51b" id="ft51b" href="#fa51b"><span class="fn">51</span></a> For the history of territorial changes in Europe, see Freeman, +<i>Historical Geography of Europe</i>, edited by Bury (Oxford), 1903; +and for the official definition of existing boundaries, see Hertslet, +<i>The Map of Europe by Treaty</i> (4 vols., London, 1875, 1891); <i>The +Map of Africa by Treaty</i> (3 vols., London, 1896). Also Lord Curzon’s +Oxford address on <i>Frontiers</i> (1907).</p> + +<p><a name="ft52b" id="ft52b" href="#fa52b"><span class="fn">52</span></a> For numerous special instances of the determining causes of +town sites, see G.G. Chisholm, “On the Distribution of Towns +and Villages in England,” <i>Geographical Journal</i> (1897), ix. 76, +x. 511.</p> + +<p><a name="ft53b" id="ft53b" href="#fa53b"><span class="fn">53</span></a> The whole subject of anthropogeography is treated in a masterly +way by F. Ratzel in his <i>Anthropogeographie</i> (Stuttgart, vol. i. 2nd +ed., 1899, vol. ii. 1891), and in his <i>Politische Geographie</i> (Leipzig, +1897). The special question of the reaction of man on his environment +is handled by G.P. Marsh in <i>Man and Nature, or Physical +Geography as modified by Human Action</i> (London, 1864).</p> + +<p><a name="ft54b" id="ft54b" href="#fa54b"><span class="fn">54</span></a> For commercial geography see G.G. Chisholm, <i>Manual of Commercial +Geography</i> (1890).</p> +</div> + + +<hr class="art" /> +<p><span class="bold">GEOID<a name="ar16" id="ar16"></a></span> (from Gr. <span class="grk" title="gê">γῆ</span>, the earth), an imaginary surface employed +by geodesists which has the property that every element +of it is perpendicular to the plumb-line where that line cuts it. +Compared with the “spheroid of reference” the surface of the +geoid is in general depressed over the oceans and raised over +the great land masses. (See <span class="sc"><a href="#artlinks">Earth, Figure of the</a></span>.)</p> + + +<hr class="art" /> +<p><span class="bold">GEOK-TEPE<a name="ar17" id="ar17"></a></span>, a former fortress of the Turkomans, in Russian +Transcaspia, in the oasis of Akhal-tekke, on the Transcaspian +railway, 28 m. N.W. of Askabad. It consisted of a walled +enclosure 1¾ m. in circuit, the wall being 18 ft. high and 20 to +30 ft. thick. In December 1880 the place was attacked by +6000 Russians under General Skobelev, and after a siege of +twenty-three days was carried by storm, although the defenders +numbered 25,000. A monument and a small museum commemorate +the event.</p> + + +<hr class="art" /> +<p><span class="bold">GEOLOGY<a name="ar18" id="ar18"></a></span> (from Gr. <span class="grk" title="gê">γῆ</span>, the earth, and <span class="grk" title="logos">λόγος</span>, science), the +science which investigates the physical history of the earth. +Its object is to trace the structural progress of our planet from +the earliest beginnings of its separate existence, through its +various stages of growth, down to the present condition of +things. It seeks to determine the manner in which the evolution +of the earth’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.</p> + +<p>But while thus culling freely from the dominions of other +sciences, geology claims as its peculiar territory the rocky +framework of the globe. In the materials composing that +framework, their composition and arrangement, the processes +of their formation, the changes which they have undergone, +and the terrestrial revolutions to which they bear witness, lie +the main data of geological history. It is the task of the geologist +to group these elements in such a way that they may be made +to yield up their evidence as to the march of events in the +evolution of the planet. He finds that they have in large +measure arranged themselves in chronological sequence,—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.</p> + +<p>But it is mainly by the remains of plants and animals imbedded +in the rocks that the geologist is guided in unravelling the +chronological succession of geological changes. He has found +that a certain order of appearance characterizes these organic +remains, that each great group of rocks is marked by its own +special types of life, and that these types can be recognized, +and the rocks in which they occur can be correlated even in +distant countries, and where no other means of comparison +would be possible. At one moment he has to deal with the bones +of some large mammal scattered through a deposit of superficial +gravel, at another time with the minute foraminifers and ostracods +of an upraised sea-bottom. Corals and crinoids crowded and +crushed into a massive limestone where they lived and died, +ferns and terrestrial plants matted together into a bed of coal +where they originally grew, the scattered shells of a submarine +sand-bank, the snails and lizards which lived and died within +a hollow-tree, the insects which have been imprisoned within +the exuding resin of old forests, the footprints of birds and +quadrupeds, the trails of worms left upon former shores—these, +and innumerable other pieces of evidence, enable the geologist +to realize in some measure what the faunas and floras of successive +periods have been, and what geographical changes the site of +every land has undergone.</p> + +<p>It is evident that to deal successfully with these varied +materials, a considerable acquaintance with different branches +of science is needful. Especially necessary is a tolerably wide +knowledge of the processes now at work in changing the surface +of the earth, and of at least those forms of plant and animal +life whose remains are apt to be preserved in geological deposits, +or which in their structure and habitat enable us to realize what +their forerunners were. It has often been insisted that the +present is the key to the past; and in a wide sense this assertion +is eminently true. Only in proportion as we understand the +present, where everything is open on all sides to the fullest investigation, +can we expect to decipher the past, where so much is +obscure, imperfectly preserved or not preserved at all. A +study of the existing economy of nature ought thus to be the +foundation of the geologist’s training.</p> + +<p>While, however, the present condition of things is thus employed, +we must obviously be on our guard against the danger +of unconsciously assuming that the phase of nature’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.</p> + +<p>In dealing with the geological record, as the accessible solid +part of the globe is called, we cannot too vividly realize that at +<span class="pagenum"><a name="page639" id="page639"></a>639</span> +the best it forms but an imperfect chronicle. Geological history +cannot be compiled from a full and continuous series of documents. +From the very nature of its origin the record is necessarily +fragmentary, and it has been further mutilated and obscured +by the revolutions of successive ages. And even where the +chronicle of events is continuous, it is of very unequal value in +different places. In one case, for example, it may present us +with an unbroken succession of deposits many thousands of +feet in thickness, from which, however, only a few meagre facts +as to geological history can be gleaned. In another instance +it brings before us, within the compass of a few yards, the +evidence of a most varied and complicated series of changes +in physical geography, as well as an abundant and interesting +suite of organic remains. These and other characteristics of +the geological record become more apparent and intelligible as +we proceed in the study of the science.</p> + +<p><i>Classification.</i>—For systematic treatment the subject may be +conveniently arranged in the following parts:—</p> + +<p>1. <i>The Historical Development of Geological Science.</i>—Here +a brief outline will be given of the gradual growth of geological +conceptions from the days of the Greeks and Romans down to +modern times, tracing the separate progress of the more important +branches of inquiry and noting some of the stages which in each +case have led up to the present condition of the science.</p> + +<p>2. <i>The Cosmical Aspects of Geology.</i>—This section embraces +the evidence supplied by astronomy and physics regarding the +form and motions of the earth, the composition of the planets +and sun, and the probable history of the solar system. The +subjects dealt with under this head are chiefly treated in separate +articles.</p> + +<p>3. <i>Geognosy.</i>—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.</p> + +<p>4. <i>Dynamical Geology</i> 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.</p> + +<p>5. <i>Geotectonic or Structural Geology</i> 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.</p> + +<p>6. <i>Palaeontological Geology.</i>—This branch of the subject, +starting from the evidence supplied by the organic forms which +are found preserved in the crust of the earth, includes such +questions as the relations between extinct and living types, +the laws which appear to have governed the distribution of life +in time and in space, the relative importance of different genera +of animals in geological inquiry, the nature and use of the +evidence from organic remains regarding former conditions +of physical geography. Some of these problems belong also to +zoology and botany, and are more fully discussed in the articles +<span class="sc"><a href="#artlinks">Palaeontology</a></span> and <span class="sc"><a href="#artlinks">Palaeobotany</a></span>.</p> + +<p>7. <i>Stratigraphical Geology.</i>—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.</p> + +<p>8. <i>Physiographical Geology</i>, proceeding from the basis of +fact laid down by stratigraphical geology regarding former +geographical changes, embraces an inquiry into the origin and +history of the features of the earth’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.</p> + +<p>Most of the detail embraced in these several sections is +relegated to separate articles, to which references are here +inserted. The following pages thus deal mainly with the general +principles and historical development of the science:—</p> + +<p class="pt2 center sc">Part I.—Historical Development</p> + +<div class="condensed"> +<p><i>Geological Ideas among the Greeks and Romans.</i>—Many geological +phenomena present themselves in so striking a form that they could +hardly fail to impress the imagination of the earliest and rudest +races of mankind. Such incidents as earthquakes and volcanic +eruptions, destructive storms on land and sea, disastrous floods and +landslips suddenly strewing valleys with ruin, must have awakened +the terror of those who witnessed them. Prominent features of +landscape, such as mountain-chains with their snows, clouds and +thunderstorms, dark river-chasms that seem purposely cleft open in +order to give passage to the torrents that rush through them, crags +with their impressive array of pinnacles and recesses must have +appealed of old, as they still do, to the awe and wonder of those +who for the first time behold them. Again, banks of sea-shells in +far inland districts would, in course of time, arrest the attention of +the more intelligent and reflective observers, and raise in their minds +some kind of surmise as to how such shells could ever have come +there. These and other conspicuous geological problems found +their earliest solution in legends and myths, wherein the more +striking terrestrial features and the elemental forces of nature were +represented to be the manifestation of the power of unseen supernatural +beings.</p> + +<p>The basin of the Mediterranean Sea was especially well adapted, +from its physical conditions, to be the birth-place of such fables. +It is a region frequently shaken by earthquakes, and contains two +distinct centres of volcanic activity, one in the Aegean Sea and one +in Italy. It is bounded on the north by a long succession of lofty +snow-capped mountain-ranges, whence copious rivers, often swollen +by heavy rains or melted snows, carry the drainage into the sea. +On the south it boasts the Nile, once so full of mystery; likewise +wide tracts of arid desert with their dreaded dust storms. The +Mediterranean itself, though an inland sea, is subject to gales, +which, on exposed coasts, raise breakers quite large enough to give a +vivid impression of the power of ocean waves. The countries that +surround this great sheet of water display in many places widely-spread +deposits full of sea shells, like those that still live in the +neighbouring bays and gulfs. Such a region was not only well fitted +to supply subjects for mythology, but also to furnish, on every side, +materials which, in their interest and suggestiveness, would appeal +to the reason of observant men.</p> + +<p>It was natural, therefore, that the early philosophers of Greece +should have noted some of these geological features, and should have +sought for other explanations of them than those to be found in the +popular myths. The opinions entertained in antiquity on these +subjects may be conveniently grouped under two heads: (1) Geological +processes now in operation, and (2) geological changes in +the past.</p> + +<p>1. <i>Contemporary Processes.</i>—The geological processes of the present +time are partly at work underground and partly on the surface of the +earth. The former, from their frequently disastrous +character, received much attention from Greek and +<span class="sidenote">Earthquakes and volcanoes.</span> +Roman authors. Aristotle, in his <i>Meteorics</i>, cites the +speculations of several of his predecessors which he rejects +in favour of his own opinion to the effect that earthquakes are due +to the generation of wind within the earth, under the influence of the +warmth of the sun and the internal heat. Wind, being the lightest +and most rapidly moving body, is the cause of motion in other +bodies, and fire, united with wind, becomes flame, which is endowed +with great rapidity of motion. Aristotle looked upon earthquakes +and volcanic eruptions as closely connected with each other, the +discharge of hot materials to the surface being the result of a severe +earthquake, when finally the wind rushes out with violence, and +sometimes buries the surrounding country under sparks and cinders, +<span class="pagenum"><a name="page640" id="page640"></a>640</span> +as had happened at Lipari. These crude conceptions of the nature +of volcanic action, and the cause of earthquakes, continued to prevail +for many centuries. They are repeated by Lucretius, who, however, +following Anaximenes, includes as one of the causes of earthquakes +the fall of mountainous masses of rock undermined by time, and the +consequent propagation of gigantic tremors far and wide through +the earth. Strabo, having travelled through the volcanic districts +of Italy, was able to recognize that Vesuvius had once been an +active volcano, although no eruption had taken place from it within +human memory. He continued to hold the belief that volcanic +energy arose from the movement of subterranean wind. He believed +that the district around the Strait of Messina, which had formerly +suffered from destructive earthquakes, was seldom visited by them +after the volcanic vents of that region had been opened, so as to +provide an escape for the subterranean fire, wind, water and burning +masses. He cites in his <i>Geography</i> a number of examples of widespread +as well as local sinkings of land, and alludes also to the uprise +of the sea-bottom. He likewise regards some islands as having been +thrown up by volcanic agency, and others as torn from the mainland +by such convulsions as earthquakes.</p> + +<p>The most detailed account of earthquake phenomena which has +come down to us from antiquity is that of Seneca in his <i>Quaestiones +Naturales</i>. This philosopher had been much interested in the +accounts given him by survivors and witnesses of the earthquake +which convulsed the district of Naples in February <span class="scs">A.D.</span> 63. He +distinguished several distinct movements of the ground: 1st, the +up and down motion (<i>succussio</i>); 2nd, the oscillatory motion (<i>inclinatio</i>); +and probably a third, that of trembling or vibration. +While admitting that some earthquakes may arise from the collapse +of the walls of subterranean cavities, he adhered to the old idea, +held by the most numerous and important previous writers, that +these commotions are caused mainly by the movements of wind +imprisoned within the earth. As to the origin of volcanic outbursts +he supposed that the subterranean wind in struggling for an outlet, +and whirling through the chasms and passages, meets with great +store of sulphur and other combustible substances, which by mere +friction are set on fire. The elder Pliny reiterates the commonly +accepted opinion as to the efficacy of wind underground. In +discussing the phenomena of earthquakes he remarks that towns +with many culverts and houses with cellars suffer less than others, +and that at Naples those houses are most shaken which stand on +hard ground. It thus appears that with regard to subterranean +geological operations, no advance was made during the time of the +Greeks and Romans as to the theoretical explanation of these phenomena; +but a considerable body of facts was collected, especially +as to the effects of earthquakes and the occurrence of volcanic +eruptions.</p> + +<p>The superficial processes of geology, being much less striking than +those of subterranean energy, naturally attracted less attention in +antiquity. The operations of rivers, however, which so +intimately affect a human population, were watched with +<span class="sidenote">Action of rivers.</span> +more or less care. Herodotus, struck by the amount of +alluvial silt brought down annually by the Nile and spread over the +flat inundated land, inferred that “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.</p> + +<p>2. <i>Past Processes.</i>—The abundant well-preserved marine shells +exposed among the upraised Tertiary and post-Tertiary deposits in +the countries bordering the Mediterranean are not infrequently +alluded to in Greek and Latin literature. +<span class="sidenote">Occurrences of fossils.</span> +Xenophanes of Colophon (614 <span class="scs">B.C.</span>) noticed the occurrence +of shells and other marine productions inland among the +mountains, and inferred from them that the land had risen out of +the sea. A similar conclusion was drawn by Xanthus the Lydian +(464 <span class="scs">B.C.</span>) from shells like scallops and cockles, which were found far +from the sea in Armenia and Lower Phrygia. Herodotus, Eratosthenes, +Strato and Strabo noted the vast quantities of fossil shells in +different parts of Egypt, together with beds of salt, as evidence that +the sea had once spread over the country. But by far the most +philosophical opinions on the past mutations of the earth’s surface +are those expressed by Aristotle in the treatise already cited. Reviewing +the evidence of these changes, he recognized that the sea +now covers tracts that were once dry land, and that land will one +day reappear where there is now sea. These alternations are to be +regarded as following each other in a certain order and periodicity. +But they are apt to escape our notice because they require successive +periods of time, which, compared with our brief existence, are of +enormous duration, and because they are brought about so imperceptibly +that we fail to detect them in progress. In a celebrated +passage in his <i>Metamorphoses</i>, Ovid puts into the mouth of the +philosopher Pythagoras an account of what was probably regarded +as the Pythagorean view of the subject in the Augustan age. It +affirms the interchange of land and sea, the erosion of valleys by +descending rivers, the washing down of mountains into the sea, the +disappearance of the rivers and the submergence of land by earthquake +movements, the separation of some islands from, and the union +of others with, the mainland, the uprise of hills by volcanic action, +the rise and extinction of burning mountains. There was a time +before Etna began to glow, and the time is coming when the mountain +will cease to burn.</p> + +<p>From this brief sketch it will be seen that while the ancients had +accumulated a good deal of information regarding the occurrence of +geological changes, their interpretations of the phenomena were to +a considerable extent mere fanciful speculation. They had acquired +only a most imperfect conception of the nature and operation of the +geological processes; and though many writers realized that the +surface of the earth has not always been, and will not always remain, +as it is now, they had no glimpse of the vast succession of changes +of that surface which have been revealed by geology. They built +hypotheses on the slenderest basis of fact, and did not realize the +necessity of testing or verifying them.</p> + +<p><i>Progress of Geological Conceptions in the Middle Ages.</i>—During the +centuries that succeeded the fall of the Western empire little progress +was made in natural science. The schoolmen in the monasteries +and other seminaries were content to take their science from the +literature of Greece and Rome. The Arabs, however, not only +collected and translated that literature, but in some departments +made original observations themselves. To one of the most illustrious +of their number, Avicenna, the translator of Aristotle, a treatise has +been ascribed, in which singularly modern ideas are expressed +regarding mountains, some of which are there stated to have been +produced by an uplifting of the ground, while others have been left +prominent, owing to the wearing away of the softer rocks around +them. In either case, it is confessed that the process would demand +long tracts of time for its completion.</p> + +<p>After the revival of learning the ancient problem presented by +fossil shells imbedded in the rocks of the interior of many countries +received renewed attention. But the conditions for its solution +were no longer what they had been in the days of the philosophers +of antiquity. Men were not now free to adopt and teach any doctrine +they pleased on the subject. The Christian church had meanwhile +arisen to power all over Europe, and adjudged as heretics all +who ventured to impugn any of her dogmas. She taught that the +land and the sea had been separated on the third day of creation, +before the appearance of any animal life, which was not created until +the fifth day. To assert that the dry land is made up in great part +of rocks that were formed in the sea, and are crowded with the +remains of animals, was plainly to impugn the veracity of the Bible. +Again, it had come to be the orthodox belief that only somewhere +about 6000 years had elapsed since the time of Adam and Eve. +If any thoughtful observer, impressed with the overwhelming force +of the evidence that the fossiliferous formations of the earth’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 +<span class="pagenum"><a name="page641" id="page641"></a>641</span> +centuries that have elapsed since that event, the whole of the fossiliferous +rocks might have been deposited. Unfortunately for this +hypothesis it ignored the fact that these rocks do not consist of +volcanic materials.</p> + +<p>So long as the fundamental question remained in dispute as to +the true character and history of the stratified portion of the earth’s +crust containing organic remains, geology as a science could not +begin its existence. The diluvialists (those who relied on the hypothesis +of the Flood) held the field during the 16th, 17th and a great +part of the 18th century. They were looked on as the champions of +orthodoxy; and, on that account, they doubtless wielded much +more influence than would have been gained by them from the +force of their arguments. Yet during those ages there were not +wanting occasional observers who did good service in combating the +prevalent misconceptions, and in preparing the way for the ultimate +triumph of truth. It was more especially in Italy, where many of +the more striking phenomena of geology are conspicuously displayed, +that the early pioneers of the science arose, and that for several +generations the most marked progress was made towards placing +the investigations of the past history of the earth upon a basis of +careful observation and scientific deduction. One of the first of +<span class="sidenote">Leonardo da Vinci; Fracastorio; Falloppio.</span> +these leaders was Leonardo da Vinci (1452-1519), who, +besides his achievements in painting, sculpture, architecture +and engineering, contributed some notable observations +regarding the great problem of the origin of fossil +shells. He ridiculed the notion that these objects could +have been formed by the influence of the stars, and maintained +that they had once belonged to living organisms, and therefore +that what is now land was formerly covered by the sea. +Girolamo Fracastorio (1483-1553) claimed that the shells could +never have been left by the Flood, which was a mere temporary +inundation, but that they proved the mountains, in which they +occur, to have been successively uplifted out of the sea. On the +other hand, even an accomplished anatomist like Gabriello Falloppio +(1523-1562) found it easier to believe that the bones of elephants, +teeth of sharks, shells and other fossils were mere earthy inorganic +concretions, than that the waters of Noah’s Flood could ever nave +reached as far as Italy.</p> + +<p>By much the most important member of this early band of Italian +writers was undoubtedly Nicolas Steno (1631-1687), who, though +born in Copenhagen, ultimately settled in Florence. +Having made a European reputation as an anatomist, +<span class="sidenote">Nicolas Steno.</span> +his attention was drawn to geological problems by finding +that the rocks of the north of Italy contained what appeared to be +sharks’ teeth closely resembling those of a dog-fish, of which he had +published the anatomy. Cautiously at first, for fear of offending +orthodox opinions, but afterwards more boldly, he proclaimed his +conviction that those objects had once been part of living animals, +and that they threw light on some of the past history of the earth. +He published in 1669 a small tract, <i>De solido intra solidum naturaliter +contento</i>, in which he developed the ideas he had formed of this +history from an attentive study of the rocks. He showed that the +stratified formations of the hills and valleys consist of such materials +as would be laid down in the form of sediment in turbid water; +that where they contain marine productions this water is proved +to have been the sea; that diversities in their composition point to +commingling of currents, carrying different kinds of sediment of +which the heaviest would first sink to the bottom. He made original +and important observations on stratification, and laid down some +of the fundamental axioms in stratigraphy. He reasoned that as +the original position of strata was approximately horizontal, when +they are found to be steeply inclined or vertical, or bent into arches, +they have been disrupted by subterranean exhalations, or by the +falling in of the roofs of underground cavernous spaces. It is to +this alteration of the original position of the strata that the inequalities +of the earth’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.</p> + +<p>To the Italian school, as especially typified in Steno, must be +assigned the honour of having thus begun to lay firmly and truly +the first foundation stones of the modern science of +geology. The same school included Antonio Vallisneri +<span class="sidenote">Lazzaro Moro.</span> +(1661-1730), who surpassed his predecessors in his wider +and more exact knowledge of the fossiliferous rocks that form the +backbone of the Italian peninsula, which he contended were formed +during a wide and prolonged submergence of the region, altogether +different from the brief deluge of Noah. There was likewise Lazzaro +Moro (1687-1740), who did good service against the diluvialists, +but the fundamental feature of his system of nature lay in the +preponderant part which, unaware of the great difference between +volcanic materials and ordinary sediment, he assigned to volcanic +action in the production of the sedimentary rocks of the earth’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.</p> + +<p><i>The Cosmogonists and Theories of the Earth.</i>—While in Italy +substantial progress was made in collecting information regarding +the fossiliferous formations of that country, and in forming conclusions +concerning them based upon more or less accurate observations, +the tendency to mere fanciful speculation, which could not be +wholly repressed in any country, reached a remarkable extravagance +in England. In proportion as materials were yet lacking from +which to construct a history of the evolution of our planet in accordance +with the teaching of the church, imagination supplied the place +of ascertained fact, and there appeared during the last twenty years +of the 18th century a group of English cosmogonists, who, by the +sensational character of their speculations, aroused general attention +both in Britain and on the continent. It may be doubted, however, +whether the effect of their writings was not to hinder the advance +of true science by diverting men from the observation of nature into +barren controversy over unrealities. It is not needful here to do +more than mention the names of Thomas Burnet, whose <i>Sacred +Theory of the Earth</i> appeared in 1681, and William Whiston, whose +New Theory of the Earth was published in 1696. Hardly less fanciful +than these writers, though his practical acquaintance with rocks +and fossils was infinitely greater, was John Woodward, whose +<i>Essay towards a Natural History of the Earth</i> dates from 1695. More +important as a contribution to science was the catalogue of the large +collection of fossils, which he had made from the rocks of England +and which he bequeathed to the university of Cambridge. This +catalogue appeared in 1728-1729 with the title of <i>An attempt towards +a Natural History of the Fossils of England</i>.</p> + +<p>A striking contrast to these cosmogonists is furnished by another +group, which arose in France and Germany, and gave to the world +the first rational ideas concerning the probable primeval +evolution of our globe. The earliest of these pioneers was +<span class="sidenote">Descartes.</span> +the illustrious philosopher René Descartes (1596-1650). He propounded +a scheme of cosmical development in which he represented +the earth, like the other planets, to have been originally a mass of +glowing material like the sun, and to have gradually cooled on the +outside, while still retaining an incandescent, self-luminous nucleus. +Yet with this noble conception, which modern science has accepted, +Descartes could not shake himself free from the time-honoured +error in regard to the origin of volcanic action. He thought that +certain exhalations within the earth condense into oil, which, when +in violent motion, enters into the subterranean cavities, where it +passes into a kind of smoke. This smoke is from time to time ignited +by a spark of fire and, pressing violently against its containing +walls, gives rise to earthquakes. If the flame breaks through to the +surface at the top of a mountain, it may escape with enormous +energy, hurling forth much earth mingled with sulphur or bitumen, +and thus producing a volcano. The mountain might burn for a +long time until at last its store of fuel in the shape of sulphur or +bitumen would be exhausted. Not only did the philosopher refrain +from availing himself of the high internal temperature of the globe +as the source of volcanic energy, he even did not make use of it as +the cause of the ignition of his supposed internal fuel, but speculated +on the kindling of the subterranean fires by the spirits or gases +setting fire to the exhalations, or by the fall of masses of rock and +the sparks produced by their friction or percussion.</p> + +<p>The ideas of Descartes regarding planetary evolution were enlarged +and made more definite by Wilhelm Gottfried Leibnitz (1646-1716), +whose teaching has largely influenced all subsequent speculation +<span class="pagenum"><a name="page642" id="page642"></a>642</span> +on the subject. In his great tract, the <i>Protogaea</i> (published in 1749, +<span class="sidenote">Leibnitz.</span> +thirty-three years after his death), he traced the probable passage +of our earth from an original condition of incandescent +vapour into that of a smooth molten globe, which, by +continuous cooling, acquired an external solid crust and rugose +surface. He thought that the more ancient rocks, such as granite +and gneiss, might be portions of the earliest outer crust; and that as +the external solidification advanced, immense subterranean cavities +were left which were filled with air and water. By the collapse of +the roofs of these caverns, valleys might be originated at the surface, +while the solid intervening walls would remain in place and form +mountains. By the disruption of the crust, enormous bodies of +water were launched over the surface of the earth, which swept vast +quantities of sediment together, and thus gave rise to sedimentary +deposits. After many vicissitudes of this kind, the terrestrial forces +calmed down, and a more stable condition of things was established.</p> + +<p>An important feature in the cosmogony of Leibnitz is the +prominent place which he assigned to organic remains in the stratified +rocks of the crust. Ridiculing the foolish attempts to account for +the presence of these objects by calling them “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.</p> + +<p>Appealing to a much wider public than Descartes or Leibnitz, and +basing his speculations on a wider acquaintance with the organic +and inorganic realms of nature, G.L.L. de Buffon (1707-1788) +was undoubtedly one of the most influential forces +<span class="sidenote">Buffon.</span> +that in Europe guided the growth of geological ideas during the +18th century. He published in 1749 a <i>Theory of the Earth</i>, in which +he adopted views similar to those of Descartes and Leibnitz as to +planetary evolution; but though he realized the importance of +fossils as records of former conditions of the earth’s surface, he +accounted for them by supposing that they had been deposited from +a universal ocean, a large part of which had subsequently been +engulfed into caverns in the interior of the globe. Thirty years +later, after having laboured with skill and enthusiasm in all branches +of natural history, he published another work, his famous <i>Époques +de la nature</i> (1778), which is specially remarkable as the first attempt +to deal with the history of the earth in a chronological manner, and +to compute, on a basis of experiment, the antiquity of the several +stages of this history. His experiments were made with globes of +cast iron, and could not have yielded results of any value for his +purpose; but in so far as his calculations were not mere random +guesses but had some kind of foundation on experiment, they +deserve respectful recognition. He divided the history of our earth +into six periods of unequal duration, the whole comprising a period +of some 70,000 or 75,000 years. He supposed that the stage of +incandescence, before the globe had consolidated to the centre, +lasted 2936 years, and that about 35,000 years elapsed before the +surface had cooled sufficiently to be touched, and therefore to be +capable of supporting living things. Terrestrial animal life, however, +was not introduced until 55,000 or 60,000 years after the beginning +of the world or about 15,000 years before our time. Looking into +the future, he foresaw that, by continued refrigeration, our globe +will eventually become colder than ice, and this fair face of nature, +with its manifold varieties of plant and animal life, will perish after +having existed for 132,000 years.</p> + +<p>Buffon’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 +Arduíno (1714-1795) in Italy, and of Johann Gottlob Lehmann +(d. 1767) and George Christian Füchsel (1722-1773) in Germany, he +would have been able to give to his “epochs” a more definite succession +of events and a greater correspondence with the facts of nature.</p> + +<p>Among the writers of the 18th century, who formed philosophical +conceptions of the system of processes by which the life of our earth +as a habitable globe is carried on, a foremost place must be assigned +to James Hutton (1726-1797). Educated for the medical profession, +<span class="sidenote">James Hutton.</span> +he studied at Edinburgh and at Paris, and took his doctor’s degree +at Leiden. But having inherited a small landed property in +Berwickshire, he took to agriculture, and after putting +his land into excellent order, let his farm and betook +himself to Edinburgh, there to gratify the scientific +tastes which he had developed early in life. He had been more +especially led to study minerals and rocks, and to meditate on the +problems which they suggest as to the constitution and history of +the earth. His journeys in Britain and on the continent of Europe +had furnished him with material for reflection; and he had gradually +evolved a system or theory in which all the scattered facts +could be arranged so as to show their mutual dependence and their +place in the orderly mechanism of the world. He used to discuss +his views with one or two of his friends, but refrained from publishing +them to the world until, on the foundation of the Royal Society of +Edinburgh, he communicated an outline of his doctrine to that +learned body in 1785. Some years later he expanded this first essay +into a larger work in two volumes, which were published in 1795 +with the title of <i>Theory of the Earth, with Proofs and Illustrations</i>.</p> + +<p>Hutton’s teaching has exercised a profound influence on modern +geology. This influence, however, has arisen less from his own +writings than from the account of his doctrines given by +his friend John Playfair in the classic work entitled +<span class="sidenote">John Playfair.</span> +<i>Illustrations of the Huttonian Theory</i>, published in 1802. +Hutton wrote in so prolix and obscure a style as rather to repel than +attract readers. Playfair, on the other hand, expressed himself in +such clear and graceful language as to command general attention, +and to gain wide acceptance for his master’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.”</p> + +<p>A distinguishing feature of the Huttonian philosophy is to be +seen in the breadth of its conceptions regarding the geological +operations continually in progress on the surface of the globe. +Hutton saw that the land is undergoing a ceaseless process of degradation, +through the influence of the air, frost, rain, rivers and the sea, +and that in course of time, if no countervailing agency should intervene, +the whole of the dry land will be washed away into the sea. +But he also perceived that this universal erosion is not everywhere +carried on at the same rate; that it is specially active along the +channels of torrents and rivers, and that, owing to this difference +these channels are gradually deepened and widened, until the +complicated valley-system of a country is carved out. He recognized +that the detritus worn away from the land must be spread out over +the floor of the sea, so as to form there strata similar to those that +compose most of the dry land. As he could detect in the structure +of land convincing evidence that former sea floors had been elevated +to form the continents and islands of to-day, he could look forward +to future ages, when the same subterranean agency which had raised +up the present land would again be employed to uplift the bed of +the existing ocean, thus to renew the surface of our earth as a +habitable globe, and to start a fresh cycle of erosion and deposition.</p> + +<p>Though Hutton was not unaware that organic remains abound in +many of the stratified rocks, he left them out of consideration in +the elaboration of his theory. It was otherwise with +one of his French contemporaries, the illustrious J.B. +<span class="sidenote">Lamarck.</span> +Lamarck (1744-1829), who, after having attained great eminence as +a botanist, turned to zoology when he was nearly fifty years of age, +and before long rose to even greater distinction in that department +<span class="pagenum"><a name="page643" id="page643"></a>643</span> +of science. His share in the classification and description of the +mollusca and in founding invertebrate palaeontology, his theory +of organic evolution and his philosophical treatment of many +biological questions have been tardily recognized, but his contributions +to geology have been less generally acknowledged. When he +accepted the “professorship of zoology; of insects, of worms and of +microscopic animals” at the Museum of Natural History, Paris, +in 1793, he at once entered with characteristic ardour and capacity +into the new field of research then opened to him. In dealing with +the mollusca he considered not merely the living but also the extinct +forms, especially the abundant, varied and well-preserved genera +and species furnished by the Tertiary deposits of the Paris basin, +of which he published descriptions and plates that proved of essential +service in the stratigraphical work of Cuvier and Alexandre +Brongniart (1770-1847). His labours among these relics of ancient +seas and lakes led him to ponder over the past history of the globe, +and as he was seldom dilatory in making known the opinions he had +formed, he communicated some of his conclusions to the National +Institute in 1799. These, including a further elaboration of his +views, he published in 1802 in a small volume entitled Hydrogéologie.</p> + +<p>This treatise, though it did not reach a second edition and has +never been reprinted, deserves an honourable place in geological +literature. Its object, the author states, was to present some important +and novel considerations, which he thought should form +the basis of a true theory of the earth. He entirely agreed with the +doctrine of the subaerial degradation of the land and the erosion of +valleys by running water. Not even Playfair could have stated this +doctrine more emphatically, and it is worthy of notice that Playfair’s +<i>Illustrations of the Huttonian Theory</i> 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.</p> + +<p>Lamarck, in reflecting upon the wide diffusion of fossil shells and +the great height above the sea at which they are found, conceived +the extraordinary idea that the ocean basin has been scoured out +by the sea, and that, by an impulse communicated to the waters +through the influence chiefly of the moon, the sea is slowly eating +away the eastern margins of the continents, and throwing up detritus +on their western coasts, and is thus gradually shifting its basin +round the globe. He would not admit the operation of cataclysms; +but insisted as strongly as Hutton on the continuity of natural +processes, and on the necessity of explaining former changes of the +earth’s surface by causes which can still be seen to be in operation. +As might be anticipated from his previous studies, he brought living +things and their remains into the forefront of his theory of the earth. +He looked upon fossils as one of the chief means of comprehending +the revolutions which the surface of the earth has undergone; +and in his little volume he again and again dwells on the vast +antiquity to which these revolutions bear witness. He acutely +argues, from the condition of fossil shells, that they must have lived +and died where their remains are now found.</p> + +<p>In the last part of his treatise Lamarck advances some peculiar +opinions in physics and chemistry, which he had broached eighteen +years before, but which had met with no acceptance among the +scientific men of his time. He believed that the tendency of all +compound substances is to decay, and thereby to be resolved into +their component constituents. Yet he saw that the visible crust +of the earth consists almost wholly of compound bodies. He therefore +set himself to solve the problem thus presented. Perceiving +that the biological action of living organisms is constantly forming +combinations of matter, which would never have otherwise come +into existence, he proceeded to draw the extraordinary conclusion +that the action of plant and animal life (the <i>Pouvoir de la vie</i>) upon +the inorganic world is so universal and so potent, that the rocks and +minerals which form the outer part of the earth’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.</p> + +<p>The last notable contributor to the cosmological literature of +geology was another illustrious Frenchman, the comparative anatomist +Cuvier (1769-1832). He was contemporary with +Lamarck, but of a very different type of mind. The +<span class="sidenote">Cuvier.</span> +brilliance of his speculations, and the charm with which he expounded +them, early gained for him a prominent place in the society of Paris. +He too was drawn by his zoological studies to investigate fossil +organic remains, and to consider the former conditions of the earth’s +surface, of which they are memorials. It was among the vertebrate +organisms of the Paris basin that he found his chief material, and +from them that he prepared the memoirs which led to him being +regarded as the founder of vertebrate palaeontology. But beyond +their biological interest, they awakened in him a keen desire to +ascertain the character and sequence of the geographical revolutions +to which they bear witness. He approached the subject from an +opposite and less philosophical point of view than that of Lamarck, +coming to it with certain preconceived notions, which affected all +his subsequent writings. While Lamarck was by instinct an evolutionist, +who sought to trace in the history of the past the operation +of the same natural processes as are still at work, Cuvier, on the +other hand, was a catastrophist, who invoked a succession of vast +cataclysms to account for the interruptions in the continuity of the +geological record.</p> + +<p>In a preliminary <i>Discourse</i> prefixed to his <i>Recherches sur les +ossemens fossiles</i> (1821) Cuvier gave an outline of what he conceived +to have been the past history of our globe, so far as he had been able +to comprehend it from his investigations of the Tertiary formations +of France. He believed that in that history evidence can be +recognized of the occurrence of many sudden and disastrous revolutions, +which, to judge from their effects on the animal life of the +time, must have exceeded in violence anything we can conceive at +the present day, and must have been brought about by other agencies +than those which are now in operation. Yet, in spite of these +catastrophes, he saw that there has been an upward progress in the +animal forms inhabiting the globe, until the series ended in the +advent of man. He could not, however, find any evidence that one +species has been developed from another, for in that case there should +have been traces of intermediate forms among the stratified formations, +where he affirmed that they had never been found. A +prominent position in the <i>Discourse</i> is given to a strenuous argument +to disprove the alleged antiquity of some nations, and to show that +the last great catastrophe occurred not more than some 5000 or +6000 years ago. Cuvier thus linked himself with those who in +previous generations had contended for the efficacy of the Deluge. +But his researches among fossil animals had given him a far wider +outlook into the geological past, and had opened up to him a succession +of deeply interesting problems in the history of life upon the +earth, which, though he had not himself material for their solution, +he could foresee would be cleared up in the future.</p> + +<p><i>Gradual Shaping of Geology into a Distinct Branch of Science.</i>—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,”<a name="fa1c" id="fa1c" href="#ft1c"><span class="sp">1</span></a> descriptive of this branch of the +investigation of nature, was not proposed until the last quarter of +the 18th century by Jean André De Luc (1727-1817) and Horace +Benedict De Saussure (1740-1749). But the science was then in a +markedly half-formed condition, theoretical speculation still in large +part supplying the place of deductions from a detailed examination +of actual fact. In 1807 a few enterprising spirits founded the +Geological Society of London for the special purpose of counteracting +the prevalent tendency and confining their intention “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.</p> + +<p>The youngest of the schools of geological thought is that of the +Evolutionists. Pointing to the whole body of evidence from inorganic +and organic nature, they maintain that the history of our +planet has been one of continual and unbroken development from +the earliest cosmical beginnings down to the present time, and that +the crust of the earth contains an abundant, though incomplete, +record of the successive stages through which the plant and animal +<span class="pagenum"><a name="page644" id="page644"></a>644</span> +kingdoms have reached their existing organization. The publication +of Darwin’s <i>Origin of Species</i> in 1859, in which evolution was made +the key to the history of the animal and vegetable kingdoms, produced +an extraordinary revolution in geological opinion. The older +schools of thought rapidly died out, and evolution became the +recognized creed of geologists all over the world.</p> + +<p><i>Development of Opinion regarding Igneous Rocks.</i>—So long as the +idea prevailed that volcanoes are caused by the combustion of +inflammable substances underground, there could be no rational +conception of volcanic action and its products. Even so late as +the middle of the 18th century, as above remarked, such a good +observer as Lazzaro Moro drew so little distinction between volcanic +and other rocks that he could believe the fossiliferous formations +to have been mainly formed of materials ejected from eruptive vents. +After his time the notion continued to prevail that all the rocks which +form the dry land were laid down under water. Even streams of +lava, which were seen to flow from an active crater, were regarded +only as portions of sedimentary or other rocks, which had been +melted by the fervent heat of the burning inflammable materials +that had been kindled underground. In spite of the speculations +of Descartes and Leibnitz, it was not yet generally comprehended +that there exists beneath the terrestrial crust a molten magma, +which, from time to time, has been injected into that crust, and has +pierced through it, so as to escape at the surface with all the energy +of an active volcano. What we now recognize to be memorials of +these former injections and propulsions were all confounded with the +rocks of unquestionably aqueous origin. The last great teacher by +whom these antiquated doctrines were formulated into a system +<span class="sidenote">Werner.</span> +and promulgated to the world was Abraham Gottlob +Werner (1749-1815), the most illustrious German mineralogist +and geognost of the second half of the 18th century. While +still under twenty-six years of age, he was appointed teacher of +mining and mineralogy at the Mining Academy of Freiberg in Saxony—a +post which he continued to fill up to the end of his life. Possessed +of great enthusiasm for his subject, clear, methodical and eloquent +in his exposition of it, he soon drew around him men from all parts +of the world, who repaired to study under the great oracle of what +he called geognosy (Gr. <span class="grk" title="gê">γῆ</span>, the earth, <span class="grk" title="gnôsis">γνῶσις</span>, knowledge) or earth-knowledge. +Reviving doctrines that had been current long before +his time, he taught that the globe was once completely surrounded +with an ocean, from which the rocks of the earth’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.”</p> + +<p>But many years before the Saxon professor began to teach, clear +evidence had been produced from central France that basalt, one +of the rocks claimed by him as a chemical precipitate and +a universal formation, is a lava which has been poured +<span class="sidenote">Origin of basalt.</span> +out in a molten state at various widely separated periods +of time and at many different places. So far back as 1752 J.E. +Guettard (1715-1786) had shown that the basaltic rocks of Auvergne +are true lavas, which have flowed out in streams from groups of +once active cones. Eleven years later the observation was confirmed +and greatly extended by Nicholas Desmarest (1725-1815), who, +during a long course of years, worked out and mapped the complicated +volcanic records of that interesting region, and demonstrated +to all who were willing impartially to examine the evidence the true +volcanic nature of basalt. These views found acceptance from some +observers, but they were vehemently opposed by the followers of +Werner, who, by the force of his genius, made his theoretical conceptions +predominate all over Europe. The controversy as to the +origin of basalt was waged with great vigour during the later decades +of the 18th century. Desmarest took no part in it. He had accumulated +such conclusive proof of the correctness of his deductions, +and had so fully expounded the clearness of the evidence in their +favour furnished by the region of Auvergne, that, when any one +came to consult him on the subject, he contented himself with giving +the advice to “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.</p> + +<p>After many years of futile controversy the first serious weakening +of the position of the dominant Neptunist school arose from the +defection of some of the most prominent of Werner’s pupils. In +particular Jean François 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.</p> + +<p>Although Desmarest had traced in Auvergne a long succession +of volcanic eruptions, of which the oldest went back to a remote +period of time, and although he had shown that this succession, +coupled with the records of contemporaneous denudation, might +be used in defining epochs of geological history, it was not until +many years after his day that volcanic action came to be recognized +as a normal part of the mechanism of our globe, which had been in +operation from the remotest past, and which had left numerous +records among the rocks of the terrestrial crust. During the progress +of the controversy between the two great opposing factions in the +later portion of the 18th and the first three decades of the 19th +century, those who espoused the Vulcanist cause were intent on +proving that certain rocks, which are intercalated among the +stratified formations and which were claimed by the Neptunists as +obviously formed by water, are nevertheless of truly igneous origin. +These observers fixed their eyes on the evidence that the material of +such rocks, instead of having been deposited from aqueous solution, +had once been actually molten, and had in that condition been thrust +between the strata, had enveloped portions of them, and had indurated +or otherwise altered them. They spoke of these masses +as “unerupted lavas”; and undoubtedly in innumerable instances +they were right. But their zeal to establish an intrusive origin led +them to overlook the proofs that some intercalated sheets of igneous +material had not been injected into the strata, but had been poured +out at the surface as truly volcanic discharges, and therefore belonged +to the ancient periods represented by the strata between which they +are interposed. It may readily be supposed that any proofs of the +contemporaneous intercalation of such sheets would be eagerly +seized upon by the Neptunists in favour of their aqueous theory. +The influence of the ancient belief that “burning mountains” +could only rise from the combustion of subterranean inflammable +materials extended even into the ranks of the Vulcanists, so far at +least as to lead to a general acquiescence in the assumption that +volcanoes appeared to belong to a late phase in the history of the +planet. It was not until after considerable progress had been made +in determining the palaeontological distinctions and order of succession +of the stratified formations of the earth’s crust that it became +possible to trace among these formations a succession of volcanic +episodes which were contemporaneous with them. In no part of +the world has an ampler record of such episodes been preserved than +in the British Isles. It was natural, therefore, that the subject +should there receive most attention. As far back as 1820 Ami Boué +(1794-1881) showed that the Old Red Sandstone of Scotland includes +a great series of volcanic rocks, and that other rocks of volcanic +origin are associated with the Carboniferous formations. H.T. +<span class="pagenum"><a name="page645" id="page645"></a>645</span> +de la Beche (1796-1855) afterwards traced proofs of contemporaneous +eruptions among the Devonian rocks of the south-west of England. +Adam Sedgwick (1785-1873) showed, first in the Lake District, +and afterwards in North Wales, the presence of abundant volcanic +sheets among the oldest divisions of the Palaeozoic series; while +Roderick Impey Murchison (1792-1871) made similar discoveries +among the Lower Silurian rocks. From the time of these pioneers +the volcanic history of the country has been worked out by many +observers until it is now known with a fulness as yet unattained +in any other region.</p> + +<p><i>Growth of Opinion regarding Earthquakes.</i>—We have seen how +crude were the conceptions of the ancients regarding the causes of +volcanic action, and that they connected volcanoes and earthquakes +as results of the commotion of wind imprisoned within subterranean +caverns and passages. One of the earliest treatises, in which the +phenomena of terrestrial movements were discussed in the spirit +of modern science, was the posthumous collection of papers by +Robert Hooke (1635-1703), entitled <i>Lectures and Discourses of +Earthquakes and Subterranean Eruptions</i>, where the probable agency +of earthquakes in upheaving and depressing land is fully considered, +but without any definite pronouncement as to the author’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 <i>Transactions of the Royal Irish Academy</i>. +From that time onward he continued to devote his energies to the +investigation, studying the effects of the Calabrian earthquake of +1857, experimenting on the transmission of waves of shock through +various materials, caused by exploding charges of gunpowder, and +collecting all the information to be obtained on the subject. His +writings, and especially his work in two volumes on <i>The First +Principles of Observational Seismology</i>, must be regarded as having +laid the foundations of this branch of modern geology (see <span class="sc"><a href="#artlinks">Earthquake</a></span>; +<span class="sc"><a href="#artlinks">Seismometer</a></span>).</p> + +<p><i>History of the Evolution of Stratigraphical Geology.</i>—Men had long +been familiar with the evidence that the present dry land once lay +under the sea, before they began to realize that the rocks, of which +the land consists, contain a record of many alternations of land and +sea, and relics of a long succession of plants and animals from early +and simple types up to the manifold and complex forms of to-day. +In countries where coal-mining had been prosecuted for generations, +it had been recognized that the rocks consist of strata superposed +on each other in a definite order, which was found to extend over +the whole of a district. As far back as 1719 John Strachey drew +attention to this fact in a communication published in the <i>Philosophical +Transactions</i>. John Michell (1760), in the paper on earthquakes +already cited, showed that he had acquired a clear understanding +of the order of succession among stratified formations, and +perceived that to disturbances of the terrestrial crust must be ascribed +the fact that the lower or older and more inclined strata form the +mountains, while the younger and more horizontal strata are spread +over the plains.</p> + +<p>In Italy G. Arduíno (1713-1795) classified the rocks in the north +of the peninsula as Primitive, Secondary, Tertiary and Volcanic. +A similar threefold order was announced for the Harz and Erzgebirge +by J.G. Lehmann in 1756. He recognized in that region an ancient +series of rocks in inclined or vertical strata, which rise to the tops +of the hills and descend to an unknown depth into the interior. +These masses, he thought, were contemporaneous with the making +of the world. Next came the Flötzgebirge, consisting of younger +sediments, disposed in flat or gently inclined sheets which overlie +the first and more disturbed series, and are full of petrified remains +of plants and animals. Lastly he included the mountains which +have from time to time been formed by local accidents. Still more +advanced were the conceptions of G.C. Füchsel, who in the year +1762 published in Latin <i>A History of the Earth and the Sea, based on +a History of the Mountains of Thuringia</i>; and in 1773, in German, +a <i>Sketch of the most Ancient History of the Earth and Man</i>. In these +works he described the stratigraphical relations and general characters +of the various geological formations in his little principality; +and taking them as indicative of a general order of succession, he +traced what he believed to have been a series of revolutions through +which the earth has passed. In interpreting this geological history, +he laid great stress on the evidence of the fossils contained in the +rocks. He recognized that the various formations differ from each +other in their enclosed organic remains, and that from these differences +the existence of former sea-bottoms and land surfaces can +be determined.</p> + +<p>The labours of these pioneers paved the way for the advent +of Werner. Though the system evolved by this teacher claimed to +discard theory and to be established on a basis of observed facts, +it rested on a succession of hypotheses, for which no better foundation +could be shown than the belief of their author in their validity. +Starting from the extremely limited stratigraphical range displayed +in the geological structure of Saxony, he took it as a type for the rest +of the globe, persuading himself and impressing upon his followers +that the rocks of that small kingdom were to be taken as examples +of his “universal formations.” The oldest portion of the series, +classed by him as “Primitive,” consisted of rocks which he maintained +had been deposited from chemical solution. Yet they +included granite, gneiss, basalt, porphyry and serpentine, which, +even in his own day, were by many observers correctly regarded +as of igneous origin. A later group of rocks, to which he gave the +name of “Transition,” comprised, in his belief, partly chemical, +partly mechanical sediments, and contained the earliest fossil +organic remains. A third group, for which he reserved Lehmann’s +name “Flötz,” was made up chiefly of mechanical detritus, while +youngest of all came the “Alluvial” series of loams, clays, sands, +gravels and peat. It was by the gradual subsidence of the ocean +that, as he believed, the general mass of the dry land emerged, the +first-formed rocks being left standing up, sometimes on end, to form +the mountains, while those of later date, less steeply inclined, +occupied successively lower levels down to the flat alluvial accumulations +of the plains. Neither Werner, nor any of his followers, +ventured to account for what became of the water as the sea-level +subsided, though, in despite of their antipathy to anything like +speculation, they could not help suggesting, as an answer to the +cogent arguments of their opponents, that “one of the celestial +bodies which sometimes approach near to the earth may have been +able to withdraw a portion of our atmosphere and of our ocean.” +Nor was any attempt made to explain the extraordinary nature of +the supposed chemical precipitates of the universal ocean. The +progress of inquiry even in Werner’s lifetime disproved some of +the fundamental portions of his system. Many of the chemical +precipitates were shown to be masses that had been erupted in a +molten state from below. His order of succession was found not +to hold good; and though he tried to readjust his sequence and to +introduce into it modifications to suit new facts, its inherent artificiality +led to its speedy decline after his death. It must be conceded, +however, that the stress which he laid upon the fact that the +rocks of the earth’s crust were deposited in a definite order had an +important influence in directing attention to this subject, and in +preparing the way for a more natural system, based not on mere +mineralogical characters, but having regard to the organic remains, +which were now being gathered in ever-increasing numbers and +variety from stratified formations of many different ages and from +all parts of the globe.</p> + +<p>It was in France and in England that the foundations of stratigraphy, +based upon a knowledge of organic remains, were first +successfully laid. Abbé J.L. Giraud-Soulavie (1752-1813), in his +<i>Histoire naturelle de la France méridionale</i>, which appeared in seven +volumes, subdivided the limestones of Vivarais into five ages, each +marked by a distinct assemblage of shells. In the lowest strata, +representing the first age, none of the fossils were believed by him +to have any living representatives, and he called these rocks “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.</p> + +<p>The rocks of the Paris basin display so clearly an orderly +arrangement, and are so distinguished for the variety and perfect +<span class="pagenum"><a name="page646" id="page646"></a>646</span> +preservation of their enclosed organic remains, that they could not +fail to attract the early notice of observers. J. É. Guettard, G.F. +Rouelle (1703-1770), N. Desmarest, A.L. Lavoisier (1743-1794) +and others made observations in this interesting district. But it +was reserved for Cuvier (1769-1832) and A. Brongniart (1770-1847) +to work out the detailed succession of the Tertiary formations, and +to show how each of these is characterized by its own peculiar +assemblage of organic remains. The later progress of investigation +has slightly corrected and greatly amplified the tabular arrangement +established by these authors in 1808, but the broad outlines of the +Tertiary stratigraphy of the Paris basin remain still as Cuvier and +Brongniart left them. The most important subsequent change +in the classification of the Tertiary formations was made by Sir +Charles Lyell, who, conceiving in 1828 the idea of a classification +of these rocks by reference to their relative proportions of living +and extinct species of shells, established, in collaboration with +G.P. Deshayes, the now universally accepted divisions Eocene, +Miocene and Pliocene.</p> + +<p>Long before Cuvier and Brongniart published an account of their +researches, another observer had been at work among the Secondary +formations of the west of England, and had independently discovered +that the component members of these formations were each +distinguished by a peculiar group of organic remains; and that this +distinction could be used to discriminate them over all the region +through which he had traced them. The remarkable man who +arrived at this far-reaching generalization was William Smith (1769-1839), +a land surveyor who, in the prosecution of his professional +business, found opportunities of traversing a great part of England, +and of putting his deductions to the test. As the result of these +journeys he accumulated materials enough to enable him to produce +a geological map of the country, on which the distribution and +succession of the rocks were for the first time delineated. Smith’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).</p> + +<p>There was one stratigraphical domain, however, into which William +Smith did not enter. He traced his sequence of rocks down into the +Coal Measures, but contented himself with only a vague reference +to what lay underneath that formation. Though some of these +underlying rocks had in various countries yielded abundant fossils, +they had generally suffered so much from terrestrial disturbances, +and their order of succession was consequently often so much +obscured throughout western Europe, that they remained but little +known for many years after the stratigraphy of the Secondary and +Tertiary series had been established. At last in 1831 Murchison +began to attack this <i>terra incognita</i> on the borders of South Wales, +working into it from the Old Red Sandstone, the stratigraphical +position of which was well known. In a few years he succeeded in +demonstrating the existence of a succession of formations, each +distinguished by its own peculiar assemblage of organic remains +which were distinct from those in any of the overlying strata. To +these formations he gave the name of Silurian (<i>q.v.</i>). From the +key which his researches supplied, it was possible to recognize in +other countries the same order of formations and the same sequence +of fossils, so that, in the course of a few years, representatives of the +Silurian system were found far and wide over the globe. While +Murchison was thus engaged, Sedgwick devoted himself to the more +difficult task of unravelling the complicated structure of North +Wales. He eventually made out the order of the several formations +there, with their vast intercalations of volcanic material. He named +them the Cambrian system (<i>q.v.</i>), and found them to contain fossils, +which, however, lay for some time unexamined by him. He at +first believed, as Murchison also did, that his rocks were all older +than any part of the Silurian series. It was eventually discovered +that a portion of them was equivalent to the lower part of that +series. The oldest of Sedgwick’s groups, containing distinctive +fossils, retain the name Cambrian, and are of high interest, as they +enclose the remains of the earliest faunas which are yet well known. +Sedgwick and Murchison rendered yet another signal service to +stratigraphical geology by establishing, in 1839, on a basis of +palaeontological evidence supplied by W. Lonsdale, the independence +of the Devonian system (<i>q.v.</i>).</p> + +<p>For many years the rocks below the oldest fossiliferous deposits +received comparatively little attention. They were vaguely described +as the “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 <span class="sc"><a href="#artlinks">Archean System</a></span>).</p> + +<p>During the latter half of the 19th century the most important +development of stratigraphical geology was the detailed working +out and application of the principle of zonal classification to the +fossiliferous formations—that is, the determination of the sequence +and distribution of organic remains in these formations, and the +arrangement of the strata into zones, each of which is distinguished +by a peculiar assemblage of fossil species (see under Part VI.). The +zones are usually named after one especially characteristic species. +This system of classification was begun in Germany with reference +to the members of the Jurassic system (<i>q.v.</i>) by A. Oppel (1856-1858) +and F.A. von Quenstedt (1858), and it has since been extended +through the other Mesozoic formations. It has even been found to +be applicable to the Palaeozoic rocks, which are now subdivided +into palaeontological zones. In the Silurian system, for example, the +graptolites have been shown by C. Lapworth to furnish a useful +basis for zonal subdivisions. The lowest fossiliferous horizon in the +Cambrian rocks of Europe and North America is known as the +<i>Olenellus</i> zone, from the prominence in it of that genus of trilobite.</p> + +<p>Another conspicuous feature in the progress of stratigraphy +during the second half of the 19th century was displayed by the rise +and rapid development of what is known as Glacial geology. The +various deposits of “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 Neuchâtel +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.</p> + +<p><i>Rise and Progress of Palaeontological Geology.</i>—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 +<span class="pagenum"><a name="page647" id="page647"></a>647</span> +noted for the number and merit of its works of this kind, such as that +of K.N. Lang (<i>Historia lapidum figuratorum Helvetiae</i>, 1708) and +those of Johann Jacob Scheuchzer (1672-1733). In England, also, +illustrated treatises were published both by men who looked on +fossils as mere freaks of nature, and by those who regarded them as +proofs of Noah’s flood. Of the former type were the works of Martin +Lister (1638-1712) and Robert Plot (<i>Natural History of Oxfordshire</i>, +1677). The Celtic scholar Edward Llwyd (1660-1709) wrote a Latin +treatise containing good plates of a thousand fossils in the Ashmolean +Museum, Oxford, and J. Woodward, in 1728-1729, published his +<i>Natural History of the Fossils of England</i>, already mentioned, wherein +he described his own extensive collection, which he bequeathed to +the University of Cambridge, where it is still carefully preserved. +The most voluminous and important of all these works, however, +appeared at a later date at Nuremberg. It was begun by G.W. +Knorr (1705-1761), who himself engraved for it a series of plates, +which for beauty and accuracy have seldom been surpassed. After +his death the work was continued by J.E.I. Walch (1725-1778), and +ultimately consisted of four massive folio volumes and nearly 300 +plates under the title of <i>Lapides diluvii universalis testes</i>. Although +the authors supposed their fossils to be relics of Noah’s flood, their +work must be acknowledged to mark a distinct onward stage in the +palaeontological department of geology.</p> + +<p>It was in France that palaeontological geology began to be cultivated +in a scientific spirit. The potter Bernard Palissy, as far back +as 1580, had dwelt on the importance of fossil shells as monuments +of revolutions of the earth’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 <i>lusus naturae</i>, 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 <i>Origin of Species</i> by Charles +Darwin were the barriers of old prejudice in this matter finally +broken down. The possibility of tracing the ancestry of living forms +back into the remotest ages was then perceived; the time-honoured +fiction that the stratified formations record a series of catastrophes +and re-creations was finally dissipated; and the earth’s crust was +seen to contain a noble, though imperfect, record of the grand +evolution of organic types of which our planet has been the theatre.</p> + +<p><i>Development of Petrographical Geology.</i>—Theophrastus, the favourite +pupil of Aristotle, wrote a treatise <i>On Stones</i>, which has come +down to our own day, and may be regarded as the earliest work on +petrography. At a subsequent period Pliny, in his <i>Natural History</i>, +collected all that was known in his day regarding the occurrence +and uses of minerals and rocks. But neither of these works is +of great scientific importance, though containing much interesting +information. Minerals from their beauty and value attracted +notice before much attention was paid to rocks, and their study +gave rise to the science of mineralogy long before geology came +into existence. When rocks began to be more particularly scrutinized, +it was chiefly from the side of their usefulness for building +and other economic purposes. The occurrence of marine shells in +many of them had early attracted attention to them. But their +varieties of composition and origin did not become the subject of +serious study until after Linnaeus and J.G. Wallerius in the 18th +century had made a beginning. The first important contribution +to this department of the science was that of Werner, who in 1786 +published a classification and description of rocks in which he +arranged them in two divisions, simple and compound, and further +distinguished them by various external characters and by their +relative age. The publication of this scheme may be said to mark +the beginning of scientific petrography. Werner’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 <i>Quarterly Journal +of the Geological Society</i> a paper “On the Microscopical Structure of +Crystals.” This essay led to a complete revolution of petrographical +methods and gave a vast impetus to the study of rocks. Petrology +entered upon a new and wider field of investigation. Not only were +the mineralogical constituents of the rocks detected, but minute +structures were revealed which shed new light on the origin and +history of these mineral masses, and opened up new paths in +theoretical geology. In the hands of H. Vogelsang, F. Zirkel, +H. Rosenbusch, and a host of other workers in all civilized countries, +the literature of this department of the science has grown to a +remarkable extent. Armed with the powerful aid of modern optical +instruments, geologists are now able with far more prospect of success +to resume the experiments begun a century before by de Saussure +and Hall. G.A. Daubrée, C. Friedel, E. Sarasin, F. Fouqué and +A. Michel Lévy in France, C. Doelter y Cisterich and E. Hussak of +Gratz, J. Morozewicz of Warsaw and others, have greatly advanced +our knowledge by their synthetical analyses, and there is every +reason to hope that further advances will be made in this field of +research.</p> + +<p><i>Rise of Physiographical Geology.</i>—Until stratigraphical geology +had advanced so far as to show of what a vast succession of rocks the +crust of the earth is built up, by what a long and complicated series +of revolutions these rocks have come to assume their present positions, +and how enormous has been the lapse of time which all these changes +represent, it was not possible to make a scientific study of the surface +features of our globe. From ancient times it had been known that +many parts of the land had once been under the sea; but down even +to the beginning of the 19th century the vaguest conceptions continued +to prevail as to the operations concerned in the submergence +and elevation of land, and as to the processes whereby the present +outlines of terrestrial topography were determined. We have seen, +for instance, that according to the teaching of Werner the oldest +rocks were first precipitated from solution in the universal ocean to +form the mountains, that the vertical position of their strata was +original, that as the waters subsided successive formations were +deposited and laid bare, and that finally the superfluous portion of the +ocean was whisked away into space by some unexplained co-operation +of another planetary body. Desmarest, in his investigation of the +volcanic history of Auvergne, was the first observer to perceive by +what a long process of sculpture the present configuration of the land +has been brought about. He showed conclusively that the valleys have +been carved out by the streams that flow in them, and that while +they have sunk deeper and deeper into the framework of the land, +the spaces of ground between them have been left as intervening +ridges and hills. De Saussure learnt a similar lesson from his studies +of the Alps, and Hutton and Playfair made it a cardinal feature in +their theory of the earth. Nevertheless the idea encountered so +much opposition that it made but little way until after the middle +of the 19th century. Geologists preferred to believe in convulsions +of nature, whereby valleys were opened and mountains were +<span class="pagenum"><a name="page648" id="page648"></a>648</span> +upheaved. That the main features of the land, such as the great +mountain-chains, had been produced by gigantic plication of the +terrestrial crust was now generally admitted, and also that minor +fractures and folds had probably initiated many of the valleys. +But those who realized most vividly the momentous results achieved +by ages of subaerial denudation perceived that, as Hutton showed, +even without the aid of underground agency, the mere flow of water +in streams across a mass of land must in course of time carve out +just such a system of valleys as may anywhere be seen. It was +J.B. Jukes who, in 1862, first revived the Huttonian doctrine, +and showed how completely it explained the drainage-lines in the +south of Ireland. Other writers followed in quick succession until, +in a few years, the doctrine came to be widely recognized as one of +the established principles of modern geology. Much help was derived +from the admirable illustrations of land-sculpture and river-erosion +supplied from the Western Territories and States of the American +Union.</p> + +<p>Another branch of physiographical geology which could only come +into existence after most of the other departments of the science +had made large progress, deals with the evolution of the framework +of each country and of the several continents and oceans of the globe. +It is now possible, with more or less confidence, to trace backward +the history of every terrestrial area, to see how sea and land have +there succeeded each other, how rivers and lakes have come and +gone, how the crust of the earth has been ridged up at widely +separated intervals, each movement determining some line of +mountains or plains, how the boundaries of the oceans have shifted +again and again in the past, and thus how, after so prolonged a series +of revolutions, the present topography of each country, and of the +globe as a whole, has been produced. In the prosecution of this +subject maps have been constructed to show what is conjectured +to have been the distribution of sea and land during the various +geological periods in different parts of the world, and thus to indicate +the successive stages through which the architecture of the land has +been gradually evolved. The most noteworthy contribution to this +department of the science is the <i>Antlitz der Erde</i> of Professor Suess +of Vienna. This important and suggestive work has been translated +into French and English.</p> +</div> + +<p class="pt2 center sc">Part II.—Cosmical Aspects</p> + +<p>Before geology had attained to the position of an inductive +science, it was customary to begin investigations into the +history of the earth by propounding or adopting some more +or less fanciful hypothesis in explanation of the origin of our +planet, or even of the universe. Such preliminary notions were +looked upon as essential to a right understanding of the manner +in which the materials of the globe had been put together. One +of the distinguishing features of Hutton’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.</p> + +<p>But there can now be little doubt that in the reaction against +those visionary and often grotesque speculations, geologists +were carried too far in an opposite direction. In allowing +themselves to believe that geology had nothing to do with +questions of cosmogony, they gradually grew up in the conviction +that such questions could never be other than mere speculation, +interesting or amusing as a theme for the employment of the +fancy, but hardly coming within the domain of sober and +inductive science. Nor would they soon have been awakened +out of this belief by anything in their own science. It is still +true that in the data with which they are accustomed to deal, +as comprising the sum of geological evidence, there can be +found no trace of a beginning, though the evidence furnished +by the terrestrial crust shows a general evolution of organic +forms from some starting-point which cannot be seen. The +oldest rocks which have been discovered on any part of the +globe have probably been derived from other rocks older than +themselves. Geology by itself has not yet revealed, and is little +likely ever to reveal, a trace of the first solid crust of our globe. +If, then, geological history is to be compiled from direct evidence +furnished by the rocks of the earth, it cannot begin at the +beginning of things, but must be content to date its first chapter +from the earliest period of which any record has been preserved +among the rocks.</p> + +<p>Nevertheless, though geology in its usual restricted sense has +been, and must ever be, unable to reveal the earliest history of +our planet, it no longer ignores, as mere speculation, what is +attempted in this subject by its sister sciences. Astronomy, +physics and chemistry have in late years all contributed to cast +light on the earlier stages of the earth’s existence, previous to +the beginning of what is commonly regarded as geological history. +But whatever extends our knowledge of the former conditions +of our globe may be legitimately claimed as part of the domain of +geology. If this branch of inquiry, therefore, is to continue +worthy of its name as the science of the earth, it must take +cognizance of these recent contributions from other sciences. +It must no longer be content to begin its annals with the records +of the oldest rocks, but must endeavour to grope its way through +the ages which preceded the formation of any rocks. Thanks +to the results achieved with the telescope, the spectroscope and +the chemical laboratory, the story of these earliest ages of our +earth is every year becoming more definite and intelligible.</p> + +<p>Up to the present time no definite light has been thrown by +physics on the origin and earliest condition of our globe. The +famous nebular theory (<i>q.v.</i>) of Kant and Laplace sketched the +supposed evolution of the solar system from a gaseous nebula, +slowly rotating round a more condensed central portion of its +mass, which eventually became the sun. As a consequence of +increased rapidity of rotation resulting from cooling and contraction, +the nebula acquired a more and more lenticular form, +until at last it threw off from its equatorial protuberance a ring +of matter. Subsequently the same process was repeated, and +other similar rings successively separated from the parent mass. +Each ring went through a corresponding series of changes until +it ultimately became a planet, with or without one or more +attendant satellites. The intimate relationship of our earth +to the sun and the other planets was, in this way, shown. But +there are some serious physical difficulties in the way of the +acceptance of the nebular hypothesis. Another explanation +is given by the meteoritic hypothesis, according to which, out +of the swarms of meteorites with which the regions of space are +crowded, the sun and planets have been formed by gradual +accretion.</p> + +<p>According to these theoretical views we should expect to find +a general uniformity of composition in the constituent matter +of the solar system. For many years the only available evidence +on this point was derived from the meteorites (<i>q.v.</i>) which so +constantly fall from outer space upon the surface of the earth. +These bodies were found to consist of elements, all of which had +been recognized as entering into the constitution of the earth. +But the discoveries of spectroscopic research have made known +a far more widely serviceable method of investigation, which +can be applied even to the luminous stars and nebulae that lie +far beyond the bounds of the solar system. By this method +information has been obtained regarding the constitution of the +sun, and many of our terrestrial metals, such as iron, nickel and +magnesium, have been ascertained to exist in the form of incandescent +vapour in the solar atmosphere. The present +condition of the sun probably represents one of the phases +through which stars and planets pass in their progress towards +becoming cool and dark bodies in space. If our globe was at +first, like its parent sun, an incandescent mass of probably +gaseous matter, occupying much more space than it now fills, +we can conceive that it has ever since been cooling and contracting +until it has reached its present form and dimensions, and that +it still retains a high internal temperature. Its oblately spheroidal +form is such as would be assumed by a rotating mass of matter +in the transition from a vaporous and self-luminous or liquid +condition to one of cool and dark solidity. But it has been +claimed that even a solid spherical globe might develop, under +the influence of protracted rotation, such a shape as the earth +at present possesses.</p> + +<p>The observed increase of temperature downwards in our +<span class="pagenum"><a name="page649" id="page649"></a>649</span> +planet has hitherto been generally accepted as a relic and proof +of an original high temperature and mobility of substance. +Recently, however, the validity of this proof has been challenged +on the ground that the ascertained amount of radium in the +rocks of the outer crust is more than sufficient to account for +the observed downward increase of temperature. Too little, +however, is known of the history and properties of what is +called radium to afford a satisfactory ground on which to +discard what has been, and still remains, the prevalent belief +on this subject.</p> + +<p>An important epoch in the geological history of the earth +was marked by the separation of the moon from its mass (see +<span class="sc"><a href="#artlinks">Tide</a></span>). Whether the severance arose from the rupture of a +surrounding ring or the gradual condensation of matter in such +a ring, or from the ejection of a single mass of matter from the +rapidly rotating planet, it has been shown that our satellite +was only a few thousand miles from the earth’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.</p> + +<p>The sun by virtue of its vast size has not yet passed out of +the condition of glowing gas, and still continues to radiate heat +beyond the farthest planet of the solar system. The earth, +however, being so small a body in comparison, would cool down +much more quickly. Underneath its hot atmosphere a crust +would conceivably begin to form over its molten surface, though +the interior might still possess a high temperature and, owing +to the feeble conducting power of rocks, would remain intensely +hot for a protracted series of ages.</p> + +<p>Full information regarding the form and size of the earth, +and its relations to the other planetary members of the solar +system, will be found in the articles <span class="sc"><a href="#artlinks">Planet</a></span> and <span class="sc"><a href="#artlinks">Solar System</a></span>. +For the purposes of geological inquiry the reader will bear in +mind that the equatorial diameter of our globe is estimated to +be about 7925 m., and the polar diameter about 7899 m.; the +difference between these two sums representing the amount of +flattening at the poles (about 26½ m.). The planet has been +compared in shape to an orange, but it resembles an orange +which has been somewhat squeezed, for its equatorial circumference +is not a regular circle but an ellipse, of which the major +axis lies in long. 8° 15′ 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.</p> + +<p>The rotation of the earth on its axis exerts an important +influence on the movements of the atmosphere, and thereby +affects the geological operations connected with these movements. +The influence of rotation is most marked in the great aerial +circulation between the poles and the equator. Currents of +air, which set out in a meridional direction from high latitudes +towards the equator, come from regions where the velocity due +to rotation is small to where it is greater, and they consequently +fall behind. Thus, in the northern hemisphere a north wind, +as it moves away from its northern source of origin, is gradually +deflected more and more towards the west and becomes a north-east +current; while in the opposite hemisphere a wind making +from high southern latitudes towards the equator becomes, +from the same cause, a south-east current. Where, on the +other hand, the air moves from the equatorial to the polar regions +its higher velocity of rotation carries it eastward, so that on the +south side of the equator it becomes a north-west current and +on the north side a south-west current. It is to this cause that +the easting and westing of the great atmospheric currents are +to be attributed, as is familiarly exemplified in the trade winds.</p> + +<p>The atmospheric circulation thus deflected influences the +circulation of the ocean. The winds which persistently blow +from the north-east on the north side of the equator, and from +the south-east on the south side, drive the superficial waters +onwards, and give rise to converging oceanic currents which +unite to form the great westerly equatorial current.</p> + +<p>A more direct effect of terrestrial rotation has been claimed +in the case of rivers which flow in a meridional direction. It has +been asserted that those, which in the northern hemisphere +flow from north to south, like the Volga, by continually passing +into regions where the velocity of rotation is increasingly greater, +are thrown more against their western than their eastern banks, +while those whose general course is in an opposite direction, like +the Irtisch and Yenesei, press more upon their eastern sides. +There cannot be any doubt that the tendency of the streams +must be in the directions indicated. But when the comparatively +slow current and constantly meandering course of most rivers +are taken into consideration, it may be doubted whether the +influence of rotation is of much practical account so far as +river-erosion is concerned.</p> + +<p>One of the cosmical relations of our planet which has been +more especially prominent in geological speculations relates to +the position of the earth’s axis of rotation. Abundant evidence +has now been obtained to prove that at a comparatively late +geological period a rich flora, resembling that of warm climates +at the present day, existed in high latitudes even within less than +9° of the north pole, where, with an extremely low temperature +and darkness lasting for half of the year, no such vegetation could +possibly now exist. It has accordingly been maintained by +many geologists that the axis of rotation must have shifted, +and that when the remarkable Arctic assemblage of fossil plants +lived the region of their growth must have lain in latitudes much +nearer to the equator of the time.</p> + +<p>The possibility of any serious displacement of the rotational +axis since a very early period in the earth’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.</p> + +<p>In recent years fresh information bearing on the minor devagations +of the pole has been obtained from a series of several +thousand careful observations made in Europe and North +America. It has thus been ascertained that the pole wanders +with a curiously irregular but somewhat spiral movement, +within an amplitude of between 40 and 50 ft., and completes +its erratic circuit in about 428 days. It was not supposed that +its movement had any geological interest, but Dr John Milne +has recently pointed out that the times of sharpest curvature +in the path of the pole coincide with the occurrence of large +earthquakes, and has suggested that, although it can hardly be +assumed that this coincidence shows any direct connexion +between earthquake frequency and changes in the position of +the earth’s axis, both effects may not improbably arise from +the same redistribution of surface material by ocean currents +and meteorological causes.</p> + +<p>If for any reason the earth’s centre of gravity were sensibly +displaced, momentous geological changes would necessarily +ensue. That the centre of gravity does not coincide with the +<span class="pagenum"><a name="page650" id="page650"></a>650</span> +centre of figure of the globe, but lies to the south of it, has long +been known. This greater aggregation of dense material in the +southern hemisphere probably dates from the early ages of the +earth’s consolidation, and it is difficult to believe that any +readjustment of the distribution of this material in the earth’s +interior is now possible. But certain rearrangements of the +hydrosphere on the surface of the globe may, from time to time, +cause a shifting of the centre of gravity, which will affect the +level of the ocean. The accumulation of enormous masses of +ice around the pole will give rise to such a displacement, and +will thus increase the body of oceanic water in the glaciated +hemisphere. Various calculations have been made of the effect +of the transference of the ice-cap from one pole to the other, a +revolution which may possibly have occurred more than once +in the past history of the globe. James Croll estimated that if +the mass of ice in the southern hemisphere be assumed to be +1000 ft. thick down to lat. 60°, its removal to the opposite +hemisphere would raise the level of the sea 80 ft. at the north pole, +while the Rev. Osmond Fisher made the rise as much as 409 ft. +The melting of the ice would still further raise the sea-level by +the addition of so large a volume of water to the ocean. To +what extent superficial changes of this kind have operated in +geological history remains an unsolved problem, but their +probable occurrence in the past has to be recognized as one of +the factors that must be considered in tracing the revolutions of +the earth’s surface.</p> + +<p><i>The Age of the Earth.</i>—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.</p> + +<p>Geologists were for many years in the habit of believing that +no limit could be assigned to the antiquity of the planet, and that +they were at liberty to make unlimited drafts on the ages of the +past. In 1862 and subsequent years, however, Lord Kelvin +(then Sir William Thomson) pointed out that these demands were +opposed to known physical facts, and that the amount of time +required for geological history was not only limited, but must +have been comprised within a comparatively narrow compass. +His argument rested on three kinds of evidence: (1) the internal +heat and rate of cooling of the earth; (2) the tidal retardation +of the earth’s rotation; and (3) the origin and age of the sun’s +heat.</p> + +<p>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.</p> + +<p>2. The argument from tidal retardation proceeds on the +admitted fact that, owing to the friction of the tide-wave, the +rotation of the earth is retarded, and is, therefore, much slower +now than it must have been at one time. Lord Kelvin affirmed +that had the globe become solid some 10,000 million years ago, +or indeed any high antiquity beyond 100 million years, the +centrifugal force due to the more rapid rotation must have given +the planet a very much greater polar flattening than it actually +possesses. He admitted, however, that, though 100 million +years ago that force must have been about 3% greater than now, +yet “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.”</p> + +<p>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.</p> + +<p>By degrees, however, there have arisen among the physicists +themselves grave doubts as to the validity of the physical +evidence on which the limitation of the earth’s age has been +founded, and at the same time greater appreciation has been +shown of the signification and <span class="correction" title="amended from stength">strength</span> of the geological proofs +of the high antiquity of our planet. In an address from the +chair of the Mathematical Section of the British Association in +1886, Professor (afterwards Sir) George Darwin reviewed the +controversy, and pronounced the following deliberate judgment +in regard to it: “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 +<span class="pagenum"><a name="page651" id="page651"></a>651</span> +certainty than that of the geologists, and the scale of geological +time remains in great measure unknown” (see also Tide, chap. +viii.).</p> + +<p>In an address to the mathematical section of the American +Association for the Advancement of Science in 1889, the vice-president +of the section, R.S. Woodward, thus expressed himself +with regard to the physical arguments brought forward by Lord +Kelvin and Professor Tait in limitation of geological time: +“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.”</p> + +<p>Still more emphatic is the protest made from the physical +side by Professor John Perry. He has attacked each of the +three lines of argument of Lord Kelvin, and has impugned the +validity of the conclusions drawn from them. The argument +from tidal retardation he dismisses as fallacious, following in +this contention the previous criticism of the Rev. Maxwell Close +and Sir George Darwin. In dealing with the argument based on +the secular cooling of the earth, he holds it to be perfectly +allowable to assume a much higher conductivity for the interior +of the globe, and that such a reasonable assumption would enable +us greatly to increase our estimate of the earth’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 +<i>probable</i> age of life on the earth is much less than any of the above +estimates; but if the palaeontologists have good reasons for +demanding much greater times, I see nothing from the physicists’ +point of view which denies them four times the greatest of these +estimates.”</p> + +<p>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 <span class="sc"><a href="#artlinks">Radioactivity</a></span>).</p> + +<p>The arguments from the geological side against the physical +contention that would limit the age of our globe to some 10 +or 20 millions of years are mainly based on the observed rates of +geological and biological changes at the present time upon land +and sea, and on the nature, physical history and organic contents +of the stratified crust of the earth. Unfortunately, actual +numerical data are not obtainable in many departments of +geological activity, and even where they can be procured they +do not yet rest on a sufficiently wide collection of accurate and +co-ordinated observations. But in some branches of dynamical +geology, material exists for, at least, a preliminary computation +of the rate of change. This is more especially the case in respect +of the wide domain of denudation. The observational records +of the action of the sea, of springs, rivers and glaciers are becoming +gradually fuller and more trustworthy. A method of making +use of these records for estimating the rate of denudation of +the land has been devised. Taking the Mississippi as a general +type of river action, it has been shown that the amount of +material conveyed by this stream into the sea in one year is +equivalent to the lowering of the general surface of the drainage +basin of the river by <span class="spp">1</span>⁄<span class="suu">6000</span> of a foot. This would amount to one +foot in 6000 years and 1000 ft. in 6 million years. So that at +the present rate of waste in the Mississippi basin a whole continent +might be worn away in a few millions of years.</p> + +<p>It is evident that as deposition and denudation are simultaneous +processes, the ascertainment of the rate at which solid +material is removed from the surface of the land supplies some +necessary information for estimating the rate at which new +sedimentary formations are being accumulated on the floor of +the sea, and for a computation of the length of time that would +be required at the present rate of change for the deposition of all +the stratified rocks that enter into the composition of the crust +of our globe. If the thickness of these rocks be assumed to be +100,000 ft., and if we could suppose them to have been laid down +over as wide an area as that of the drainage basins from the +waste of which they were derived, then at the present rate of +denudation their accumulation would require some 600 millions +of years. But, as Dr A.R. Wallace has justly pointed out, the +tract of sea-floor over which the material derived from the waste +of the terrestrial surface is laid down is at present much less than +that from which this material is worn away. We have no means, +however, of determining what may have been the ratio between +the two areas in past time. Certainly ancient marine sedimentary +rocks cover at the present day a much more extensive area than +that in which they are now being elaborated. If we take the +ratio postulated by Dr Wallace—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.</p> + +<p>But no one who has made himself familiar with the actual +composition of these formations and the detailed structure of the +terrestrial crust can fail to recognize how vague, imperfect and +misleading are the data on which such computations are founded. +It requires no prolonged acquaintance with the earth’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 +<span class="pagenum"><a name="page652" id="page652"></a>652</span> +follow each other in apparently uninterrupted sequence, and +might be thought to have been deposited continuously at the +same general rate, and without the intervention of any pause, it +can be demonstrated that sometimes an inch or two of sediment +<span class="correction" title="amended from much">might</span>, on certain horizons, represent the deposit of an enormously +longer period than a hundred or a thousand times the same +amount of sediment on other horizons. A prolonged study of +these questions leads to a profound conviction that in many +parts of the geological record the time represented by sedimentary +deposits may be vastly less than the time which is not +so represented.</p> + +<p>It has often been objected that the present rate of geological +change ought not to be taken as a measure of the rate in past +time, because the total sum of terrestrial energy has been steadily +diminishing, and geological processes must consequently have +been more vigorous in former ages than they are now. Geologists +do not pretend to assert that there has been no variation +or diminution in the activities of the various processes which +they have to study. What they do insist on is that the +present rate of change is the only one which we can watch and +measure, and which will thus supply a statistical basis for any +computations on the subject. But it has been dogmatically +affirmed that because terrestrial energy has been diminishing +therefore all kinds of geological work must have been more +vigorously and more rapidly carried on in former times than +now; that there were far more abundant and more stupendous +volcanoes, more frequent and more destructive earthquakes, +more gigantic upheavals and subsidences, more powerful oceanic +waves and tides, more violent atmospheric disturbances with +heavier rainfall and more active denudation.</p> + +<p>It is easy to make these assertions, and they look plausible; +but, after all, they rest on nothing stronger than assumption. +They can be tested by an appeal to the crust of the earth, in +which the geological history of our planet has been so fully recorded. +Had such portentous manifestations of geological +activity ever been the normal condition of things since the +beginning of that history, there ought to be a record of them in +the rocks. But no evidence for them has been found there, +though it has been diligently sought for in all quarters of the +globe. We may confidently assert that while geological changes +may quite possibly have taken place on a gigantic scale in the +earliest ages of the earth’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.</p> + +<p>That this conclusion is warranted even with regard to the innate +energy of the globe itself will be seen if we institute a comparison +between the more ancient and the more recent manifestations of +that energy. Take, for example, the proofs of gigantic plication, +fracture and displacement within the terrestrial crust. These, +as they have affected the most ancient rocks of Europe, have +been worked out in great detail in the north-west of Scotland. +But they are not essentially different from or on a greater scale +than those which have been proved to have affected the Alps, +and to have involved strata of so recent a date as the older +Tertiary formations. On the contrary, it may be doubted +whether any denuded core of an ancient mountain-chain reveals +traces of such stupendous disturbances of the crust as those +which have given rise to the younger mountain-chains of the +globe. It may, indeed, quite well have been the rule that instead +of diminishing in intensity of effect, the consequences of terrestrial +contraction have increased in magnitude, the augmenting +thickness of the crust offering greater resistance to the stresses, +and giving rise to vaster plications, faults, thrust-planes and +metamorphism, as this growing resistance had to be overcome.</p> + +<p>The assertion that volcanic action must have been more +violent and more persistent in ancient times than it is now has +assuredly no geological evidence in its support. It is quite true +that there are vastly more remains of former volcanoes scattered +over the surface of the globe than there are active craters now, +and that traces of copious eruptions of volcanic material can be +followed back into some of the oldest parts of the geological +record. But we have no proof that ever at any one time in +geological history there have been more or larger or more vigorous +volcanoes than those of recent periods. It may be said that the +absence of such proof ought not to invalidate the assertion until +a far wider area of the earth’s surface has been geologically +studied. But most assuredly, as far as geological investigation +has yet gone, there is an overwhelming body of evidence to show +that from the earliest epochs in geological history, as registered +in the stratified rocks, volcanic action has manifested itself very +much as it does now, but on a less rather than on a greater scale. +Nowhere can this subject be more exhaustively studied than in +the British Isles, where a remarkably complete series of volcanic +eruptions has been chronicled ranging from the earliest Palaeozoic +down to older Tertiary time. The result of a prolonged study +of British volcanic geology has demonstrated that, even to +minute points of detail, there has been a singular uniformity in +the phenomena from beginning to end. The oldest lavas and +ashes differ in no essential respect from the youngest. Nor have +they been erupted more copiously or more frequently. Many +successive volcanic periods have followed each other after prolonged +intervals of repose, each displaying the same general +sequence of phenomena and similar evidence of gradual diminution +and extinction. The youngest, instead of being the feeblest, +were the most extensive outbursts in the whole of this prolonged +series.</p> + +<p>If now we turn for evidence of the alleged greater activity +of all the epigene or superficial forces, and especially for proofs +of more rapid denudation and deposition on the earth’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 +<span class="pagenum"><a name="page653" id="page653"></a>653</span> +together, just as iron-sand may be seen gathered into thin sheets +on sandy beaches at the present day. Again, the same series +of primeval sediments includes intercalations of fine silt, which +has been deposited as regularly and intermittently there as it +has been among the most recent formations. These bands of +shale have been diligently searched for fossils, as yet without +success; but they may eventually disclose organic remains older +than any hitherto found in Europe.</p> + +<p>We now come to the consideration of the palaeontological +evidence as to the value of geological time. Here the conclusions +derived from a study of the structure of the sedimentary formations +are vastly strengthened and extended. In the first place, +the organization of the most ancient plants and animals furnishes +no indication that they had to contend with any greater violence +of storm, flood, wave or ocean-current than is familiar to their +modern descendants. The oldest trees, shrubs, ferns and +club-mosses display no special structures that suggest a difference +in the general conditions of their environment. The most +ancient crinoids, sponges, crustaceans, arachnids and molluscs +were as delicately constructed as those of to-day, and their +remains are often found in such perfect preservation as to show +that neither during their lifetime nor after their death were they +subject to any greater violence of the elements than their living +representatives now experience. Of much more cogency, +however, is the evidence supplied by the grand upward succession +of organic forms, from the most ancient stratified rocks up to +the present day. No biologist now doubts for a moment that +this marvellous succession is the result of a gradual process of +evolution from lower to higher types of organization. There +may be differences of opinion as to the causes which have governed +this process and the order of the steps through which it has +advanced, but no one who is conversant with the facts will now +venture to deny that it has taken place, and that, on any possible +explanation of its progress, it must have demanded an enormous +lapse of time. In the Cambrian or oldest fossiliferous formations +there is already a large and varied fauna, in which the leading +groups of invertebrate life are represented. On no tenable +hypothesis can these be regarded as the first organisms that +came into being on our planet. They must have had a long +ancestry, and as Darwin first maintained, the time required for +their evolution may have been “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.</p> + +<p>The impressive character of the evidence furnished by the +sequence of organic forms throughout the great series of fossiliferous +strata can hardly be fully realized without a detailed and +careful study of the subject. Professor E.B. Poulton, in an +address to the zoological section of the British Association at +the Liverpool Meeting in 1896, showed how overwhelming are +the demands which this evidence makes for long periods of time, +and how impossible it is of comprehension unless these demands +be conceded. The history of life upon the earth, though it will +probably always be surrounded with great and even insuperable +difficulties, becomes broadly comprehensible in its general +progress when sufficient time is granted for the evolution +which it records; but it remains unintelligible on any other +conditions.</p> + +<p>Taken then as a whole, the body of evidence, geological and +palaeontological, in favour of the high antiquity of our globe +is so great, so manifold, and based on such an ever-increasing +breadth of observation and reflection, that it may be confidently +appealed to in answer to the physical arguments which would +seek to limit that antiquity to ten or twenty millions of years. +In the present state of science it is out of our power to state +positively what must be the lowest limit of the age of the earth. +But we cannot assume it to be much less, and it may possibly +have been much more, than the 100 millions of years which Lord +Kelvin was at one time willing to concede.<a name="fa2c" id="fa2c" href="#ft2c"><span class="sp">2</span></a></p> + +<p class="pt2 center sc">Part III.—Geognosy. The Investigation of the Nature +and Composition of the Materials of which the +Earth Consists</p> + +<p>This division of the science is devoted to a description of the +parts of the earth—of the atmosphere and ocean that surround +the planet, and more especially of the solid materials that underlie +these envelopes and extend downwards to an unknown distance +into the interior. These various constituents of the globe are +here considered as forms of matter capable of being analysed, +and arranged according to their composition and the place they +take in the general composition of the globe.</p> + +<p>Viewed in the simplest way the earth may be regarded as +made up of three distinct parts, each of which ever since an +early period of planetary history has been the theatre of important +geological operations. (1) An envelope of air, termed +the <i>atmosphere</i>, which surrounds the whole globe; (2) A lower +and less extensive envelope of water, known as the <i>hydrosphere</i> +(Gr. <span class="grk" title="hydôr">ὕδωρ</span>, water) which, constituting the oceans and seas, +covers nearly three-fourths of the underlying solid surface of the +planet; (3) A globe, called the <i>lithosphere</i> (Gr. <span class="grk" title="lithos">λίθος</span>, stone), +the external part of which, consisting of solid stone, forms the +<i>crust</i>, while underneath, and forming the vast mass of the +interior, lies the <i>nucleus</i>, regarding the true constitution of +which we are still ignorant.</p> + +<p>1. <i>The Atmosphere.</i>—The general characters of the atmosphere +are described in separate articles (see especially <span class="sc"><a href="#artlinks">Atmosphere</a></span>; +<span class="sc"><a href="#artlinks">Meteorology</a></span>). Only its relations to geology have here to be +considered. As this gaseous envelope encircles the whole +globe it is the most universally present and active of all the +agents of geological change. Its efficacy in this respect arises +partly from its composition, and the chemical reactions which +it effects upon the surface of the land, partly from its great +variations in temperature and moisture, and partly from its +movements.</p> + +<div class="condensed"> +<p>Many speculations have been made regarding the chemical +composition of the atmosphere during former geological periods. +There can indeed be little doubt that it must originally have differed +greatly from its present condition. If the whole mass of the planet +originally existed in a gaseous state, there would be practically no +atmosphere. The present outer envelope of air may be considered +to be the surviving relic of this condition, after all the other constituents +have been incorporated into the hydrosphere and lithosphere. +The oxygen, which now forms fully a half of the outer +crust of the earth, was doubtless originally, whether free or in +combination, part of the atmosphere. So, too, the vast beds of coal +found all over the world, in geological formations of many different +ages, represent so much carbonic acid once present in the air. The +chlorides and other salts in the sea may likewise partly represent +materials carried down out of the atmosphere in the primitive +condensation of the aqueous vapour, though they have been continually +increased ever since by contributions from the drainage of +the land. It has often been suggested that, during the Carboniferous +period, the atmosphere must have been warmer and more charged +with aqueous vapour and carbon dioxide than at the present day, +to admit of so luxuriant a flora as that from which the coal-seams +were formed. There seems, however, to be at present no method +of arriving at any certainty on this subject. Lastly, the amount of +carbonic acid absorbed in the weathering of rocks at the surface, and +the consequent production of carbonates, represents an enormous +abstraction of this gas.</p> + +<p>As at present constituted, the atmosphere is regarded as a +<span class="pagenum"><a name="page654" id="page654"></a>654</span> +mechanical mixture of nearly four volumes of nitrogen and one of +oxygen, together with an average of 3.5 parts of carbon dioxide in +every 10,000 parts of air, and minute quantities of various other +gases and solid particles. Of the vapours contained in it by far the +most important is that of water which, although always present, +varies greatly in amount according to variations in temperature. +By condensation the water vapour appears in visible form as dew, +mist, cloud, rain, hail, snow and ice, and in these forms includes and +carries down some of the other vapours, gases and solid particles +present in the air. The circulation of water from the atmosphere to +the land, from the land to the sea, and again from the sea to the +land, forms the great geological process whereby the habitable +condition of the planet is maintained and the surface of the land +is sculptured (Part IV.).</p> +</div> + +<p>2. <i>The Hydrosphere.</i>—The water envelope covers nearly +three-fourths of the surface of the earth, and forms the various +oceans and seas which, though for convenience of reference +distinguished by separate names, are all linked together in one +great body. The physical characters of this vast envelope are +discussed in separate articles (see <span class="sc"><a href="#artlinks">Ocean</a></span> and <span class="sc"><a href="#artlinks">Oceanography</a></span>). +Viewed from the geological standpoint, the features of the +sea that specially deserve attention are first the composition of +its waters, and secondly its movements.</p> + +<div class="condensed"> +<p>Sea-water is distinguished from that of ordinary lakes and rivers +by its greater specific gravity and its saline taste. Its average +density is about 1.026, but it varies even within the same ocean, +being least where large quantities of fresh water are added from +rain or melting snow and ice, and greatest where evaporation is most +active. That sea-water is heavier than fresh arises from the greater +proportion of salts which it contains in solution. These salts constitute +about three and a half parts in every hundred of water. +They consist mainly of chlorides of sodium and magnesium, the +sulphates of magnesium, calcium and potassium, with minuter +quantities of magnesium bromide and calcium carbonate. Still +smaller proportions of other substances have been detected, gold for +example having been found in the proportion of 1 part in 15,180,000.</p> + +<p>That many of the salts have existed in the sea from the time of +its first condensation out of the primeval atmosphere appears to +be probable. It is manifest, however, that, whatever may have +been the original composition of the oceans, they have for a vast +section of geological time been constantly receiving mineral matter +in solution from the land. Every spring, brook and river removes +various salts from the rocks over which it moves, and these substances, +thus dissolved, eventually find their way into the sea. +Consequently sea-water ought to contain more or less traceable +proportions of every substance which the terrestrial waters can +remove from the land, in short, of probably every element present +in the outer shell of the globe, for there seems to be no constituent +of this earth which may not, under certain circumstances, be held +in solution in water. Moreover, unless there be some counteracting +process to remove these mineral ingredients, the ocean water ought +to be growing, insensibly perhaps, but still assuredly, <span class="correction" title="amended from salter">saltier</span>, for the +supply of saline matter from the land is incessant.</p> + +<p>To the geologist the presence of mineral solutions in sea-water is +a fact of much importance, for it explains the origin of a considerable +part of the stratified rocks of the earth’s crust. By evaporation +the water has given rise to deposits of rock-salt, gypsum and other +materials. The lime contained in solution, whether as sulphate or +carbonate, has been extracted by many tribes of marine animals, +which have thus built up out of their remains vast masses of solid +limestone, of which many mountain-chains largely consist.</p> + +<p>Another important geological feature of the sea is to be seen +in the fact that its basins form the great receptacles for the detritus +worn away from the land. Besides the limestones, the visible parts +of the terrestrial crust are, in large measure, composed of sedimentary +rocks which were originally laid down on the sea-bottom. Moreover, +by its various movements, the sea occupies a prominent place +among the epigene or superficial agents which produce geological +changes on the surface of the globe.</p> +</div> + +<p>3. <i>The Lithosphere.</i>—Beneath the gaseous and liquid envelopes +lies the solid part of the planet, which is conveniently regarded +as consisting of two parts,—(<i>a</i>) the crust, and (<i>b</i>) the interior +or nucleus.</p> + +<p>It was for a long time a prevalent belief that the interior of the +globe is a molten mass round which an outer shell has gradually +formed through cooling. Hence the term “crust” +was applied to this external solid envelope, which +<span class="sidenote">The crust.</span> +was variously computed to be 10, 20, or more miles in thickness. +The portion of this crust accessible to human observation was +seen to afford abundant evidence of vast plications and corrugations +of its substance, which were regarded as only explicable +on the supposition of a thin solid collapsible shell floating on a +denser liquid interior. When, however, physical arguments +were adduced to show the great rigidity of the earth as a whole, +the idea of a thin crust enclosing a molten nucleus was reluctantly +abandoned by geologists, who found the problem of the earth’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.</p> + +<p>Though we cannot hope ever to have direct acquaintance with +more than the mere outside skin of our planet, we may be led +to infer the irregular distribution of materials within +the crust from the present distribution of land and +<span class="sidenote">The interior.</span> +water, and the observed differences in the amount of +deflection of the plumb-line near the sea and near mountain-chains. +The fact that the southern hemisphere is almost wholly +covered with water appears explicable only on the assumption +of an excess of density in the mass of that portion of the planet. +The existence of such a vast sheet of water as that of the Pacific +Ocean is to be accounted for, as Archdeacon J.H. Pratt pointed +out, by the presence of “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.</p> + +<p>In our ignorance regarding the chemical constitution of the +nucleus of our planet, an argument has sometimes been based +upon the known fact that the specific gravity of the globe +as a whole is about double that of the crust. This has been +held by some writers to prove that the interior must consist of +much heavier material and is therefore probably metallic. But +the effect of pressure ought to make the density of the nucleus +much higher, even if the interior consisted of matter no heavier +than the crust. That the total density of the planet does not +greatly exceed its observed amount seems only explicable on +the supposition that some antagonistic force counteracts the +effects of pressure. The only force we can suppose capable of so +acting is heat. But comparatively little is yet known regarding +the compression of gases, liquids and solids under such vast +pressures as must exist within the nucleus.</p> + +<p>That the interior of the earth possesses a high temperature +is inferred from the evidence of various sources. (1) Volcanoes, +which are openings that constantly, or intermittently, give out +hot vapours and molten lava from reservoirs beneath the crust. +Besides active volcanoes, it is known that former eruptive vents +<span class="pagenum"><a name="page655" id="page655"></a>655</span> +have been abundantly and widely distributed over the globe +from the earliest geological periods down to our own day. +(2) Hot springs are found in many parts of the globe, with +temperatures varying up to the boiling point of water. (3) +From mines, tunnels and deep borings into the earth it has +been ascertained that in all quarters of the globe below the +superficial zone of invariable temperature, there is a progressive +increase of heat towards the interior. The rate of this increase +varies, being influenced, among other causes, by the varying +conductivity of the rocks. But the average appears to be +about 1° Fahr. for every 50 or 60 ft. of descent, as far down as +observations have extended. Though the increase may not +advance in the same proportion at great depths, the inference +has been confidently drawn that the temperature of the nucleus +must be exceedingly high.</p> + +<p>The probable condition of the earth’s interior has been a fruitful +source of speculation ever since geology came into existence; +but no general agreement has been arrived at on the subject. +Three chief hypotheses have been propounded: (1) that the +nucleus is a molten mass enclosed within a solid shell; (2) that, +save in local vesicular spaces which may be filled with molten +or gaseous material, the globe is solid and rigid to the centre; +(3) that the great body of the nucleus consists of incandescent +vapours and gases, especially vaporous iron, which under the +gigantic pressure within the earth are so compressed as to confer +practical rigidity on the globe as a whole, and that outside this +main part of the nucleus the gases pass into a shell of molten +magma, which, in turn, shades off outwards into the comparatively +thin, cool solidified crust. Recent seismological observations +have led to the inference that the outer crust, some 30 to +45 m. thick, must rapidly merge into a fairly homogeneous +nucleus which, whatever be its constitution, transmits undulatory +movements through its substance with uniform velocity and is +believed to possess a high rigidity.</p> + +<p>The origin of the earth’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.</p> + +<div class="condensed"> +<p><i>Constitution of the Earth’s Crust.</i>—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.</p> + +<p>The combinations of the elements which are of most importance +in the constitution of the terrestrial crust consist of oxides. From +the mean of a large number of analyses of the rocks of the lower or +primitive portion of the crust, it has been ascertained that silica +(SiO<span class="su">2</span>) forms almost 60% and alumina (Al<span class="su">2</span>O<span class="su">3</span>) upwards of 15% of +the whole. The other combinations in order of importance are +lime (CaO) 4.90%, magnesia (MgO) 4.36, soda (Na<span class="su">2</span>O) 3.55, ferrous +oxide (FeO) 3.52, potash (K<span class="su">2</span>O) 2.80, ferric oxide (Fe<span class="su">2</span>O<span class="su">3</span>) 2.63, water +(H<span class="su">2</span>O) 1.52, titanium oxide (TiO<span class="su">2</span>) 0.60, phosphoric acid (P<span class="su">2</span>O<span class="su">5</span>) +0.22; the other combinations of elements thus form less than 1% +of the crust.</p> + +<p>These different combinations of the elements enter into further +combinations with each other so as to produce the wide assortment +of simple minerals (see <span class="sc"><a href="#artlinks">Mineralogy</a></span>). Thus, silica and alumina are +combined to form the aluminous silicates, which enter so largely +into the composition of the crust of the earth. The silicates of +magnesia, potash and soda constitute other important families of +minerals. A mass of material composed of one, but more usually +of more than one mineral, is known as a <i>rock</i>. Under this term +geologists are accustomed to class not only solid stone, such as +granite and limestone, but also less coherent materials such as clay, +peat and even loose sand. The accessible portion of the earth’s +crust consists of various kinds of rocks, which differ from each other +in structure, composition and origin, and are therefore susceptible +of diverse classifications according to the point of view from which +they are considered. The details of this subject will be found in +the article <span class="sc"><a href="#artlinks">Petrology</a></span>.</p> + +<p><i>Classification of Rocks.</i>—Various systems of classification of rocks +have been proposed, but none of them is wholly satisfactory. The +most useful arrangement for most purposes of the geologist is one +based on the broad differences between them in regard to their mode +of origin. From this point of view they may be ranged in three +divisions:</p> + +<p>1. In the first place, a large number of rocks may be described +as original or underived, for it is not possible to trace them back to +any earlier source. They belong to the primitive constitution of the +planet, and, as they have all come up from below through the crust, +they serve to show the nature of the material which lies immediately +below the outer parts of that crust. They include the numerous +varieties of lava, which have been poured out in a molten state from +volcanic vents, also a great series of other rocks which, though they +may never have been erupted to the surface, have been forced +upward in a melted condition into the other rocks of the crust and +have solidified there. From their mode of origin this great class of +rocks has been called “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.</p> + +<p>Though the igneous rocks occupy extensive areas in some countries, +they nevertheless cover a much smaller part of the whole surface of +the land than is taken up by the second division or stratified rocks. +But they increase in quantity downwards and probably extend +continuously round the globe below the other rocks. This important +series brings before us the relations of the molten magma within the +earth to the overlying crust and to the outer surface. On the one +hand, it includes the oldest and most deep-seated extravasations +of that magma, which have been brought to light by ruptures and +upheavals of the crust and prolonged denudation. On the other, +it presents to our study the varied outpourings of molten and +fragmentary materials in the discharges of modern and ancient +<span class="pagenum"><a name="page656" id="page656"></a>656</span> +volcanoes. Between these two extremes of position and age, we +find that the crust has been, as it were, riddled with injections of +the magma from below. These features will be further noticed in +Part V. of this article.</p> + +<p>2. The “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: +(<i>a</i>) By far the largest proportion of them consists of different kinds +of sediment derived from the disintegration of pre-existing rocks. +In this “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. +(<i>b</i>) Another group consists of materials that have been deposited by +chemical precipitation from solution in water. The white sinter +laid down by calcareous springs is a familiar example on a small +scale. Beds of rock-salt, gypsum and dolomite have, in some +regions, been accumulated to a thickness of many thousand feet, +by successive precipitations of the salt contained in the water of +inland seas. (<i>c</i>) An abundant and highly important series of sedimentary +formations has been formed from the remains of plants and +animals. Such accumulations may arise either from the transport +and deposit of these remains, as in the case of sheets of drift-wood, +and banks of drifted sea-shells, or from the growth and decay of +the organisms on the spot, as happens in peat bogs and in coral-reefs.</p> + +<p>As the sedimentary rocks have for the most part been laid down +under water, and more especially on the sea-floor, they are often +spoken of as “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.</p> + +<p>It is from the sedimentary rocks that the main portion of geological +history is derived. They have been deposited one over another +in successive strata from a remote period in the development of +the globe down to the present time. From this arrangement they +have been termed “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.</p> + +<p>3. In the third place comes a series of rocks which are not now +in their original condition, but have undergone such alteration as +to have acquired new characters that more or less conceal their +first structures. Some of them can be readily recognized as altered +igneous masses; others are as manifestly of sedimentary origin; +while of many it is difficult to decide what may have been their +pristine character. To this series the term “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 <span class="sc"><a href="#artlinks">Metamorphism</a></span>).</p> + +<p>Among the less altered metamorphic rocks of sedimentary origin, +the successive laminae of deposit of the original sediment can be +easily observed; but they are also traversed by a new set of divisional +planes, along which they split across the original bedding. +Together with this superinduced cleavage there have been developed +in them minute hairs, scales and rudimentary crystals. Further +stages of alteration are marked by the increase of micaceous scales, +garnets and other minerals, especially along the planes of cleavage, +until the whole rock becomes crystalline, and displays its chief +component minerals in successive discontinuous folia which merge +into each other, and are often crumpled and puckered. Massive +igneous rocks can be observed to have undergone intense crushing +and cleavage, and to have ultimately assumed a crystalline foliated +character. Rocks which present this aspect are known as schists +(<i>q.v.</i>). They range from the finest silky slates, or phyllites, up to the +coarsest gneisses, which in hand-specimens can hardly be distinguished +from granites. There is indeed every reason to believe +that such gneisses were probably originally true granites, and that +their foliation and recrystallization have been the result of metamorphism.</p> + +<p>The schists are more especially to be found in the heart of +mountain-chains, and in regions where the lowest and oldest parts +of the earth’s crust have, in the course of geological revolutions, +been exposed to the light of day. They have been claimed by some +writers to be part of the original or primitive surface of our globe +that first consolidated on the molten nucleus. But the progress of +investigation all over the world has shown that this supposition +cannot be sustained. The oldest known rocks present none of the +characters of molten material that has cooled and hardened in the +air, like the various forms of recent lava. On the contrary, they +possess many of the features characteristic of bodies of eruptive +material that have been injected into the crust at some depth underground, +and are now visible at the surface, owing to the removal +by denudation of the rocks under which they consolidated. In their +less foliated portions they can be recognized as true eruptive rocks. +In many places gneisses that possess a thoroughly typical foliation +have been found to pierce ancient sedimentary formations as intrusive +bosses and veins.</p> +</div> + +<p class="pt2 center sc">Part IV.—Dynamical Geology</p> + +<p>This section of the science includes the investigation of those +processes of change which are at present in progress upon the +earth, whereby modifications are made on the structure and +composition of the crust, on the relations between the interior +and the surface, as shown by volcanoes, earthquakes and other +terrestrial disturbances, on the distribution of oceans and +continents, on the outlines of the land, on the form and depth +of the sea-bottom, on climate, and on the races of plants and +animals by which the earth is tenanted. It brings before us, +in short, the whole range of activities which it is the province of +geology to study, and leads us to precise notions regarding their +relations to each other and the results which they achieve. A +knowledge of this branch of the subject is thus the essential +groundwork of a true and fruitful acquaintance with the principles +of geology, seeing that it necessitates a study of the present order +of nature, and thus provides a key for the interpretation of the +past.</p> + +<p>The whole range of operations included within the scope of +inquiry in this branch of the science may be regarded as a vast +cycle of change, into which we may break at any point, and +round which we may travel, only to find ourselves brought +back to our starting-point. It is a matter of comparatively +small moment at what part of the cycle we begin our inquiries. +We shall always find that the changes we see in action have +resulted from some that preceded, and give place to others +which follow them.</p> + +<p>At an early time in the earth’s history, anterior to any of the +periods of which a record remains in the visible rocks, the chief +sources of geological action probably lay within the earth itself. +If, as is generally supposed, the planet still retained a great +store of its initial heat, it was doubtless the theatre of great +chemical changes, giving rise, perhaps, to manifestations of +volcanic energy somewhat like those which have so marvellously +roughened the surface of the moon. As the outer layers of the +globe cooled, and the disturbances due to internal heat and +chemical action became less marked, the conditions would +arise in which the materials for geological history were accumulated. +The influence of the sun, which must always have +operated, would then stand out more clearly, giving rise to that +wide circle of superficial changes wherein variations of temperature +and the circulation of air and water over the surface of the +earth come into play.</p> + +<p>In the pursuit of his inquiries into the past history and into +the present <i>régime</i> of the earth, the geologist must needs keep +his mind ever open to the reception of evidence for kinds +and especially for degrees of action which he had not before +imagined. Human experience has been too short to allow him +to assume that all the causes and modes of geological change +have been definitively ascertained. On the earth itself there may +remain for future discovery evidence of former operations by +heat, magnetism, chemical change or otherwise, which may +explain many of the phenomena with which geology has to deal. +Of the influences, so many and profound, which the sun exerts +upon our planet, we can as yet only perceive a little. Nor can +we tell what other cosmical influences may have lent their aid in +the evolution of geological changes.</p> + +<p>Much useful information regarding many geological processes +has been obtained from experimental research in laboratories +and elsewhere, and much more may be confidently looked for +<span class="pagenum"><a name="page657" id="page657"></a>657</span> +from future extensions of this method of inquiry. The early +experiments of Sir James Hall, already noticed, formed the +starting-point for numerous subsequent researches, which have +elucidated many points in the origin and history of rocks. It +is true that we cannot hope to imitate those operations of nature +which demand enormous pressures and excessively high temperatures +combined with a long lapse of time. But experience +has shown that in regard to a large number of processes, it is +possible to imitate nature’s working with sufficient accuracy +to enable us to understand them, and so to modify and control +the results as to obtain a satisfactory solution of some geological +problems.</p> + +<p>In the present state of our knowledge, all the geological +energy upon and within the earth must ultimately be traced +back to the primeval energy of the parent nebula or sun. There +is, however, a certain propriety and convenience in distinguishing +between that part of it which is due to the survival of some of +the original energy of the planet and that part which arises +from the present supply of energy received day by day from the +sun. In the former case we have to deal with the interior of +the earth, and its reaction upon the surface; in the latter, we +deal with the surface of the earth and to some extent with its +reaction on the interior. This distinction allows of a broad +treatment of the subject under two divisions:</p> + +<p>I. Hypogene or Plutonic Action: The changes within the +earth caused by internal heat, mechanical movement and +chemical rearrangements.</p> + +<p>II. Epigene or Surface Action: The changes produced on the +superficial parts of the earth, chiefly by the circulation of air +and water set in motion by the sun’s heat.</p> + +<p class="pt2 center"><i>DIVISION I.—HYPOGENE OR PLUTONIC ACTION</i></p> + +<p>In the discussion of this branch of the subject we must carry +in our minds the conception of a globe still possessing a high +internal temperature, radiating heat into space and consequently +contracting in bulk. Portions of molten rocks from inside are +from time to time poured out at the surface. Sudden shocks +are generated by which destructive earthquakes are propagated +through the diameter of the globe as well as to and along +its surface. Wide geographical areas are pushed up or sink +down. In the midst of these movements remarkable changes +are produced upon the rocks of the crust; they are plicated, +fractured, crushed, rendered crystalline and even fused.</p> + +<div class="condensed"> +<p class="pt2 center">(A) <i>Volcanoes and Volcanic Action.</i></p> + +<p>This subject is discussed in the article <span class="sc"><a href="#artlinks">Volcano</a></span>, and only a +general view of its main features will be given here. Under the term +volcanic action (vulcanism, vulcanicity) are embraced all the +phenomena connected with the expulsion of heated materials from +the interior of the earth to the surface. A volcano may be defined +as a conical hill or mountain, built up wholly or mainly of materials +which have been ejected from below, and which have accumulated +around the central vent of eruption. As a rule its truncated summit +presents a cup-shaped cavity, termed the crater, at the bottom of +which is the opening of the main funnel or pipe whereby communication +is maintained with the heated interior. From time to +time, however, in large volcanoes rents are formed on the sides of +the cone, whence steam and other hot vapours and also streams of +molten lava are poured forth. On such rents smaller or parasitic +cones are often formed, which imitate the operations of the parent +cone and, after repeated eruptions, may rise to hills hundreds of +feet in height. In course of centuries the result of the constant +outpouring of volcanic materials may be to build up a large mountain +like Etna, which towers above the sea to a height of 10,840 feet, and +has some 200 minor cones along its flanks.</p> + +<p>But all volcanic eruptions do not proceed from central orifices. +In Iceland it has been observed that, from fissures opened in the +ground and extending for long distances, molten material has issued +in such abundance as to be spread over the surrounding country +for many miles, while along the lines of fissure small cones or hillocks +of fragmentary material have accumulated round more active parts +of the rent. There is reason to believe that in the geological past +this fissure-type of eruption has repeatedly been developed, as well +as the more common form of central cones like Vesuvius or Etna.</p> + +<p>In the operations of existing volcanoes only the superficial manifestations +of volcanic action are observable. But when the rocks of +the earth’s crust are studied, they are found to enclose the relics +of former volcanic eruptions. The roots of ancient volcanoes have +thus been laid bare by geological revolutions; and some of the +subterranean phases of volcanic action are thereby revealed which +are wholly concealed in an active volcano. Hence to obtain as +complete a conception as possible of the nature and history of +volcanic action, regard must be had, not merely to modern volcanoes, +but to the records of ancient eruptions which have been preserved +within the crust.</p> + +<p>The substances discharged from volcanic vents consist of—(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.</p> + +<p>Nothing is yet known as to the determining cause of any particular +volcanic eruption. Some vents, like that of Stromboli, in the +Mediterranean, are continually active, and have been so ever since +man has observed them. Others again have been only intermittently +in eruption, with intervals of centuries between their periods of +activity. We are equally in the dark as to what has determined +the sites on which volcanic action has manifested itself. There is +reason, indeed, to believe that extensive fractures of the terrestrial +crust have often provided passages up which the vapours, imprisoned +in the internal magma, have been able to make their way, accompanied +by other products. Where chains of volcanoes rise along +definite lines, like those of Sumatra, Java, and many other tracts +both in the Old and the New World, there appears to be little doubt +that their linear distribution should be attributed to this cause. +But where a volcano has appeared by itself, in a region previously +exempt from volcanic action, the existence of a contributing fissure +cannot be so confidently presumed. The study of certain ancient +volcanoes, the roots of which have been exposed by long denudation, +has shown an absence of any visible trace of their having availed +themselves of fractures in the crust. The inference has been drawn +that volcanic energy is capable of itself drilling an orifice through the +crust, probably at some weaker part, and ejecting its products at +the surface. The source of this energy is to be sought in the enormous +expansive force of the vapours and gases dissolved in the magma. +They are kept in solution by the enormous pressure within the earth; +but as the lava approaches the surface and this pressure is relieved +these dissolved vapours and gases rush out with explosive violence, +blowing the upper part of the lava column into dust, and allowing +portions of the liquid mass below to rise and escape, either from the +crater or from some fissure which the vigour of explosion has opened +on the side of the cone. So gigantic is the energy of these pent-up +vapours, that, after a long period of volcanic quiescence, they +sometimes burst forth with such violence as to blow off the whole of +the upper part or even one side of a large cone. The history of +Vesuvius, and the great eruptions of Krakatoa in 1883 and of +Bandaizan in 1888 furnish memorable examples of great volcanic +convulsions. It has been observed that such stupendous discharges +of aeriform and fragmentary matter may be attended with the +emission of little or no lava. On the other hand, some of the largest +outflows of lava have been accompanied by comparatively little +fragmentary material. Thus, the great lava-floods of Iceland in +1783 spread for 40 m. away from their parent fissure, which was +marked only by a line of little cones of slag.</p> + +<p>The temperature of lava as it issues from underground has been +measured more or less satisfactorily, and affords an indication of +that existing within the earth. At Vesuvius it has been ascertained +to be more than 2000° Fahr. At first the molten rock glows with a +white light, which rapidly reddens, and disappears under the rugged +brown and black crust that forms on the surface. Underneath this +badly conducting crust, the lava cools so slowly that columns of +steam have been noticed rising from its surface more than 80 years +after its eruption.</p> + +<p>Considerable alteration in the topography of volcanic regions +may be produced by successive eruptions. The fragmentary +materials are sometimes discharged in such abundance as to cover +the ground for many miles around with a deposit of loose ashes, +cinders and slag. Such a deposit accumulating to a depth of many +<span class="pagenum"><a name="page658" id="page658"></a>658</span> +feet may completely bury valleys and water-courses, and thus +greatly affect the drainage. The coarsest materials accumulate +nearest to the vent that emits them. The finer dust is not infrequently +hurled forth with such an impetus as to be carried for +thousands of feet into the tracks of upper air-currents, whereby it +may be borne for hundreds of miles away from the vent so as ultimately +to fall to the ground in countries far removed from any active +volcano. Outflows of lava, from their greater solidity and durability, +produce still more serious and lasting changes in the external features +of the ground over which they flow. As they naturally seek the +lowest levels, they find their way into the channels of streams. +If they keep along the channels, they seal them up under a mass of +compact stone which the running water, if not wholly diverted +elsewhere, will take many long centuries to cut through. If, on the +other hand, the lava crosses a stream, it forms a massive dam, +above which the water is ponded back so as to form a lake.</p> + +<p>As the result of prolonged activity a volcanic cone is gradually +built up by successive outflows of lava and showers of dust and +stones. These materials are arranged in beds, or sheets, inclined +outwards from the central vent. On surrounding level ground the +alternating beds are flat. In course of time, deep gullies are cut on +the outer slopes of the cone by rain, and by the heavy showers that +arise from the condensation of the copious discharges of steam +during eruptions. Along the sides of these ravines instructive +sections may be studied of the volcanic strata. The larger rivers of +some volcanic regions have likewise eroded vast gorges in the more +horizontal lavas and ashes of the flatter country, and have thus laid +bare stupendous cliffs, along which the successive volcanic sheets +can be seen piled above each other for many hundred feet. On a +small scale, some of these features are well displayed among the +rivers that drain the volcanic tracts of central France; on a great +scale, they are presented in the course of the Snake river, and other +streams that traverse the great volcanic country of western North +America. Similar volcanic scenery has been produced in western +Europe by the action of denudation in dissecting the flat Tertiary +lavas of Scotland, the Faeroe Isles and Iceland.</p> + +<p>Of special interest to the geologist are those volcanoes which have +taken their rise on the sea-bottom; for the volcanic intercalations +among the stratified formations of the earth’s crust are almost +entirely of submarine origin. Many active volcanoes situated on +islands have begun their eruptions below sea-level. Both Vesuvius +and Etna sprang up on the floor of the Mediterranean sea, and have +gradually built up their cones into conspicuous parts of the dry land. +Examples of a similar history are to be found among the volcanic +islands of the Pacific Ocean. In some of these cases a movement +of elevation has carried the submarine lavas, tuffs and agglomerates +above sea-level, and has furnished opportunities of comparing these +materials with those of recent subaerial origin, and also with the +ancient records of submarine eruptions which have been preserved +among the stratified formations. From the evidence thus supplied, +it can be shown that the materials ejected from modern submarine +volcanic vents closely resemble those accumulated by subaerial +volcanoes; that the dust, ashes and stones become intermingled or +interstratified with coral-mud, or other non-volcanic deposit of the +sea-bottom, that vesicular lavas may be intercalated among them +as on land, and that between the successive sheets of volcanic +origin, layers of limestone may be laid down which are composed +chiefly, or wholly, of the remains of calcareous marine organisms.</p> + +<p>Though active volcanoes are widely distributed over the globe, +and are especially abundant around the vast basin of the Pacific +Ocean, they afford an incomplete picture of the extent to which +volcanic action has displayed itself on the surface of our planet. +When the rocks of the land are attentively studied they disclose +proofs of that action in many districts where there is now no outward +sign of it. Not only so, but they reveal that volcanoes have been in +eruption in some of these districts during many different periods of +the past, back to the beginnings of geological history. The British +Islands furnish a remarkable example of such a series of ancient +eruptions. From the Cambrian period all through Palaeozoic times +there rose at intervals in that country a succession of volcanic centres +from some of which thousands of feet of lavas and tuffs were discharged. +Again in older Tertiary times the same region witnessed +a stupendous outpouring of basalt, the surviving relics of which +are more than 3000 ft. thick, and cover many hundreds of square +miles. Similar evidence is supplied in other countries both in the +Old and the New world. Hence it is proved that, in the geological +past, volcanic action has been vigorous at long intervals on the same +sites during a vast series of ages, though no active vents are to be +seen there now. The volcanoes now active form but a small proportion +of the total number which has appeared on the surface of +the earth.</p> + +<p>With regard to the cause of volcanic action much has been +speculated, but little can be confidently affirmed. That water in +the form of occluded gas plays the chief part in forcing the lava +column up a volcanic chimney, and in the violent explosions that +accompany the rise of the molten material, is generally admitted. +But opinions differ as to the source of this water. According to +some investigators, it should be regarded as in large measure of +meteoric origin, derived from the descent of rain into the earth, and +its absorption by the molten magma in the interior. Others, contending +that the supply so furnished, even if it could reach and be +dissolved in the magma, would yet be insufficient to furnish the +prodigious quantity of aqueous vapour discharged during an eruption, +maintain that the water belongs to the magma itself. They point +to the admitted fact that many substances, particularly metals in +a state of fusion, can absorb large quantities of vapours and gases +without chemical combination, and on cooling discharge them with +eruptive phenomena somewhat like those of volcanoes. This +question must be regarded as one of the still unsolved problems of +geology.</p> + +<p class="pt2 center">(B) <i>Movements of the Earth’s Crust.</i></p> + +<p>Among the hypogene forces in geological dynamics an important +place must be assigned to movements of the terrestrial crust. Though +the expression “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 <span class="sc"><a href="#artlinks">Seismometer</a></span>), it has +been ascertained that the ground beneath our feet is subject to +continual slight tremors, and feeble pulsations of longer duration, +some of which may be due to daily or seasonal variations of temperature, +atmospheric pressure or other meteorological causes. +The establishment of self-recording seismometers all over the world +has led to the detection of many otherwise imperceptible shocks, +over and above the appreciable earth-waves propagated from earthquake +centres of disturbance. Moreover, it has been ascertained +that some parts of the surface of the land are slowly rising, while +others are falling with reference to the sea-level. From time to +time the surface suffers calamitous devastation from earthquakes, +when portions of the crust under great strain suddenly give way. +Lastly, at intervals, probably separated from each other by vast +periods of time, the terrestrial crust undergoes intense plication +and fracture, and is consequently ridged up into mountain-chains. +No event of this kind has been witnessed since man began to record +his experiences. But from the structure of mountains, as laid open +by prolonged denudation, it is possible to form a vivid conception +of the nature and effects of these most stupendous of all geological +revolutions.</p> + +<p>In considering this department of geological inquiry it will be +convenient to treat it under the following heads: (1) Slow depression +and upheaval; (2) Earthquakes; (3) Mountain-making; (4) +Metamorphism of rocks.</p> + +<p>1. <i>Slow Depression and Upheaval.</i>—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.</p> + +<p>The uprise of land has been detected in various parts of the world. +One of the most celebrated instances is that of the shores of the Gulf +of Bothnia, where, at Stockholm, the elevation, between the years +1774 and 1875, appears to have been 48 centimetres (18½ in.) in +a century. But on the west side of Sweden, fronting the Skager Rak, +the coast, between the years 1820 and 1870, rose 30 centimetres, +which is at the rate of 60 centimetres, or nearly 2 ft. in a century. +In the region of the Great Lakes in the interior of Canada and the +United States it has been ascertained that the land is undergoing a +slow tilt towards the south-west, of which the mean rate appears to +be rather less than 6 in. in a century. If this rate of change should +continue the waters of Lake Michigan, owing to the progress of the +tilt, will, in some 500 or 600 years, submerge the city of Chicago, +and eventually the drainage of the lakes will be diverted into the +basin of the Mississippi. Proof of recent emergence of land is supplied +by what are called “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.</p> + +<p>The cause of these slow upward and downward movements of the +crust of the earth is still imperfectly understood. Upheaval might +conceivably be produced by an ascent of the internal magma, and the +consequent expansion of the overlying crust by heat; while depression +might follow any subsidence of the magma, or its displacement +<span class="pagenum"><a name="page659" id="page659"></a>659</span> +to another district. If, as is generally believed, the globe is still +contracting, the shrinkage of the surface may cause both these +movements. Subsidence will be in excess, but between subsiding +tracts lateral thrust may suffice to push upward intervening more +solid and stable ground; but no solution of the problem yet proposed +is wholly satisfactory.</p> + +<p>2. <i>Earthquakes.</i>—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.</p> + +<p>From their disastrous effects on buildings and human lives, an +exaggerated importance has been imputed to earthquakes as agents +of geological change. Experience shows that even after a severe +shock which may have destroyed numerous towns and villages, +together with thousands of their inhabitants, the face of the country +has suffered scarcely any perceptible change, and that, in the course +of a year or two, when the ruined houses and prostrate trees have +been cleared away, little or no obvious trace of the catastrophe may +remain. Among the more enduring records of a great earthquake +may be enumerated (<i>a</i>) landslips, which lay bare hillsides, and sometimes +pond back the drainage of valleys so as to give rise to lakes; +(<i>b</i>) alterations of the topography, as in fissuring of the ground, or in +the production of inequalities whereby the drainage is affected; +new valleys and new lakes may thus be formed, while previously +existing lakes may be emptied; (<i>c</i>) permanent changes of level, +either in an upward or downward direction.</p> + +<p>3. <i>Mountain-making.</i>—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.</p> + +<p>4. <i>Metamorphism of Rocks</i> (see <span class="sc"><a href="#artlinks">Metamorphism</a></span>).—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.</p> + +<p>Regional metamorphism has arisen in the heart of mountain-chains, +and in any other district where the deformation of the crust +has been sufficiently intense. There is another type of alteration +termed “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.</p> +</div> + +<p class="pt2 center"><i>DIVISION II.—EPIGENE OR SUPERFICIAL ACTION</i></p> + +<p>It is on the surface of the globe, and by the operation of agents +working there, that at present the chief amount of visible geological +change is effected. In considering this branch of inquiry, +we are not involved in a preliminary difficulty regarding the very +nature of the agencies as is the case in the investigation of +plutonic action. On the contrary, the surface agents are carrying +on their work under our very eyes. We can watch it in all its +stages, measure its progress, and mark in many ways how +accurately it represents similar changes which, for long ages +previously, must have been effected by the same means. But +in the systematic treatment of this subject we encounter a +difficulty of another kind. We discover that while the operations +to be discussed are numerous and readily observable, they are so +interwoven into one great network that any separation of them +under different subdivisions is sure to be more or less artificial +and to convey an erroneous impression. While, therefore, under +the unavoidable necessity of making use of such a classification +of subjects, we must always bear in mind that it is employed +merely for convenience, and that in nature superficial geological +action must be continually viewed as a whole, since the work of +each agent has constant reference to that of the others, and is +not properly intelligible unless that connexion be kept in view.</p> + +<p>The movements of the air; the evaporation from land and +sea; the fall of rain, hail and snow; the flow of rivers and +glaciers; the tides, currents and waves of the ocean; the growth +and decay of organized existence, alike on land and in the depths +of the sea;—in short, the whole circle of movement, which is +continually in progress upon the surface of our planet, are the +subjects now to be examined. It is desirable to adopt some +general term to embrace the whole of this range of inquiry. For +this end the word epigene (Gr. <span class="grk" title="epi">ἐπί</span>, upon) has been suggested as +a convenient term, and antithetical to hypogene (Gr. <span class="grk" title="hypo">ὑπό</span>, under), +or subterranean action.</p> + +<p>A simple arrangement of this part of Geological Dynamics is +in three sections:</p> + +<div class="list"> +<p>A. <i>Air.</i>—The influence of the atmosphere in destroying and +forming rocks.</p> + +<p>B. <i>Water.</i>—The geological functions of the circulation of +water through the air and between sea and land, and the +action of the sea.</p> + +<p><span class="pagenum"><a name="page660" id="page660"></a>660</span></p> + +<p>C. <i>Life.</i>—The part taken by plants and animals in preserving, +destroying or reproducing geological formations.</p> +</div> + +<p>The words destructive, reproductive and conservative, +employed in describing the operations of the epigene agents, do +not necessarily imply that anything useful to man is destroyed, +reproduced or preserved. On the contrary, the destructive +action of the atmosphere may turn barren rock into rich soil, +while its reproductive effects sometimes turn rich land into +barren desert. Again, the conservative influence of vegetation +has sometimes for centuries retained as barren morass what +might otherwise have become rich meadow or luxuriant woodland. +The terms, therefore, are used in a strictly geological +sense, to denote the removal and re-deposition of material, and +its agency in preserving what lies beneath it.</p> + +<div class="condensed"> +<p class="pt2 center">(A) <i>The Air.</i></p> + +<p>As a geological agent, the air brings about changes partly by its +component gases and partly by its movements. Its destructive +action is both chemical and mechanical. The chemical changes are +probably mainly, if not entirely, due to the moisture of the air, +and particularly to the gases, vapours and organic matter which +the moisture contains. Dry air seems to have little or no appreciable +influence in promoting these reactions. As the changes in question +are similar to those much more abundantly brought about by rain +they are described in the following section under the division on rain.</p> + +<p>Among the more recognizable mechanical changes effected in +the atmosphere, one of considerable importance is to be seen in the +result of great and rapid changes of temperature. Heat expands +rocks, while cold contracts them. In countries with a great annual +range of temperature, considerable difficulty is sometimes experienced +in selecting building materials liable to be little affected by the +alternate expansion and contraction, which prevents the joints of +masonry from remaining close and tight. In dry tropical climates, +where the days are intensely hot and the nights extremely cold, the +rapid nocturnal contraction produces a strain so great as to rival +frost in its influence upon the surface of exposed rocks, disintegrating +them into sand, or causing them to crack or peel off in skins or +irregular pieces. Dr Livingstone found in Africa (12° S. lat., 34° E. +long.) that surfaces of rock which during the day were heated up to +137° Fahr., cooled so rapidly by radiation at night that, unable to +sustain the strain of contraction, they split and threw off sharp +angular fragments from a few ounces to 100 or 200 ℔ in weight. +In temperate regions this action, though much less pronounced, +still makes itself felt. In these climates, however, and still more in +high latitudes, somewhat similar results are brought about by frost.</p> + +<p>By its motion in wind the air drives loose sand over rocks, and in +course of time abrades and smoothes them. “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.</p> + +<p>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.</p> + +<p class="pt2 center">(B) <i>Water.</i></p> + +<p>In treating of the epigene action of water in geological processes +it will be convenient to deal first with its operations in traversing +the land, and then with those which it performs in the sea. The +circulation of water from land to sea and again from sea to land +constitutes the fundamental cause of most of the daily changes by +which the surface of the land is affected.</p> + +<p>1. <i>Rain.</i>—Rain effects two kinds of changes upon the surface of +the land. It acts <i>chemically</i> upon soils and stones, and sinking under +ground continues a great series of similar reactions there. It acts +<i>mechanically</i>, by washing away loose materials, and thus powerfully +affecting the contours of the land. Its chemical action depends +mainly upon the nature and proportion of the substances which, in +descending to the earth, it abstracts from the atmosphere. Rain +always absorbs a little air, which, in addition to its nitrogen and +oxygen, contains carbonic acid, and in minute proportions, sodium +chloride, sulphuric acid and other ingredients, especially inorganic +dust, organic particles and living germs. Probably the most generally +efficient of these constituents are oxygen, carbonic acid and organic +matter. Armed with these reagents, rain effects a chemical decomposition +of the rocks on which it falls, and through which it sinks +underground. The principal changes thus produced are as follows: +(<i>a</i>) Oxidation.—Owing to the prominence of oxygen in rain-water, +and its readiness to unite with any substance which can contain +more of it, a thin oxidized pellicle is formed on the surface of many +rocks on which rain falls, and this oxidized layer if not at once +washed off, sinks deeper until a crust is formed over the stone. A +familiar illustration of this action is afforded by the rust, or oxide, +which forms on iron when exposed to moisture, though this iron +may be kept long bright if allowed to remain screened from moist +air and rain. (<i>b</i>) Deoxidation.—Organic matter having an affinity +for more oxygen decomposes peroxides by depriving them of some +part of their share of that element and reducing them to protoxides. +These changes are especially noticeable among the iron oxides so +abundantly diffused among rocks. Hence rain-water, in sinking +through soil and obtaining such organic matter, becomes thereby +a reducing agent. (<i>c</i>) Solution.—This may take place either by the +simple action of the water, as in the solution of rock-salt, or by the +influence of the carbonic acid present in the rain. (<i>d</i>) Formation of +Carbonates.—A familiar example of the action of carbonic acid +in rain is to be seen in the corrosion of exposed marble slabs. The +carbonic acid dissolves some of the lime, which, as a bicarbonate, +is held in solution in the carbonated water, but is deposited again +when the water loses its carbonic acid or evaporates. It is not +merely carbonates, however, which are liable to this kind of destruction. +Even silicates of lime, potash and soda, combinations existing +abundantly as constituents of rocks, are attacked; their silica is +liberated, and their alkalis or alkaline earths, becoming carbonates, +are removed in solution. (<i>e</i>) Hydration.—Some minerals, containing +little or no water, and therefore called anhydrous, when exposed to +the action of the atmosphere, absorb water, or become hydrous, +and are then usually more prone to further change. Hence the rocks +of which they form part become disintegrated.</p> + +<p>Besides the reactions here enumerated, a considerable amount of +decay may be observed as the result of the presence of sulphuric +and nitric acid in the air, especially in that of large towns and +manufacturing districts, where much coal is consumed. Metallic +surfaces, as well as various kinds of stone, are there corroded, while +the mortar of walls may often be observed to be slowly swelling out +and dropping off, owing to the conversion of the lime into sulphate. +Great injury is likewise done from a similar cause to marble monuments +in exposed graveyards.</p> + +<p>The general result of the disintegrating action of the air and of +rain, including also that of plants and animals, to be noticed in the +sequel, is denoted by the term “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 +<span class="pagenum"><a name="page661" id="page661"></a>661</span> +picked out by the weather, minute and frail fossils which are wholly +invisible on a freshly broken surface of the stone. Many rocks +weather with a thick crust, or even decay inwards for many feet or +yards. Basalt, for example, often shows a yellowish-brown ferruginous +layer on its surface, formed by the conversion of its felspar +into kaolin, and the removal of its calcium silicate as carbonate, +by the hydration of its olivine and augite and their conversion into +serpentine, or some other hydrous magnesian silicate, and by the +conversion of its magnetite into limonite. Granite sometimes shows +in a most remarkable way the distance to which weathering can +reach. It may occasionally be dug into for a depth of 20 or 30 ft., +the quartz crystals and veins retaining their original positions, while +the felspar is completely kaolinized. It is to the endlessly varied +effects of weathering that the abundant fantastic shapes assumed +by crags and other rocky masses are due. Most varieties of rock +have their own characteristic modes of weathering, whereby they +may be recognized even from a distance. To some of these features +reference will be made in Part VIII.</p> + +<p>The mechanical action of rain, which is intimately bound up with +its chemical action, consists in washing off the fine superficial +particles of rocks which have been corroded and loosened by the +process of weathering, and in thus laying open fresh portions to the +same influences of decay. The detritus so removed is partly carried +down into the soil which is thereby enriched, partly held in suspension +in the little runnels into which the rain-drops gather as they begin +to flow over the land, partly pushed downwards along the surface +of sloping ground. A good deal of it finds its way into the nearest +brooks and rivers, which are consequently made muddy by heavy +rain.</p> + +<p>It is natural that a casual consideration of the subject should lead +to an impression that, though the general result of the fall of rain +upon a land-surface must lead to some amount of disintegration and +lowering of that surface, the process must be so slow and slight as +hardly to be considered of much importance among geological +operations. But further attention will show such an impression to +be singularly erroneous. It loses sight of the fact that a change +which may be hardly appreciable within a human lifetime, or even +within the comparatively brief span of geological time embraced in +the compass of human history, may nevertheless become gigantic +in its results in the course of immensely protracted periods. An +instructive lesson in the erosive action of rain may be found in the +pitted and channelled surface of ground lying under the drip of the +eaves of a cottage. The fragments of stone and pebbles of gravel +that form part of the soil can there be seen sticking out of the ground, +because being hard they resist the impetus of the falling drops, +protecting for a time the earth beneath them, while that which +surrounded and covered them is washed away. From this familiar +illustration the observer may advance through every stage in the +disappearance of material which once covered the surface, until he +comes to examples where once continuous and thick sheets of solid +rock have been reduced to a few fragments or have been entirely +removed. Since the whole land surface over which rain falls is +exposed to this waste, the superficial covering of decayed rock or +soil, as Hutton insisted, is constantly, though imperceptibly, travelling +outward and downward to the sea. In this process of transport +rain is an important carrying agent, while at the same time it serves +to connect the work of the other disintegrating forces, and to make +it conducive to the general degradation of the land. Though this +decay is general and constant, it is obviously not uniform. In some +places where, from the nature of the rock, from the flatness of the +ground, or from other causes, rain works under great difficulties, +the rate of waste may be extremely slow. In other places it may +be rapid enough to be appreciable from year to year. A survey of +this department of geological activity shows how unequal wasting +by rain, combined with the operations of brooks and rivers, has +produced the details of the present relief of the land, those tracts +where the destruction has been greatest forming hollows and valleys, +others, where it has been less, rising into ridges and hills (Part VIII.).</p> + +<p>Rain-action is not merely destructive, but is accompanied with +reproductive effects, chief of which is the formation of soil. In +favourable situations it has gathered together accumulations of loam +and earth from neighbouring higher ground, such as the “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.</p> + +<p>2. <i>Underground Water.</i>—Of the rain which falls upon the land +one portion flows off into brooks and rivers by which the water is +conducted back to the ocean; the larger part, however, sinks into +the ground and disappears. It is this latter part which has now +to be considered. Over and above the proportion of the rainfall +which is absorbed by living vegetation and by the soil, there is a +continual filtering down of the water from the surface into the rocks +that lie below, where it partly lodges in pores and interstices, and +partly finds its way into subterranean joints and fissures, in which +it performs an underground circulation, and ultimately issues once +more at the surface in the form of springs (<i>q.v.</i>). In the course of +this circulation the water performs an important geological task. +Not only carrying down with it the substances which the rain has +abstracted from the air, but obtaining more acids and organic +matter from the soil, it is enabled to effect chemical changes in the +rocks underneath, and especially to dissolve limestone and other +calcareous formations. So considerable is the extent of this solution +in some places that the springs which come to the surface, and begin +there to evaporate and lose some of their carbonic acid, contain more +dissolved lime than they can hold. They consequently deposit it +in the form of calcareous tuff or sinter (<i>q.v.</i>). Other subterranean +waters issue with a large proportion of iron-salts in solution which +form deposits of ochre. The various mineral springs so largely +made use of for the mitigation or cure of diseases owe their properties +to the various salts which they have dissolved out of rocks +underground. As the result of prolonged subterranean solution in +limestone districts, passages and caves (<i>q.v.</i>), sometimes of great +width and length, are formed. When these lie near the surface their +roofs sometimes fall in and engulf brooks and rivers, which then +flow for some way underground until the tunnels conduct them back +again to daylight on some lower ground.</p> + +<p>Besides its chemical activity water exerts among subterranean +rocks a mechanical influence which leads to important changes in +the topography of the surface. In removing the mineral matter, +either in solution or as fine sediment, it sometimes loosens the support +of overlying masses of rock which may ultimately give way on sloping +ground, and rush down the declivities in the form of landslips. +These destructive effects are specially frequent on the sides of valleys +in mountainous countries and on lines of sea-cliff.</p> + +<p>3. <i>Brooks and Rivers.</i>—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.</p> + +<p>The mechanical action of rivers consists (<i>a</i>) in transporting mud, +sand, gravel and blocks of stone from higher to lower levels; (<i>b</i>) +in using these loose materials to widen and deepen their channels +by erosion; (<i>c</i>) in depositing their load of detritus wherever possible +and thus to make new geological formations.</p> + +<p>(<i>a</i>) <i>Transporting Power.</i>—River-water is distinguished from that +of springs by being less transparent, because it contains more or less +mineral matter in suspension, derived mainly from what is washed +down by rain, or carried in by brooks, but partly also from the +abrasion of the water-channels by the erosive action of the rivers +themselves. The progress of this burden of detritus may be instructively +followed from the mountain-tributaries of a river down to +the mouth of the main stream. In the high grounds the water-courses +may be observed to be choked with large fragments of rock +disengaged from the cliffs and crags on either side. Traced downwards +the blocks are seen to become gradually smaller and more rounded. +They are ground against each other, and upon the rocky sides and +bottom of the channel, getting more and more reduced as they +descend, and at the same time abrading the rocks over or against +which they are driven. Hence a great deal of débris is produced, +and is swept along by the onward and downward movement of the +water. The finer portions, such as mud and fine sand, are carried +in suspension, and impart the characteristic turbidity to river-water; +the coarser sand and gravel are driven along the river-bottom. +The proportion of suspended mineral matter has been +ascertained with more or less precision for a number of rivers. As +an illustrative example of a river draining a vast area with different +climates, forms of surface and geological structure the Mississippi +may be cited. The average proportion of sediment in its water was +ascertained by Humphreys and Abbot to be <span class="spp">1</span>⁄<span class="suu">1500</span> by weight or +<span class="spp">1</span>⁄<span class="suu">2900</span> by volume. These engineers found that, in addition to this +suspended material, coarse detritus is constantly being pushed +forward along the bed of the river into the Gulf of Mexico, to an +amount which they estimated at about 750,000,000 cubic ft. of +sand, earth and gravel; they concluded that the Mississippi carries +into the gulf every year an amount of mechanically transported +sediment sufficient to make a prism one square mile in area and +268 ft. in height.</p> + +<p>(<i>b</i>) <i>Excavating Power.</i>—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. +<span class="pagenum"><a name="page662" id="page662"></a>662</span> +The uneven bed of a swiftly flowing stream may in this way be +honeycombed with such eroded basins which coalesce and thus +appreciably lower the surface of the bed. The steeper the channel, +other conditions being equal, the more rapid will be the erosion. +Geological structure also affects the character and rate of the excavation. +Where the rocks are so arranged as to favour the formation +and persistence of a waterfall, a long chasm may be hollowed out +like that of the Niagara below the falls, where a hard thick bed of +nearly flat limestone lies on softer and more easily eroded shales. +The latter are scooped out from underneath the limestone, which +from time to time breaks off in large masses and the waterfall +gradually retreats up stream, while the ravine is proportionately +lengthened. To the excavating power of rivers the origin of the +valley systems of the dry land must be mainly assigned (see Part VIII.).</p> + +<p>(<i>c</i>) <i>Reproductive Power.</i>—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 (<i>q.v.</i>). On land, +every lake in mountain districts furnishes instances of this kind of +alluvium. But the most important deltas are those formed in the +sea at the mouths of the larger rivers of the globe. Off many coast-lines +the detritus washed from the land gathers into bars, which +enclose long strips of water more or less completely separated from +the sea outside and known as lagoons. A chain of such lagoon-barriers +stretches for hundreds of miles round the Gulf of Mexico +and the eastern shores of the United States.</p> + +<p>4. <i>Lakes.</i>—These sheets of water, considered as a whole, do not +belong to the normal system of drainage on the land whereby valleys +are excavated. On the contrary they are exceptional to it; for +the constant tendency of running water is to fill them up, or to drain +them by wearing down the barriers that contain them at their +outflow. Some of them are referable to movements of the terrestrial +crust whereby depressions arise on the surface of the land, as has +been noted after earthquakes. Others have arisen from solution +such as that of rock-salt or of limestone, the removal of which by +underground water causes a subsidence of the ground above. A +third type of lake-basin occurs in regions that are now or have once +been subject to the erosive action of glaciers (see under next subdivision, +<i>Terrestrial Ice</i>). Many small lakes or tarns have been +caused by the deposit of débris across a valley as by landslips or +moraines. Considered from a geological point of view, lakes perform +an important function in regulating the drainage of the ground below +their outfall and diminishing the destructive effects of floods, in +filtering the water received from their affluent streams, and in +providing undisturbed areas of deposit in which thick and extensive +lacustrine formations may be accumulated. In the inland basins +of some dry climates the lakes are salt, owing to excess of evaporation, +and their bottoms become the sites of chemical deposits, particularly +of chlorides of sodium and magnesium, and calcium sulphate and +carbonate.</p> + +<p>5. <i>Terrestrial Ice.</i>—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.</p> + +<p>But the most striking geological work performed by terrestrial +ice is that achieved by glaciers (<i>q.v.</i>) and ice-sheets. These vast +masses of moving ice, when they descend from mountains where the +steeper rocks are clear of snow, receive on their surface the débris +detached by frost from the declivities above, and bear these materials +to lower levels or to the sea. Enormous quantities of rock-rubbish +are thus transported in the Alps and other high mountain ranges. +When the ice retreats the boulders carried by it are dropped where +it melts, and left there as memorials of the former extension of the +glaciers. Evidence of this nature proves the much wider extent of +the Alpine ice at a comparatively recent geological date. It can +also be shown that detritus from Scandinavia has been ice-borne to +the south-east of England and far into the heart of Europe.</p> + +<p>The ice, by means of grains of sand and pieces of stone which it +drags along, scores, scratches and polishes the surfaces of rock +underneath it, and, in this way, produces the abundant fine sediment +that gives the characteristic milky appearance to the rivers that +issue from the lower ends of glaciers. By such long-continued +attrition the rocks are worn down, portions of them of softer nature, +or where the ice acts with especial vigour, are hollowed out into +cavities which, on the disappearance of the ice, may be filled with +water and become tarns or lakes. Rocks over which land-ice has +passed are marked by a peculiar smooth, flowing outline, which +forms a contrast to the more rugged surface produced by ordinary +weathering. They are covered with groovings, which range from +the finest striae left by sharp grains of sand to deep ruts ground out +by blocks of stone. The trend of these markings shows the direction +in which the ice flowed. By their evidence the position and movement +of former glaciers in countries from which the ice has entirely +vanished may be clearly determined (see <span class="sc"><a href="#artlinks">Glacial Period</a></span>).</p> + +<p>6. <i>The Sea.</i>—The physical features of the sea are discussed in +separate articles (see <span class="sc"><a href="#artlinks">Ocean and Oceanography</a></span>). The sea must +be regarded as the great regulator of temperature and climate over +the globe, and as thus exerting a profound influence on the distribution +of plant and animal life. Its distinctly geological work is partly +erosive and partly reproductive. As an eroding agent it must to +some extent effect chemical decompositions in the rocks and sediments +over which it spreads; but these changes have not yet been +satisfactorily studied. Undoubtedly, its chief destructive power +is of a mechanical kind, and arises from the action of its waves in +beating upon shore-cliffs. By the alternate compression and +expansion of the air in crevices of the rocks on which heavy breakers +fall, and by the hydraulic pressure which these masses of sea-water +exert on the walls of the fissures into which they rush, large masses +of rock are loosened and detached, and caves and tunnels are drilled +along the base of sea-cliffs. Probably still more efficacious are the +blows of the loose shingle, which, caught up and hurled forward by +the waves, falls with great force upon the shore rocks, battering +them as with a kind of artillery until they are worn away. The +smooth surfaces of the rocks within reach of the waves contrasted +with their angular forms above that limit bear witness to the amount +of waste, while the rounded forms of the boulders and shingle show +that they too are being continually reduced in size. Thus the sea, +by its action on the coasts, produces much sediment, which is swept +away by its waves and currents and strewn over its floor. Besides +this material, it is constantly receiving the fine silt and sand carried +down by rivers. As the floor of the ocean is thus the final receptacle +for the waste of the land, it becomes the chief era on the surface of +the globe for the accumulation of new stratified formations. And +such has been one of its great functions since the beginning of +geological time, as is proved by the rocks that form the visible part +of the earth’s crust, and consist in great part of marine deposits. +Chemical precipitates take place more especially in enclosed parts +of the sea, where concentration of the water by evaporation can take +place, and where layers of sodium chloride, calcium sulphate and +carbonate, and other salts are laid down. But the chief marine +accumulations are of detrital origin. Near the land and for a variable +distance extending sometimes to 200 or 300 m. from shore the +deposits consist chiefly of sediments derived from the waste of the +land, the finer silts being transported farthest from their source. +At greater depths and distances the ocean floor receives a slow deposit +of exceedingly fine clay, which is believed to be derived from the +decomposition of pumice and volcanic dust from insular or submarine +volcanoes. Wide tracts of the bottom are covered with +various forms of ooze derived from the accumulation of the remains +of minute organisms.</p> + +<p class="pt2 center">(C) <i>Life.</i></p> + +<p>Among the agents by which geological changes are carried on +upon the surface of the globe living organisms must be enumerated. +Both plants and animals co-operate with the inorganic agents in +promoting the degradation of the land. In some cases, on the other +hand, they protect rocks from decay, while, by the accumulation of +their remains, they give rise to extensive formations both upon the +land and in the sea. Their operations may hence be described as +alike destructive, conservative and reproductive. Under this heading +also the influence of Man as a geological agent deserves notice.</p> + +<p>(<i>a</i>) <i>Plants.</i>—Vegetation promotes the disintegration of rocks and +soil in the following ways: (1) By keeping the surfaces of stone +moist, and thus promoting both mechanical and chemical dissolution, +as is especially shown by liverworts, mosses and other moisture-loving +plants. (2) By producing through their decay carbonic and +<span class="pagenum"><a name="page663" id="page663"></a>663</span> +other acids, which, together with decaying organic matter taken up +by passing moisture, become potent in effecting the chemical decomposition +of rocks and in promoting the disintegration of soils. (3) +By inserting their roots or branches between joints of rock, which +are thereby loosened, so that large slices may be eventually wedged +off. (4) By attracting rain, as thick woods, forests and peat-mosses +do, and thus accelerating the general waste of a country by running +water. (5) By promoting the decay of diseased and dead plants and +animals, as when fungi overspread a damp rotting tree or the carcase +of a dead animal.</p> + +<p>That plants also exert a conservative influence on the surface of +the land is shown in various ways. (1) The formation of a stratum +of turf protects the soil and rocks underneath from being rapidly +disintegrated and washed away by atmospheric action. (2) Many +plants, even without forming a layer of turf, serve by their roots or +branches to protect the loose sand or soil on which they grow from +being removed by wind. The common sand-carex and other arenaceous +plants bind the loose sand-dunes of our coasts, and give them a +permanence, which would at once be destroyed were the sand laid +bare again to storms. The growth of shrubs and brushwood along +the course of a stream not only keeps the alluvial banks from being +so easily undermined and removed as would otherwise be the case, +but serves to arrest the sediment in floods, filtering the water and +thereby adding to the height of the flood plain. (3) Some marine +plants, like the calcareous nullipores, afford protection to shore +rocks by covering them with a hard incrustation. The tangles and +smaller Fuci which grow abundantly on the littoral zone break the +force of the waves or diminish the effects of ground swell. (4) +Forests and brushwood protect the soil, especially on slopes, from +being washed away by rain or ploughed up by avalanches.</p> + +<p>Plants contribute by the aggregation of their remains to the +formation of stratified deposits. Some marine algae which secrete +carbonate of lime not only encrust rocks but give rise to sheets of +submarine limestone. An analogous part is played in fresh-water +lakes by various lime-secreting plants, such as <i>Chara</i>. Long-continued +growth of vegetation has, in some regions, produced thick +accumulations of a dark loam, as in the black cotton soil (<i>regur</i>) of +India, and the black earth (<i>tchernozom</i>) of Russia. Peat-mosses +are formed in temperate and arctic climates by the growth of marsh-loving +plants, sometimes to a thickness of 40 or 50 ft. In tropical +regions the mangrove swamps on low moist shores form a dense +jungle, sometimes 20 m. broad, which protects these shores from the +sea until, by the arrest of sediment and the constant contribution of +decayed vegetation, the spongy ground is at last turned into firm +soil. Some plants (diatoms) can abstract silica and build it into +their framework, so that their remains form a siliceous deposit or +ooze which covers spaces of the deep sea-floor estimated at more +than ten millions of square miles in extent.</p> + +<p>(<i>b</i>) <i>Animals.</i>—These exert a destructive influence in the following +ways: (1) By seriously affecting the composition and arrangement +of the vegetable soil. Worms bring up the lower portions of the +soil to the surface, and while thus promoting its fertility increase +its liability to be washed away by rain. Burrowing animals, by +throwing up the soil and subsoil, expose these to be dried and blown +away by the wind. At the same time their subterranean passages +serve to drain off the superficial water and to injure the stability +of the surface of the ground above them. In Britain the mole and +rabbit are familiar examples. (2) By interfering with or even diverting +the flow of streams. Thus beaver-dams check the current of +water-courses, intercept floating materials, and sometimes turn +streams into new channels. The embankments of the Mississippi +are sometimes weakened to such an extent by the burrowings of the +cray-fish as to give way and allow the river to inundate the surrounding +country. Similar results have happened in Europe from +subterranean operations of rats. (3) Some mollusca bore into stone +or wood and by the number of contiguous perforations greatly +weaken the material. (4) Many animals exercise a ruinously +destructive influence upon vegetation. Of the numerous plagues +of this kind the locust, phylloxera and Colorado beetle may be cited.</p> + +<p>The most important geological function performed by animals is +the formation of new deposits out of their remains. It is chiefly by +the lower grades of the animal kingdom that this work is accomplished, +especially by molluscs, corals and foraminifera. Shell-banks +are formed abundantly in such comparatively shallow and enclosed +basins as that of the North Sea, and on a much more extensive scale +on the floor of the West Indian seas. By the coral polyps thick +masses of limestones have been built up in the warmer seas of the +globe (see <span class="sc"><a href="#artlinks">Coral Reefs</a></span>). The floor of the Atlantic and other oceans +is covered with a fine calcareous ooze derived mainly from the +remains of foraminifera, while in other regions the bottom shows a +siliceous ooze formed almost entirely of radiolaria. Vertebrate +animals give rise to phosphatic deposits formed sometimes of their +excrement, as in guano and coprolites, sometimes of an accumulation +of their bones.</p> + +<p>(<i>c</i>) <i>Man.</i>—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.</p> + +<p>Still more obvious are the results of human interference with the +flow of water: (1) By increasing or diminishing the rainfall man +directly affects the volume of rivers. (2) By his drainage operations +he makes the rain to run off more rapidly than before, and thereby +increases the magnitude of floods and of the destruction caused by +them. (3) By wells, bores, mines, or other subterranean works he +interferes with the underground waters, and consequently with the +discharge of springs. (4) By embanking rivers he confines them to +narrow channels, sometimes increasing their scour, and enabling +them to carry their sediment further seaward, sometimes causing +them to deposit it over the plains and raise their level. (5) By his +engineering operations for water-supply he abstracts water from its +natural basins and depletes the streams.</p> + +<p>In many ways man alters the aspect of a country: (1) By changing +forest into bare mountain, or clothing bare mountains with forest. +(2) By promoting the growth or causing the removal of peat-mosses. +(3) By heedlessly uncovering sand-dunes, and thereby setting in +motion a process of destruction which may convert hundreds of +acres of fertile land into waste sand, or by prudently planting the +dunes with sand-loving vegetation and thus arresting their landward +progress. (4) By so guiding the course of rivers as to make them +aid him in reclaiming waste land, and bringing it under cultivation. +(5) By piers and bulwarks, whereby the ravages of the sea are +stayed, or by the thoughtless removal from the beach of stones +which the waves had themselves thrown up, and which would have +served for a time to protect the land. (6) By forming new deposits +either designedly or incidentally. The roads, bridges, canals, +railways, tunnels, villages and towns with which man has covered +the surface of the land will in many cases form a permanent record +of his presence. Under his hand the whole surface of civilized +countries is very slowly covered with a stratum, either formed +wholly by him or due in great measure to his operations and containing +many relics of his presence. The soil of ancient towns has +been increased to a depth of many feet by their successive destructions +and renovations.</p> + +<p>Perhaps the most subtle of human influences are to be seen in the +distribution of plant and animal life upon the globe. Some of man’s +doings in this domain are indeed plain enough, such as the extirpation +of wild animals, the diminution or destruction of some forms of +vegetation, the introduction of plants and animals useful to himself, +and especially the enormous predominance given by him to the +cereals and to the spread of sheep and cattle. But no such extensive +disturbance of the normal conditions of the distribution of life can +take place without carrying with it many secondary effects, and +setting in motion a wide cycle of change and of reaction in the +animal and vegetable <span class="correction" title="amended from kindgoms">kingdoms</span>. For example, the incessant +warfare waged by man against birds and beasts of prey in districts +given up to the chase leads sometimes to unforeseen results. The +weak game is allowed to live, which would otherwise be killed off +and give more room for the healthy remainder. Other animals +which feed perhaps on the same materials as the game are by the +same cause permitted to live unchecked, and thereby to act as a +further hindrance to the spread of the protected species. But the +indirect results of man’s interference with the régime of plants and +animals still require much prolonged observation.</p> +</div> + +<p class="pt2 center"><span class="sc">Part V.—Geotectonic or Structural Geology</span></p> + +<p>From a study of the nature and composition of minerals and +rocks, and an investigation of the different agencies by which +they are formed and modified, the geologist proceeds to inquire +how these materials have been put together so as to build up the +visible part of the earth’s crust. He soon ascertains that they +have not been thrown together wholly at random, but that they +show a recognizable order of arrangement. Some of them, +especially those of most recent growth, remain in their original +condition and position, but, in proportion to their antiquity, +they generally present increasing alteration, until it may no +longer be possible to tell what was their pristine state. As by +far the largest accessible portion of the terrestrial crust consists of +stratified rocks, and as these furnish clear evidence of most of the +modifications to which they have been subjected in the long +course of geological history, it is convenient to take them into +<span class="pagenum"><a name="page664" id="page664"></a>664</span> +consideration first. They possess a number of structures which +belong to the original conditions in which they were accumulated. +They present in addition other structures which have been superinduced +upon them, and which they share with the unstratified +or igneous rocks.</p> + +<p class="pt2 center sc">1. Original Structures</p> + +<p>(<i>a</i>) <i>Stratified Rocks.</i>—This extensive and important series is +above all distinguished by possessing a prevailing stratified +arrangement. Their materials have been laid down in laminae, +layers and strata, or beds, pointing generally to the intermittent +deposition of the sediments of which they consist. As this +stratification was, as a rule, originally nearly or quite horizontal, +it serves as a base from which to measure any subsequent disturbance +which the rocks have undergone. The occurrence of +false-bedding, <i>i.e.</i> bands of inclined layers between the normal +planes of stratification, does not form any real exception; but +indicates the action of shifting currents whereby the sediment +was transported and thrown down. Other important records of +the original conditions of deposit are supplied by ripple-marks, +sun-cracks, rain-prints and concretions.</p> + +<div class="condensed"> +<p>From the nature of the material further light is cast on the geographical +conditions in which the strata were accumulated. Thus, +conglomerates indicate the proximity of old shore-lines, sandstones +mark deposits in comparatively shallow water, clays and shales +point to the tranquil accumulation of fine silt at a greater depth +and further from land, while fossiliferous limestones bear witness to +clearer water in which organisms flourished at some distance from +deposits of sand and mud. Again, the alternation of different kinds +of sediment suggests a variability in the conditions of deposition, +such as a shifting of the sediment-bearing currents and of the areas +of muddy and clear water. A thick group of conformable strata, +that is, a series of deposits which show no discordance in their +stratification, may usually be regarded as having been laid down on +a sea-floor that was gently sinking. Here and there evidence is +obtainable of the limits or of the progress of the subsidence by what +is called “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.</p> + +<p>But in all speculations of this kind we must bear in mind that the +length of time represented by a given depth of strata is not to be +estimated merely from their thickness or lithological character. +The interval between the deposit of two successive laminae of shale +may have been as long as, or even longer than, that required for +the formation of one of the laminae. In like manner the interval +needed for the transition from one stratum or kind of strata to +another may often have been more than equal to the time required +for the formation of the strata on either side. But the relative +chronological importance of the bars or lines in the geological +record can seldom be satisfactorily discussed merely on lithological +grounds. This must mainly be decided on the evidence of organic +remains, as shown in Part VI., where the grouping of the stratified +rocks into formations and systems is described.</p> +</div> + +<p>(<i>b</i>) <i>Igneous Rocks.</i>—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.</p> + +<p>Considered as part of the architecture of the crust of the earth, +igneous rocks are conveniently divisible into two great series: +(1) those bodies of material which have been injected into the +crust and have solidified there, and (2) those which have reached +the surface and have been ejected there, either in a molten state +as lava or in a fragmental form as dust, ashes and scoriae. The +first of these divisions represents the plutonic, intrusive or +subsequent phase of eruptivity; the second marks the volcanic, +interstratified or contemporaneous phase.</p> + +<div class="condensed"> +<p>1. The plutonic or intrusive rocks, which have been forced into +the crust and have consolidated there, present a wide range of texture +from the most coarse-grained granites to the most perfect natural +glass. Seeing that they have usually cooled with extreme slowness +underground, they are as a general rule more largely crystalline +than the volcanic series. The form assumed by each individual +body of intrusive material has depended upon the shape of the space +into which it has been injected, and where it has cooled and become +solid. This shape has been determined by the local structure of +the earth’s crust on the one hand and by the energy of the eruptive +force on the other. It offers a convenient basis for the classification +of the intrusive rocks, which, as part of the framework of the crust, +may thus be grouped according to the shape of the cavity which +received them, as bosses, sills, dikes and necks.</p> + +<p>Bosses, or stocks, are the largest and most shapeless extravasations +of erupted material. They include the great bodies of granite which, +in most countries of the world, have risen for many miles through +the stratified formations and have altered the rocks around them +by contact-metamorphism. Sills, or intrusive sheets, are bed-like +masses which have been thrust between the planes of sedimentary +or even of igneous rocks. The term laccolite has been applied to +sills which are connected with bosses. Intrusive sheets are distinguishable +from true contemporaneously intercalated lavas by not +keeping always to the same platform, but breaking across and +altering the contiguous strata, and by the closeness of their texture +where they come in contact with the contiguous rocks, which, being +cold, chilled the molten material and caused it to consolidate on its +outer margins more rapidly than in its interior. Dikes or veins +are vertical walls or ramifying branches of intrusive material which +has consolidated in fissures or irregular clefts of the crust. Necks +are volcanic chimneys which have been filled up with erupted +material, and have now been exposed at the surface after prolonged +denudation has removed not only the superficial volcanic masses +originally associated with them, but also more or less of the upper +part of the vents. Plutonic rocks do not present evidence of their +precise geological age. All that can be certainly affirmed from +them is that they must be younger than the rocks into which they +have been intruded. From their internal structure, however, and +from the evidence of the rocks associated with them, some more or +less definite conjectures may be made as to the limits of time within +which they were probably injected.</p> + +<p>2. The interstratified or volcanic series is of special importance +in geology, inasmuch as it contains the records of volcanic action +during the past history of the globe. It was pointed out in Part I. +that while towards the end of the 18th and in the beginning of the +19th century much attention was paid by Hutton and his followers +to the proofs of intrusion afforded by what they called the “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 +<span class="pagenum"><a name="page665" id="page665"></a>665</span> +means of the evidence which they supply, it has been ascertained +that volcanic action has been manifested in the globe since the +earliest geological periods. In the British Isles, for example, the +volcanic record is remarkably full for the long series of ages from +Cambrian to Permian time, and again for the older Tertiary period.</p> +</div> + +<p class="pt2 center sc">2. <span class="sc">Subsequently induced Structures</span></p> + +<p>After their accumulation, whether as stratified or eruptive +masses, all kinds of rocks have been subject to various changes, +and have acquired in consequence a variety of superinduced +structures. It has been pointed out in the part of this article +dealing with dynamical geology that one of the most important +forms of energy in the evolution of geological processes is to be +found in the movements that take place within the crust of the +earth. Some of these movements are so slight as to be only +recognizable by means of delicate instruments; but from this +inferior limit they range up to gigantic convulsions by which +mountain-chains are upheaved. The crust must be regarded as +in a perpetual state of strain, and its component materials are +therefore subject to all the effects which flow from that condition. +It is the one great object of the geotectonic division of geology to +study the structures which have been developed in consequence +of earth-movements, and to discover from this investigation the +nature of the processes whereby the rocks of the crust have been +brought into the condition and the positions in which we now +find them. The details of this subject will be found in separate +articles descriptive of each of the technical terms applied to the +several kinds of superinduced structures. All that need be +offered here is a general outline connecting the several portions +of the subject together.</p> + +<div class="condensed"> +<p>One of the most universal of these later structures is to be seen +in the divisional planes, usually vertical or highly inclined, by which +rocks are split into quadrangular or irregularly shaped blocks. +To these planes the name of joints has been given. They are of +prime importance from an industrial point of view, seeing that the +art of quarrying consists mainly in detecting and making proper +use of them. Their abundance in all kinds of rocks, from those of +recent date up to those of the highest antiquity, affords a remarkable +testimony to the strains which the terrestrial crust has suffered. +They have arisen sometimes from tension, such as that caused by +contraction from the drying and consolidation of an aqueous sediment +or from the cooling of a molten mass; sometimes from torsion +during movements of the crust.</p> + +<p>Although the stratified rocks were originally deposited in a more +or less nearly horizontal position on the floor of the sea, where now +visible on the dry land they are seldom found to have retained their +flatness. On the contrary, they are seen to have been generally +tilted up at various angles, sometimes even placed on end (crop, +dip, strike). When a sufficiently large area of ground is examined, +the inclination into which the strata have been thrown may be +observed not to continue far in the same direction, but to turn over +to the opposite or another quarter. It can then be seen that in +reality the rocks have been thrown into undulations. From the +lowest and flattest arches where the departure from horizontality +may be only trifling, every step may be followed up to intense +curvature, where the strata have been compressed and plicated as +if they had been piles of soft carpets (anticline, syncline, monocline, +geo-anticline, geo-syncline, isoclinal, plication, curvature, quaquaversal). +It has further happened abundantly all over the surface of +the globe that relief from internal strain in the crust has been obtained +by fracture, and the consequent subsidence or elevation of one or +both sides of the fissure. The differential movement between the +two sides may be scarcely perceptible in the feeblest dislocation, +but in the extreme cases it may amount to many thousand feet +(fault, fissure, dislocation, hade, slickensides). The great faults in a +country are among its most important structural features, and as +they not infrequently continue to be lines of weakness in the crust +along which sudden slipping may from time to time take place, they +become the lines of origin of earthquakes. The San Francisco +earthquake of 1906, already cited, affords a memorable illustration +of this connexion.</p> + +<p>It is in a great mountain-chain that the extraordinary complication +of plicated and faulted structures in the crust of the earth can +be most impressively beheld. The combination of overturned folds +with rupture has been already referred to as a characteristic feature +in the Alps (Part IV.). The gigantic folds have in many places been +pushed over each other so as to lie almost flat, while the upper limb +has not infrequently been driven for many miles beyond the lower +by a rupture along the axis. In this way successive slices of a thick +series of formations have been carried northwards on the northern +slope of the Alps, and have been piled so abnormally above each +other that some of their oldest members recur several times on +different thrust-planes, the whole being underlain by Tertiary +strata (see <span class="sc"><a href="#artlinks">Alps</a></span>). Further proof of the colossal compression to +which the rocks have been subjected is afforded by their intense +crumpling and corrugation, and by the abundantly faulted and +crushed condition to which they have been reduced. Similar +evidence as to stresses in the terrestrial crust and the important +changes which they produce among the rocks may also be obtained +on a smaller scale in many non-mountainous countries.</p> + +<p>Another marked result of the compression of the terrestrial crust +has been induced in some rocks by the production of the fissile +structure which is typically shown in roofing-slate (cleavage). +Closely connected with this internal rearrangement has been the +development of microscopic microlites or crystals (rutile, mica, &c.) +in argillaceous slates which were undoubtedly originally fine marine +mud and silt. From this incipient form of metamorphism successive +stages may be traced through the various kinds of argillite and +phyllite into mica-schist, and thence into more crystalline gneissoid +varieties (foliation, slate, mica-schist, gneiss). The Alps afford +excellent illustrations of these transformations.</p> + +<p>The fissures produced in the crust are sometimes clean, sharply +defined divisional planes, like cracks across a pane of glass. Much +more usually, however, the rocks on either side have been broken up +by the friction of movement, and the fault is marked by a variable +breadth of this broken material. Sometimes the walls have separated +and molten rock has risen from below and solidified between them +as a dike. Occasionally the fissures have opened to the surface, +and have been filled in from above with detritus, as in the sandstone-dikes +of Colorado and California. In mineral districts the fissures +have been filled with various spars and ores, forming what are known +as mineral veins.</p> + +<p>Where one series of rocks is covered by another without any +break or discordance in the stratification they are said to be conformable. +But where the older series has been tilted up or visibly +denuded before being overlain by the younger, the latter is termed +unconformable. This relation is one of the greatest value in +structural geology, for it marks a gap in the geological record, which +may represent a vast lapse of time not there recorded by strata.</p> +</div> + +<p class="pt2 center sc">Part VI.—Paleontological Geology</p> + +<p>This division of the science deals with fossils, or the traces +of plants and animals preserved in the rocks of the earth’s crust, +and endeavours to gather from them information as to the history +of the globe and its inhabitants. The term “fossil” (Lat. +<i>fossilis</i>, from <i>fodere</i>, 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.</p> + +<div class="condensed"> +<p>A study of the land-surfaces and sea-floors of the present time +shows that there are so many chances against the conservation +of the remains of either terrestrial or marine animals and plants +that if, as is probable, the same conditions existed in former geological +periods, we should regard the occurrence of organic remains among +the stratified formations of the earth’s crust as generally the result +of various fortunate accidents.</p> + +<p>Let us consider, in the first place, the chances for the preservation +of remains of the present fauna and flora of a country. The surface +of the land may be densely clothed with forest and abundantly +peopled with animal life. But the trees die and moulder into soil. +<span class="pagenum"><a name="page666" id="page666"></a>666</span> +The animals, too, disappear, generation after generation, and leave +few or no perceptible traces of their existence. If we were not aware +from authentic records that central and northern Europe were +covered with vast forests at the beginning of our era, how could we +know this fact? What has become of the herds of wild oxen, the +bears, wolves and other denizens of primeval Europe? How could +we prove from the examination of the surface soil of any country +that those creatures had once abounded there? The conditions for +the preservation of any relics of the plant and animal life of a terrestrial +surface must obviously be always exceptional. They are +supplied only where the organic remains can be protected from the +air and superficial decay. Hence they may be observed in (1) the +deposits on the floors of lakes; (2) in peat-mosses; (3) in deltas at +river-mouths; and (4) under the stalagmite of caverns in limestone +districts. But in these and other favourable places a mere infinitesimal +fraction of the fauna or flora of a land-surface is likely to be +entombed or preserved.</p> + +<p>In the second place, although in the sea the conditions for the +preservation of organic remains are in many respects more favourable +than on land, they are apt to be frustrated by many adverse circumstances. +While the level of the land remains stationary, there can +be but little effective entombment of marine organisms in littoral +deposits; for only a limited accumulation of sediment will be formed +until subsidence of the sea-floor takes place. In the trifling beds of +sand or gravel thrown up on a stationary shore, only the harder and +more durable forms of life, such as gastropods and lamellibranchs, +which can withstand the triturating effects of the beach waves, are +likely to remain uneffaced.</p> + +<p>Below tide-marks, along the margin of the land where sediment +is gradually deposited, the conditions are more favourable for the +preservation of marine organisms. In the sheets of sand and mud +there laid down the harder parts of many forms of life may be +entombed and protected from decay. But only a small proportion +of the total marine fauna may be expected to appear in such deposits. +At the best, merely littoral and shallow-water forms will occur, and, +even under the most favourable conditions, they will represent but +a fraction of the whole assemblage of life in these juxta-terrestrial +parts of the ocean. As we recede from the land the rate of deposition +of sediment on the sea-floor must become feebler, until, in the remote +central abysses, it reaches a hardly appreciable minimum. Except, +therefore, where some kind of ooze or other deposit is accumulating +in these more pelagic regions, the conditions must be on the whole +unfavourable for the preservation of any adequate representation +of the deep-sea fauna. Hard durable objects, such as teeth and +bones, may slowly accumulate, and be protected by a coating of +peroxide of manganese, or of some of the silicates now forming here +and there over the deep-sea bottom; or the rate of growth of the +abysmal deposit may be so tardy that most of the remains of at +least the larger animals will disappear, owing to decay, before they +can be covered up and preserved. Any such deep-sea formation, +if raised into land, would supply but a meagre picture of the whole +life of the sea.</p> + +<p>It would thus appear that the portion of the sea-floor best suited +for receiving and preserving the most varied assemblage of marine +organic remains is the area in front of the land, to which rivers and +currents bring continual supplies of sediment. The most favourable +conditions for the accumulation of a thick mass of marine fossiliferous +strata will arise when the area of deposit is undergoing a gradual +subsidence. If the rate of depression and that of deposit were equal, +or nearly so, the movement might proceed for a vast period without +producing any great apparent change in marine geography, and even +without seriously affecting the distribution of life over the sea-floor +within the area of subsidence. Hundreds or thousands of feet of +sedimentary strata might in this way be heaped up round the continents, +containing a fragmentary series of organic remains belonging +to those forms of comparatively shallow-water life which had hard +parts capable of preservation. There can be little doubt that such +has, in fact, been the history of the main mass of stratified formations +in the earth’s crust. By far the largest proportion of these piles +of marine strata has unquestionably been laid down in water of no +great depth within the area of deposit of terrestrial sediment. +The enormous thickness to which they attain seems only explicable +by prolonged and repeated movements of subsidence, interrupted, +however, as we know, by other movements of a contrary kind.</p> + +<p>Since the conditions for the preservation of organic remains exist +more favourably under the sea than on land, marine organisms must +be far more abundantly conserved than those of the land. This is +true to-day, and has, as far as known, been true in all past geological +time. Hence for the purposes of the geologist the fossil remains of +marine forms of life far surpass all others in value. Among them +there will necessarily be a gradation of importance, regulated chiefly +by their relative abundance. Now, of all the marine tribes which +live within the juxta-terrestrial belt of sedimentation, unquestionably +the Mollusca stand in the place of pre-eminence as regards their +aptitude for becoming fossils. They almost all possess a hard, durable +shell, capable of resisting considerable abrasion and readily passing +into a mineralized condition. They are extremely abundant both as +to individuals and genera. They occur on the shore within tide +mark, and range thence down into the abysses. Moreover, they +appear to have possessed these qualifications from early geological +times. In the marine Mollusca, therefore, we have a common ground +of comparison between the stratified formations of different periods. +They have been styled the alphabet of palaeontological inquiry.</p> +</div> + +<p>There are two main purposes to which fossils may be put in +geological research: (1) to throw light upon former conditions +of physical geography, such as the presence of land, rivers, +lakes and seas, in places where they do not now exist, changes +of climate, and the former distribution of plants and animals; +and (2) to furnish a guide in geological chronology whereby +rocks may be classified according to relative date, and the facts +of geological history may be arranged and interpreted as a +connected record of the earth’s progress.</p> + +<div class="condensed"> +<p>1. As examples of the first of these two directions of inquiry +reference may be made to (<i>a</i>) former land-surfaces revealed by the +occurrence of layers of soil with tree-stumps and roots still in the +position of growth (see <span class="sc"><a href="#artlinks">Purbeckian</a></span>); (<i>b</i>) ancient lakes proved by +beds of marl or limestone full of lacustrine shells; (<i>c</i>) old sea-bottoms +marked by the occurrence of marine organisms; (<i>d</i>) variations in +the quality of the water, such as freshness or saltness, indicated by +changes in the size and shape of the fossils; (<i>e</i>) proximity to former +land, suggested by the occurrence of abundant drift-wood in the +strata; (<i>f</i>) former conditions of climate, different from the present, +as evidenced by such organisms as tropical types of plants and +animals intercalated among the strata of temperate or northern +countries.</p> + +<p>2. In applying fossils to the determination of geological chronology +it is first necessary to ascertain the order of superposition of the +rocks. Obviously, in a continuous series of undisturbed sedimentary +deposits the lowest must necessarily be the oldest, and the plants or +animals which they contain must have lived and died before any of +the organisms that occur in the overlying strata. This order of +superposition having been settled in a series of formations, it is +found that the fossils at the bottom are not quite the same as those +at the top of the series. Tracing the beds upward, we discover that +species after species of the lowest platforms disappears, until perhaps +not one of them is found. With the cessation of these older species +others make their entrance. These, in turn, are found to die out, +and to be replaced by newer forms. After patient examination of +the rocks, it has been ascertained that every well-marked “formation,” +or group of strata, is characterized by its own species or +genera, or by a general assemblage, or <i>facies</i>, of organic forms. +Such a generalization can only, of course, be determined by actual +practical experience over an area of some size. When the typical +fossils of a formation are known, they serve to identify that formation +in its progress across a country. Thus, in tracts where the true +order of superposition cannot be determined, owing to the want of +sections or to the disturbed condition of the rocks, fossils serve as a +means of identification and furnish a guide to the succession of the +rocks. They even demonstrate that in some mountainous ground +the beds have been turned completely upside down, where it +can be shown that the fossils in what are now the uppermost +strata ought properly to lie underneath those in the beds below +them.</p> + +<p>It is by their characteristic fossils that the stratified rocks of the +earth’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 <i>homotaxis</i>, a term which expresses the fact that the order +in which the leading types of organized existence have appeared +upon the earth has been similar even in widely separated regions. +It is evident that, in this way, a reliable method of comparison +is furnished, whereby the stratified formations of different parts of +the earth’s crust can be brought into relation with each other. +Had the geologist continued to remain, as in the days of Werner, +hampered by the limitations imposed by a reliance on mere lithological +characters, he would have made little or no progress in +deciphering the record of the successive phases of the history of +the globe chronicled in the crust. Just as, at the present time, +sheets of gravel in one place are contemporaneous with sheets of +mud at another, so in the past all kinds of sedimentation have been +in progress simultaneously, and those of one period may not be +distinguishable in themselves from those of another. Little or no +reliance can be placed upon lithological resemblances or differences +in comparing the sedimentary formations of different countries.</p> + +<p>In making use of fossil evidence for the purpose of subdividing +the stratified rocks of the earth’s crust, it is found to be applicable +to the smaller details of stratigraphy as well as to the definition of +large groups of strata. Thus a particular stratum may be marked +by the occurrence in it of various fossils, one or more of which may +be distinctive, either from occurring in no other bed above and +below or from special abundance in that stratum. One or more of +these species is therefore used as a guide to the occurrence of the bed +<span class="pagenum"><a name="page667" id="page667"></a>667</span> +in question, which is called by the name of the most abundant +species. In this way what is called a “geological horizon,” or +“zone,” is marked off, and its exact position in the series of formations +is fixed.</p> + +<p>Perhaps the most distinctive feature in the progress of palaeontological +geology during the last half century has been the recognition +and wide application of this method of zonal stratigraphy, which, +in itself, was only a further development of William Smith’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 <i>Die Juraformation Englands, Frankreichs und +des südwestlichen Deutschlands</i> (1856-1858) marked an epoch in the +development of stratigraphical geology. Combining what had been +done by various observers with his own laborious researches in +France, England, Württemberg and Bavaria, he drew up a classification +of the Jurassic system, grouping its several formations into zones, +each characterized by some distinctly predominant fossil after which +it was named (see <span class="sc"><a href="#artlinks">Lias</a></span>). The same method of classification was +afterwards extended to the Cretaceous series by A.D. d’Orbigny, +E. Hébert and others, until the whole Mesozoic rocks from the +Trias to the top of the Chalk has now been partitioned into zones, +each named after some characteristic species or genus of fossils. +More recently the principle has been extended to the Palaeozoic +formations, though as yet less fully than to the younger parts of the +geological record. It has been successfully applied by Professor C. +Lapworth to the investigation of the Silurian series (see <span class="sc"><a href="#artlinks">Silurian</a></span>; +<span class="sc"><a href="#artlinks">Ordovician System</a></span>). He found that the species of graptolites +have each a comparatively narrow vertical range, and they may +consequently be used for stratigraphical purposes. Applying the +method, in the first instance, to the highly plicated Silurian rocks of +the south of Scotland, he found that by means of graptolites he was +able to work out the structure of the ground. Each great group of +strata was seen to possess its own graptolitic zones, and by their +means could be identified not only in the original complex Scottish +area, but in England and Wales and in Ireland. It was eventually +ascertained that the succession of zones in Great Britain could be +recognized on the Continent, in North America and even in Australia. +The brachiopods and trilobites have likewise been made use of for +zonal purposes among the oldest sedimentary formations. The +most ancient of the Palaeozoic systems has as its fitting base the +<i>Olenellus</i> zone.</p> + +<p>Within undefined and no doubt variable geographical limits +palaeontological zones have been found to be remarkably persistent. +They follow each other in the same general order, but not always +with equal definiteness. The type fossil may appear in some districts +on a higher or a lower platform than it does in others. Only to a +limited degree is there any coincidence between lithological variations +in the strata and the sequence of the zones. In the Jurassic formations, +indeed, where frequent alternations of different sedimentary +materials are to be met with, it is in some cases possible to trace a +definite upward or downward limit for a zone by some abrupt +change in the sedimentation, such as from limestone to shale. But +such a precise demarcation is impossible where no distinct bands of +different sediments are to be seen. The zones can then only be +vaguely determined by finding their characteristic fossils, and noting +where these begin to appear in the strata and where they cease. +It would seem, therefore, that the sequence of palaeontological +zones, or life-horizons, has not depended merely upon changes in +the nature of the conditions under which the organisms lived. We +should naturally expect that these changes would have had a marked +influence; that, for instance, a difference should be perceptible +between the character of the fossils in a limestone and that of those +in a shale or a sandstone. The environment, when a limestone was +in course of deposition, would generally be one of clear water, +favourable for a more vigorous and more varied fauna than where +a shale series was accumulating, when the water would be discoloured, +and only such animals would continue to live in it, or on +the bottom, as could maintain themselves in the midst of mud. +But no such lithological reason, betokening geographical changes +that would affect living creatures, can be adduced as a universally +applicable explanation of the occurrence and limitation of palaeontological +zones. One of these zones may be only a few inches, or +feet or yards in vertical extent, and no obvious lithological or other +cause can be seen why its specially characteristic fossils should +not be found just as frequently in the similar strata above and +below. There is often little or no evidence of any serious change +in the conditions of sedimentation, still less of any widespread +physical disturbance, such as the catastrophes by which the +older geologists explained the extinction of successive types of +life.</p> + +<p>It has been suggested that, where the life-zones are well defined, +sedimentation has been extremely slow, and that though these zones +follow each other with no break in the sedimentation, they were +really separated by prolonged intervals of time during which organic +evolution could come effectively into play. But it is not easy to +explain how, for example in the Lower Lias, there could have been +a succession of prodigious intervals, when practically no sediment +was laid down, and yet that the strata should show no sign of contemporaneous +disturbance or denudation, but succeed each other +as if they had been accumulated by one continuous process of +deposit. It must be admitted that the problem of life-zones in +stratigraphical geology has not yet been solved.</p> + +<p>As Darwin first cogently showed, the history of life has been very +imperfectly registered in the stratified parts of the earth’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.</p> + +<p>While the mere fact that one series of rocks lies unconformably +on another proves the lapse of a considerable interval between their +respective dates, the relative length of this interval may sometimes +be proved by means of fossil evidence, and by this alone. Let us +suppose, for example, that a certain group of formations has been +disturbed, upraised, denuded and covered unconformably by a +second group. In lithological characters the two may closely resemble +each other, and there may be nothing to show that the gap represented +by their unconformability is of an important character. In +many cases, indeed, it would be quite impossible to pronounce any +well-grounded judgment as to the amount of interval, even measured +by the vague relative standards of geological chronology. But if +each group contains a well-preserved suite of organic remains, it +may not only be possible, but easy, to say exactly how much of the +geological record has been left out between the two sets of formations. +By comparing the fossils with those obtained from regions where the +geological record is more complete, it may be ascertained, perhaps, +that the lower rocks belong to a certain platform or stage in geological +history which for our present purpose we may call D, and that the +upper rocks can in like manner be paralleled with stage H. It would +be then apparent that at this locality the chronicles of three great +geological periods E, F, and G were wanting, which are elsewhere +found to be intercalated between D and H. The lapse of time represented +by this unconformability would thus be equivalent to that +required for the accumulation of the three missing formations in +those regions where sedimentation was more continuous.</p> + +<p>Fossil evidence may be made to prove the existence of gaps which +are not otherwise apparent. As has been already remarked, changes +in organic forms must, on the whole, have been extremely slow in +the geological past. The whole species of a sea-floor could not pass +entirely away, and be replaced by other forms, without the lapse +of long periods of time. If then among the conformable stratified +formations of former ages we encounter sudden and abrupt changes +in the <i>facies</i> of the fossils, we may be certain that these must mark +omissions in the record, which we may hope to fill in from a more +perfect series elsewhere. The complete biological contrasts between +the fossil contents of unconformable strata are sufficiently explicable. +It is not so easy to give a satisfactory account of those which occur +where the beds are strictly conformable, and where no evidence can +be observed of any considerable change of physical conditions at the +time of deposit. A group of strata having the same general lithological +characters throughout may be marked by a great discrepance +between the fossils above and below a certain line. A few species +may pass from the one into the other, or perhaps every species may +be different. In cases of this kind, when proved to be not merely +local but persistent over wide areas, we must admit, notwithstanding +the apparently undisturbed and continuous character of the original +deposition of the strata, that the abrupt transition from the one <i>facies</i> +of fossils to the other represents a long interval of time which has not +been recorded by the deposit of strata. A.C. Ramsay, who called +attention to these gaps, termed them “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.</p> + +<p>From all these facts it is clear that the geological record, as it now +exists, is at the best but an imperfect chronicle of geological history. +In no country is it complete. The lacunae of one region must be +supplied from another. Yet in proportion to the geographical +distance between the localities where the gaps occur and those +whence the missing intervals are supplied, the element of uncertainty +in our reading of the record is increased. The most desirable +method of research is to exhaust the evidence for each area or +province, and to compare the general order of its succession as a +whole with that which can be established for other provinces.</p> +</div> + +<p><span class="pagenum"><a name="page668" id="page668"></a>668</span></p> + +<p class="pt2 center sc">Part VII.—Stratigraphical Geology</p> + +<p>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.</p> + +<p>The leading principles of stratigraphy may be summed up +as follows:</p> + +<p>1. In every stratigraphical research the fundamental requisite +is to establish the order of superposition of the strata. Until +this is accomplished it is impossible to arrange the dates, and +make out the sequence of geological history.</p> + +<p>2. The stratified portion of the earth’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 <i>zone</i> or <i>horizon</i>, +and usually bears the name of one of its more characteristic +fossils, as the <i>Planorbis</i>-zone of the Lower Lias, which is so +called from the prevalence in it of the ammonite <i>Psiloceras +planorbis</i>. Two or more such zones related to each other by the +possession of a number of the same characteristic species or +genera have been designated <i>beds</i> or an <i>assise</i>. Two or more +sets of beds or assises similarly related form a <i>group</i> or <i>stage</i>; a +number of groups or stages make a <i>series</i>, <i>formation</i> or <i>section</i>, +and a succession of formations may be united into a <i>system</i>.</p> + +<p>3. Some living species of plants and animals can be traced +downwards through the more recent geological formations; +but the number which can be so followed grows smaller as the +examination is pursued into more ancient deposits. With their +disappearance other species or genera present themselves which +are no longer living. These in turn may be traced backward into +earlier formations, till they too cease and their places are taken by +yet older forms. It is thus shown that the stratified rocks contain +the records of a gradual progression of organic forms. A species +which has once died out does not seem ever to have reappeared.</p> + +<p>4. When the order of succession of organic remains among the +stratified rocks has been determined, they become an invaluable +guide in the investigation of the relative age of rocks and the +structure of the land. Each zone and formation, being characterized +by its own species or genera, may be recognized by their +means, and the true succession of strata may thus be confidently +established even in a country wherein the rocks have been +shattered by dislocation, folded, inverted or metamorphosed.</p> + +<p>5. Though local differences exist in regard to the precise zone +in which a given species of organism may make its first appearance, +the general order of succession of the organic forms found in the +rocks is never inverted. The record is nowhere complete in any +region, but the portions represented, even though extremely +imperfect, always follow each other in their proper chronological +order, unless where disturbance of the crust has intervened to +destroy the original sequence.</p> + +<p>6. The relative chronological value of the divisions of the +geological record is not to be measured by mere depth of strata. +While it may be reasonably assumed that, in general, a great +thickness of stratified rock must mark the passage of a long +period of time, it cannot safely be affirmed that a much less +thickness elsewhere must represent a correspondingly diminished +period. The need for this caution may sometimes be made +evident by an unconformability between two sets of rocks, as +has already been explained. The total depth of both groups +together may be, say 1000 ft. Elsewhere we may find a single +unbroken formation reaching a depth of 10,000 ft.; but it would +be unwarrantable to assume that the latter represents ten times +the length of time indicated by the former two. So far from +this being the case, it might not be difficult to show that the +minor thickness of rock really denotes by far the longer geological +interval. If, for instance, it could be proved that the upper +part of both the sections lies on one and the same geological +platform, but that the lower unconformable series in the one +locality belongs to a far lower and older system of rocks than the +base of the thick conformable series in the other, then it would +be clear that the gap marked by the unconformability really +indicates a longer period than the massive succession of deposits.</p> + +<p>7. Fossil evidence furnishes the chief means of comparing the +relative value of formations and groups of rock. A “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 <i>facies</i> or general character of the fossils on each side. +If, for example, in one case we find every species to be dissimilar +above and below a certain horizon, while in another locality only +half of the species on each side are peculiar, we naturally infer, +if the total number of species seems large enough to warrant +the inference, that the interval marked by the former break +was much longer than that marked by the second. But we may +go further and compare by means of fossil evidence the relation +between breaks in the succession of organic remains and the +depth of strata between them.</p> + +<div class="condensed"> +<p>Three formations of fossiliferous strata, A, C, and H, may occur +conformably above each other. By a comparison of the fossil +contents of all parts of A, it may be ascertained that, while some +species are peculiar to its lower, others to its higher portions, yet the +majority extend throughout the formation. If now it is found that +of the total number of species in the upper portion of A only one-third +passes up into C, it may be inferred with some plausibility that the +time represented by the break between A and C was really longer +than that required for the accumulation of the whole of the formation +A. It might even be possible to discover elsewhere a thick intermediate +formation B filling up the gap between A and C. In like +manner were it to be discovered that, while the whole of the formation +C is characterized by a common suite of fossils, not one of the species +and only one half of the genera pass up into H, the inference could +hardly be resisted that the gap between the two formations marks +the passage of a far longer interval than was needed for the deposition +of the whole of C. And thus we reach the remarkable conclusion +that, thick though the stratified formations of a country may be, +in some cases they may not represent so long a total period of time +as do the gaps in their succession,—in other words, that non-deposition +was more frequent and prolonged than deposition, or that the +intervals of time which have been recorded by strata have not been +so long as those which have not been so recorded.</p> +</div> + +<p>In all speculations of this nature, however, it is necessary +to reason from as wide a basis of observation as possible, seeing +that so much of the evidence is negative. Especially needful +is it to bear in mind that the cessation of one or more species +at a certain line among the rocks of a particular district may +mean nothing more than that, onward from the time marked +by that line, these species, owing to some change in the conditions +of life, were compelled to migrate or became locally extinct or, +from some alteration in the conditions of fossilization, were no +longer imbedded and preserved as fossils. They may have +continued to flourish abundantly in neighbouring districts for +a long period afterward. Many examples of this obvious +truth might be cited. Thus in a great succession of mingled +marine, brackish-water and terrestrial strata, like that of the +Carboniferous Limestone series of Scotland, corals, crinoids +<span class="pagenum"><a name="page669" id="page669"></a>669</span> +and brachiopods abound in the limestones and accompanying +shales, but disappear as the sandstones, ironstones, clays, coals +and bituminous shales supervene. An observer meeting for the +first time with an instance of this disappearance, and remembering +what he had read about breaks in succession, might be +tempted to speculate about the extinction of these organisms, +and their replacement by other and later forms of life, such as +the ferns, lycopods, estuarine or fresh-water shells, ganoid +fishes and other fossils so abundant in the overlying strata. +But further research would show him that high above the plant-bearing +sandstones and coals other limestones and shales might +be observed, once more charged with the same marine fossils +as before, and still farther overlying groups of sandstones, coals +and carbonaceous beds followed by yet higher marine limestones. +He would thus learn that the same organisms, after being +locally exterminated, returned again and again to the same +area. After such a lesson he would probably pause before too +confidently asserting that the highest bed in which we can +detect certain fossils marks their final appearance in the history +of life. Some breaks in the succession may thus be extremely +local, one set of organisms having been driven to a different part +of the same region, while another set occupied their place until +the first was enabled to return.</p> + +<p>8. The geological record is at the best but an imperfect +chronicle of the geological history of the earth. It abounds +in gaps, some of which have been caused by the destruction of +strata owing to metamorphism, denudation or otherwise, others +by original non-deposition, as above explained. Nevertheless +from this record alone can the progress of the earth be traced. +It contains the registers of the appearance and disappearance +of tribes of plants and animals which have from time to time +flourished on the earth. Only a small proportion of the total +number of species which have lived in past time have been thus +chronicled, yet by collecting the broken fragments of the record +an outline at least of the history of life upon the earth can be +deciphered.</p> + +<p>It cannot be too frequently stated, nor too prominently kept +in view, that, although gaps occur in the succession of organic +remains as recorded in the rocks, they do not warrant the conclusion +that any such blank intervals ever interrupted the progress +of plant and animal life upon the globe. There is every reason +to believe that the march of life has been unbroken, onward and +upward. Geological history, therefore, if its records in the +stratified formations were perfect, ought to show a blending +and gradation of epoch with epoch. But the progress has been +constantly interrupted, now by upheaval, now by volcanic +outbursts, now by depression. These interruptions serve as +natural divisions in the chronicle, and enable the geologist to +arrange his history into periods. As the order of succession +among stratified rocks was first made out in Europe, and as many +of the gaps in that succession were found to be widespread over +the European area, the divisions which experience established +for that portion of the globe came to be regarded as typical, +and the names adopted for them were applied to the rocks of +other and far distant regions. This application has brought out +the fact that some of the most marked breaks in the European +series do not exist elsewhere, and, on the other hand, that some +portions of that series are much more complete than the corresponding +sections in other regions. Hence, while the general +similarity of succession may remain, different subdivisions and +nomenclature are required as we pass from continent to continent.</p> + +<p>The nomenclature adopted for the subdivisions of the geological +record bears witness to the rapid growth of geology. It is a +patch-work in which no system nor language has been adhered +to, but where the influences by which the progress of the science +has been moulded may be distinctly traced. Some of the earliest +names are lithological, and remind us of the fact that mineralogy +and petrography preceded geology in the order of birth—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.</p> + +<p>But whatever be the name chosen to designate a particular +group of strata, it soon comes to be used as a chronological or +homotaxial term, apart altogether from the stratigraphical +character of the strata to which it is applied. Thus we speak +of the Chalk or Cretaceous system, and embrace under that +term formations which may contain no chalk; and we may +describe as Silurian a series of strata utterly unlike in lithological +characters to the formations in the typical Silurian country. +In using these terms we unconsciously allow the idea of relative +date to arise prominently before us. Hence such a word as +“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.</p> + +<p>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.</p> + +<p class="pt2 center sc">Part VIII.—Physiographical Geology</p> + +<p>This department of geological inquiry investigates the origin +and history of the present topographical features of the land. +As these features must obviously be related to those of earlier +time which are recorded in the rocks of the earth’s crust, they +cannot be satisfactorily studied until at least the main outlines +of the history of these rocks have been traced. Hence physiographical +research comes appropriately after the other branches +of the science have been considered.</p> + +<p>From the stratigraphy of the terrestrial crust we learn that +by far the largest part of the area of dry land is built up of marine +formations; and therefore that the present land is not an +aboriginal portion of the earth’s surface, but has been overspread +by the sea in which its rocks were mainly accumulated. We +further discover that this submergence of the land did not +happen once only, but again and again in past ages and in all +parts of the world. Yet although the terrestrial areas varied +much from age to age in their extent and in their distribution, +being at one time more continental, at another more insular, +there is reason to believe that these successive diminutions and +expansions have on the whole been effected within, or not far +outside, the limits of the existing continents. There is no +evidence that any portion of the present land ever lay under the +deeper parts of the ocean. The abysmal deposits of the ocean-floor +have no true representatives among the sedimentary +formations anywhere visible on the land. Nor, on the other +hand, can it be shown that any part of the existing ocean +abysses ever rose above sea-level into dry land. Hence geologists +have drawn the inference that the ocean basins have probably +been always where they now are; and that although the continental +areas have often been narrowed by submergence and by +denudation, there has probably seldom or never been a complete +disappearance of land. The fact that the sedimentary formations +of each successive geological period consist to so large an +extent of mechanically formed terrigenous detritus, affords +good evidence of the coexistence of tracts of land as well as of +extensive denudation.</p> + +<p><span class="pagenum"><a name="page670" id="page670"></a>670</span></p> + +<p class="pt2 center"><i>The Geological Record or Order of Succession of the Stratified +Formations of the Earth’s Crust.</i></p> + +<table class="nobctr f90" summary="Contents"> +<tr><td class="tccm allb"> </td> <td class="tccm allb"> </td> <td class="tccm allb" colspan="2">Europe.</td> <td class="tccm allb">North America.</td></tr> + +<tr><td class="tccm allb" rowspan="2">Quaternary<br />or<br />Post-Tertiary.</td> + +<td class="tccm allb">Recent,<br />Post-glacial<br />or Human.</td> + +<td class="tcl rb bb" style="width: 40%;" colspan="2"><p>Historic, up to the present time.</p> +<p>Prehistoric, comprising deposits of the Iron, Bronze, and later Stone Ages.</p> +<p>Neolithic—alluvium, peat, lake-dwellings, loess, &c.</p> +<p>Palaeolithic—river-gravels, cave-deposits, &c.</p></td> + +<td class="tcl rb bb" style="width: 40%;"><p>Similar to the European development, but with scantier traces of the presence of man.</p></td></tr> + +<tr><td class="tccm allb">Pleistocene or Glacial.</td> + +<td class="tcl rb bb" colspan="2"><p>Older Loess and valley-gravels; cave-deposits.</p> +<p>Strand-lines or raised beaches; youngest moraines.</p> +<p>Upper Boulder-clays; eskers; marine sands and clays.</p> +<p>Interglacial deposits.</p> +<p>Lower boulder-clay or Till, with striated rock-surfaces below.</p></td> + +<td class="tcl rb bb"><p>As in Europe, it is hardly possible to assign a definite chronological place to +each of the various deposits of this period, terrestrial and marine. They generally resemble the +European series. The characteristic marine, fluviatile and lacustrine terraces, which +overlie the older drifts, have been classed as the Champlain Group.</p></td></tr> + +<tr><td class="tccm allb" rowspan="4">Cainozoic or Tertiary.</td> + +<td class="tccm allb">Pliocene.</td> + +<td class="tcl allb" colspan="2"><p>Newer:—English Forest-Bed Group; Red and Norwich Crag; Amstelian and Scaldesian groups + of Belgium and Holland; Sicilian and Astian of France and Italy.</p> +<p>Older:—English Coralline Crag; Diestian of Belgium; Plaisancian of southern France and Italy.</p></td> + +<td class="tcl allb"><p>On the Atlantic border represented by the marine Floridian series; in the interior +by a subaerial and lacustrine series; and on the Pacific border by the thick marine series of San Francisco.</p></td></tr> + +<tr><td class="tccm allb">Miocene.</td> + +<td class="tcl allb" colspan="2"><p>Wanting in Britain; well developed in France, S. E. Europe and Italy; divisible +into the following groups in descending order: (1) Pontian; (2) Sarmatian; (3) Tortonian; (4) Helvetian; +(5) Langhian (Burdigalian).</p></td> + +<td class="tcl allb"><p>Represented in the Eastern States by a marine series (Yorktown or Chesapeake, Chipola +and Chattahoochee groups), and in the interior by the lacustrine Loup Fork (Nebraska), Deep +River, and John Day groups.</p></td></tr> + +<tr><td class="tccm allb">Oligocene.</td> + +<td class="tcl allb" colspan="2"><p>In Britain the “fluvio-marine series” of the Isle of Wight; +also the volcanic plateaux of Antrim and Inner Hebrides and those of the Faeroe Isles and Iceland. In +continental Europe the following subdivisions have been established in descending order: +(1) Aquitanian, (2) Stampian (Rupelian), (3) Tongrain (Sannoisian).</p></td> + +<td class="tcl allb"><p>On the Atlantic border no equivalents have been satisfactorily +recognised, but on the Pacific side there are marine deposits in N. W. Oregon, which +may represent this division. In the interior the equivalent is believed to be the fresh-water +White River series, including (1) <i>Protoceras</i> beds, (2) <i>Oreodon</i> beds, +and (3) <i>Titanothervum</i> beds.</p></td></tr> + +<tr><td class="tccm allb">Eocene.</td> + +<td class="tcl allb" colspan="2"><p>Barton sands and clays; Ludian series of France.</p> +<p>Bracklesham Beds; Lutetian (Calcaire grossier and Caillasses) of Paris basin.</p> +<p>London clay, Woolwich and Reading Beds; Thanet sands; Ypresian or Londinian of N. France and Belgium; + Sparnacian and Thanetian groups</p></td> + +<td class="tcl allb"><p>Woodstock and Aquia Creek groups of Potomac River; Vicksburg, Jackson, + Claiborne, Buhrstone, and Lignitic groups of Mississippi.</p> +<p>In the interior a thick series of fresh-water formations, comprising, in descending order, + the Uinta, Bridger, Wind River, Wasatch, Torrejon, and Puerco groups.</p> +<p>On the Pacific side the marine Tejon series of Oregon and California.</p></td></tr> + +<tr><td class="tccm allb" rowspan="5">Mesozoic or Secondary.</td> + +<td class="tccm cl bb">Cretaceous. Upper.</td> + +<td class="tcl bb" colspan="2"><p>Danian—wanting in Britain; uppermost limestone of Denmark.</p> +<p>Senonian—Upper Chalk with Flints of England; Aturian and Emscherian stages on the European continent.</p> +<p>Turonian—Middle Chalk with few flints, and comprising the Angoumian and Ligerian stages.</p> +<p>Cenomanian—Lower Chalk and Chalk Marl.</p> +<p>Albian—Upper Greensand and Gault.</p></td> + +<td class="tcl allb" rowspan="3"><p>On the Atlantic border both marine strata and others containing a + terrestrial flora represent the Cretaceous series of formations.</p> +<p>In the interior there is also a commingling of marine with lacustrine deposits. At the top lies the + Laramie or Lignitic series with an abundant terrestrial flora, passing down into the lacustrine + and brackish-water Montana series. Of older date, the Colorado series contains an abundant + marine fauna, yet includes also some Niobrara marls and limestones are likewise of marine + origin, but the lower members of the series (Benton and Dakota) show another great representation of + fresh-water sedimentation with lignites and coals.</p> +<p>In California a vast succession of marine deposits (Shasta-Chico) represents the Cretaceous system; + and in western British N. America coal-seams also occur.</p></td></tr> + +<tr><td class="tccm cl">Cretaceous. Lower.</td> + +<td class="tcl" colspan="2"><p>Aptian—Lower Greensand; Marls and limestones of Provence, &c.</p> +<p>Urgonian (Barremian)—Atherfield clay; massive Hippurite limestones of southern France.</p> +<p>Neocomian—Weald clay and Hastings sand; Hauterivian and Valanginian sub-stages of + Switzerland and France.</p></td></tr> + +<tr><td colspan="2"> </td></tr> + +<tr><td class="tccm allb">Jurassic.</td> + +<td class="tcl allb" colspan="2"><p>Purbeckian—Purbeck beds; Münder Mergel; largely present in Westphalia.</p> +<p>Portlandian—Portland group of England, represented in S. France by the thick Tithonian limestones.</p> +<p>Kimmeridgian— Kimmeridge Clay of England; Virgulian and Pterocerian groups of + N. France; represented by thick limestones in the Mediterranean basin.</p> +<p>Corallian—Coral Rag, Coralline Oolite; Sequanian stages of the Continent, + comprising the sub-stages of Astartian and Rauracian.</p> +<p>Oxfordian—Oxford Clay; Axgovian and Neuvizyan stages.</p> +<p>Callovian—Kellaways Rock, Divesian sub-stage of N. France.</p> +<p>Bathonian—series of English strata from Cornbrash down to Fuller’s Earth.</p> +<p>Bajocian—Inferior Oolite of England.</p> +<p>Lassic—divisible into (1) Upper Lias or Toarcian, (2) Middle Lias, Marlstone or Charmouthian, (3) Lower + Lias of Sinemurian and Hettangian.</p></td> + +<td class="tcl allb"><p>Representatives of the Middle and lower Jurassic formations have been found in + California and Oregon, and farther north among the Arctic islands.</p> +<p>Strata containing Lower Jurassic marine fossils appear in Wyoming and Dakota; and above them come + the <i>Atlantosaurus</i> and <i>Baptanodon</i> beds, which have yielded so large a + variety of deinosaurs and other vertebrates, and especially the remains of a number of genera + of small mammals.</p></td></tr> + +<tr><td class="tccm allb">Triassic.</td> + +<td class="tcl allb" colspan="2"><p>In Germany and western Europe this division represents the deposits of + inland seas or lagoons, and is divisible into the following stages in descending + order: (1) Rhaetic, (2) Keuper, (3) Muschelkalk, (4) Bunter. In the eastern Alps and the Mediterranean + basin the contemporaneous sedimentary formations are those of open clear sea, in which a thickness of many + thousand feet of strata was accumulated.</p></td> + +<td class="tcl allb"><p>In New York, Connecticut, New Brunswick, and Nova Scotia a series of red sandstone + (Newark series) contains land-plants and labyrinthodonts like the lagoon type of central + and western Europe. On the Pacific slope, however, marine equivalents occur, representing + the pelagic type of south-eastern Europe.</p></td></tr> + +<tr><td class="tccm allb" rowspan="9">Palaeozoic or Primary.</td> + +<td class="tccm allb">Permian.</td> + +<td class="tcl allb" colspan="2"><p>Thuringian—Zechstein, Magnesian Limestone; named from its development + in Thuringia; well represented also in Saxony, Bavaria and Bohemia.</p> +<p>Saxonian—Rothliegendes Group; Red Sandstones, &c.</p> +<p>Autunian—where the strata present the lagoon facies, well displayed at Autun + in France; where the marine type is predominant, as in Russia, the group has been termed Artinskian.</p></td> + +<td class="tcl allb"><p>To this division of the geological record the Upper Barren + Measures of the coal-fields of Pennsylvania, Prince Edward Island, Nova Scotia and + New Brunswick have been assigned.</p> +<p>Farther south in Kansas, Texas, and Nebraska the representatives of the division have an + abundant marine fauna.</p></td></tr> + +<tr><td class="tccm allb">Carboniferous.</td> + +<td class="tcl allb" colspan="2"><p>Stephanian or Uralian—represented in Russia by marine formations, and in + central and western Europe by numerous small basins containing a peculiar + flora and in some places a great variety of insects.</p> +<p>Westphalian or Moscovian—Coal-measures, Millstone Grit.</p> +<p>Culm or Dinantian—Carboniferous Limestone and Calciferous Sandstone series.</p></td> + +<td class="tcl allb"><p>Upper productive Coal-measures.</p> +<p>Lower Barren measures.</p> +<p>Lower productive Coal-measures.</p> +<p>Pottsville conglomerate.</p> +<p>Mauch Chunk shales; limestones of Chester, St Louis, &c.</p> +<p>Pocono series; Kinderhook limestone.</p></td></tr> + +<tr><td class="tccm allb" rowspan="4">Devonian and Old Red Sandstone.</td> + +<td class="tccm allb">Devonian type.</td> + +<td class="tccm allb">Old Red Sandstone type.</td> + +<td class="rb"> </td></tr> + +<tr><td class="tclm rb cl"><p>Upper</p> + <p>    Famennian.</p> + <p>    Frasnian.</p></td> + +<td class="tclm rb cl"><p>Yellow and red sandstone with <i>Holoptychius</i>, + <i>Bothriolepis</i>, &c.</p></td> + +<td class="tcl rb cl"><p>Catskill red sandstone; Old Red Sandstone type: the strata below show the + Devonian type.</p> +<p>Chemung Group.</p> +<p>Genesee Group.</p></td></tr> + +<tr><td class="tclm rb">Middle<br /> +    Givetian.<br /> +    Eifelian.</td> + +<td class="tcl rb"><p>Caithness Flagstones with <i>Osteolepus</i>, <i>Dipterus</i>, + <i>Homosteus</i>, &c.</p></td> + +<td class="tclm rb"><p>Hamilton Group.</p> +<p>Marcellus Group.</p></td></tr> + +<tr><td class="tclm rb cl"><p>Lower</p> + <p>    Coblentizian.</p> + <p>    Gedinnian.</p></td> + +<td class="tclm rb cl"><p>Red and purple sandstones and conglomerates with <i>Cephalaspis</i>, + <i>Pteraspis</i>,</p></td> + +<td class="tcl rb cl"><p>Corniferous Limestone.</p> +<p>Onondaga Limestone.</p> + <p>    Upper Helderberg Group.</p> +<p>Oriskany Sandstone.</p></td></tr> + +<tr><td class="tccm allb" rowspan="2">Silurian.</td> + +<td class="tclm rb tb" colspan="2">Upper<br /> +    Ludlow Group.<br /> +    Wenlock Group.<br /> +    Llandovery Group.</td> + +<td class="tclm rb tb"><p>Lower Helderberg Group.</p> +<p>Water-Lime.</p> +<p>Niagara Shale and Limestone.</p> +<p>Clinton Group.</p> +<p>Medina Group.</p></td></tr> + +<tr><td class="tclm rb cl" colspan="2">Lower (Ordovician)<br /> +    Ludlow Group.<br /> +    Wenlock Group.<br /> +    Llandovery Group.</td> + +<td class="tclm rb cl"><p>Cincinnati Group.</p> +<p>Utica Group.</p> +<p>Trenton Group.</p> +<p>Chazy Group.</p> +<p>Calciferous Group.</p></td></tr> + +<tr><td class="tccm allb">Cambrian.</td> + +<td class="tcl allb" colspan="2"><p>Upper or <i>Olenus</i> series—Tremadoc slates and <i>Lingula</i> Flags.</p> +<p>Middle or <i>Pardoxides</i> series—Menevian Group.</p> +<p>Lower or <i>Olenellus</i> series—Llanberis and Harlech Group, and <i>Olenellus</i>-zone.</p></td> + +<td class="tcl allb"><p>Upper or Potsdam series with <i>Olenus</i> and <i>Dicelocephalus</i> fauna.</p> +<p>Middle or Acadian series with <i>Paradoxides</i> fauna.</p> +<p>Lower or Georgian series with <i>Olenellus</i> fauna.</p></td></tr> + +<tr><td class="tccm allb">Archean, Pre-Cambrian Eozoic.</td> + +<td class="tccm allb"> </td> + +<td class="tcl allb" colspan="2"><p>In Scotland, underneath the Cambrian Olenellus group, lies unconformably + a mass of red sandstone and conglomerate (Torridonian) 8000 or 10,000 ft. thick, which rests with a strong + gneisses and schists (Lewisian). A thick series of slates and phyllites lies below the oldest Palaeozoic rocks + in central Europe, with coarse gneisses below.</p></td> + +<td class="tcl allb"><p>In Canada and the Lake Superior region of the United States a vast succession of + rocks of Pre-Cambrian age has been grouped into the following subdivisions in descending order: (1) Keweenwan, + lying unconformably on (2) Animikie, separated by a strong unconformability from + (3) Upper Huronian, (4) Lower Huronian with an unconformable base, (5) Goutchiching, + (6) Laurentian. In the eastern part of Canada, Newfoundland, &c., and also in Montana, + sedimentary formations of great thickness below the lowest Cambrian zone have + been found to contain some obscure organisms.</p></td></tr> + +</table> + +<p><span class="pagenum"><a name="page671" id="page671"></a>671</span></p> + +<p class="pt2">From these general considerations we proceed to inquire how +the existing topographical features of the land arose. Obviously +the co-operation of the two great geological agencies of hypogene +and epigene energy, which have been at work from the beginning +of our globe’s decipherable history, must have been the cause +to which these features are to be assigned; and the task of the +geologist is to ascertain, if possible, the part that has been taken +by each. There is a natural tendency to see in a stupendous +piece of scenery, such as a deep ravine, a range of hills, a line of +precipice or a chain of mountains, evidence only of subterranean +convulsion; and before the subject was taken up as a matter +of strict scientific induction, an appeal to former cataclysms +was considered a sufficient solution of the problems presented +by such features of landscape. The rise of the modern +Huttonian school, however, led to a more careful examination +of these problems. The important share taken by erosion in the +determination of the present features of landscape was then +recognized, while a fuller appreciation of the relative parts +played by the hypogene and epigene causes has gradually been +reached.</p> + +<p>1. The study of the progress of denudation at the present +time has led to the conclusion that even if the rate of waste +were not more rapid than it is to-day, it would yet suffice in a +comparatively brief geological period to reduce the dry land to +below the sea-level. But not only would the area of the land be +diminished by denudation, it could hardly fail to be more or +less involved in those widespread movements of subsidence, +during which the thick sedimentary formations of the crust +appear to have been accumulated. It is thus manifest that there +must have been from time to time during the history of our +globe upward movements of the crust, whereby the balance +between land and sea was redressed. Proofs of such movements +have been abundantly preserved among the stratified formations. +We there learn that the uplifts have usually followed each other +at long intervals between which subsidence prevailed, and thus +that there has been a prolonged oscillation of the crust over the +great continental areas of the earth’s surface.</p> + +<p>An examination of that surface leads to the recognition of two +great types of upheaval. In the one, the sea-floor, with all its +thick accumulations of sediment, has been carried upwards, +sometimes for several thousand feet, so equably that the strata +retain their original flatness with hardly any sensible disturbance +for hundreds of square miles. In the other type the solid crust +has been plicated, corrugated and dislocated, especially along +particular lines, and has attained its most stupendous disruption +in lofty chains of mountains. Between these two phases of uplift +many intermediate stages have been developed, according to +the direction and intensity of the subterranean force and the +varying nature and disposition of the rocks Of the crust.</p> + +<p>(<i>a</i>) Where the uplift has extended over wide spaces, without +appreciable deformation of the crust, the flat strata have given +rise to low plains, or if the amount of uprise has been great +enough, to high plains, plateaux or tablelands. The plains of +Russia, for example, lie for the most part on such tracts of +equably uplifted strata. The great plains of the western interior +of the United States form a great plateau or tableland, 5000 or +6000 ft. above the sea, and many thousands of square miles in +extent, on which the Rocky Mountains have been ridged up.</p> + +<p>(<i>b</i>) It is in a great mountain-chain that the complicated +structures developed during disturbances of the earth’s crust +can best be studied (see Parts IV. and V. of this article), and +where the influence of these structures on the topography of the +surface is most effectively displayed. Such a chain may be the +result of one colossal disturbance; but those of high geological +antiquity usually furnish proofs of successive uplifts with more +or less intervening denudation. Formed along lines of continental +displacement in the crust, they have again and again given +relief from the strain of compression by fresh crumpling, fracture +and uprise. The chief guide in tracing these successive stages +of growth is supplied by unconformability. If, for example, a +mountain-range consists of upraised Silurian rocks, upon the +upturned and denuded edges of which the Carboniferous Limestone +lies transgressively, it is clear that its original upheaval +must have taken place in the period of geological time represented +by the interval between the Silurian and the Carboniferous +Limestone formations. If, as the range is followed along its +course, the Carboniferous Limestone is found to be also highly +inclined and covered unconformably by the Upper Coal-measures, +a second uplift of that portion of the ground can be proved to +have taken place between the time of the Limestone and that of +the Upper Coal-measures. By this simple and obvious kind of +evidence the relative ages of different mountain-chains may +be compared. In most great chains, however, the rocks have +been so intensely crumpled, and even inverted, that much +labour may be required before their true relations can be determined.</p> + +<p>The Alps furnish an instructive example of the long series of +revolutions through which a great mountain-system may have +passed before reaching its present development. The first +beginnings of the chain may have been upraised before the +oldest Palaeozoic formations were laid down. There are at +least traces of land and shore-lines in the Carboniferous period. +Subsequent submergences and uplifts appear to have occurred +during the Mesozoic periods. There is evidence that thereafter +the whole region sank deep under the sea, in which the older +Tertiary sediments were accumulated, and which seems to +have spread right across the heart of the Old World. But after +the deposition of the Eocene formations came the gigantic +disruptions whereby all the rocks of the Alpine region were +folded over each other, crushed, corrugated, fractured and +displaced, some of their older portions, including the fundamental +gneisses and schists, being squeezed up, torn off, and pushed +horizontally for many miles over the younger rocks. But this +upheaval, though the most momentous, was not the last which +the chain has undergone, for at a later epoch in Tertiary time +renewed disturbance gave rise to a further series of ruptures +and plications. The chain thus successively upheaved has +been continuously exposed to denudation and has consequently +lost much of its original height. That it has been left in a state +of instability is indicated by the frequent earthquakes of the +Alpine region, which doubtless arise from the sudden snapping +of rocks under intense strain.</p> + +<p>A distinct type of mountain due to direct hypogene action is +to be seen in a volcano. It has been already pointed out (Part IV. +sect. 1) that at the vents which maintain a communication +between the molten magma of the earth’s interior and the +surface, eruptions take place whereby quantities of lava and +fragmentary materials are heaped round each orifice of +discharge. A typical volcanic mountain takes the form of a +perfect cone, but as it grows in size and its main vent is choked, +while the sides of the cone are unable to withstand the force of +the explosions or the pressure of the ascending column of lava, +eruptions take place laterally, and numerous parasitic cones +arise on the flanks of the parent mountain. Where lava flows +out from long fissures, it may pile up vast sheets of rock, and +bury the surrounding country under several thousand feet of +solid stone, covering many hundreds of square miles. In this +way volcanic tablelands have been formed which, attacked by +the denuding forces, are gradually trenched by valleys and +ravines, until the original level surface of the lava-field may be +almost or wholly lost. As striking examples of this physiographical +type reference may be made to the plateau of Abyssinia, +the Ghats of India, the plateaux of Antrim, the Inner Hebrides +and Iceland, and the great lava-plains of the western territories +of the United States.</p> + +<p>2. But while the subterranean movements have upraised +portions of the surface of the lithosphere above the level of the +ocean, and have thus been instrumental in producing the existing +tracts of land, the detailed topographical features of a landscape +<span class="pagenum"><a name="page672" id="page672"></a>672</span> +are not solely, nor in general even chiefly, attributable to these +movements. From the time that any portion of the sea-floor +appears above sea-level, it undergoes erosion by the various +epigene agents. Each climate and geological region has its own +development of these agents, which include air, aridity, rapid and +frequent alternations of wetness and dryness or of heat and +cold, rain, springs, frosts, rivers, glaciers, the sea, plant and +animal life. In a dry climate subject to great extremes of +temperature the character and rate of decay will differ from +those of a moist or an arctic climate. But it must be remembered +that, however much they may vary in activity and in the results +which they effect, the epigene forces work without intermission, +while the hypogene forces bring about the upheaval of land only +after long intervals. Hence, trifling as the results during a +human life may appear, if we realize the multiplying influence +of time we are led to perceive that the apparently feeble superficial +agents can, in the course of ages, achieve stupendous +transformations in the aspect of the land. If this efficacy may +be deduced from what can be seen to be in progress now, it +may not less convincingly be shown, from the nature of the +sedimentary rocks of the earth’s crust, to have been in progress +from the early beginnings of geological history. Side by side +with the various upheavals and subsidences, there has been a +continuous removal of materials from the land, and an equally +persistent deposit of these materials under water, with the +consequent growth of new rocks. Denudation has been aptly +compared to a process of sculpturing wherein, while each of the +implements employed by nature, like a special kind of graving +tool, produces its own characteristic impress on the land, they +all combine harmoniously towards the achievement of their +one common task. Hence the present contours of the land +depend partly on the original configuration of the ground, and +the influence it may have had in guiding the operations of the +erosive agents, partly on the vigour with which these agents +perform their work, and partly on the varying structure and +powers of resistance possessed by the rocks on which the erosion +is carried on.</p> + +<p>Where a new tract of land has been raised out of the sea +by such an energetic movement as broke up the crust and +produced the complicated structure and tumultuous external +forms of a great mountain chain, the influence of the hypogene +forces on the topography attains its highest development. +But even the youngest existing chain has suffered so greatly +from denudation that the aspect which it presented at the time +of its uplift can only be dimly perceived. No more striking +illustration of this feature can be found than that supplied by +the Alps, nor one where the geotectonic structures have been +so fully studied in detail. On the outer flanks of these mountains +the longitudinal ridges and valleys of the Jura correspond with +lines of anticline and syncline. Yet though the dominant +topographical elements of the region have obviously been +produced by the plication of the stratified formations, each +ridge has suffered so large an amount of erosion that the younger +rocks have been removed from its crest where the older members +of the series are now exposed to view, while on every slope +proofs may be seen of extensive denudation. If from these +long wave-like undulations of the ground, where the relations +between the disposition of the rocks below and the forms of +the surface are so clearly traceable, the observer proceeds +inwards to the main chain, he finds that the plications and +displacements of the various formations assume an increasingly +complicated character; and that although proofs of great +denudation continue to abound, it becomes increasingly difficult +to form any satisfactory conjecture as to the shape of the ground +when the upheaval ended or any reliable estimate of the amount +of material which has since then been removed. Along the +central heights the mountains lift themselves towards the sky +like the storm-swept crests of vast earth-billows. The whole +aspect of the ground suggests intense commotion, and the +impression thus given is often much intensified by the twisted +and crumpled strata, visible from a long distance, on the crags +and crests. On this broken-up surface the various agents of +denudation have been ceaselessly engaged since it emerged +from the sea. They have excavated valleys, sometimes along +depressions provided for them by the subterranean disturbances, +sometimes down the slopes of the disrupted blocks of ground. +So powerful has been this erosion that valleys cut out along +lines of anticline, which were natural ridges, have sometimes +become more important than those in lines of syncline, which +were structurally depressions. The same subaerial forces have +eroded lake-basins, dug out corries or cirques, notched the +ridges, splintered the crests and furrowed the slopes, leaving +no part of the original surface of the uplifted chain +unmodified.</p> + +<p>It has often been noted with surprise that features of +underground structure which, it might have been confidently +anticipated, should have exercised a marked influence on the +topography of the surface have not been able to resist the +levelling action of the denuding agents, and do not now affect +the surface at all. This result is conspicuously seen in coal-fields +where the strata are abundantly traversed by faults. These +dislocations, having sometimes a displacement of several hundred +feet, might have been expected to break up the surface into +a network of cliffs and plains; yet in general they do not modify +the level character of the ground above. One of the most +remarkable faults in Europe is the great thrust which bounds +the southern edge of the Belgian coal-field and brings the +Devonian rocks above the Coal-measures. It can be traced +across Belgium into the Boulonnais, and may not improbably +run beneath the Secondary and Tertiary rocks of the south of +England. It is crossed by the valleys of the Meuse and other +northerly-flowing streams. Yet so indistinctly is it marked +in the Meuse valley that no one would suspect its existence from +any peculiarity in the general form of the ground, and even an +experienced geologist, until he had learned the structure of the +district, would scarcely detect any fault at all.</p> + +<p>Where faults have influenced the superficial topography, +it is usually by giving rise to a hollow along which the subaerial +agents and especially running water can act effectively. Such +a hollow may be eventually widened and deepened into a valley. +On bare crags and crests, lines of fault are apt to be marked by +notches or clefts, and they thus help to produce the pinnacles +and serrated outlines of these exposed uplands.</p> + +<p>It was cogently enforced by Hutton and Playfair, and independently +by Lamarck, that no co-operation of underground +agency is needed to produce such topography as may be seen +in a great part of the world, but that if a tract of sea-floor were +upraised into a wide plain, the fall of rain and the circulation +of water over its surface would in the end carve out such a system +of hills and valleys as may be seen on the dry land now. No +such plain would be a dead-level. It would have inequalities +on its surface which would serve as channels to guide the drainage +from the first showers of rain. And these channels would be +slowly widened and deepened until they would become ravines +and valleys, while the ground between them would be left projecting +as ridges and hills. Nor would the erosion of such a system +of water-courses require a long series of geological periods for +its accomplishment. From measurements and estimates of the +amount of erosion now taking place in the basin of the Mississippi +river it has been computed that valleys 800 ft. deep might be +carved out in less than a million years. In the vast tablelands +of Colorado and other western regions of the United States an +impressive picture is presented of the results of mere subaerial +erosion on undisturbed and nearly level strata. Systems of +stream-courses and valleys, river gorges unexampled elsewhere +in the world for depth and length, vast winding lines of escarpment, +like ranges of sea-cliffs, terraced slopes rising from plateau +to plateau, huge buttresses and solitary stacks standing like +islands out of the plains, great mountain-masses towering into +picturesque peaks and pinnacles cleft by innumerable gullies, +yet everywhere marked by the parallel bars of the horizontal +strata out of which they have been carved—these are the orderly +symmetrical characteristics of a country where the scenery is +due entirely to the action of subaerial agents on the one hand and +<span class="pagenum"><a name="page673" id="page673"></a>673</span> +the varying resistance of perfectly regular stratified rocks on the +other.</p> + +<p>The details of the sculpture of the land have mainly depended +on the nature of the materials on which nature’s erosive tools +have been employed. The joints by which all rocks are traversed +have been especially serviceable as dominant lines down which +the rain has filtered, up which the springs have risen and into +which the frost wedges have been driven. On the high bare +scarps of a lofty mountain the inner structure of the mass is laid +open, and there the system of joints even more than faults is +seen to have determined the lines of crest, the vertical walls of +cliff and precipice, the forms of buttress and recess, the position +of cleft and chasm, the outline of spire and pinnacle. On the +lower slopes, even under the tapestry of verdure which nature +delights to hang where she can over her naked rocks, we may +detect the same pervading influence of the joints upon the forms +assumed by ravines and crags. Each kind of stone, too, gives +rise to its own characteristic form of scenery. Massive crystalline +rocks, such as granite, break up along their joints and often +decay into sand or earth along their exposed surfaces, giving +rise to rugged crags with long talus slopes at their base. The +stratified rocks besides splitting at their joints are especially +distinguished by parallel ledges, cornices and recesses, produced +by the irregular decay of their component strata, so that they +often assume curiously architectural types of scenery. But +besides this family feature they display many minor varieties of +aspect according to their lithological composition. A range of +sandstone hills, for example, presents a marked contrast to one +of limestone, and a line of chalk downs to the escarpments +formed by alternating bands of harder and softer clays and +shales.</p> + +<p>It may suffice here merely to allude to a few of the more +important parts of the topography of the land in their relation +to physiographical geology. A true mountain-chain, viewed +from the geological side, is a mass of high ground which owes its +prominence to a ridging-up of the earth’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.”</p> + +<p>Tablelands, as already pointed out, may be produced either +by the upheaval of tracts of horizontal strata from the sea-floor +into land; or by the uprise of plains of denudation, where rocks +of various composition, structure and age have been levelled +down to near or below the level of the sea by the co-operation +of the various erosive agents. Most of the great tablelands +of the globe are platforms of little-disturbed strata which have +been upraised bodily to a considerable elevation. No sooner, +however, are they placed in that position than they are attacked +by running water, and begin to be hollowed out into systems of +valleys. As the valleys sink, the platforms between them grow +into narrower and more definite ridges, until eventually the +level tableland is converted into a complicated network of hills +and valleys, wherein, nevertheless, the key to the whole arrangement +is furnished by a knowledge of the disposition and effects +of the flow of water. The examples of this process brought to +light in Colorado, Wyoming, Nevada and the other western +regions by Newberry, King, Hayden, Powell and other explorers, +are among the most striking monuments of geological operations +in the world.</p> + +<p>Examples of ancient and much decayed tablelands formed by +the denudation of much disturbed rocks are furnished by the +Highlands of Scotland and of Norway. Each of these tracts of +high ground consists of some of the oldest and most dislocated +formations of Europe, which at a remote period were worn down +into a plain, and in that condition may have lain long submerged +under the sea and may possibly have been overspread there +with younger formations. Having at a much later time been +raised several thousand feet above sea-level the ancient platforms +of Britain and Scandinavia have been since exposed to denudation, +whereby each of them has been so deeply channeled into +glens and fjords that it presents to-day a surface of rugged +hills, either isolated or connected along the flanks, while only +fragments of the general surface of the tableland can here and +there be recognized amidst the general destruction.</p> + +<p>Valleys have in general been hollowed out by the greater +erosive action of running water along the channels of drainage. +Their direction has been probably determined in the great +majority of cases by irregularities of the surface along which +the drainage flowed on the first emergence of the land. Sometimes +these irregularities have been produced by folds of the +terrestrial crust, sometimes by faults, sometimes by the irregularities +on the surface of an uplifted platform of deposition or of +denudation. Two dominant trends may be observed among +them. Some are longitudinal and run along the line of flexures +in the upraised tract of land, others are transverse where the +drainage has flowed down the slopes of the ridges into the longitudinal +valleys or into the sea. The forms of valleys have been +governed partly by the structure and composition of the rocks, +and partly by the relative potency of the different denuding +agents. Where the influence of rain and frost has been slight, +and the streams, supplied from distant sources, have had +sufficient declivity, deep, narrow, precipitous ravines or gorges +have been excavated. The canyons of the arid region of the +Colorado are a magnificent example of this result. Where, on +the other hand, ordinary atmospheric action has been more +rapid, the sides of the river channels have been attacked, and +open sloping glens and valleys have been hollowed out. A +gorge or defile is usually due to the action of a waterfall, which, +beginning with some abrupt declivity or precipice in the course +of the river when it first commenced to flow, or caused by some +hard rock crossing the channel, has eaten its way backward.</p> + +<p>Lakes have been already referred to, and their modes of origin +have been mentioned. As they are continually being filled up +with the detritus washed into them from the surrounding +regions they cannot be of any great geological antiquity, unless +where by some unknown process their basins are from time to +time widened and deepened.</p> + +<p>In the general subaerial denudation of a country, innumerable +minor features are worked out as the structure of the rocks +controls the operations of the eroding agents. Thus, among +comparatively undisturbed strata, a hard bed resting upon +others of a softer kind is apt to form along its outcrop a line of +cliff or escarpment. Though a long range of such cliffs resembles +a coast that has been worn by the sea, it may be entirely due to +mere atmospheric waste. Again, the more resisting portions of +a rock may be seen projecting as crags or knolls. An igneous +mass will stand out as a bold hill from amidst the more decomposable +strata through which it has risen. These features, +often so marked on the lower grounds, attain their most conspicuous +development among the higher and barer parts of the +mountains, where subaerial disintegration is most rapid. The +torrents tear out deep gullies from the sides of the declivities. +Corries or cirques are scooped out on the one hand and naked +precipices are left on the other. The harder bands of rock +project as massive ribs down the slopes, shoot up into prominent +<i>aiguilles</i>, or help to give to the summits the notched saw-like +outlines they so often present.</p> + +<p>The materials worn from the surface of the higher are spread +out over the lower grounds. The streams as they descend begin +to drop their freight of sediment when, by the lessening of their +declivity, their carrying power is diminished. The great plains +of the earth’s surface are due to this deposit of gravel, sand and +loam. They are thus monuments at once of the destructive and +reproductive processes which have been in progress unceasingly +since the first land rose above the sea and the first shower of rain +fell. Every pebble and particle of their soil, once part of the +distant mountains, has travelled slowly and fitfully to lower +levels. Again and again have these materials been shifted, +ever moving downward and sea-ward. For centuries, perhaps, +they have taken their share in the fertility of the plains and +<span class="pagenum"><a name="page674" id="page674"></a>674</span> +have ministered to the nurture of flower and tree, of the bird of +the air, the beast of the field and of man himself. But their +destiny is still the great ocean. In that bourne alone can they +find undisturbed repose, and there, slowly accumulating in +massive beds, they will remain until, in the course of ages, +renewed upheaval shall raise them into future land, there once +more to pass through the same cycle of change.</p> +<div class="author">(A. Ge.)</div> + +<div class="condensed"> +<p><span class="sc">Literature.</span>—<i>Historical</i>: The standard work is Karl A. von +Zittel’s <i>Geschichte der Geologie und Paläontologie</i> (1899), of which +there is an abbreviated, but still valuable, English translation; +D’Archiac, <i>Histoire des progrès de la géologie</i>, deals especially with +the period 1834-1850; Keferstein, <i>Geschichte und Literatur der +Geognosie</i>, gives a summary up to 1840; while Sir A. Geikie’s +<i>Founders of Geology</i> (1897; 2nd ed., 1906) deals more particularly +with the period 1750-1820. General treatises: Sir Charles Lyell’s +<i>Principles of Geology</i> is a classic. Of modern English works, Sir A. +Geikie’s <i>Text Book of Geology</i> (4th ed., 1903) occupies the first place; +the work of T.C. Chamberlin and R.D. Salisbury, <i>Geology</i>; <i>Earth +History</i> (3 vols., 1905-1906), is especially valuable for American +geology. A. de Lapparent’s <i>Traité de géologie</i> (5th ed., 1906), is the +standard French work. H. Credner’s <i>Elemente der Geologie</i> has gone +through several editions in Germany. Dynamical and physiographical +geology are elaborately treated by E. Suess, <i>Das Antlitz +der Erde</i>, translated into English, with the title <i>The Face of the Earth</i>. +The practical study of the science is treated of by F. von Richthofen, +<i>Führer für Forschungsreisende</i> (1886); G.A. Cole, <i>Aids in Practical +Geology</i> (5th ed., 1906); A. Geikie, <i>Outlines of Field Geology</i> (5th ed., +1900). The practical applications of Geology are discussed by +J.V. Elsden, <i>Applied Geology</i> (1898-1899). The relations of Geology +to scenery are dealt with by Sir A. Geikie, <i>Scenery of Scotland</i> (3rd ed., +1901); J.E. Marr, <i>The Scientific Study of Scenery</i> (1900); Lord +Avebury, <i>The Scenery of Switzerland</i> (1896); <i>The Scenery of England</i> +(1902); and J. Geikie, <i>Earth Sculpture</i> (1898). A detailed bibliography +is given in Sir A. Geikie’s <i>Text Book of Geology</i>. See also +the separate articles on geological subjects for special references to +authorities.</p> +</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1c" id="ft1c" href="#fa1c"><span class="fn">1</span></a> In De Luc’s <i>Lettres physiques et morales sur les montagnes</i> (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.</p> + +<p><a name="ft2c" id="ft2c" href="#fa2c"><span class="fn">2</span></a> The subject of the age of the earth has also been discussed by +Professor J. Joly and Professor W.J. Sollas. The former geologist, +approaching the question from a novel point of view, has estimated +the total quantity of sodium in the water of the ocean and the +quantity of that element received annually by the ocean from the +denudation of the land. Dividing the one sum by the other, he +arrives at the result that the probable age of the earth is between +90 and 100 millions of years (<i>Trans. Roy. Dublin Soc.</i> ser. ii. vol. vii., +1899, p. 23: <i>Geol. Mag.</i>, 1900, p. 220). Professor Sollas believes +that this limit exceeds what is required for the evolution of geological +history, that the lower limit assigned by Lord Kelvin falls short of +what the facts demand, and that geological time will probably be +found to have been comprised within some indeterminate period +between these limits. (Address to Section C, <i>Brit. Assoc. Report</i>, +1900; <i>Age of the Earth</i>, London, 1905.)</p> +</div> + + +<hr class="art" /> +<p><span class="bold">GEOMETRICAL CONTINUITY.<a name="ar19" id="ar19"></a></span> In a report of the Institute +prefixed to Jean Victor Poncelet’s <i>Traité des propriétés projectives +des figures</i> (Paris, 1822), it is said that he employed “ce +qu’il appelle le principe de continuité.” The law or principle +thus named by him had, he tells us, been tacitly assumed as +axiomatic by “les plus savans géomètres.” It had in fact been +enunciated as “lex continuationis,” and “la loi de la continuité,” +by Gottfried Wilhelm Leibnitz (Oxf. N.E.D.), and previously +under another name by Johann Kepler in cap. iv. 4 of his <i>Ad +Vitellionem paralipomena quibus astronomiae pars optica traditur</i> +(Francofurti, 1604). Of sections of the cone, he says, there are +five species from the “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 +<i>at infinity</i> on the axis <i>within or without the curve</i>. The line from it +to any point of the section is parallel to the axis. To carry out +the analogy we must speak paradoxically, and say that the line-pair +likewise has foci, which in this case coalesce as in the circle +and fall upon the lines themselves; for our geometrical terms +should be subject to analogy. Kepler dearly loves analogies, his +most trusty teachers, acquainted with all the secrets of nature, +“<i>omnium naturae arcanorum conscios</i>.” 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.</p> + +<p>Here, then, we find formulated by Kepler the doctrine of the +concurrence of parallels at a single point at infinity and the +principle of continuity (under the name analogy) in relation to the +infinitely great. Such conceptions so strikingly propounded in +a famous work could not escape the notice of contemporary +mathematicians. Henry Briggs, in a letter to Kepler from +Merton College, Oxford, dated “10 Cal. Martiis 1625,” suggests +improvements in the <i>Ad Vitellionem paralipomena</i>, and gives +the following construction: Draw a line CBADC, and let an +ellipse, a parabola, and a hyperbola have B and A for focus and +vertex. Let CC be the other foci of the ellipse and the hyperbola. +Make AD equal to AB, and with centres CC and radius in each +case equal to CD describe circles. Then any point of the ellipse +is equidistant from the focus B and one circle, and any point of +the hyperbola from the focus B and the other circle. Any point +P of the parabola, in which the second focus is missing or infinitely +distant, is equidistant from the focus B and the line +through D which we call the directrix, this taking the place of +either circle when its centre C is at infinity, and every line CP +being then parallel to the axis. Thus Briggs, and we know not +how many “savans géomètres” 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 +<i>Méthode universelle de mettre en perspective les objets donnés +réellement ou en devis</i> (Paris, 1636); but “Le privilége étoit de +1630.” (Poudra, <i>Œuvres de Des.</i>, i. 55). Kepler as a modern +geometer is best known by his <i>New Stereometry of Wine Casks</i> +(Lincii, 1615), in which he replaces the circuitous Archimedean +method of exhaustion by a direct “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.</p> + +<p>To complete the theory of continuity, the one thing needful +was the idea of imaginary points implied in the algebraical +geometry of René Descartes, in which equations between variables +representing co-ordinates were found often to have imaginary +roots. Newton, in his two sections on “Inventio orbium” +(<i>Principia</i> i. 4, 5), shows in his brief way that he is familiar with +the principles of modern geometry. In two propositions he uses +an auxiliary line which is supposed to cut the conic in X and Y, +but, as he remarks at the end of the second (prop. 24), it may not +cut it at all. For the sake of brevity he passes on at once with the +observation that the required constructions are evident from the +case in which the line cuts the trajectory. In the scholium +appended to prop. 27, after saying that an asymptote is a tangent +at infinity, he gives an unexplained general construction for the +axes of a conic, which seems to imply that it has asymptotes. +In all such cases, having equations to his loci in the background, +he may have thought of elements of the figure as passing into the +imaginary state in such manner as not to vitiate conclusions +arrived at on the hypothesis of their reality.</p> + +<p>Roger Joseph Boscovich, a careful student of Newton’s works, +has a full and thorough discussion of geometrical continuity in +the third and last volume of his <i>Elementa universae matheseos</i> +(ed. prim. Venet, 1757), which contains <i>Sectionum conicarum +elementa nova quadam methodo concinnata et dissertationem de +transformatione locorum geometricorum, ubi de continuitatis +lege, et de quibusdam infiniti mysteriis</i>. His first principle is +that all varieties of a defined locus have the same properties, so +that what is demonstrable of one should be demonstrable in like +manner of all, although some artifice may be required to bring +out the underlying analogy between them. The opposite +extremities of an infinite straight line, he says, are to be regarded +as joined, as if the line were a circle having its centre at the +infinity on either side of it. This leads up to the idea of a <i>veluti +plus quam infinita extensio</i>, a line-circle containing, as we say, +the line infinity. Change from the real to the imaginary state is +contingent upon the passage of some element of a figure through +zero or infinity and never takes place <i>per saltum</i>. Lines being +some positive and some negative, there must be negative rectangles +and negative squares, such as those of the exterior +diameters of a hyperbola. Boscovich’s first principle was that +of Kepler, by whose <i>quantumvis absurdis locutionibus</i> the boldest +<span class="pagenum"><a name="page675" id="page675"></a>675</span> +applications of it are covered, as when we say with Poncelet +that all concentric circles in a plane touch one another in two +imaginary fixed points at infinity. In G.K. Ch. von Staudt’s +<i>Geometrie der Lage and Beiträge zur G. der L.</i> (Nürnberg, 1847, +1856-1860) the geometry of position, including the extension of +the field of pure geometry to the infinite and the imaginary, is +presented as an independent science, “welche des Messens nicht +bedarf.” (See <span class="sc"><a href="#artlinks">Geometry</a></span>: <i>Projective</i>.)</p> + +<p>Ocular illusions due to distance, such as Roger Bacon notices +in the <i>Opus majus</i> (i. 126, ii. 108, 497; Oxford, 1897), lead up to +or illustrate the mathematical uses of the infinite and its reciprocal +the infinitesimal. Specious objections can, of course, be +made to the anomalies of the law of continuity, but they are +inherent in the higher geometry, which has taught us so much +of the “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 <i>Transactions of the Cambridge Philosophical +Society</i> (1900). It had been generally overlooked, until attention +was called to it by the present writer in a note read in 1880 (<i>Proc. +C.P.S.</i> iv. 14-17), and shortly afterwards in <i>The Ancient and +Modern Geometry of Conics, with Historical Notes and Prolegomena</i> +(Cambridge 1881).</p> +<div class="author">(C. T.*)</div> + + +<hr class="art" /> +<p><span class="bold">GEOMETRY,<a name="ar20" id="ar20"></a></span> the general term for the branch of mathematics +which has for its province the study of the properties of +space. From experience, or possibly intuitively, we characterize +existent space by certain fundamental qualities, termed axioms, +which are insusceptible of proof; and these axioms, in conjunction +with the mathematical entities of the point, straight line, +curve, surface and solid, appropriately defined, are the premises +from which the geometer draws conclusions. The geometrical +axioms are merely conventions; on the one hand, the system +may be based upon inductions from experience, in which case +the deduced geometry may be regarded as a branch of physical +science; or, on the other hand, the system may be formed by +purely logical methods, in which case the geometry is a phase +of pure mathematics. Obviously the geometry with which we +are most familiar is that of existent space—the three-dimensional +space of experience; this geometry may be termed Euclidean, +after its most famous expositor. But other geometries exist, +for it is possible to frame systems of axioms which definitely +characterize some other kind of space, and from these axioms +to deduce a series of non-contradictory propositions; such +geometries are called non-Euclidean.</p> + +<p>It is convenient to discuss the subject-matter of geometry +under the following headings:</p> + +<p>I. <i>Euclidean Geometry</i>: a discussion of the axioms of existent +space and of the geometrical entities, followed by a synoptical +account of Euclid’s Elements.</p> + +<p>II. <i>Projective Geometry</i>: primarily Euclidean, but differing +from I. in employing the notion of geometrical continuity (<i>q.v.</i>)—points +and lines at infinity.</p> + +<p>III. <i>Descriptive Geometry</i>: the methods for representing upon +planes figures placed in space of three dimensions.</p> + +<p>IV. <i>Analytical Geometry</i>: the representation of geometrical +figures and their relations by algebraic equations.</p> + +<p>V. <i>Line Geometry</i>: an analytical treatment of the line regarded +as the space element.</p> + +<p>VI. <i>Non-Euclidean Geometry</i>: a discussion of geometries +other than that of the space of experience.</p> + +<p>VII. <i>Axioms of Geometry</i>: a critical analysis of the foundations +of geometry.</p> + +<div class="condensed"> +<p>Special subjects are treated under their own headings: <i>e.g.</i> +<span class="sc"><a href="#artlinks">Projection</a></span>, <span class="sc"><a href="#artlinks">Perspective</a></span>; <span class="sc"><a href="#artlinks">Curve</a></span>, <span class="sc"><a href="#artlinks">Surface</a></span>; <span class="sc"><a href="#artlinks">Circle</a></span>, <span class="sc"><a href="#artlinks">Conic +Section</a></span>; <span class="sc"><a href="#artlinks">Triangle</a></span>, <span class="sc"><a href="#artlinks">Polygon</a></span>, <span class="sc"><a href="#artlinks">Polyhedron</a></span>; there are also +articles on special curves and figures, <i>e.g.</i> <span class="sc"><a href="#artlinks">Ellipse</a></span>, <span class="sc"><a href="#artlinks">Parabola</a></span>, +<span class="sc"><a href="#artlinks">Hyperbola</a></span>; <span class="sc"><a href="#artlinks">Tetrahedron</a></span>, <span class="sc"><a href="#artlinks">Cube</a></span>, <span class="sc"><a href="#artlinks">Octahedron</a></span>, <span class="sc"><a href="#artlinks">Dodecahedron</a></span>, +<span class="sc"><a href="#artlinks">Icosahedron</a></span>; <span class="sc"><a href="#artlinks">Cardioid</a></span>, <span class="sc"><a href="#artlinks">Catenary</a></span>, <span class="sc"><a href="#artlinks">Cissoid</a></span>, <span class="sc"><a href="#artlinks">Conchoid</a></span>, <span class="sc"><a href="#artlinks">Cycloid</a></span>, +<span class="sc"><a href="#artlinks">Epicycloid</a></span>, <span class="sc"><a href="#artlinks">Limaçon</a></span>, <span class="sc"><a href="#artlinks">Oval</a></span>, <span class="sc"><a href="#artlinks">Quadratrix</a></span>, <span class="sc"><a href="#artlinks">Spiral</a></span>, &c.</p> +</div> + +<p><i>History.</i>—The origin of geometry (Gr. <span class="grk" title="gê">γῆ</span>, earth, <span class="grk" title="metron">μέτρον</span>, a +measure) is, according to Herodotus, to be found in the etymology +of the word. Its birthplace was Egypt, and it arose from the +need of surveying the lands inundated by the Nile floods. In +its infancy it therefore consisted of a few rules, very rough and +approximate, for computing the areas of triangles and quadrilaterals; +and, with the Egyptians, it proceeded no further, the +geometrical entities—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.<a name="fa1d" id="fa1d" href="#ft1d"><span class="sp">1</span></a> The +first step towards its elevation to the rank of a science was made +by Thales (<i>q.v.</i>) of Miletus, who transplanted the elementary +Egyptian mensuration to Greece. Thales clearly abstracted +the notions of points and lines, founding the geometry of the +latter unit, and discovering <i>per saltum</i> many propositions concerning +areas, the circle, &c. The empirical rules of the Egyptians +were corrected and developed by the Ionic School which he +founded, especially by Anaximander and Anaxagoras, and in +the 6th century <span class="scs">B.C.</span> passed into the care of the Pythagoreans. +From this time geometry exercised a powerful influence on +Greek thought. Pythagoras (<i>q.v.</i>), seeking the key of the +universe in arithmetic and geometry, investigated logically the +principles underlying the known propositions; and this resulted +in the formulation of definitions, axioms and postulates which, +in addition to founding a <i>science</i> of geometry, permitted a +crystallization, fractional, it is true, of the amorphous collection +of material at hand. Pythagorean geometry was essentially a +geometry of areas and solids; its goal was the regular solids—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>i.e.</i> the geometry of constructions +made by the ruler and compasses, exercised considerable +influence in other directions. The first problem led to the +discovery of the method of <i>exhaustion</i> for determining areas. +Antiphon inscribed a square in a circle, and on each side an +isosceles triangle having its vertex on the circle; on the sides +of the octagon so obtained, isosceles triangles were again constructed, +the process leading to inscribed polygons of 8, 16 and +32 sides; and the areas of these polygons, which are easily +determined, are successive approximations to the area of the +circle. Bryson of Heraclea took an important step when he +circumscribed, in addition to inscribing, polygons to a circle, +but he committed an error in treating the circle as the mean of +the two polygons. The method of Antiphon, in assuming that +by continued division a polygon can be constructed coincident +with the circle, demanded that magnitudes are not infinitely +divisible. Much controversy ranged about this point; Aristotle +supported the doctrine of infinite divisibility; Zeno attempted +to show its absurdity. The mechanical tracing of loci, a principle +initiated by Archytas of Tarentum to solve the last two problems, +was a frequent subject for study, and several mechanical curves +were thus discovered at subsequent dates (cissoid, conchoid, +quadratrix). Mention may be made of Hippocrates, who, +besides developing the known methods, made a study of similar +figures, and, as a consequence, of proportion. This step is +important as bringing into line discontinuous number and +continuous magnitude.</p> + +<p>A fresh stimulus was given by the succeeding Platonists, who, +accepting in part the Pythagorean cosmology, made the study +of geometry preliminary to that of philosophy. The many +discoveries made by this school were facilitated in no small +measure by the clarification of the axioms and definitions, the +logical sequence of propositions which was adopted, and, more +especially, by the formulation of the analytic method, <i>i.e.</i> of +assuming the truth of a proposition and then reasoning to a +<span class="pagenum"><a name="page676" id="page676"></a>676</span> +known truth. The main strength of the Platonist geometers +lies in stereometry or the geometry of solids. The Pythagoreans +had dealt with the sphere and regular solids, but the pyramid, +prism, cone and cylinder were but little known until the Platonists +took them in hand. Eudoxus established their mensuration, +proving the pyramid and cone to have one-third the content +of a prism and cylinder on the same base and of the same height, +and was probably the discoverer of a proof that the volumes of +spheres are as the cubes of their radii. The discussion of sections +of the cone and cylinder led to the discovery of the three curves +named the parabola, ellipse and hyperbola (see <span class="sc"><a href="#artlinks">Conic Section</a></span>); +it is difficult to over-estimate the importance of this discovery; +its investigation marks the crowning achievement of Greek +geometry, and led in later years to the fundamental theorems +and methods of modern geometry.</p> + +<p>The presentation of the subject-matter of geometry as a connected +and logical series of propositions, prefaced by <span class="grk" title="Horoi">Ὅροι</span> or +foundations, had been attempted by many; but it is to Euclid +that we owe a complete exposition. Little indeed in the <i>Elements</i> +is probably original except the arrangement; but in this Euclid +surpassed such predecessors as Hippocrates, Leon, pupil of +Neocleides, and Theudius of Magnesia, devising an apt logical +model, although when scrutinized in the light of modern mathematical +conceptions the proofs are riddled with fallacies. According +to the commentator Proclus, the <i>Elements</i> were written with +a twofold object, first, to introduce the novice to geometry, and +secondly, to lead him to the regular solids; conic sections found +no place therein. What Euclid did for the line and circle, +Apollonius did for the conic sections, but there we have a discoverer +as well as editor. These two works, which contain the greatest +contributions to ancient geometry, are treated in detail in +Section I. <i>Euclidean Geometry</i> and the articles <span class="sc"><a href="#artlinks">Euclid</a></span>; <span class="sc"><a href="#artlinks">Conic +Section</a></span>; <span class="sc"><a href="#artlinks">Appolonius</a></span>. Between Euclid and Apollonius there +flourished the illustrious Archimedes, whose geometrical discoveries +are mainly concerned with the mensuration of the +circle and conic sections, and of the sphere, cone and cylinder, +and whose greatest contribution to geometrical method is the +elevation of the method of exhaustion to the dignity of an instrument +of research. Apollonius was followed by Nicomedes, the +inventor of the conchoid; Diocles, the inventor of the cissoid; +Zenodorus, the founder of the study of isoperimetrical figures; +Hipparchus, the founder of trigonometry; and Heron the elder, +who wrote after the manner of the Egyptians, and primarily +directed attention to problems of practical surveying.</p> + +<p>Of the many isolated discoveries made by the later Alexandrian +mathematicians, those of Menelaus are of importance. He +showed how to treat spherical triangles, establishing their +properties and determining their congruence; his theorem on +the products of the segments in which the sides of a triangle +are cut by a line was the foundation on which Carnot erected +his theory of transversals. These propositions, and also those +of Hipparchus, were utilized and developed by Ptolemy (<i>q.v.</i>), +the expositor of trigonometry and discoverer of many isolated +propositions. Mention may be made of the commentator Pappus, +whose <i>Mathematical Collections</i> is valuable for its wealth of +historical matter; of Theon, an editor of Euclid’s <i>Elements</i> and +commentator of Ptolemy’s <i>Almagest</i>; of Proclus, a commentator +of Euclid; and of Eutocius, a commentator of Apollonius and +Archimedes.</p> + +<p>The Romans, essentially practical and having no inclination +to study science <i>qua</i> science, only had a geometry which sufficed +for surveying; and even here there were abundant inaccuracies, +the empirical rules employed being akin to those of the Egyptians +and Heron. The Hindus, likewise, gave more attention to +computation, and their geometry was either of Greek origin or +in the form presented in trigonometry, more particularly connected +with arithmetic. It had no logical foundations; each +proposition stood alone; and the results were empirical. The +Arabs more closely followed the Greeks, a plan adopted as a +sequel to the translation of the works of Euclid, Apollonius, +Archimedes and many others into Arabic. Their chief contribution +to geometry is exhibited in their solution of algebraic +equations by intersecting conics, a step already taken by the +Greeks in isolated cases, but only elevated into a <i>method</i> by Omar +al Hayyami, who flourished in the 11th century. During the +middle ages little was added to Greek and Arabic geometry. +Leonardo of Pisa wrote a <i>Practica geometriae</i> (1220), wherein +Euclidean methods are employed; but it was not until the 14th +century that geometry, generally Euclid’s <i>Elements</i>, became +an essential item in university curricula. There was, however, +no sign of original development, other branches of mathematics, +mainly algebra and trigonometry, exercising a greater fascination +until the 16th century, when the subject again came into favour.</p> + +<p>The extraordinary mathematical talent which came into being +in the 16th and 17th centuries reacted on geometry and gave rise +to all those characters which distinguish modern from ancient +geometry. The first innovation of moment was the formulation +of the principle of geometrical continuity by Kepler. The notion +of infinity which it involved permitted generalizations and +systematizations hitherto unthought of (see <span class="sc"><a href="#artlinks">Geometrical +Continuity</a></span>); and the method of indefinite division applied to +rectification, and quadrature and cubature problems avoided +the cumbrous method of exhaustion and provided more accurate +results. Further progress was made by Bonaventura Cavalieri, +who, in his <i>Geometria indivisibilibus continuorum</i> (1620), devised +a method intermediate between that of exhaustion and +the infinitesimal calculus of Leibnitz and Newton. The logical +basis of his system was corrected by Roberval and Pascal; and +their discoveries, taken in conjunction with those of Leibnitz, +Newton, and many others in the fluxional calculus, culminated +in the branch of our subject known as differential geometry +(see <span class="sc"><a href="#artlinks">Infinitesimal Calculus</a></span>; <span class="sc"><a href="#artlinks">Curve</a></span>; <span class="sc"><a href="#artlinks">Surface</a></span>).</p> + +<p>A second important advance followed the recognition that +conics could be regarded as projections of a circle, a conception +which led at the hands of Desargues and Pascal to modern +<i>projective geometry</i> and <i>perspective</i>. A third, and perhaps the +most important, advance attended the application of algebra to +geometry by Descartes, who thereby founded <i>analytical geometry</i>. +The new fields thus opened up were diligently explored, but the +calculus exercised the greatest attraction and relatively little +progress was made in geometry until the beginning of the 19th +century, when a new era opened.</p> + +<p>Gaspard Monge was the first important contributor, stimulating +analytical and differential geometry and founding <i>descriptive +geometry</i> in a series of papers and especially in his lectures at the +École polytechnique. Projective geometry, founded by Desargues, +Pascal, Monge and L.N.M. Carnot, was crystallized by +J.V. Poncelet, the creator of the modern methods. In his +<i>Traité des propriétés des figures</i> (1822) the line and circular points +at infinity, imaginaries, polar reciprocation, homology, cross-ratio +and projection are systematically employed. In Germany, +A.F. Möbius, J. Plücker and J. Steiner were making far-reaching +contributions. Möbius, in his <i>Barycentrische Calcul</i> (1827), +introduced homogeneous co-ordinates, and also the powerful +notion of geometrical transformation, including the special +cases of collineation and duality; Plücker, in his <i>Analytisch-geometrische +Entwickelungen</i> (1828-1831), and his <i>System der +analytischen Geometrie</i> (1835), introduced the abridged notation, +line and plane co-ordinates, and the conception of generalized +space elements; while Steiner, besides enriching geometry in +numerous directions, was the first to systematically generate +figures by projective pencils. We may also notice M. Chasles, +whose <i>Aperçu historique</i> (1837) is a classic. Synthetic geometry, +characterized by its fruitfulness and beauty, attracted most +attention, and it so happened that its originally weak logical +foundations became replaced by a more substantial set of axioms. +These were found in the anharmonic ratio, a device leading to +the liberation of synthetic geometry from metrical relations, +and in involution, which yielded rigorous definitions of imaginaries. +These innovations were made by K.J.C. von Staudt. +Analytical geometry was stimulated by the algebra of invariants, +a subject much developed by A. Cayley, G. Salmon, S.H. Aronhold, +L.O. Hesse, and more particularly by R.F.A. Clebsch.</p> + +<p>The introduction of the line as a space element, initiated by +<span class="pagenum"><a name="page677" id="page677"></a>677</span> +H. Grassmann (1844) and Cayley (1859), yielded at the hands of +Plücker a new geometry, termed <i>line geometry</i>, a subject +developed more notably by F. Klein, Clebsch, C.T. Reye and +F.O.R. Sturm (see Section V., <i>Line Geometry</i>).</p> + +<p><i>Non-euclidean geometries</i>, having primarily their origin in the +discussion of Euclidean parallels, and treated by Wallis, Saccheri +and Lambert, have been especially developed during the 19th +century. Four lines of investigation may be distinguished:—the +naïve-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 Civittà, +and the Germans Pasch and Hilbert.</p> +<div class="author">(C. E.*)</div> + +<p class="pt2 center sc">I. Euclidean Geometry</p> + +<p>This branch of the science of geometry is so named since its +methods and arrangement are those laid down in Euclid’s +<i>Elements</i>.</p> + +<p>§ 1. <i>Axioms.</i>—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.</p> + +<p>§ 2. <i>Definitions.</i>—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.</p> + +<p>Euclid gives the essence of these statements as definitions:—</p> + +<div class="condensed list"> +<p>Def. 1, I. <i>A point is that which has no parts, or which has no magnitude.</i></p> + +<p>Def. 2, I. <i>A line is length without breadth.</i></p> + +<p>Def. 5, I. <i>A superficies is that which has only length and breadth.</i></p> + +<p>Def. 1, XI. <i>A solid is that which has length, breadth and thickness.</i></p> +</div> + +<p>It is to be noted that the synthetic method is adopted by +Euclid; the analytical derivation of the successive ideas of +“surface,” “line,” and “point” from the experimental realization +of a “solid” does not find a place in his system, although +possessing more advantages.</p> + +<p>If we allow motion in geometry, we may generate these +entities by moving a point, a line, or a surface, thus:—</p> + +<table class="reg f90" summary="poem"><tr><td> <div class="poemr"> +<p>The path of a moving point is a line.</p> + +<p>The path of a moving line is, in general, a surface.</p> + +<p>The path of a moving surface is, in general, a solid.</p> +</div> </td></tr></table> + +<p>And we may then assume that the lines, surfaces and solids, +as defined before, can all be generated in this manner. From +this generation of the entities it follows again that the boundaries—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.</p> + +<p>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.</p> + +<p>§ 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.</p> + +<p>Whilst for the straight line the verbal definition and axiom +are kept apart, Euclid mixes them up in the case of the plane. +Here the Definition 7, I., includes an axiom. It defines a plane +as a surface which has the property that every straight line +which joins any two points in it lies altogether in the surface. +But if we take a straight line and a point in such a surface, and +draw all straight lines which join the latter to all points in the +first line, the surface will be fully determined. This construction +is therefore sufficient as a definition. That every other straight +line which joins any two points in this surface lies altogether +in it is a further property, and to assume it gives another axiom.</p> + +<p>Thus a number of Euclid’s axioms are hidden among his first +definitions. A still greater confusion exists in the present +editions of Euclid between the postulates and axioms so called, +but this is due to later editors and not to Euclid himself. The +latter had the last three axioms put together with the postulates +(<span class="grk" title="aitêmata">αἰτήματα</span>), so that these were meant to include all assumptions +relating to space. The remaining assumptions, which relate to +magnitudes in general, viz. the first eight “axioms” in modern +editions, were called “common notions” (<span class="grk" title="koivai ennoiai">κοιναὶ ἔννοιαι</span>). +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 +<i>Elements</i>, we have therefore to take the three postulates, the +last three axioms as generally given, a few axioms hidden in the +definitions, and an axiom used by Euclid in the proof of Prop. +4, I, and on a few other occasions, viz. that figures may be +moved in space without change of shape or size.</p> + +<p>§ 4. <i>Postulates.</i>—The assumptions actually made by Euclid +may be stated as follows:—</p> + +<div class="condensed"> +<p>(1) Straight lines exist which have the property that any one of +them may be produced both ways without limit, that through any +two points in space such a line may be drawn, and that any two of +them coincide throughout their indefinite extensions as soon as two +points in the one coincide with two points in the other. (This +gives the contents of Def. 4, part of Def. 35, the first two Postulates, +and Axiom 10.)</p> + +<p>(2) Plane surfaces or planes exist having the property laid down +in Def. 7, that every straight line joining any two points in such a +surface lies altogether in it.</p> + +<p>(3) Right angles, as defined in Def. 10, are possible, and all right +angles are equal; that is to say, wherever in space we take a plane, +and wherever in that plane we construct a right angle, all angles +thus constructed will be equal, so that any one of them may be made +to coincide with any other. (Axiom 11.)</p> + +<p>(4) The 12th Axiom of Euclid. This we shall not state now, but +only introduce it when we cannot proceed any further without it.</p> + +<p>(5) Figures maybe freely moved in space without change of shape +or size. This is assumed by Euclid, but not stated as an axiom.</p> + +<p>(6) In any plane a circle may be described, having any point in +that plane as centre, and its distance from any other point in that +plane as radius. (Postulate 3.)</p> +</div> + +<p>The definitions which have not been mentioned are all +“nominal definitions,” that is to say, they fix a name for a +thing described. Many of them overdetermine a figure.</p> + +<p>§ 5. Euclid’s <i>Elements</i> (see <span class="sc"><a href="#artlinks">Euclid</a></span>) are contained in thirteen +books. Of these the first four and the sixth are devoted to +“plane geometry,” as the investigation of figures in a plane is +generally called. The 5th book contains the theory of proportion +<span class="pagenum"><a name="page678" id="page678"></a>678</span> +which is used in Book VI. The 7th, 8th and 9th books are purely +arithmetical, whilst the 10th contains a most ingenious treatment +of geometrical irrational quantities. These four books will be +excluded from our survey. The remaining three books relate to +figures in space, or, as it is generally called, to “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 +<i>Elements</i>. In the first four and in the 6th book it is to be understood +that all figures are drawn in a plane.</p> + +<div class="condensed"> +<p class="pt2 center sc">Book I. of Euclid’s “Elements.”</p> + +<p>§ 6. According to the third postulate it is possible to draw in +any plane a circle which has its centre at any given point, and its +radius equal to the distance of this point from any other point +given in the plane. This makes it possible (Prop. 1) to construct +on a given line AB an equilateral triangle, by drawing first a circle +with A as centre and AB as radius, and then a circle with B as +centre and BA as radius. The point where these circles intersect—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.</p> + +<p>§ 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:—</p> + +<p><i>If two triangles have two sides of the one equal to two sides of the +other, each to each, and have also the angles contained by those sides +equal to one another, they shall also have their bases or third sides +equal; and the two triangles shall be equal; and their other angles +shall be equal, each to each, namely, those to which the equal sides are +opposite.</i></p> + +<p>That is to say, the triangles are “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.</p> + +<p>If the last proposition be applied to an isosceles triangle, which +has two sides equal, we obtain the theorem (Prop. 5), <i>if two sides +of a triangle are equal, then the angles opposite these sides are equal</i>.</p> + +<p>Euclid’s proof is somewhat complicated, and a stumbling-block +to many schoolboys. The proof becomes much simpler if we consider +the isosceles triangle ABC (AB = AC) twice over, once as a triangle +BAC, and once as a triangle CAB; and now remember that AB, AC +in the first are equal respectively to AC, AB in the second, and the +angles included by these sides are equal. Hence the triangles are +equal, and the angles in the one are equal to those in the other, viz. +those which are opposite equal sides, <i>i.e.</i> angle ABC in the first +equals angle ACB in the second, as they are opposite the equal +sides AC and AB in the two triangles.</p> + +<p>There follows the converse theorem (Prop. 6). <i>If two angles in +a triangle are equal, then the sides opposite them are equal</i>,—<i>i.e.</i> the +triangle is isosceles. The proof given consists in what is called a +<i>reductio ad absurdum</i>, a kind of proof often used by Euclid, and +principally in proving the converse of a previous theorem. It +assumes that the theorem to be proved is wrong, and then shows +that this assumption leads to an absurdity, <i>i.e.</i> to a conclusion +which is in contradiction to a proposition proved before—that +therefore the assumption made cannot be true, and hence that +the theorem is true. It is often stated that Euclid invented this +kind of proof, but the method is most likely much older.</p> + +<p>§ 8. It is next proved that <i>two triangles which have the three sides +of the one equal respectively to those of the other are identically equal, +hence that the angles of the one are equal respectively to those of the +other, those being equal which are opposite equal sides</i>. This is Prop. 8, +Prop. 7 containing only a first step towards its proof.</p> + +<p>These theorems allow now of the solution of a number of problems, +viz.:—</p> + +<p><i>To bisect a given angle</i> (Prop. 9).</p> + +<p><i>To bisect a given finite straight line</i> (Prop. 10).</p> + +<p><i>To draw a straight line perpendicularly to a given straight line +through a given point in it</i> (Prop. 11), <i>and also through a given point +not in it</i> (Prop. 12).</p> + +<p>The solutions all depend upon properties of isosceles triangles.</p> + +<p>§ 9. The next three theorems relate to angles only, and might have +been proved before Prop. 4, or even at the very beginning. The +first (Prop. 13) says, <i>The angles which one straight line makes with +another straight line on one side of it either are two right angles or +are together equal to two right angles</i>. This theorem would have +been unnecessary if Euclid had admitted the notion of an angle +such that its two limits are in the same straight line, and had besides +defined the sum of two angles.</p> + +<p>Its converse (Prop. 14) is of great use, inasmuch as it enables us +in many cases to prove that two straight lines drawn from the same +point are one the continuation of the other. So also is</p> + +<p>Prop. 15. <i>If two straight lines cut one another, the vertical or opposite +angles shall be equal.</i></p> + +<p>§ 10. Euclid returns now to properties of triangles. Of great +importance for the next steps (though afterwards superseded by a +more complete theorem) is</p> + +<p>Prop. 16. <i>If one side of a triangle be produced, the exterior angle +shall be greater than either of the interior opposite angles.</i></p> + +<p>Prop. 17. <i>Any two angles of a triangle are together less than two +right angles, is an immediate consequence of it.</i> By the aid of these +two, the following fundamental properties of triangles are easily +proved:—</p> + +<p>Prop. 18. <i>The greater side of every triangle has the greater angle +opposite to it</i>;</p> + +<p>Its converse, Prop. 19. <i>The greater angle of every triangle is subtended +by the greater side, or has the greater side opposite to it</i>;</p> + +<p>Prop. 20. <i>Any two sides of a triangle are together greater than the +third side</i>;</p> + +<p>And also Prop. 21. <i>If from the ends of the side of a triangle there +be drawn two straight lines to a point within the triangle, these shall +be less than the other two sides of the triangle, but shall contain a greater +angle.</i></p> + +<p>§ 11. Having solved two problems (Props. 22, 23), he returns to two +triangles which have two sides of the one equal respectively to two +sides of the other. It is known (Prop. 4) that if the included angles +are equal then the third sides are equal; and conversely (Prop. 8), +if the third sides are equal, then the angles included by the first +sides are equal. From this it follows that if the included angles are +not equal, the third sides are not equal; and conversely, that if the +third sides are not equal, the included angles are not equal. Euclid +now completes this knowledge by proving, that “<i>if the included +angles are not equal, then the third side in that triangle is the greater +which contains the greater angle</i>”; and conversely, that “<i>if the third +sides are unequal, that triangle contains the greater angle which contains +the greater side</i>.” These are Prop. 24 and Prop. 25.</p> + +<p>§ 12. The next theorem (Prop. 26) says that <i>if two triangles have +one side and two angles of the one equal respectively to one side and +two angles of the other, viz. in both triangles either the angles adjacent +to the equal side, or one angle adjacent and one angle opposite it, then +the two triangles are identically equal</i>.</p> + +<p>This theorem belongs to a group with Prop. 4 and Prop. 8. Its +first case might have been given immediately after Prop. 4, but the +second case requires Prop. 16 for its proof.</p> + +<p>§ 13. We come now to the investigation of parallel straight lines, +<i>i.e.</i> of straight lines which lie in the same plane, and cannot be made +to meet however far they be produced either way. The investigation +which starts from Prop. 16, will become clearer if a few names be +explained which are not all used by Euclid. If two straight lines +be cut by a third, the latter is now generally called a “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.”</p> + +<p>We may now state Prop. 16 thus:—<i>If two straight lines which +meet are cut by a transversal, their alternate angles are unequal</i>. For +the lines will form a triangle, and one of the alternate angles will +be an exterior angle to the triangle, the other interior and opposite +to it.</p> + +<p>From this follows at once the theorem contained in Prop. 27. +<i>If two straight lines which are cut by a transversal make alternate +angles equal, the lines cannot meet, however far they be produced, +hence they are parallel.</i> This proves the existence of parallel +lines.</p> + +<p>Prop. 28 states the same fact in different forms. <i>If a straight +line, falling on two other straight lines, make the exterior angle equal +to the interior and opposite angle on the same side of the line, or make</i> +<span class="pagenum"><a name="page679" id="page679"></a>679</span> +<i>the interior angles on the same side together equal to two right angles, +the two straight lines shall be parallel to one another</i>.</p> + +<p>Hence we know that, “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.”</p> + +<p>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?”</p> + +<p>The answer to either of these two questions implies the answer +to the other. But it has been found impossible to prove that the +negation or the affirmation of either is true.</p> + +<p>The difficulty which thus arises is overcome by Euclid assuming +that the first question has to be answered in the affirmative. This +gives his last axiom (12), which we quote in his own words.</p> + +<p>Axiom 12.—<i>If a straight line meet two straight lines, so as to make +the two interior angles on the same side of it taken together less than +two right angles, these straight lines, being continually produced, shall +at length meet on that side on which are the angles which are less than +two right angles.</i></p> + +<p>The answer to the second of the above questions follows from this, +and gives the theorem Prop. 29:—<i>If a straight line fall on two parallel +straight lines, it makes the alternate angles equal to one another, and +the exterior angle equal to the interior and opposite angle on the same +side, and also the two interior angles on the same side together equal +to two right angles</i>.</p> + +<p>§ 14. With this a new part of elementary geometry begins. The +earlier propositions are independent of this axiom, and would be +true even if a wrong assumption had been made in it. They all +relate to figures in a plane. But a plane is only one among an infinite +number of conceivable surfaces. We may draw figures on any one +of them and study their properties. We may, for instance, take a +sphere instead of the plane, and obtain “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.</p> + +<p>For spherical triangles, therefore, all the important propositions +4, 8, 26; 5 and 6; and 18, 19 and 20 will hold good.</p> + +<p>This remark will be sufficient to show the impossibility of proving +Euclid’s last axiom, which would mean proving that this axiom is +a consequence of the others, and hence that the theory of parallels +would hold on a spherical surface, where the other axioms do hold, +whilst parallels do not even exist.</p> + +<p>It follows that the axiom in question states an inherent difference +between the plane and other surfaces, and that the plane is only +fully characterized when this axiom is added to the other assumptions.</p> + +<p>§ 15. The introduction of the new axiom and of parallel lines leads +to a new class of propositions.</p> + +<p>After proving (Prop. 30) that “<i>two lines which are each parallel +to a third are parallel to each other</i>,” we obtain the new properties +of triangles contained in Prop. 32. Of these the second part is the +most important, viz. the theorem, <i>The three interior angles of every +triangle are together equal to two right angles</i>.</p> + +<p>As easy deductions not given by Euclid but added by Simson +follow the propositions about the angles in polygons, they are given +in English editions as corollaries to Prop. 32.</p> + +<p>These theorems do not hold for spherical figures. The sum of the +interior angles of a spherical triangle is always greater than two +right angles, and increases with the area.</p> + +<p>§ 16. The theory of parallels as such may be said to be finished +with Props. 33 and 34, which state properties of the parallelogram, +<i>i.e.</i> of a quadrilateral formed by two pairs of parallels. They are—</p> + +<p>Prop. 33. <i>The straight lines which join the extremities of two equal +and parallel straight lines towards the same parts are themselves equal +and parallel</i>; and</p> + +<p>Prop. 34. <i>The opposite sides and angles of a parallelogram are +equal to one another, and the diameter (diagonal) bisects the parallelogram, +that is, divides it into two equal parts.</i></p> + +<p>§ 17. The rest of the first book relates to areas of figures.</p> + +<p>The theory is made to depend upon the theorems—</p> + +<p>Prop. 35. <i>Parallelograms on the same base and between the same +parallels are equal to one another</i>; and</p> + +<p>Prop. 36. <i>Parallelograms on equal bases and between the same +parallels are equal to one another</i>.</p> + +<p>As each parallelogram is bisected by a diagonal, the last theorems +hold also if the word parallelogram be replaced by “triangle,” as is +done in Props. 37 and 38.</p> + +<p>It is to be remarked that Euclid proves these propositions only +in the case when the parallelograms or triangles have their bases in +the same straight line.</p> + +<p>The theorems converse to the last form the contents of the next +three propositions, viz.: Props, 40 and 41.—<i>Equal triangles, on +the same or on equal bases, in the same straight line, and on the same +side of it, are between the same parallels</i>.</p> + +<p>That the two cases here stated are given by Euclid in two separate +propositions proved separately is characteristic of his method.</p> + +<p>§ 18. To compare areas of other figures, Euclid shows first, in +Prop. 42, how <i>to draw a parallelogram which is equal in area to a +given triangle, and has one of its angles equal to a given angle</i>. If the +given angle is right, then the problem is solved <i>to draw a “rectangle” +equal in area to a given triangle</i>.</p> + +<p>Next this parallelogram is transformed into another parallelogram, +<i>which has one of its sides equal to a given straight line</i>, whilst its angles +remain unaltered. This may be done by aid of the theorem in</p> + +<p>Prop. 43. <i>The complements of the parallelograms which are about +the diameter of any parallelogram are equal to one another.</i></p> + +<p>Thus the problem (Prop. 44) is solved to <i>construct a parallelogram +on a given line, which is equal in area to a given triangle, and which +has one angle equal to a given angle</i> (generally a right angle).</p> + +<p>As every polygon can be divided into a number of triangles, we +can now construct a parallelogram having a given angle, say a +right angle, and being equal in area to a given polygon. For each +of the triangles into which the polygon has been divided, a parallelogram +may be constructed, having one side equal to a given straight +line and one angle equal to a given angle. If these parallelograms +be placed side by side, they may be added together to form a single +parallelogram, having still one side of the given length. This is +done in Prop. 45.</p> + +<p>Herewith a means is found to compare areas of different polygons. +We need only construct two rectangles equal in area to the given +polygons, and having each one side of given length. By comparing +the unequal sides we are enabled to judge whether the areas are +equal, or which is the greater. Euclid does not state this consequence, +but the problem is taken up again at the end of the second book, +where it is shown how to construct a square equal in area to a given +polygon.</p> + +<p>Prop. 46 is: <i>To describe a square on a given straight line</i>.</p> + +<p>§ 19. The first book concludes with one of the most important +theorems in the whole of geometry, and one which has been celebrated +since the earliest times. It is stated, but on doubtful authority, +that Pythagoras discovered it, and it has been called by his name. +If we call that side in a right-angled triangle which is opposite the +right angle the hypotenuse, we may state it as follows:—</p> + +<p>Theorem of Pythagoras (Prop. 47).—<i>In every right-angled triangle +the square on the hypotenuse is equal to the sum of the squares of the +other sides.</i></p> + +<p>And conversely—</p> + +<p>Prop. 48. <i>If the square described on one of the sides of a triangle be +equal to the squares described on the other sides, then the angle contained +by these two sides is a right angle.</i></p> + +<p>On this theorem (Prop. 47) almost all geometrical measurement +depends, which cannot be directly obtained.</p> + +<p class="pt2 center sc">Book II.</p> + +<p>§ 20. The propositions in the second book are very different in +character from those in the first; they all relate to areas of rectangles +and squares. Their true significance is best seen by stating them in +an algebraic form. This is often done by expressing the lengths of +lines by aid of numbers, which tell how many times a chosen unit +is contained in the lines. If there is a unit to be found which is contained +an exact number of times in each side of a rectangle, it is +easily seen, and generally shown in the teaching of arithmetic, that +the rectangle contains a number of unit squares equal to the product +of the numbers which measure the sides, a unit square being the +square on the unit line. If, however, no such unit can be found, +this process requires that connexion between lines and numbers +which is only established by aid of ratios of lines, and which is therefore +at this stage altogether inadmissible. But there exists another +way of connecting these propositions with algebra, based on modern +notions which seem destined greatly to change and to simplify +mathematics. We shall introduce here as much of it as is required +for our present purpose.</p> + +<p>At the beginning of the second book we find a definition according +to which “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.</p> + +<p>We shall denote a finite straight line by a single small letter, +a, b, c, ... x, and the area of the rectangle contained by two lines +a and b by ab, and this we shall call the product of the two lines a +and b. It will be understood that this definition has nothing to do +with the definition of a product of numbers.</p> + +<p>We define as follows:—</p> + +<p>The <i>sum</i> of two straight lines a and b means a straight line c which +may be divided in two parts equal respectively to a and b. This sum +is denoted by a + b.</p> + +<p>The <i>difference</i> of two lines a and b (in symbols, a-b) means a line +c which when added to b gives a; that is,</p> + +<p class="center">a − b = c if b + c = a.</p> + +<p>The <i>product</i> of two lines a and b (in symbols, ab) means the area +<span class="pagenum"><a name="page680" id="page680"></a>680</span> +of the rectangle contained by the lines a and b. For aa, which +means the square on the line a, we write a².</p> + +<p>§ 21. The first ten of the fourteen propositions of the second book +may then be written in the form of formulae as follows:—</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc">Prop.</td> <td class="tcr">1.</td> <td class="tcl">a (b + c + d + ... ) = ab + ac + ad + ...</td></tr> + +<tr><td class="tcc">”</td> <td class="tcr">2.</td> <td class="tcl">ab + ac = a² if b + c = a.</td></tr> + +<tr><td class="tcc">”</td> <td class="tcr">3.</td> <td class="tcl">a (a + b) = a² + ab.</td></tr> + +<tr><td class="tcc">”</td> <td class="tcr">4.</td> <td class="tcl">(a + b)² = a² + 2ab + b².</td></tr> + +<tr><td class="tcc">”</td> <td class="tcr">5.</td> <td class="tcl">(a + b)(a − b) + b² = a².</td></tr> + +<tr><td class="tcc">”</td> <td class="tcr">6.</td> <td class="tcl">(a + b)(a − b) + b² = a².</td></tr> + +<tr><td class="tcc">”</td> <td class="tcr">7.</td> <td class="tcl">a² + (a − b)² = 2a (a − b) + b².</td></tr> + +<tr><td class="tcc">”</td> <td class="tcr">8.</td> <td class="tcl">4(a + b)a + b² = (2a + b)².</td></tr> + +<tr><td class="tcc">”</td> <td class="tcr">9.</td> <td class="tcl">(a + b)² + (a − b)² = 2a² + 2b².</td></tr> + +<tr><td class="tcc">”</td> <td class="tcr">10.</td> <td class="tcl">(a + b)² + (a − b)² = 2a² + 2b².</td></tr> +</table> + +<p>It will be seen that 5 and 6, and also 9 and 10, are identical. In +Euclid’s statement they do not look the same, the figures being +arranged differently.</p> + +<p>If the letters a, b, c, ... denoted numbers, it follows from algebra +that each of these formulae is true. But this does not prove them in +our case, where the letters denote lines, and their products areas +without any reference to numbers. To prove them we have to +discover the laws which rule the operations introduced, viz. addition +and multiplication of segments. This we shall do now; and we shall +find that these laws are the same with those which hold in algebraical +addition and multiplication.</p> + +<p>§ 22. In a sum of numbers we may change the order in which +the numbers are added, and we may also add the numbers together +in groups and then add these groups. But this also holds for the +sum of segments and for the sum of rectangles, as a little consideration +shows. That the sum of rectangles has always a meaning +follows from the Props. 43-45 in the first book. These laws about +addition are reducible to the two—</p> + +<p class="center">a + b = b + a</p> +<div class="author">(1),</div> + +<p class="center">a + (b + c) = a + b + c</p> +<div class="author">(2);</div> + +<p class="noind">or, when expressed for rectangles,</p> + +<p class="center">ab + ed = ed + ab</p> +<div class="author">(3),</div> + +<p class="center">ab + (cd + ef) = ab + cd + ef</p> +<div class="author">(4).</div> + +<p class="noind">The brackets mean that the terms in the bracket have been added +together before they are added to another term. The more general +cases for more terms may be deduced from the above.</p> + +<p>For the product of two numbers we have the law that it remains +unaltered if the factors be interchanged. This also holds for our +geometrical product. For if ab denotes the area of the rectangle +which has a as base and b as altitude, then ba will denote the area +of the rectangle which has b as base and a as altitude. But in a +rectangle we may take either of the two lines which contain it as +base, and then the other will be the altitude. This gives</p> + +<p class="center">ab = ba</p> +<div class="aut">(5).</div> + +<p>In order further to multiply a sum by a number, we have in algebra +the rule:—Multiply each term of the sum, and add the products +thus obtained. That this holds for our geometrical products is +shown by Euclid in his first proposition of the second book, where +he proves that the area of a rectangle whose base is the sum of a +number of segments is equal to the sum of rectangles which have +these segments separately as bases. In symbols this gives, in the +simplest case,</p> + +<p class="center">a(b + c) = ab + ac</p> + +<p class="noind">and</p> + +<p class="center">(b + c)a = ba + ca</p> +<div class="aut">(6).</div> + +<p class="noind">To these laws, which have been investigated by Sir William Hamilton +and by Hermann Grassmann, the former has given special names. +He calls the laws expressed in</p> + +<table class="reg" summary="poem"><tr><td> <div class="poemr"> +<p>(1) and (3) the commutative law for addition;</p> + +<p class="i3">(5) the commutative law for multiplication;</p> + +<p>(2) and (4) the associative laws for addition;</p> + +<p class="i3">(6) the distributive law.</p> +</div> </td></tr></table> + +<p>§ 23. Having proved that these six laws hold, we can at once +prove every one of the above propositions in their algebraical form.</p> + +<p>The first is proved geometrically, it being one of the fundamental +laws. The next two propositions are only special cases of the first. +Of the others we shall prove one, viz. the fourth:—</p> + +<p class="center">(a + b)² = (a + b)(a + b) = (a + b)a + (a + b)b</p> +<div class="aut">by (6).</div> + +<p class="noind">But</p> +<p class="center">(a + b)a = aa + ba</p> +<div class="aut">by (6),</div> + +<p class="center">= aa + ab</p> +<div class="aut">by (5);</div> + +<p class="noind">and</p> + +<p class="center">(a + b)b = ab + bb</p> +<div class="aut">by (6).</div> + +<p class="noind">Therefore</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcr">(a + b)²</td> <td class="tcl">= aa + ab + (ab + bb)</td></tr> +<tr><td class="tcr"> </td> <td class="tcl">= aa + (ab + ab) + bb</td></tr> +<tr><td class="tcr"> </td> <td class="tcl">= aa + 2ab + bb</td></tr> +</table> + +<div class="aut">by (4).</div> + +<p>This gives the theorem in question.</p> + +<p>In the same manner every one of the first ten propositions is +proved.</p> + +<p>It will be seen that the operations performed are exactly the same +as if the letters denoted numbers.</p> + +<p>Props. 5 and 6 may also be written thus—</p> + +<p class="center">(a + b)(a − b) = a² − b².</p> + +<p>Prop. 7, which is an easy consequence of Prop. 4, may be transformed. +If we denote by c the line a + b, so that</p> + +<p class="center">c = a + b, a = c − b,</p> + +<p class="noind">we get</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcr">c² + (c − b)²</td> <td class="tcl">= 2c(c − b) + b²</td></tr> +<tr><td class="tcr"> </td> <td class="tcl">= 2c² − 2bc + b².</td></tr> +</table> + +<p class="noind">Subtracting c² from both sides, and writing a for c, we get</p> + +<p class="center">(a − b)² = a² − 2ab + b².</p> + +<p>In Euclid’s <i>Elements</i> this form of the theorem does not appear, +all propositions being so stated that the notion of subtraction does +not enter into them.</p> + +<p>§ 24. The remaining two theorems (Props. 12 and 13) connect +the square on one side of a triangle with the sum of the squares on +the other sides, in case that the angle between the latter is acute or +obtuse. They are important theorems in trigonometry, where it is +possible to include them in a single theorem.</p> + +<p>§ 25. There are in the second book two problems, Props. 11 and 14.</p> + +<p>If written in the above symbolic language, the former requires to +find a line x such that a(a − x) = x². Prop. 11 contains, therefore, +the solution of a quadratic equation, which we may write x² + ax = a². +The solution is required later on in the construction of a regular +decagon.</p> + +<p>More important is the problem in the last proposition (Prop. 14). +It requires the construction of a square equal in area to a given +rectangle, hence a solution of the equation</p> + +<p class="center">x² = ab.</p> + +<p>In Book I., 42-45, it has been shown how a rectangle may be constructed +equal in area to a given figure bounded by straight lines. +By aid of the new proposition we may therefore now determine a +line such that the square on that line is equal in area to any given +rectilinear figure, or we can <i>square</i> any such figure.</p> + +<p>As of two squares that is the greater which has the greater side, +it follows that now the comparison of two areas has been reduced +to the comparison of two lines.</p> + +<p>The problem of reducing other areas to squares is frequently met +with among Greek mathematicians. We need only mention the +problem of squaring the circle (see <span class="sc"><a href="#artlinks">Circle</a></span>).</p> + +<p>In the present day the comparison of areas is performed in a +simpler way by reducing all areas to rectangles having a common +base. Their altitudes give then a measure of their areas.</p> + +<p>The construction of a rectangle having the base u, and being equal +in area to a given rectangle, depends upon Prop. 43, I. This therefore +gives a solution of the equation</p> + +<p class="center">ab = ux,</p> + +<p class="noind">where x denotes the unknown altitude.</p> + +<p class="pt2 center sc">Book III.</p> + +<p>§ 26. The third book of the <i>Elements</i> relates exclusively to properties +of the circle. A circle and its circumference have been defined +in Book I., Def. 15. We restate it here in slightly different words:—</p> + +<p><i>Definition</i>.—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.</p> + +<p>Of the new definitions, of which eleven are given at the beginning +of the third book, a few only require special mention. The first, +which says that circles with equal radii are equal, is in part a theorem, +but easily proved by applying the one circle to the other. Or it +may be considered proved by aid of Prop. 24, equal circles not being +used till after this theorem.</p> + +<p>In the second definition is explained what is meant by a line +which “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.</p> + +<p>For the same reason we shall call a straight line joining two points +on the circumference of a circle a “chord.”</p> + +<p>Definitions 4 and 5 may be replaced with a slight generalization +by the following:—</p> + +<p><i>Definition</i>.—By the distance of a point from a line is meant the +length of the perpendicular drawn from the point to the line.</p> + +<p>§ 27. From the definition of a circle it follows that every circle +has a centre. Prop. 1 requires to find it when the circle is given, +<i>i.e.</i> when its circumference is drawn.</p> + +<p>To solve this problem a chord is drawn (that is, any two points in +the circumference are joined), and through the point where this is +bisected a perpendicular to it is erected. Euclid then proves, first, +that no point off this perpendicular can be the centre, hence that the +centre must lie in this line; and, secondly, that of the points on the +perpendicular one only can be the centre, viz. the one which bisects +the parts of the perpendicular bounded by the circle. In the second +part Euclid silently assumes that the perpendicular there used does +cut the circumference in two, and only in two points. The proof +therefore is incomplete. The proof of the first part, however, is +exact. By drawing two non-parallel chords, and the perpendiculars +which bisect them, the centre will be found as the point where these +perpendiculars intersect.</p> + +<p>§ 28. In Prop. 2 it is proved that a chord of a circle lies altogether +within the circle.</p> + +<p><span class="pagenum"><a name="page681" id="page681"></a>681</span></p> + +<p>What we have called the first part of Euclid’s solution of Prop. 1 +may be stated as a theorem:—</p> + +<p><i>Every straight line which bisects a chord, and is at right angles to it, +passes through the centre of the circle.</i></p> + +<p>The converse to this gives Prop. 3, which may be stated thus:—</p> + +<p><i>If a straight line through the centre of a circle bisect a chord, then +it is perpendicular to the chord, and if it be perpendicular to the chord +it bisects it.</i></p> + +<p>An easy consequence of this is the following theorem, which is +essentially the same as Prop. 4:—</p> + +<p><i>Two chords of a circle, of which neither passes through the centre, +cannot bisect each other.</i></p> + +<p>These last three theorems are fundamental for the theory of the +circle. It is to be remarked that Euclid never proves that a straight +line cannot have more than two points in common with a circumference.</p> + +<p>§ 29. The next two propositions (5 and 6) might be replaced by +a single and a simpler theorem, viz:—</p> + +<p><i>Two circles which have a common centre, and whose circumferences +have one point in common, coincide.</i></p> + +<p>Or, more in agreement with Euclid’s form:—</p> + +<p><i>Two different circles, whose circumferences have a point in common, +cannot have the same centre.</i></p> + +<p>That Euclid treats of two cases is characteristic of Greek mathematics.</p> + +<p>The next two propositions (7 and 8) again belong together. They +may be combined thus:—</p> + +<p><i>If from a point in a plane of a circle, which is not the centre, straight +lines be drawn to the different points of the circumference, then of all +these lines one is the shortest, and one the longest, and these lie both in +that straight line which joins the given point to the centre. Of all the +remaining lines each is equal to one and only one other, and these +equal lines lie on opposite sides of the shortest or longest, and make +equal angles with them.</i></p> + +<p>Euclid distinguishes the two cases where the given point lies within +or without the circle, omitting the case where it lies in the circumference.</p> + +<p>From the last proposition it follows that if from a point more +than two equal straight lines can be drawn to the circumference, +this point must be the centre. This is Prop. 9.</p> + +<p>As a consequence of this we get</p> + +<p><i>If the circumferences of the two circles have three points in common +they coincide.</i></p> + +<p>For in this case the two circles have a common centre, because +from the centre of the one three equal lines can be drawn to points +on the circumference of the other. But two circles which have a +common centre, and whose circumferences have a point in common, +coincide. (Compare above statement of Props. 5 and 6.)</p> + +<p>This theorem may also be stated thus:—</p> + +<p><i>Through three points only one circumference may be drawn; or, +Three points determine a circle.</i></p> + +<p>Euclid does not give the theorem in this form. He proves, however, +<i>that the two circles cannot cut another in more than two points</i> +(Prop. 10), and <i>that two circles cannot touch one another in more points +than one</i> (Prop. 13).</p> + +<p>§ 30. Propositions 11 and 12 assert that <i>if two circles touch, then +the point of contact lies on the line joining their centres</i>. This gives +two propositions, because the circles may touch either internally +or externally.</p> + +<p>§ 31. Propositions 14 and 15 relate to the length of chords. The +first says <i>that equal chords are equidistant from the centre, and that +chords which are equidistant from the centre are equal</i>;</p> + +<p>Whilst Prop. 15 compares unequal chords, viz. <i>Of all chords the +diameter is the greatest, and of other chords that is the greater which +is nearer to the centre</i>; and conversely, <i>the greater chord is nearer to +the centre</i>.</p> + +<p>§ 32. In Prop. 16 the tangent to a circle is for the first time introduced. +The proposition is meant to show that the straight line +at the end point of the diameter and at right angles to it is a tangent. +The proposition itself does not state this. It runs thus:—</p> + +<p>Prop. 16. <i>The straight line drawn at right angles to the diameter +of a circle, from the extremity of it, falls without the circle; and no +straight line can be drawn from the extremity, between that straight +line and the circumference, so as not to cut the circle.</i></p> + +<p><i>Corollary</i>.—The straight line at right angles to a diameter drawn +through the end point of it touches the circle.</p> + +<p>The statement of the proposition and its whole treatment show +the difficulties which the tangents presented to Euclid.</p> + +<p>Prop. 17 solves the problem <i>through a given point, either in the +circumference or without it, to draw a tangent to a given circle</i>.</p> + +<p>Closely connected with Prop. 16 are Props. 18 and 19, which +state (Prop. 18), <i>that the line joining the centre of a circle to the point +of contact of a tangent is perpendicular to the tangent</i>; and conversely +(Prop. 19), <i>that the straight line through the point of contact +of, and perpendicular to, a tangent to a circle passes through the centre +of the circle</i>.</p> + +<p>§ 33. The rest of the book relates to angles connected with a +circle, viz. angles which have the vertex either at the centre or +on the circumference, and which are called respectively angles +at the centre and angles at the circumference. Between these +two kinds of angles exists the important relation expressed as +follows:—</p> + +<p>Prop. 20. <i>The angle at the centre of a circle is double of the angle +at the circumference on the same base, that is, on the same arc.</i></p> + +<p>This is of great importance for its consequences, of which the +two following are the principal:—</p> + +<p>Prop. 21. <i>The angles in the same segment of a circle are equal to +one another</i>;</p> + +<p>Prop. 22. <i>The opposite angles of any quadrilateral figure inscribed +in a circle are together equal to two right angles.</i></p> + +<p>Further consequences are:—</p> + +<p>Prop. 23. <i>On the same straight line, and on the same side of it, there +cannot be two similar segments of circles, not coinciding with one +another</i>;</p> + +<p>Prop. 24. <i>Similar segments of circles on equal straight lines are +equal to one another.</i></p> + +<p>The problem Prop. 25. <i>A segment of a circle being given to describe +the circle of which it is a segment</i>, may be solved much more easily +by aid of the construction described in relation to Prop. 1, III., +in § 27.</p> + +<p>§ 34. There follow four theorems connecting the angles at the +centre, the arcs into which they divide the circumference, and the +chords subtending these arcs. They are expressed for angles, arcs +and chords in equal circles, but they hold also for angles, arcs and +chords in the same circle.</p> + +<p>The theorems are:—</p> + +<p>Prop. 26. <i>In equal circles equal angles stand on equal arcs, whether +they be at the centres or circumferences</i>;</p> + +<p>Prop. 27. (converse to Prop. 26). <i>In equal circles the angles which +stand on equal arcs are equal to one another, whether they be at the +centres or the circumferences</i>;</p> + +<p>Prop. 28. <i>In equal circles equal straight lines</i> (equal chords) <i>cut +off equal arcs, the greater equal to the greater, and the less equal to +the less</i>;</p> + +<p>Prop. 29 (converse to Prop. 28). <i>In equal circles equal arcs are +subtended by equal straight lines.</i></p> + +<p>§ 35. Other important consequences of Props. 20-22 are:—</p> + +<p>Prop. 31. <i>In a circle the angle in a semicircle is a right angle; +but the angle in a segment greater than a semicircle is less than a right +angle; and the angle in a segment less than a semicircle is greater than +a right angle</i>;</p> + +<p>Prop. 32. <i>If a straight line touch a circle, and from the point of +contact a straight line be drawn cutting the circle, the angles which +this line makes with the line touching the circle shall be equal to the +angles which are in the alternate segments of the circle.</i></p> + +<p>§ 36. Propositions 30, 33, 34, contain problems which are solved +by aid of the propositions preceding them:—</p> + +<p>Prop. 30. <i>To bisect a given arc, that is, to divide it into two equal +parts</i>;</p> + +<p>Prop. 33. <i>On a given straight line to describe a segment of a circle +containing an angle equal to a given rectilineal angle</i>;</p> + +<p>Prop. 34. <i>From a given circle to cut off a segment containing an +angle equal to a given rectilineal angle</i>.</p> + +<p>§ 37. If we draw chords through a point A within a circle, they +will each be divided by A into two segments. Between these segments +the law holds that the rectangle contained by them has the +same area on whatever chord through A the segments are taken. +The value of this rectangle changes, of course, with the position +of A.</p> + +<p>A similar theorem holds if the point A be taken without the circle. +On every straight line through A, which cuts the circle in two points +B and C, we have two segments AB and AC, and the rectangles +contained by them are again equal to one another, and equal to the +square on a tangent drawn from A to the circle.</p> + +<p>The first of these theorems gives Prop. 35, and the second Prop. +36, with its corollary, whilst Prop. 37, the last of Book III., gives +the converse to Prop. 36. The first two theorems may be combined +in one:—</p> + +<p><i>If through a point A in the plane of a circle a straight line be drawn +cutting the circle in B and C, then the rectangle AB.AC has a constant +value so long as the point A be fixed; and if from A a tangent AD can +be drawn to the circle, touching at D, then the above rectangle equals the +square on AD.</i></p> + +<p>Prop. 37 may be stated thus:—</p> + +<p><i>If from a point A without a circle a line be drawn cutting the circle +in B and C, and another line to a point D on the circle, and AB.AC = +AD², then the line AD touches the circle at D.</i></p> + +<p>It is not difficult to prove also the converse to the general proposition +as above stated. This proposition and its converse may be +expressed as follows:—</p> + +<p><i>If four points ABCD be taken on the circumference of a circle, and +if the lines AB, CD, produced if necessary, meet at E, then</i></p> + +<p class="center">EA·EB = EC·ED;</p> + +<p class="noind"><i>and conversely, if this relation holds then the four points lie on a circle, +that is, the circle drawn through three of them passes through the +fourth.</i></p> + +<p>That a circle may always be drawn through three points, provided +that they do not lie in a straight line, is proved only later on in +Book IV.</p> + +<p><span class="pagenum"><a name="page682" id="page682"></a>682</span></p> + +<p class="pt2 center sc">Book IV.</p> + +<p>§ 38. The fourth book contains only problems, all relating to +the construction of triangles and polygons inscribed in and circumscribed +about circles, and of circles inscribed in or circumscribed +about triangles and polygons. They are nearly all given for their +own sake, and not for future use in the construction of figures, as +are most of those in the former books. In seven definitions at the +beginning of the book it is explained what is understood by figures +inscribed in or described about other figures, with special reference +to the case where one figure is a circle. Instead, however, of saying +that one figure is described about another, it is now generally said +that the one figure is circumscribed about the other. We may then +state the definitions 3 or 4 thus:—</p> + +<p><i>Definition.</i>—A polygon is said to be inscribed in a circle, and the +circle is said to be circumscribed about the polygon, if the vertices +of the polygon lie in the circumference of the circle.</p> + +<p>And definitions 5 and 6 thus:—</p> + +<p><i>Definition.</i>—A polygon is said to be circumscribed about a circle, +and a circle is said to be inscribed in a polygon, if the sides of the +polygon are tangents to the circle.</p> + +<p>§ 39. The first problem is merely constructive. It requires to +draw in a given circle a chord equal to a given straight line, which +is not greater than the diameter of the circle. The problem is not +a determinate one, inasmuch as the chord may be drawn from any +point in the circumference. This may be said of almost all problems +in this book, especially of the next two. They are:—</p> + +<p>Prop. 2. <i>In a given circle to inscribe a triangle equiangular to a +given triangle;</i></p> + +<p>Prop. 3. <i>About a given circle to circumscribe a triangle equiangular +to a given triangle.</i></p> + +<p>§ 40. Of somewhat greater interest are the next problems, where +the triangles are given and the circles to be found.</p> + +<p>Prop. 4. <i>To inscribe a circle in a given triangle.</i></p> + +<p>The result is that the problem has always a solution, viz. the +centre of the circle is the point where the bisectors of two of the +interior angles of the triangle meet. The solution shows, though +Euclid does not state this, that the problem has but one solution; +and also,</p> + +<p><i>The three bisectors of the interior angles of any triangle meet in a +point, and this is the centre of the circle inscribed in the triangle.</i></p> + +<p>The solutions of most of the other problems contain also theorems. +Of these we shall state those which are of special interest; Euclid +does not state any one of them.</p> + +<p>§ 41. Prop. 5. <i>To circumscribe a circle about a given triangle.</i></p> + +<p>The one solution which always exists contains the following:—</p> + +<p><i>The three straight lines which bisect the sides of a triangle at right +angles meet in a point, and this point is the centre of the circle circumscribed +about the triangle.</i></p> + +<p>Euclid adds in a corollary the following property:—</p> + +<p>The centre of the circle circumscribed about a triangle lies within, +on a side of, or without the triangle, according as the triangle is +acute-angled, right-angled or obtuse-angled.</p> + +<p>§ 42. Whilst it is always possible to draw a circle which is inscribed +in or circumscribed about a given triangle, this is not the case with +quadrilaterals or polygons of more sides. Of those for which this +is possible the regular polygons, <i>i.e.</i> polygons which have all their +sides and angles equal, are the most interesting. In each of them a +circle may be inscribed, and another may be circumscribed about it.</p> + +<p>Euclid does not use the word regular, but he describes the polygons +in question as <i>equiangular</i> and <i>equilateral</i>. We shall use the name +regular polygon. The regular triangle is equilateral, the regular +quadrilateral is the square.</p> + +<p>Euclid considers the regular polygons of 4, 5, 6 and 15 sides. +For each of the first three he solves the problems—(1) to inscribe +such a polygon in a given circle; (2) to circumscribe it about a +given circle; (3) to inscribe a circle in, and (4) to circumscribe a +circle about, such a polygon.</p> + +<p>For the regular triangle the problems are not repeated, because +more general problems have been solved.</p> + +<p>Props. 6, 7, 8 and 9 solve these problems for the square.</p> + +<p>The general problem of inscribing in a given circle a regular +polygon of n sides depends upon the problem of dividing the circumference +of a circle into n equal parts, or what comes to the same +thing, of drawing from the centre of the circle n radii such that the +angles between consecutive radii are equal, that is, to divide the +space about the centre into n equal angles. Thus, if it is required +to inscribe a square in a circle, we have to draw four lines from the +centre, making the four angles equal. This is done by drawing +two diameters at right angles to one another. The ends of these +diameters are the vertices of the required square. If, on the other +hand, tangents be drawn at these ends, we obtain a square circumscribed +about the circle.</p> + +<p>§ 43. To construct a <i>regular pentagon</i>, we find it convenient first +to construct a <i>regular decagon</i>. This requires to divide the space +about the centre into ten equal angles. Each will be <span class="spp">1</span>⁄<span class="suu">10</span>th of a right +angle, or <span class="spp">1</span>⁄<span class="suu">5</span>th of two right angles. If we suppose the decagon constructed, +and if we join the centre to the end of one side, we get an +isosceles triangle, where the angle at the centre equals <span class="spp">1</span>⁄<span class="suu">5</span>th of two +right angles; hence each of the angles at the base will be <span class="spp">2</span>⁄<span class="suu">5</span>ths of +two right angles, as all three angles together equal two right angles. +Thus we have to construct an isosceles triangle, having the angle at +the vertex equal to half an angle at the base. This is solved in +Prop. 10, by aid of the problem in Prop. 11 of the second book. If +we make the sides of this triangle equal to the radius of the given +circle, then the base will be the side of the regular decagon inscribed +in the circle. This side being known the decagon can be constructed, +and if the vertices are joined alternately, leaving out half their +number, we obtain the regular pentagon. (Prop. 11.)</p> + +<p>Euclid does not proceed thus. He wants the pentagon before +the decagon. This, however, does not change the real nature of +his solution, nor does his solution become simpler by not mentioning +the decagon.</p> + +<p>Once the regular pentagon is inscribed, it is easy to circumscribe +another by drawing tangents at the vertices of the inscribed pentagon. +This is shown in Prop. 12.</p> + +<p>Props. 13 and 14 teach how a circle may be inscribed in or circumscribed +about any given regular pentagon.</p> + +<p>§ 44. The <i>regular hexagon</i> is more easily constructed, as shown +in Prop. 15. The result is that the side of the regular hexagon +inscribed in a circle is equal to the radius of the circle.</p> + +<p>For this polygon the other three problems mentioned are not +solved.</p> + +<p>§ 45. The book closes with Prop. 16. To inscribe a regular +quindecagon in a given circle. If we inscribe a regular pentagon +and a regular hexagon in the circle, having one vertex in common, +then the arc from the common vertex to the next vertex of the +pentagon is <span class="spp">1</span>⁄<span class="suu">5</span>th of the circumference, and to the next vertex of the +hexagon is <span class="spp">1</span>⁄<span class="suu">6</span>th of the circumference. The difference between these +arcs is, therefore, <span class="spp">1</span>⁄<span class="suu">5</span> − <span class="spp">1</span>⁄<span class="suu">6</span> = <span class="spp">1</span>⁄<span class="suu">30</span>th of the circumference. The latter may, +therefore, be divided into thirty, and hence also in fifteen equal parts, +and the regular quindecagon be described.</p> + +<p>§ 46. We conclude with a few theorems about regular polygons +which are not given by Euclid.</p> + +<p><i>The straight lines perpendicular to and bisecting the sides of any +regular polygon meet in a point. The straight lines bisecting the angles +in the regular polygon meet in the same point. This point is the centre +of the circles circumscribed about and inscribed in the regular polygon.</i></p> + +<p>We can bisect any given arc (Prop. 30, III.). Hence we can divide +a circumference into 2n equal parts as soon as it has been divided +into n equal parts, or as soon as a regular polygon of n sides has been +constructed. Hence—</p> + +<p><i>If a regular polygon of n sides has been constructed, then a regular +polygon of 2n sides, of 4n, of 8n sides, &c., may also be constructed.</i> +Euclid shows how to construct regular polygons of 3, 4, 5 and 15 +sides. It follows that we can construct regular polygons of</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcr">3,</td> <td class="tcr">6,</td> <td class="tcr">12,</td> <td class="tcr">24</td> <td class="tcc">sides</td></tr> +<tr><td class="tcr">4,</td> <td class="tcr">8,</td> <td class="tcr">16,</td> <td class="tcr">32</td> <td class="tcc">”</td></tr> +<tr><td class="tcr">5,</td> <td class="tcr">10,</td> <td class="tcr">20,</td> <td class="tcr">40</td> <td class="tcc">”</td></tr> +<tr><td class="tcr">15,</td> <td class="tcr">30,</td> <td class="tcr">60,</td> <td class="tcr">120</td> <td class="tcc">”</td></tr> +</table> + +<p>The construction of any new regular polygon not included in one +of these series will give rise to a new series. Till the beginning of the +19th century nothing was added to the knowledge of regular polygons +as given by Euclid. Then Gauss, in his celebrated <i>Arithmetic</i>, +proved that every regular polygon of 2<span class="sp">n</span> + 1 sides may be constructed +if this number 2<span class="sp">n</span> + 1 be prime, and that no others except those +with 2<span class="sp">m</span> (2<span class="sp">n</span> + 1) sides can be constructed by elementary methods. +This shows that regular polygons of 7, 9, 13 sides cannot thus be +constructed, but that a regular polygon of 17 sides is possible; for +17 = 2<span class="sp">4</span> + 1. The next polygon is one of 257 sides. The construction +becomes already rather complicated for 17 sides.</p> + +<p class="pt2 center sc">Book V.</p> + +<p>§ 47. The fifth book of the <i>Elements</i> is not exclusively geometrical. +It contains the theory of ratios and proportion of quantities in +general. The treatment, as here given, is admirable, and in every +respect superior to the algebraical method by which Euclid’s theory +is now generally replaced. We shall treat the subject in order to +show why the usual algebraical treatment of proportion is not really +sound. We begin by quoting those definitions at the beginning of +Book V. which are most important. These definitions have given +rise to much discussion.</p> + +<p>The only definitions which are essential for the fifth book are +Defs. 1, 2, 4, 5, 6 and 7. Of the remainder 3, 8 and 9 are more +than useless, and probably not Euclid’s, but additions of later editors, +of whom Theon of Alexandria was the most prominent. Defs. 10 +and 11 belong rather to the sixth book, whilst all the others are +merely nominal. The really important ones are 4, 5, 6 and 7.</p> + +<p>§ 48. To define a magnitude is not attempted by Euclid. The +first two definitions state what is meant by a “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.</p> + +<p>Def. 3, which is probably due to Theon, professes to define a ratio, +but is as meaningless as it is uncalled for, for all that is wanted is +given in Defs. 5 and 7.</p> + +<p>In Def. 5 it is explained what is meant by saying that two magnitudes +have the same ratio to one another as two other magnitudes, +<span class="pagenum"><a name="page683" id="page683"></a>683</span> +and in Def. 7 what we have to understand by a greater or a less ratio. +The 6th definition is only nominal, explaining the meaning of the +word <i>proportional</i>.</p> + +<p>Euclid represents magnitudes by lines, and often denotes them +either by single letters or, like lines, by two letters. We shall use +only single letters for the purpose. If a and b denote two magnitudes +of the same kind, their ratio will be denoted by a : b; if c and d are +two other magnitudes of the same kind, but possibly of a different +kind from a and b, then if c and d have the same ratio to one another +as a and b, this will be expressed by writing—</p> + +<p class="center">a : b :: c : d.</p> + +<p>Further, if m is a (whole) number, ma shall denote the multiple +of a which is obtained by taking it m times.</p> + +<p>§ 49. The whole theory of ratios is based on Def. 5.</p> + +<p>Def. 5. <i>The first of four magnitudes is said to have the same ratio +to the second that the third has to the fourth when, any equimultiples +whatever of the first and the third being taken, and any equimultiples +whatever of the second and the fourth, if the multiple of the first be less +than that of the second, the multiple of the third is also less than that of +the fourth; and if the multiple of the first is equal to that of the second, +the multiple of the third is also equal to that of the fourth; and if the +multiple of the first is greater than that of the second, the multiple of +the third is also greater than that of the fourth.</i></p> + +<p>It will be well to show at once in an example how this definition +can be used, by proving the first part of the first proposition in the +sixth book. <i>Triangles of the same altitude are to one another as +their bases</i>, or if a and b are the bases, and α and β the areas, of two +triangles which have the same altitude, then a : b :: α : β.</p> + +<p>To prove this, we have, according to Definition 5, to show—</p> + +<table class="reg" summary="poem"><tr><td> <div class="poemr"> + <p>if ma > nb, then mα > nβ,</p> + <p>if ma = nb, then mα = nβ,</p> + <p>if ma < nb, then mα < nβ.</p> +</div> </td></tr></table> + +<p class="noind">That this is true is in our case easily seen. We may suppose that +the triangles have a common vertex, and their bases in the same +line. We set off the base a along the line containing the bases +m times; we then join the different parts of division to the vertex, +and get m triangles all equal to α. The triangle on ma as base equals, +therefore, mα. 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 nβ, 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 mα = nβ if ma = nb, and if their bases +are unequal, then that has the greater area which is on the greater +base; in other words, mα is greater than, equal to, or less than +nβ, according as ma is greater than, equal to, or less than nb, which +was to be proved.</p> + +<p>§ 50. It will be seen that even in this example it does not become +evident what a ratio really is. It is still an open question whether +ratios are magnitudes which we can compare. We do not know +whether the ratio of two lines is a magnitude of the same kind as the +ratio of two areas. Though we might say that Def. 5 defines <i>equal +</i>ratios, still we do not know whether they are equal in the sense of +the axiom, that two things which are equal to a third are equal to +one another. That this is the case requires a proof, and until this +proof is given we shall use the :: instead of the sign = , which, however, +we shall afterwards introduce.</p> + +<p>As soon as it has been established that all ratios are like magnitudes, +it becomes easy to show that, in some cases at least, they +are numbers. This step was never made by Greek mathematicians. +They distinguished always most carefully between continuous +magnitudes and the discrete series of numbers. In modern times +it has become the custom to ignore this difference.</p> + +<p>If, in determining the ratio of two lines, a common measure can +be found, which is contained m times in the first, and n times in +the second, then the ratio of the two lines equals the ratio of the +two numbers m : n. This is shown by Euclid in Prop. 5, X. But the +ratio of two numbers is, as a rule, a fraction, and the Greeks did +not, as we do, consider fractions as numbers. Far less had they +any notion of introducing irrational numbers, which are neither +whole nor fractional, as we are obliged to do if we wish to say that +all ratios are numbers. The incommensurable numbers which are +thus introduced as ratios of incommensurable quantities are nowadays +as familiar to us as fractions; but a proof is generally omitted +that we may apply to them the rules which have been established +for rational numbers only. Euclid’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.</p> + +<p>§. 51. Using the notation explained above we express the first +propositions as follows:—</p> + +<p>Prop. 1. If</p> + +<p class="center">a = ma′, b = mb′, c = mc′,</p> + +<p class="noind">then</p> + +<p class="center">a + b + c = m(a′ + b′ + c′).</p> + +<p>Prop. 2. If</p> + +<table class="reg" summary="poem"><tr><td> <div class="poemr"> +<p>a = mb, and c = md,</p> +<p>e = nb, and f = nd,</p> +</div> </td></tr></table> + +<p class="noind">then a + e is the same multiple of b as c + f is of d, viz.:—</p> + +<p class="center">a + e = (m + n)b, and c + f = (m + n)d.</p> + +<p>Prop. 3. If a = mb, c = md, then is na the same multiple of b +that nc is of d, viz. na = nmb, nc = nmd.</p> + +<p>Prop. 4. If</p> + +<p class="center">a : b :: c : d,</p> + +<p class="noind">then</p> + +<p class="center">ma : nb :: mc : nd.</p> + +<p>Prop. 5. If</p> + +<p class="center">a = mb, and c = md,</p> + +<p class="noind">then</p> + +<p class="center">a − c = m(b − d).</p> + +<p>Prop. 6. If</p> + +<p class="center">a = mb, c = md,</p> + +<p class="noind">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.</p> + +<p>All these propositions relate to <i>equimultiples</i>. Now follow propositions +about ratios which are compared as to their magnitude.</p> + +<p>§ 52. Prop. 7. If a = b, then a : c :: b : c and c : a :: c : b.</p> + +<p>The proof is simply this. As a = b we know that ma = mb; therefore +if</p> + +<p class="center">ma > nc, then mb > nc,</p> + +<p class="noind">if</p> + +<p class="center">ma = nc, then mb = nc,</p> + +<p class="noind">if</p> + +<p class="center">ma < nc, then mb < nc,</p> + +<p class="noind">therefore the first proportion holds by Definition 5.</p> + +<p>Prop. 8. If</p> + +<p class="center">a > b, then a : c > b : c,</p> + +<p class="noind">and</p> + +<p class="center">c : a < c : b.</p> + +<p class="noind">The proof depends on Definition 7.</p> + +<p>Prop. 9 (converse to Prop. 7). If</p> + +<p class="center">a : c :: b : c,</p> + +<p class="noind">or if</p> + +<p class="center">c : a :: c : b, then a = b.</p> + +<p>Prop. 10 (converse to Prop. 8). If</p> + +<p class="center">a : c > b : c, then a > b,</p> + +<p class="noind">and if</p> + +<p class="center">c : a < c : b, then a < b.</p> + +<p>Prop. 11. If</p> + +<p class="center">a : b :: c : d,</p> + +<p class="noind">and</p> + +<p class="center">a : b :: e : f,</p> + +<p class="noind">then</p> + +<p class="center">c : d :: e : f.</p> + +<p>In words, <i>if too ratios are equal to a third, they are equal to one +another</i>. After these propositions have been proved, we have a +right to consider a ratio as a <i>magnitude</i>, for only now can we consider +a ratio as something for which the axiom about magnitudes +holds: things which are equal to a third are equal to one another.</p> + +<p>We shall indicate this by writing in future the sign = instead +of ::. The remaining propositions, which explain themselves, may +then be stated as follows:</p> + +<p>§ 53. Prop. 12. If</p> + +<p class="center">a : b = c : d = e : f,</p> + +<p class="noind">then</p> + +<p class="center">a + c + e : b + d + f = a : b.</p> + +<p>Prop. 13. If</p> + +<p class="center">a : b = c : d and c : d > e : f,</p> + +<p class="noind">then</p> + +<p class="center">a : b > e : f.</p> + +<p>Prop. 14. If</p> + +<p class="center">a : b = c : d, and a > c, then b > d.</p> + +<p>Prop. 15. Magnitudes have the same ratio to one another that +their equimultiples have—</p> + +<p class="center">ma : mb = a : b.</p> + +<p>Prop. 16. If a, b, c, d are magnitudes of the same kind, and if</p> + +<p class="center">a : b = c : d,</p> + +<p class="noind">then</p> + +<p class="center">a : c = b : d.</p> + +<p>Prop. 17. If</p> + +<p class="center">a + b : b = c + d : d,</p> + +<p class="noind">then</p> + +<p class="center">a : b = c : d.</p> + +<p>Prop. 18 (converse to 17). If</p> + +<p class="center">a : b = c : d</p> + +<p class="noind">then</p> + +<p class="center">a + b : b = c + d : d.</p> + +<p>Prop. 19. If a, b, c, d are quantities of the same kind, and if</p> + +<p class="center">a : b = c : d,</p> + +<p class="noind">then</p> + +<p class="center">a − c : b − d = a : b.</p> + +<p>§ 54. Prop. 20. <i>If there be three magnitudes, and another three, +which have the same ratio, taken two and two, then if the first be greater +than the third, the fourth shall be greater than the sixth: and if equal, +equal; and if less, less.</i></p> + +<p>If we understand by</p> + +<p class="center">a : b : c : d : e : ... = a′ : b′ : c′ : d′ : e′ : ...</p> + +<p class="noind">that the ratio of any two consecutive magnitudes on the first side +equals that of the corresponding magnitudes on the second side, +we may write this theorem in symbols, thus:—</p> + +<p>If a, b, c be quantities of one, and d, e, f magnitudes of the same +or any other kind, such that</p> + +<p class="center">a : b : c = d : e : f,</p> + +<p class="noind">and if</p> + +<p class="center">a > c, then d > f,</p> + +<p class="noind">but if</p> + +<p class="center">a = c, then d = f,</p> + +<p class="noind">and if</p> + +<p class="center">a < c, then d < f.</p> + +<p>Prop. 21. If</p> + +<p class="center">a : b = e : f and b : c = d : e,</p> + +<p class="noind">or if</p> + +<p class="center">a : b : c = 1/f : 1/e : 1/d,</p> + +<span class="pagenum"><a name="page684" id="page684"></a>684</span> + +<p class="noind">and if</p> + +<p class="center">a > c, then d > f,</p> + +<p class="noind">but if</p> + +<p class="center">a = c, then d = f,</p> + +<p class="noind">and if</p> + +<p class="center">a < c, then d < f.</p> + +<p>By aid of these two propositions the following two are proved.</p> + +<p>§ 55. Prop. 22. <i>If there be any number of magnitudes, and as +many others, which have the same ratio, taken two and two in order, +the first shall have to the last of the first magnitudes the same ratio +which the first of the others has to the last.</i></p> + +<p>We may state it more generally, thus:</p> + +<p class="noind">If</p> + +<p class="center">a : b : c : d : e: ... = a′ : b′ : c′ : d′ : e′ : ... ,</p> + +<p class="noind">then not only have two consecutive, but any two magnitudes on +the first side, the same ratio as the corresponding magnitudes on +the other. For instance—</p> + +<p class="center">a : c = a′ : c′; b : e = b′ : e′, &c.</p> + +<p>Prop. 23 we state only in symbols, viz.:—</p> + +<p class="center">a : b : c : d : e : ... = 1/a′ : 1/b′ : 1/c′ : 1/d′ : 1/e′ ...,</p> + +<p class="noind">then</p> + +<p class="center">a : c = c′ : a′,<br /> +b : e = e′ : b′,</p> + +<p class="noind">and so on.</p> + +<p>Prop. 24 comes to this: If a : b = c : d and e : b = f : d, then</p> + +<p class="center">a + e : b = c + f : d.</p> + +<p>Some of the proportions which are considered in the above propositions +have special names. These we have omitted, as being of +no use, since algebra has enabled us to bring the different operations +contained in the propositions under a common point of view.</p> + +<p>§ 56. The last proposition in the fifth book is of a different +character.</p> + +<p>Prop. 25. <i>If four magnitudes of the same kind be proportional, +the greatest and least of them together shall be greater than the other +two together.</i> In symbols—</p> + +<p>If a, b, c, d be magnitudes of the same kind, and if a : b = c : d, +and if a is the greatest, hence d the least, then a + d > b + c.</p> + +<p>§ 57. We return once again to the question. What is a ratio? +We have seen that we may treat ratios as magnitudes, and that all +ratios are magnitudes of the same kind, for we may compare any +two as to their magnitude. It will presently be shown that ratios +of lines may be considered as <i>quotients</i> of lines, so that a ratio appears +as answer to the question, How often is one line contained in another? +But the answer to this question is given by a number, at least in +some cases, and in all cases if we admit incommensurable numbers. +Considered from this point of view, we may say the fifth book of the +<i>Elements</i> shows that some of the simpler algebraical operations +hold for incommensurable numbers. In the ordinary algebraical +treatment of numbers this proof is altogether omitted, or given by +a process of limits which does not seem to be natural to the subject.</p> + +<p class="pt2 center sc">Book VI.</p> + +<p>§ 58. The sixth book contains the theory of similar figures. +After a few definitions explaining terms, the first proposition gives +the first application of the theory of proportion.</p> + +<p>Prop. 1. <i>Triangles and parallelograms of the same altitude are to +one another as their bases.</i></p> + +<p>The proof has already been considered in § 49.</p> + +<p>From this follows easily the important theorem</p> + +<p>Prop. 2. <i>If a straight line be drawn parallel to one of the sides +of a triangle it shall cut the other sides, or those sides produced, proportionally; +and if the sides or the sides produced be cut proportionally, +the straight line which joins the points of section shall be parallel to +the remaining side of the triangle.</i></p> + +<p>§ 59. The next proposition, together with one added by Simson +as Prop. A, may be expressed more conveniently if we introduce a +modern phraseology, viz. if in a line AB we assume a point C between +A and B, we shall say that C divides AB internally in the ratio +AC : CB; but if C be taken in the line AB produced, we shall say +that AB is divided externally in the ratio AC : CB.</p> + +<p>The two propositions then come to this:</p> + +<p>Prop. 3. <i>The bisector of an angle in a triangle divides the opposite +side internally in a ratio equal to the ratio of the two sides including +that angle;</i> and conversely, <i>if a line through the vertex of a triangle +divide the base internally in the ratio of the two other sides, then that +line bisects the angle at the vertex</i>.</p> + +<p>Simson’s Prop. A. <i>The line which bisects an exterior angle of a +triangle divides the opposite side externally in the ratio of the other +sides;</i> and conversely, <i>if a line through the vertex of a triangle divide +the base externally in the ratio of the sides, then it bisects an exterior +angle at the vertex of the triangle</i>.</p> + +<p>If we combine both we have—</p> + +<p><i>The two lines which bisect the interior and exterior angles at one +vertex of a triangle divide the opposite side internally and externally +in the same ratio, viz. in the ratio of the other two sides.</i></p> + +<p>§ 60. The next four propositions contain the theory of similar +triangles, of which four cases are considered. They may be stated +together.</p> + +<p><i>Two triangles are similar</i>,—</p> + +<p>1. (Prop. 4). <i>If the triangles are equiangular:</i></p> + +<p>2. (Prop. 5). <i>If the sides of the one are proportional to those of +the other</i>;</p> + +<p>3. (Prop. 6). <i>If two sides in one are proportional to two sides in +the other, and if the angles contained by these sides are equal</i>;</p> + +<p>4. (Prop. 7). <i>If two sides in one are proportional to two sides in +the other, if the angles opposite homologous sides are equal, and if +the angles opposite the other homologous sides are both acute, both right +or both obtuse; homologous sides being in each case those which are +opposite equal angles</i>.</p> + +<p>An important application of these theorems is at once made to +a right-angled triangle, viz.:—</p> + +<p>Prop. 8. <i>In a right-angled triangle, if a perpendicular be drawn +from the right angle to the base, the triangles on each side of it are +similar to the whole triangle, and to one another</i>.</p> + +<p><i>Corollary.</i>—From this it is manifest that the perpendicular +drawn from the right angle of a right-angled triangle to the base +is a mean proportional between the segments of the base, and also +that each of the sides is a mean proportional between the base and +the segment of the base adjacent to that side.</p> + +<p>§ 61. There follow four propositions containing problems, in +language slightly different from Euclid’s, viz.:—</p> + +<p>Prop. 9. <i>To divide a straight line into a given number of equal +parts</i>.</p> + +<p>Prop. 10. <i>To divide a straight line in a given ratio</i>.</p> + +<p>Prop. 11. <i>To find a third proportional to two given straight lines</i>.</p> + +<p>Prop. 12. <i>To find a fourth proportional to three given straight +lines</i>.</p> + +<p>Prop. 13. <i>To find a mean proportional between two given straight +lines</i>.</p> + +<p>The last three may be written as equations with one unknown +quantity—viz. if we call the given straight lines a, b, c, and the +required line x, we have to find a line x so that</p> + +<p>Prop. 11.</p> + +<p class="center">a : b = b : x;</p> + +<p>Prop. 12.</p> + +<p class="center">a : b = c : x;</p> + +<p>Prop. 13.</p> + +<p class="center">a : x = x : b.</p> + +<p>We shall see presently how these may be written without the +signs of ratios.</p> + +<p>§ 62. Euclid considers next proportions connected with parallelograms +and triangles which are equal in area.</p> + +<p>Prop. 14. <i>Equal parallelograms which have one angle of the one +equal to one angle of the other have their sides about the equal angles +reciprocally proportional; and parallelograms which have one angle +of the one equal to one angle of the other, and their sides about the equal +angles reciprocally proportional, are equal to one another</i>.</p> + +<p>Prop. 15. <i>Equal triangles which have one angle of the one equal +to one angle of the other, have their sides about the equal angles reciprocally +proportional; and triangles which have one angle of the one equal +to one angle of the other, and their sides about the equal angles reciprocally +proportional, are equal to one another</i>.</p> + +<table class="flt" style="float: right; width: 320px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:269px; height:167px" src="images/img684.jpg" alt="" /></td></tr></table> + +<p>The latter proposition is really the same as the former, for if, as +in the accompanying diagram, +in the figure belonging to the +former the two equal parallelograms +AB and BC be bisected +by the lines DF and EG, and +if EF be drawn, we get the +figure belonging to the latter.</p> + +<p>It is worth noticing that +the lines FE and DG are +parallel. We may state therefore +the theorem—</p> + +<p><i>If two triangles are equal in +area, and have one angle in the one vertically opposite to one angle +in the other, then the two straight lines which join the remaining two +vertices of the one to those of the other triangle are parallel</i>.</p> + +<p>§ 63. A most important theorem is</p> + +<p><i>Prop. 16. If four straight lines be proportionals, the rectangle +contained by the extremes is equal to the rectangle contained by the +means; and if the rectangle contained by the extremes be equal to the +rectangle contained by the means, the four straight lines are proportionals</i>.</p> + +<p>In symbols, if a, b, c, d are the four lines, and</p> + +<p class="noind">if</p> + +<p class="center">a : b = c : d,</p> + +<p class="noind">then</p> + +<p class="center">ad = bc;</p> + +<p class="noind">and conversely, if</p> + +<p class="center">ad = bc,</p> + +<p class="noind">then</p> + +<p class="center">a : b = c : d,</p> + +<p class="noind">where ad and bc denote (as in § 20), the areas of the rectangles +contained by a and d and by b and c respectively.</p> + +<p>This allows us to transform every proportion between four lines +into an equation between two products.</p> + +<p>It shows further that the operation of forming a product of two +lines, and the operation of forming their ratio are each the inverse +of the other.</p> + +<p>If we now define a quotient a/b of two lines as the <i>number</i> which +multiplied into b gives a, so that</p> + +<table class="math0" summary="math"> +<tr><td>a</td> +<td rowspan="2">b = a,</td></tr> +<tr><td class="denom">b</td></tr></table> + +<p><span class="pagenum"><a name="page685" id="page685"></a>685</span></p> + +<p class="noind">we see that from the equality of two quotients</p> + +<table class="math0" summary="math"> +<tr><td>a</td> +<td rowspan="2">=</td> <td>c</td></tr> +<tr><td class="denom">b</td> <td class="denom">d</td></tr></table> + +<p class="noind">follows, if we multiply both sides by bd,</p> + +<table class="math0" summary="math"> +<tr><td>a</td> +<td rowspan="2">b·d =</td> <td>c</td> +<td rowspan="2">d·b,</td></tr> +<tr><td class="denom">b</td> <td class="denom">d</td></tr></table> + +<p class="center">ad = cb.</p> + +<p>But from this it follows, according to the last theorem, that</p> + +<p class="center">a : b = c : d.</p> + +<p>Hence we conclude that the quotient a/b and the ratio a : b are +different forms of the same magnitude, only with this important +difference that the quotient a/b would have a meaning only if a and +b have a common measure, until we introduce incommensurable +numbers, while the ratio a : b has always a meaning, and thus gives +rise to the introduction of incommensurable numbers.</p> + +<p>Thus it is really the theory of ratios in the fifth book which enables +us to extend the geometrical calculus given before in connexion +with Book II. It will also be seen that if we write the ratios in +Book V. as quotients, or rather as fractions, then most of the theorems +state properties of quotients or of fractions.</p> + +<p>§ 64. Prop. 17. <i>If three straight lines are proportional the rectangle +contained by the extremes is equal to the square on the mean;</i> and +conversely, is only a special case of 16. After the problem, Prop. +18, <i>On a given straight line to describe a rectilineal figure similar +and similarly situated to a given rectilineal figure</i>, there follows another +fundamental theorem:</p> + +<p>Prop. 19. <i>Similar triangles are to one another in the duplicate +ratio of their homologous sides.</i> In other words, the areas of similar +triangles are to one another as the squares on homologous sides. +This is generalized in:</p> + +<p>Prop. 20. <i>Similar polygons may be divided into the same number +of similar triangles, having the same ratio to one another that the +polygons have; and the polygons are to one another in the duplicate +ratio of their homologous sides.</i></p> + +<p>§ 65. Prop. 21. <i>Rectilineal figures which are similar to the same +rectilineal figure are also similar to each other</i>, is an immediate consequence +of the definition of similar figures. As similar figures +may be said to be equal in “shape” but not in “size,” we may state +it also thus:</p> + +<p>“Figures which are equal in shape to a third are equal in shape +to each other.”</p> + +<p>Prop. 22. <i>If four straight lines be proportionals, the similar +rectilineal figures similarly described on them shall also be proportionals; +and if the similar rectilineal figures similarly described on four +straight lines be proportionals, those straight lines shall be proportionals.</i></p> + +<p>This is essentially the same as the following:—</p> + +<p class="noind"><i>If</i></p> + +<p class="center">a : b = c : d,</p> + +<p class="noind"><i>then</i></p> + +<p class="center">a² : b² = c² : d².</p> + +<p>§ 66. Now follows a proposition which has been much discussed +with regard to Euclid’s exact meaning in saying that a ratio is +<i>compounded</i> of two other ratios, viz.:</p> + +<p>Prop. 23. <i>Parallelograms which are equiangular to one another, +have to one another the ratio which is compounded of the ratios of their +sides.</i></p> + +<p>The proof of the proposition makes its meaning clear. In symbols +the ratio a : c is compounded of the two ratios a : b and b : c, and if +a : b = a′ : b′, b : c = b″ : c″, then a : c is compounded of a′ : b′ and +b″ : c″.</p> + +<p>If we consider the ratios as numbers, we may say that the one +ratio is the product of those of which it is compounded, or in symbols,</p> + +<table class="math0" summary="math"> +<tr><td>a</td> +<td rowspan="2">=</td> <td>a</td> +<td rowspan="2">·</td> <td>b</td> +<td rowspan="2">=</td> <td>a′</td> +<td rowspan="2">·</td> <td>b″</td> +<td rowspan="2">, if</td> <td>a</td> +<td rowspan="2">=</td> <td>a′</td> +<td rowspan="2">and</td> <td>b</td> +<td rowspan="2">=</td> <td>b″</td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">c</td> <td class="denom">b</td> +<td class="denom">c</td> <td class="denom">b′</td> +<td class="denom">c″</td> <td class="denom">b</td> +<td class="denom">b′</td> <td class="denom">c</td> +<td class="denom">c″</td></tr></table> + +<p>The theorem in Prop. 23 is the foundation of all mensuration of +areas. From it we see at once that two rectangles have the ratio +of their areas compounded of the ratios of their sides.</p> + +<p>If A is the area of a rectangle contained by a and b, and B that +of a rectangle contained by c and d, so that A = ab, B = cd, then +A : B = ab : cd, and this is, the theorem says, compounded of the +ratios a : c and b : d. In forms of quotients,</p> + +<table class="math0" summary="math"> +<tr><td>a</td> +<td rowspan="2">·</td> <td>b</td> +<td rowspan="2">=</td> <td>ab</td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">c</td> <td class="denom">d</td> +<td class="denom">cd</td></tr></table> + +<p>This shows how to multiply quotients in our geometrical calculus.</p> + +<p>Further, <i>Two triangles have the ratios of their areas compounded +of the ratios of their bases and their altitude.</i> For a triangle is equal +in area to half a parallelogram which has the same base and the +same altitude.</p> + +<p>§ 67. To bring these theorems to the form in which they are usually +given, we assume a straight line u as our unit of length (generally +an inch, a foot, a mile, &c.), and determine the number α which +expresses how often u is contained in a line a, so that α denotes the +ratio a : u whether commensurable or not, and that a = αu. We +call this number α the numerical value of a. If in the same manner +β be the numerical value of a line b we have</p> + +<p class="center">a : b = α : β;</p> + +<p class="noind">in words: <i>The ratio of two lines (and of two like quantities in general) +is equal to that of their numerical values.</i></p> + +<p>This is easily proved by observing that a = αu, b = βu, therefore +a : b = αu : βu, and this may without difficulty be shown to equal α : β.</p> + +<p>If now a, b be base and altitude of one, a′, b′ those of another +parallelogram, α, β and α′, β′ their numerical values respectively, +and A, A′ their areas, then</p> + +<table class="math0" summary="math"> +<tr><td>A</td> +<td rowspan="2">=</td> <td>a</td> +<td rowspan="2">·</td> <td>b</td> +<td rowspan="2">=</td> <td>α</td> +<td rowspan="2">·</td> <td>β</td> +<td rowspan="2">=</td> <td>αβ</td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">A′</td> <td class="denom">a′</td> +<td class="denom">b′</td> <td class="denom">α′</td> +<td class="denom">β′</td> <td class="denom">α′β′</td></tr></table> + +<p class="noind">In words: <i>The areas of two parallelograms are to each other as the +products of the numerical values of their bases and altitudes.</i></p> + +<p>If especially the second parallelogram is the unit square, <i>i.e.</i> a +square on the unit of length, then α′ = β′ = 1, A′ = u², and we have</p> + +<table class="math0" summary="math"> +<tr><td>A</td> +<td rowspan="2">= αβ or A = αβ·u².</td></tr> +<tr><td class="denom">A′</td></tr></table> + +<p>This gives the theorem: The number of unit squares contained in +a parallelogram equals the product of the numerical values of base +and altitude, and similarly the number of unit squares contained in +a triangle equals half the product of the numerical values of base +and altitude.</p> + +<p>This is often stated by saying that the area of a parallelogram is +equal to the product of the base and the altitude, meaning by this +product the product of the numerical values, and not the product as +defined above in § 20.</p> + +<p>§ 68. Propositions 24 and 26 relate to parallelograms about +diagonals, such as are considered in Book I., 43. They are—</p> + +<p>Prop. 24. <i>Parallelograms about the diameter of any parallelogram +are similar to the whole parallelogram and to one another</i>; and its +converse (Prop. 26), <i>If two similar parallelograms have a common +angle, and be similarly situated, they are about the same diameter.</i></p> + +<p>Between these is inserted a problem.</p> + +<p>Prop. 25. <i>To describe a rectilineal figure which shall be similar to +one given rectilinear figure, and equal to another given rectilineal +figure</i>.</p> + +<p>§ 69. Prop. 27 contains a theorem relating to the theory of +maxima and minima. We may state it thus:</p> + +<p>Prop. 27. <i>If a parallelogram be divided into two by a straight line +cutting the base, and if on half the base another parallelogram be constructed +similar to one of those parts, then this third parallelogram is +greater than the other part.</i></p> + +<p>Of far greater interest than this general theorem is a special case +of it, where the parallelograms are changed into rectangles, and +where one of the parts into which the parallelogram is divided is +made a square; for then the theorem changes into one which is +easily recognized to be identical with the following:—</p> + +<p><i>Of all rectangles which have the same perimeter the square has the +greatest area.</i></p> + +<p>This may also be stated thus:—</p> + +<p><i>Of all rectangles which have the same area the square has the least +perimeter.</i></p> + +<p>§ 70. The next three propositions contain problems which may +be said to be solutions of quadratic equations. The first two are, +like the last, involved in somewhat obscure language. We transcribe +them as follows:</p> + +<p><i>Problem</i>.—To describe on a given base a parallelogram, and to +divide it either internally (Prop. 28) or externally (Prop. 29) from +a point on the base into two parallelograms, of which the one has +a given size (is equal in area to a given figure), whilst the other +has a given shape (is similar to a given parallelogram).</p> + +<p>If we express this again in symbols, calling the given base a, the +one part x, and the altitude y, we have to determine x and y in the +first case from the equations</p> + +<p class="center">(a − x)y = k²,</p> + +<table class="math0" summary="math"> +<tr><td>x</td> +<td rowspan="2">=</td> <td>p</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">y</td> <td class="denom">q</td></tr></table> + +<p class="noind">k² being the given size of the first, and p and q the base and altitude +of the parallelogram which determine the shape of the second of the +required parallelograms.</p> + +<p>If we substitute the value of y, we get</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">(a − x)x =</td> <td>pk²</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">q</td></tr></table> + +<p class="noind">or,</p> + +<p class="center">ax − x² = b²,</p> + +<p class="noind">where a and b are known quantities, taking b² = pk²/q.</p> + +<p>The second case (Prop. 29) gives rise, in the same manner, to the +quadratic</p> + +<p class="center">ax + x² = b².</p> + +<p>The next problem—</p> + +<p>Prop. 30. <i>To cut a given straight line in extreme and mean ratio</i>, +leads to the equation</p> + +<p class="center">ax + x² = a².</p> + +<p><span class="pagenum"><a name="page686" id="page686"></a>686</span></p> + +<p>This is, therefore, only a special case of the last, and is, besides, +an old acquaintance, being essentially the same problem as that +proposed in II. 11.</p> + +<p>Prop. 30 may therefore be solved in two ways, either by aid of +Prop. 29 or by aid of II. 11. Euclid gives both solutions.</p> + +<p>§ 71. Prop. 31 (Theorem). <i>In any right-angled triangle, any +rectilineal figure described on the side subtending the right angle is +equal to the similar and similarly-described figures on the sides containing +the right angle</i>,—is a pretty generalization of the theorem of +Pythagoras (I. 47).</p> + +<p>Leaving out the next proposition, which is of little interest, we +come to the last in this book.</p> + +<p>Prop. 33. <i>In equal circles angles, whether at the centres or the +circumferences, have the same ratio which the arcs on which they stand +have to one another; so also have the sectors</i>.</p> + +<p>Of this, the part relating to angles at the centre is of special +importance; it enables us to measure angles by arcs.</p> + +<p>With this closes that part of the <i>Elements</i> which is devoted to +the study of figures in a plane.</p> + +<p class="pt2 center sc">Book XI.</p> + +<p>§ 72. In this book figures are considered which are not confined +to a plane, viz. first relations between lines and planes in space, +and afterwards properties of solids.</p> + +<p>Of new definitions we mention those which relate to the perpendicularity +and the inclination of lines and planes.</p> + +<p>Def. 3. <i>A straight line is perpendicular, or at right angles, to a +plane when it makes right angles with every straight line meeting it +in that plane</i>.</p> + +<p>The definition of perpendicular planes (Def. 4) offers no difficulty. +Euclid defines the inclination of lines to planes and of planes to +planes (Defs. 5 and 6) by aid of plane angles, included by straight +lines, with which we have been made familiar in the first books.</p> + +<p>The other important definitions are those of parallel planes, +which never meet (Def. 8), and of solid angles formed by three or +more planes meeting in a point (Def. 9).</p> + +<p>To these we add the definition of a line parallel to a plane as a +line which does not meet the plane.</p> + +<p>§ 73. Before we investigate the contents of Book XI., it will be +well to recapitulate shortly what we know of planes and lines from +the definitions and axioms of the first book. There a plane has +been defined as a surface which has the property that every straight +line which joins two points in it lies altogether in it. This is equivalent +to saying that a straight line which has two points in a plane +has all points in the plane. Hence, a straight line which does not +lie in the plane cannot have more than one point in common with +the plane. This is virtually the same as Euclid’s Prop. 1, viz.:—</p> + +<p>Prop. 1. <i>One part of a straight line cannot be in a plane and another +part without it</i>.</p> + +<p>It also follows, as was pointed out in § 3, in discussing the definitions +of Book I., that a plane is determined already by one straight +line and a point without it, viz. if all lines be drawn through the +point, and cutting the line, they will form a plane.</p> + +<p>This may be stated thus:—</p> + +<p><i>A plane is determined</i>—</p> + +<p>1st, <i>By a straight line and a point which does not lie on it;</i></p> + +<p>2nd, <i>By three points which do not lie in a straight line</i>; for if two +of these points be joined by a straight line we have case 1;</p> + +<p>3rd, <i>By two intersecting straight lines</i>; for the point of intersection +and two other points, one in each line, give case 2;</p> + +<p>4th, <i>By two parallel lines</i> (Def. 35, I.).</p> + +<p>The third case of this theorem is Euclid’s</p> + +<p>Prop. 2. <i>Two straight lines which cut one another are in one plane, +and three straight lines which meet one another are in one plane</i>.</p> + +<p>And the fourth is Euclid’s</p> + +<p>Prop. 7. <i>If two straight lines be parallel, the straight line drawn +from any point in one to any point in the other is in the same plane +with the parallels</i>. From the definition of a plane further follows</p> + +<p>Prop. 3. <i>If two planes cut one another, their common section is a +straight line</i>.</p> + +<p>§ 74. Whilst these propositions are virtually contained in the +definition of a plane, the next gives us a new and fundamental +property of space, showing at the same time that it is possible to +have a straight line perpendicular to a plane, according to Def. 3. +It states—</p> + +<p>Prop. 4. <i>If a straight line is perpendicular to two straight lines +in a plane which it meets, then it is perpendicular to all lines in the plane +which it meets, and hence it is perpendicular to the plane</i>.</p> + +<p>Def. 3 may be stated thus: If a straight line is perpendicular +to a plane, then it is perpendicular to every line in the plane which +it meets. The converse to this would be</p> + +<p><i>All straight lines which meet a given straight line in the same point, +and are perpendicular to it, lie in a plane which is perpendicular to +that line</i>.</p> + +<p>This Euclid states thus:</p> + +<p>Prop. 5. <i>If three straight lines meet all at one point, and a straight +line stands at right angles to each of them at that point, the three straight +lines shall be in one and the same plane</i>.</p> + +<p>§ 75. There follow theorems relating to the theory of parallel +lines in space, viz.:—</p> + +<p>Prop. 6. <i>Any two lines which are perpendicular to the same plane +are parallel to each other;</i> and conversely</p> + +<p>Prop. 8. <i>If of two parallel straight lines one is perpendicular to a +plane, the other is so also.</i></p> + +<p>Prop. 7. <i>If two straight lines are parallel, the straight line which +joins any point in one to any point in the other is in the same plane as +the parallels.</i> (See above, § 73.)</p> + +<p>Prop. 9. <i>Two straight lines which are each of them parallel to the +same straight line, and not in the same plane with it, are parallel to +one another</i>; where the words, “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</p> + +<p>Prop. 10. <i>If two angles in different planes have the two limits of +the one parallel to those of the other, then the angles are equal.</i> That +their planes are parallel is shown later on in Prop. 15.</p> + +<p>This theorem is not necessarily true, for the angles in question +may be supplementary; but then the one angle will be equal to +that which is adjacent and supplementary to the other, and this +latter angle will also have its limits parallel to those of the first.</p> + +<p>From this theorem it follows that if we take any two straight +lines in space which do not meet, and if we draw through any point +P in space two lines parallel to them, then the angle included by +these lines will always be the same, whatever the position of the +point P may be. This angle has in modern times been called the +angle between the given lines:—</p> + +<p><i>By the angles between two not intersecting lines we understand the +angles which two intersecting lines include that are parallel respectively +to the two given lines.</i></p> + +<p>§ 76. It is now possible to solve the following two problems:—</p> + +<p><i>To draw a straight line perpendicular to a given plane from a given +point which lies</i></p> + +<p>1. <i>Not in the plane</i> (Prop. 11).</p> + +<p>2. <i>In the plane</i> (Prop. 12).</p> + +<p>The second case is easily reduced to the first—viz. if by aid of +the first we have drawn any perpendicular to the plane from some +point without it, we need only draw through the given point in the +plane a line parallel to it, in order to have the required perpendicular +given. The solution of the first part is of interest in itself. It depends +upon a construction which may be expressed as a theorem.</p> + +<p><i>If from a point A without a plane a perpendicular AB be drawn to the +plane, and if from the foot B of this perpendicular another perpendicular +BC be drawn to any straight line in the plane, then the straight line +joining A to the foot C of this second perpendicular will also be perpendicular +to the line in the plane.</i></p> + +<p>The theory of perpendiculars to a plane is concluded by the +theorem—</p> + +<p>Prop. 13. <i>Through any point in space, whether in or without a +plane, only one straight line can be drawn perpendicular to the plane.</i></p> + +<p>§ 77. The next four propositions treat of parallel planes. It is +shown <i>that planes which have a common perpendicular are parallel</i> +(Prop. 14); <i>that two planes are parallel if two intersecting straight +lines in the one are parallel respectively to two straight lines in the +other plane</i> (Prop. 15); <i>that parallel planes are cut by any plane in +parallel straight lines</i> (Prop. 16); and lastly, <i>that any two straight +lines are cut proportionally by a series of parallel planes</i> (Prop. 17).</p> + +<p>This theory is made more complete by adding the following +theorems, which are easy deductions from the last: <i>Two parallel +planes have common perpendiculars</i> (converse to 14); and <i>Two +planes which are parallel to a third plane are parallel to each other.</i></p> + +<p>It will be noted that Prop. 15 at once allows of the solution of +the problem: “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.</p> + +<p>§ 78. We come now to planes which are perpendicular to one +another. Two theorems relate to them.</p> + +<p>Prop. 18. <i>If a straight line be at right angles to a plane, every +plane which passes through it shall be at right angles to that plane.</i></p> + +<p>Prop. 19. <i>If two planes which cut one another be each of them +perpendicular to a third plane, their common section shall be perpendicular +to the same plane.</i></p> + +<p>§ 79. If three planes pass through a common point, and if they +bound each other, a solid angle of three faces, or a <i>trihedral</i> angle, +is formed, and similarly by more planes a solid angle of more faces, +or a <i>polyhedral</i> angle. These have many properties which are quite +analogous to those of triangles and polygons in a plane. Euclid +states some, viz.:—</p> + +<p>Prop. 20. <i>If a solid angle be contained by three plane angles, any +two of them are together greater than the third.</i></p> + +<p>But the next—</p> + +<p>Prop. 21. <i>Every solid angle is contained by plane angles, which +are together less than four right angles</i>—has no analogous theorem +in the plane.</p> + +<p>We may mention, however, that the theorems about triangles +contained in the propositions of Book I., which do not depend +upon the theory of parallels (that is all up to Prop. 27), have their +corresponding theorems about trihedral angles. The latter are +formed, if for “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 <i>faces</i>. We get, for instance, from I. 4, the +<span class="pagenum"><a name="page687" id="page687"></a>687</span> +theorem, <i>If two trihedral angles have the angles of two faces in the one +equal to the angles of two faces in the other, and have likewise the angles +included by these faces equal, then the angles in the remaining faces are +equal, and the angles between the other faces are equal each to each, viz. +those which are opposite equal faces.</i> The solid angles themselves are +not necessarily equal, for they may be only symmetrical like the +right hand and the left.</p> + +<p>The connexion indicated between triangles and trihedral angles +will also be recognized in</p> + +<p>Prop. 22. <i>If every two of three plane angles be greater than the +third, and if the straight lines which contain them be all equal, a triangle +may be made of the straight lines that join the extremities of those equal +straight lines.</i></p> + +<p>And Prop. 23 solves the problem, <i>To construct a trihedral angle +having the angles of its faces equal to three given plane angles, any two +of them being greater than the third.</i> It is, of course, analogous to the +problem of constructing a triangle having its sides of given length.</p> + +<p>Two other theorems of this kind are added by Simson in his +edition of Euclid’s <i>Elements</i>.</p> + +<p>§ 80. These are the principal properties of lines and planes in +space, but before we go on to their applications it will be well to +define the word <i>distance</i>. In geometry distance means always +“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.</p> + +<p>§ 81. <i>Parallelepipeds</i>.—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.:—</p> + +<p>Prop. 24. <i>If a solid be contained by six planes two and two of +which are parallel, the opposite planes are similar and equal parallelograms.</i></p> + +<p>Euclid calls this solid henceforth a parallelepiped, though he +never defines the word. Either face of it may be taken as <i>base</i>, +and its distance from the opposite face as <i>altitude</i>.</p> + +<p>Prop. 25. <i>If a solid parallelepiped be cut by a plane parallel to +two of its opposite planes, it divides the whole into two solids, the base +of one of which shall be to the base of the other as the one solid is to the +other.</i></p> + +<p>This theorem corresponds to the theorem (VI. 1) that parallelograms +between the same parallels are to one another as their bases. +A similar analogy is to be observed among a number of the remaining +propositions.</p> + +<p>§ 82. After solving a few problems we come to</p> + +<p>Prop. 28. <i>If a solid parallelepiped be cut by a plane passing +through the diagonals of two of the opposite planes, it shall be cut in +two equal parts.</i></p> + +<p>In the proof of this, as of several other propositions, Euclid +neglects the difference between solids which are symmetrical like +the right hand and the left.</p> + +<p>Prop. 31. <i>Solid parallelepipeds, which are upon equal bases, and +of the same altitude, are equal to one another.</i></p> + +<p>Props. 29 and 30 contain special cases of this theorem leading up +to the proof of the general theorem.</p> + +<p>As consequences of this fundamental theorem we get</p> + +<p>Prop. 32. <i>Solid parallelepipeds, which have the same altitude, are +to one another as their bases;</i> and</p> + +<p>Prop. 33. <i>Similar solid parallelepipeds are to one another in the +triplicate ratio of their homologous sides.</i></p> + +<p>If we consider, as in § 67, the ratios of lines as numbers, we may +also say—</p> + +<p><i>The ratio of the volumes of similar parallelepipeds is equal to the +ratio of the third powers of homologous sides.</i></p> + +<p>Parallelepipeds which are not similar but equal are compared by +aid of the theorem</p> + +<p>Prop. 34. <i>The bases and altitudes of equal solid parallelepipeds +<span class="correction" title="amended from and">are</span> reciprocally proportional; and if the bases and altitudes be reciprocally +proportional, the solid parallelepipeds are equal.</i></p> + +<p>§ 83. Of the following propositions the 37th and 40th are of +special interest.</p> + +<p>Prop. 37. <i>If four straight lines be proportionals, the similar solid +parallelepipeds, similarly described from them, shall also be proportionals; +and if the similar parallelepipeds similarly described +from four straight lines be proportionals, the straight lines shall be +proportionals.</i></p> + +<p>In symbols it says—</p> + +<p class="center">If a : b = c : d, then a³ : b³ = c³: d³.</p> + +<p>Prop. 40 teaches how to compare the volumes of triangular +prisms with those of parallelepipeds, by proving <i>that a triangular +prism is equal in volume to a parallelepiped, which has its altitude +and its base equal to the altitude and the base of the triangular +prism.</i></p> + +<p>§ 84. From these propositions follow all results relating to the +mensuration of volumes. We shall state these as we did in the case +of areas. The starting-point is the “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.”</p> + +<p>If we take a certain line u as unit length, then the square on u is +the unit of area, and the cube on u the unit of volume, that is to +say, if we wish to measure a volume we have to determine how +many unit cubes it contains.</p> + +<p>A rectangular parallelepiped has, as a rule, the three edges unequal, +which meet at a point. Every other edge is equal to one +of them. If a, b, c be the three edges meeting at a point, then we +may take the rectangle contained by two of them, say by b and c, +as base and the third as altitude. Let V be its volume, V′ 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,</p> + +<p><i>Rectangular parallelepipeds on equal bases are proportional to their +altitudes.</i></p> + +<p>If we have two rectangular parallelepipeds, of which the first has +the volume V and the edges a, b, c, and the second, the volume V′ +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 V<span class="su">1</span> and V<span class="su">2</span>. We then have</p> + +<p class="center">V : V<span class="su">1</span> = a : a′; V<span class="su">1</span> : V<span class="su">2</span> = b : b′, V<span class="su">2</span> : V′ = c : c′.</p> + +<p>Compounding these, we have</p> + +<p class="center">V : V′ = (a : a′) (b : b′) (c : c′),</p> + +<p class="noind">or</p> + +<table class="math0" summary="math"> +<tr><td>V</td> +<td rowspan="2">=</td> <td>a</td> +<td rowspan="2">·</td> <td>b</td> +<td rowspan="2">·</td> <td>c</td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">V′</td> <td class="denom">a′</td> +<td class="denom">b′</td> <td class="denom">c′</td></tr></table> + +<p class="noind">Hence, as a special case, making V′ equal to the unit cube U on u +we get</p> + +<table class="math0" summary="math"> +<tr><td>V</td> +<td rowspan="2">=</td> <td>a</td> +<td rowspan="2">·</td> <td>b</td> +<td rowspan="2">·</td> <td>c</td> +<td rowspan="2">= α·β·γ,</td></tr> +<tr><td class="denom">U</td> <td class="denom">u</td> +<td class="denom">u</td> <td class="denom">u</td></tr></table> + +<p class="noind">where α, β, γ are the numerical values of a, b, c; that is, <i>The number +of unit cubes in a rectangular parallelepiped</i> is equal to the product +of the numerical values of its three edges. This is generally expressed +by saying the volume of a rectangular parallelepiped is +measured by the product of its sides, or by the product of its base +into its altitude, which in this case is the same.</p> + +<p>Prop. 31 allows us to extend this to any parallelepipeds, and Props. +28 or 40, to triangular prisms.</p> + +<p><i>The volume of any parallelepiped, or of any triangular prism, is +measured by the product of base and altitude.</i></p> + +<p>The consideration that any polygonal prism may be divided into +a number of triangular prisms, which have the same altitude and +the sum of their bases equal to the base of the polygonal prism, +shows further that the same holds for any prism whatever.</p> + +<p class="pt2 center sc">Book XII.</p> + +<p>§ 85. In the last part of Book XI. we have learnt how to compare +the volumes of parallelepipeds and of prisms. In order to determine +the volume of any solid bounded by plane faces we must determine +the volume of pyramids, for every such solid may be decomposed +into a number of pyramids.</p> + +<p>As every pyramid may again be decomposed into triangular +pyramids, it becomes only necessary to determine their volume. +This is done by the</p> + +<p><i>Theorem.</i>—Every triangular pyramid is equal in volume to one +third of a triangular prism having the same base and the same +altitude as the pyramid.</p> + +<p>This is an immediate consequence of Euclid’s</p> + +<p>Prop. 7. <i>Every prism having a triangular base may be divided +into three pyramids that have triangular bases, and are equal to one +another.</i></p> + +<p>The proof of this theorem is difficult, because the three triangular +pyramids into which the prism is divided are by no means equal in +shape, and cannot be made to coincide. It has first to be proved +that two triangular pyramids have equal volumes, if they have +equal bases and equal altitudes. This Euclid does in the following +manner. He first shows (Prop. 3) that a triangular pyramid may +be divided into four parts, of which two are equal triangular pyramids +similar to the whole pyramid, whilst the other two are equal triangular +prisms, and further, that these two prisms together are +greater than the two pyramids, hence more than half the given +pyramid. He next shows (Prop. 4) that if two triangular pyramids +are given, having equal bases and equal altitudes, and if each be +divided as above, then the two triangular prisms in the one are +equal to those in the other, and each of the remaining pyramids in +the one has its base and altitude equal to the base and altitude of +the remaining pyramids in the other. Hence to these pyramids the +same process is again applicable. We are thus enabled to cut out +of the two given pyramids equal parts, each greater than half the +original pyramid. Of the remainder we can again cut out equal +parts greater than half these remainders, and so on as far as we like. +This process may be continued till the last remainder is smaller +than any assignable quantity, however small. It follows, so we +should conclude at present, that the two volumes must be equal, for +they cannot differ by any assignable quantity.</p> + +<p>To Greek mathematicians this conclusion offers far greater +<span class="pagenum"><a name="page688" id="page688"></a>688</span> +difficulties. They prove elaborately, by a <i>reductio ad absurdum</i>, +that the volumes cannot be unequal. This proof must be read in +the <i>Elements.</i> We must, however, state that we have in the above +not proved Euclid’s Prop. 5, but only a special case of it. Euclid +does not suppose that the bases of the two pyramids to be compared +are equal, and hence he proves that the volumes are as the bases. +The reasoning of the proof becomes clearer in the special case, from +which the general one may be easily deduced.</p> + +<p>§ 86. Prop. 6 extends the result to pyramids with polygonal +bases. From these results follow again the rules at present given +for the mensuration of solids, viz. a pyramid is the third part of a +triangular prism having the same base and the same altitude. But +a triangular prism is equal in volume to a parallelepiped which +has the same base and altitude. Hence if B is the base and h the +altitude, we have</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">Volume of prism</td> <td class="tcl">= Bh,</td></tr> +<tr><td class="tcl">Volume of pyramid</td> <td class="tcl">= <span class="spp">1</span>⁄<span class="suu">3</span>Bh,</td></tr> +</table> + +<p class="noind">statements which have to be taken in the sense that B means the +number of square units in the base, h the number of units of length +in the altitude, or that B and h denote the numerical values of base +and altitude.</p> + +<p>§ 87. A method similar to that used in proving Prop. 5 leads to +the following results relating to solids bounded by simple curved +surfaces:—</p> + +<p>Prop. 10. <i>Every cone is the third part of a cylinder which has the +same base, and is of an equal altitude with it.</i></p> + +<p>Prop. 11. <i>Cones or cylinders of the same altitude are to one another +as their bases.</i></p> + +<p>Prop. 12. <i>Similar cones or cylinders have to one another the triplicate +ratio of that which the diameters of their bases have.</i></p> + +<p>Prop. 13. <i>If a cylinder be cut by a plane parallel to its opposite +planes or bases, it divides the cylinder into two cylinders, one of which +is to the other as the axis of the first to the axis of the other;</i> which +may also be stated thus:—</p> + +<p><i>Cylinders on the same base are proportional to their altitudes.</i></p> + +<p>Prop. 14. <i>Cones or cylinders upon equal bases are to one another +as their altitudes.</i></p> + +<p>Prop. 15. <i>The bases and altitudes of equal cones or cylinders are +reciprocally proportional, and if the bases and altitudes be reciprocally +proportional, the cones or cylinders are equal to one another.</i></p> + +<p>These theorems again lead to formulae in mensuration, if we +compare a cylinder with a prism having its base and altitude equal to +the base and altitude of the cylinder. This may be done by the +method of exhaustion. We get, then, the result that their bases are +equal, and have, if B denotes the numerical value of the base, and +h that of the altitude,</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">Volume of cylinder</td> <td class="tcl">= Bh,</td></tr> +<tr><td class="tcl">Volume of cone</td> <td class="tcl">= <span class="spp">1</span>⁄<span class="suu">3</span>Bh.</td></tr> +</table> + +<p>§ 88. The remaining propositions relate to circles and spheres. +Of the sphere only one property is proved, viz.:—</p> + +<p>Prop. 18. <i>Spheres have to one another the triplicate ratio of that +which their diameters have.</i> The mensuration of the sphere, like +that of the circle, the cylinder and the cone, had not been settled +in the time of Euclid. It was done by Archimedes.</p> + +<p class="pt2 center sc">Book XIII.</p> + +<p>§ 89. The 13th and last book of Euclid’s <i>Elements</i> is devoted to +the regular solids (see <span class="sc"><a href="#artlinks">Polyhedron</a></span>). It is shown that there are +five of them, viz.:—</p> + +<p>1. The regular <i>tetrahedron</i>, with 4 triangular faces and 4 vertices;</p> + +<p>2. The <i>cube</i>, with 8 vertices and 6 square faces;</p> + +<p>3. The <i>octahedron</i>, with 6 vertices and 8 triangular faces;</p> + +<p>4. The <i>dodecahedron</i>, with 12 pentagonal faces, 3 at each of the +20 vertices;</p> + +<p>5. The <i>icosahedron</i>, with 20 triangular faces, 5 at each of the +12 vertices.</p> + +<p>It is shown how to inscribe these solids in a given sphere, and +how to determine the lengths of their edges.</p> + +<p>§ 90. The 13th book, and therefore the <i>Elements</i>, conclude with +the scholium, “that no other regular solid exists besides the five +ones enumerated.”</p> + +<p>The proof is very simple. Each face is a regular polygon, hence +the angles of the faces at any vertex must be angles in equal regular +polygons, must be together less than four right angles (XI. 21), and +must be three or more in number. Each angle in a regular triangle +equals two-thirds of one right angle. Hence it is possible to form +a solid angle with three, four or five regular triangles or faces. +These give the solid angles of the tetrahedron, the octahedron and +the icosahedron. The angle in a square (the regular quadrilateral) +equals one right angle. Hence three will form a solid angle, that +of the cube, and four will not. The angle in the regular pentagon +equals <span class="spp">6</span>⁄<span class="suu">5</span> of a right angle. Hence three of them equal <span class="spp">18</span>⁄<span class="suu">5</span> (<i>i.e.</i> less +than 4) right angles, and form the solid angle of the dodecahedron. +Three regular polygons of six or more sides cannot form a solid +angle. Therefore no other regular solids are possible.</p> +</div> +<div class="author">(O. H.)</div> + +<p class="pt2 center sc">II. Projective Geometry</p> + +<p>It is difficult, at the outset, to characterize projective geometry +as compared with Euclidean. But a few examples will at least +indicate the practical differences between the two.</p> + +<p>In Euclid’s <i>Elements</i> almost all propositions refer to the <i>magnitude</i> +of lines, angles, areas or volumes, and therefore to measurement. +The statement that an angle is right, or that two straight +lines are parallel, refers to measurement. On the other hand, +the fact that a straight line does or does not cut a circle is independent +of measurement, it being dependent only upon the +mutual “position” of the line and the circle. This difference +becomes clearer if we project any figure from one plane to another +(see <span class="sc"><a href="#artlinks">Projection</a></span>). By this the length of lines, the magnitude +of angles and areas, is altered, so that the projection, or shadow, +of a square on a plane will not be a square; it will, however, +be some quadrilateral. Again, the projection of a circle will not +be a circle, but some other curve more or less resembling a circle. +But one property may be stated at once—no straight line can cut +the projection of a circle in more than two points, because no +straight line can cut a circle in more than two points. There +are, then, some properties of figures which do not alter by +projection, whilst others do. To the latter belong nearly all +properties relating to measurement, at least in the form in which +they are generally given. The others are said to be projective +properties, and their investigation forms the subject of projective +geometry.</p> + +<p>Different as are the kinds of properties investigated in the old +and the new sciences, the methods followed differ in a still +greater degree. In Euclid each proposition stands by itself; +its connexion with others is never indicated; the leading ideas +contained in its proof are not stated; general principles do not +exist. In the modern methods, on the other hand, the greatest +importance is attached to the leading thoughts which pervade +the whole; and general principles, which bring whole groups of +theorems under one aspect, are given rather than separate propositions. +The whole tendency is towards generalization. +A straight line is considered as given in its entirety, extending +both ways to infinity, while Euclid never admits anything but +finite quantities. The treatment of the infinite is in fact another +fundamental difference between the two methods: Euclid avoids +it; in modern geometry it is systematically introduced.</p> + +<p>Of the different modern methods of geometry, we shall treat +principally of the methods of projection and correspondence which +have proved to be the most powerful. These have become independent +of Euclidean Geometry, especially through the <i>Geometrie +der Lage</i> of V. Staudt and the <i>Ausdehnungslehre</i> of Grassmann.</p> + +<p>For the sake of brevity we shall presuppose a knowledge of +Euclid’s <i>Elements</i>, although we shall use only a few of his propositions.</p> + +<div class="condensed"> +<p>§ 1. <i>Geometrical Elements.</i> We consider space as filled with points, +lines and planes, and these we call the elements out of which our +figures are to be formed, calling any combination of these elements a +“figure.”</p> + +<p>By a line we mean a straight line in its entirety, extending both +ways to infinity; and by a plane, a plane surface, extending in all +directions to infinity.</p> + +<p>We accept the three-dimensional space of experience—the space +assumed by Euclid—which has for its properties (among others):—</p> + +<p>Through any two points in space one and only one line may be +drawn;</p> + +<p>Through any three points which are not in a line, one and only one +plane may be placed;</p> + +<p>The intersection of two planes is a line;</p> + +<p>A line which has two points in common with a plane lies in the +plane, hence the intersection of a line and a plane is a single point; and</p> + +<p>Three planes which do not meet in a line have one single point in +common.</p> + +<p>These results may be stated differently in the following form:—</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p>I. A plane is determined—</p></td> +<td class="tcl" style="width: 50%;"><p>A point is determined—</p></td></tr> + +<tr><td class="tcl1 rb3"><p>1. By three points which do not lie in a line;</p> +<p>2. By two intersecting lines;</p> +<p>3. By a line and a point which does not lie in it.</p></td> + +<td class="tcl1"><p>1. By three planes which do not pass through a line;</p> +<p>2. By two intersecting lines</p> +<p>3. By a plane and a line which does not lie in it.</p></td></tr> + +<tr><td class="tcl rb3"><p>A line is determined—</p></td> +<td class="tcl"> </td></tr> + +<tr><td class="tcl1 rb3"><p>1. By two points;</p></td> +<td class="tcl1"><p>2. By two planes.</p></td></tr></table> + +<p><span class="pagenum"><a name="page689" id="page689"></a>689</span></p> + +<p>It will be observed that not only are planes determined by points, +but also points by planes; that therefore the planes may be considered +as elements, like points; and also that in any one of the +above statements we may interchange the words point and plane, +and we obtain again a correct statement, provided that these +statements themselves are true. As they stand, we ought, in +several cases, to add “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.</p> + +<table class="flt" style="float: right; width: 350px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:294px; height:205px" src="images/img689.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 1.</span></td></tr></table> + +<p>§ 2. <i>Parallels. Point at Infinity.</i>—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: <i>Every</i> line through S which joins it to any point +at an infinite distance in p is parallel to p. But by Euclid’s 12th +axiom there is but one line parallel to p through S. The difficulty in +which we are thus involved is due to the fact that we try to reason +about infinity as if we, with our finite capabilities, could comprehend +the infinite. To overcome this difficulty, we may say that all points +at infinity in a line <i>appear</i> to us as one, and may be replaced by a +single “ideal” point.</p> + +<p>We may therefore now give the following definitions and axiom:—</p> + +<p><i>Definition.</i>—Lines which meet at infinity are called parallel.</p> + +<p><i>Axiom.</i>—All points at an infinite distance in a line may be considered +as one single point.</p> + +<p><i>Definition.</i>—This ideal point is called the <i>point at infinity</i> in the +line.</p> + +<p>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.</p> + +<p>This point at infinity in a line is reached whether we move a +point in the one or in the opposite direction of a line to infinity. +A line thus appears closed by this point, and we speak as if we +could move a point along the line from one position A to another +B in two ways, either through the point at infinity or through finite +points only.</p> + +<p>It must never be forgotten that this point at infinity is ideal; +in fact, the whole notion of “infinity” is only a mathematical +conception, and owes its introduction (as a method of research) to +the working generalizations which it permits.</p> + +<p>§ 3. <i>Line and Plane at Infinity.</i>—Having arrived at the notion of +replacing all points at infinity in a line by one ideal point, there is no +difficulty in replacing all points at infinity in a plane by one ideal +line.</p> + +<p>To make this clear, let us suppose that a line p, which cuts two +fixed lines a and b in the points A and B, moves parallel to itself +to a greater and greater distance. It will at last cut both a and +b at their points at infinity, so that a line which joins the two points +at infinity in two intersecting lines lies altogether at infinity. Every +other line in the plane will meet it therefore at infinity, and thus it +contains all points at infinity in the plane.</p> + +<p><i>All points at infinity in a plane lie in a line, which is called the</i> line +at infinity <i>in the plane.</i></p> + +<p>It follows that parallel planes must be considered as planes +having a common line at infinity, for any other plane cuts them in +parallel lines which have a point at infinity in common.</p> + +<p>If we next take two intersecting planes, then the point at infinity +in their line of intersection lies in both planes, so that their lines +at infinity meet. Hence every line at infinity meets every other +line at infinity, and they are therefore all in one plane.</p> + +<p><i>All points at infinity in space may be considered as lying in one +ideal plane, which is called the</i> plane at infinity.</p> + +<p>§ 4. <i>Parallelism.</i>—We have now the following definitions:—</p> + +<p>Parallel lines are lines which meet at infinity;</p> + +<p>Parallel planes are planes which meet at infinity;</p> + +<p>A line is parallel to a plane if it meets it at infinity.</p> + +<p>Theorems like this—Lines (or planes) which are parallel to a third +are parallel to each other—follow at once.</p> + +<p>This view of parallels leads therefore to no contradiction of +Euclid’s <i>Elements.</i></p> + +<p>As immediate consequences we get the propositions:—</p> + +<p>Every line meets a plane in one point, or it lies in it;</p> + +<p>Every plane meets every other plane in a line;</p> + +<p>Any two lines in the same plane meet.</p> + +<p>§ 5. <i>Aggregates of Geometrical Elements.</i>—We have called points, +lines and planes the elements of geometrical figures. We also say +that an element of one kind contains one of the other if it lies in it +or passes through it.</p> + +<p>All the elements of one kind which are contained in one or two +elements of a different kind form aggregates which have to be +enumerated. They are the following:—</p> + +<p>I. Of one dimension.</p> + +<div class="list1"> +<p>1. The <i>row</i>, or range, <i>of points</i> formed by all points in a line, + which is called its base.</p> + +<p>2. The <i>flat pencil</i> formed by all the lines through a point in + a plane. Its base is the point in the plane.</p> + +<p>3. The <i>axial pencil</i> formed by all planes through a line + which is called its base or axis.</p> +</div> + +<p>II. Of two dimensions.</p> + +<div class="list1"> +<p>1. The field of points and lines—that is, a plane with all its + points and all its lines.</p> + +<p>2. The pencil of lines and planes—that is, a point in space + with all lines and all planes through it.</p> +</div> + +<p>III. Of three dimensions.</p> + +<div class="list1"> +<p>The space of points—that is, all points in space.</p> + +<p>The space of planes—that is, all planes in space.</p> +</div> + +<p>IV. Of four dimensions.</p> + +<div class="list1"> +<p>The space of lines, or all lines in space.</p> +</div> + +<p>§ 6. <i>Meaning of “Dimensions.”</i>—The word dimension in the above +needs explanation. If in a plane we take a row p and a pencil with +centre Q, then through every point in p one line in the pencil will +pass, and every ray in Q will cut p in one point, so that we are +entitled to say a row contains as many points as a flat pencil lines, +and, we may add, as an axial pencil planes, because an axial pencil +is cut by a plane in a flat pencil.</p> + +<p>The number of elements in the row, in the flat pencil, and in the +axial pencil is, of course, infinite and indefinite too, but the same in +all. This number may be denoted by ∞. Then a plane contains +∞² points and as many lines. To see this, take a flat pencil in a +plane. It contains ∞ lines, and each line contains ∞ 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 ∞ lines and contains ∞ points; hence there are ∞² +lines in a plane.</p> + +<p>A pencil in space contains as many lines as a plane contains +points and as many planes as a plane contains lines, for any plane +cuts the pencil in a field of points and lines. Hence a pencil contains +∞² lines and ∞² planes. <i>The field and the pencil are of two +dimensions.</i></p> + +<p>To count the number of points in space we observe that each +point lies on some line in a pencil. But the pencil contains ∞² +lines, and each line ∞ points; hence space contains ∞³ points. +Each plane cuts any fixed plane in a line. But a plane contains +∞² lines, and through each pass ∞ planes; therefore space contains +∞³ planes.</p> + +<p>Hence space contains as many planes as points, but it contains +an infinite number of times more lines than points or planes. To +count them, notice that every line cuts a fixed plane in one point. +But ∞² lines pass through each point, and there are ∞² points in the +plane. Hence there are ∞<span class="sp">4</span> lines in space. <i>The space of points +and planes is of three dimensions, but the space of lines is of four +dimensions.</i></p> + +<p>A field of points or lines contains an infinite number of rows and +flat pencils; a pencil contains an infinite number of flat pencils +and of axial pencils; space contains a triple infinite number of +pencils and of fields, ∞<span class="sp">4</span> rows and axial pencils and ∞<span class="sp">5</span> flat pencils—or, +in other words, each point is a centre of ∞² flat pencils.</p> + +<p>§ 7. The above enumeration allows a classification of figures. +Figures in a row consist of groups of points only, and figures in +the flat or axial pencil consist of groups of lines or planes. In the +plane we may draw polygons; and in the pencil or in the point, +solid angles, and so on.</p> + +<p>We may also distinguish the different measurements We have—</p> + +<table class="reg" summary="poem"><tr><td> <div class="poemr"> +<p>In the row, length of segment;</p> +<p>In the flat pencil, angles;</p> +<p>In the axial pencil, dihedral angles between two planes;</p> +<p>In the plane, areas;</p> +<p>In the pencil, solid angles;</p> +<p>In the space of points or planes, volumes.</p> +</div> </td></tr></table> + +<p class="pt2 center sc">Segments of a Line</p> + +<p>§ 8. Any two points A and B in space determine on the line through +them a finite part, which may be considered as being described by +a point moving from A to B. This we shall denote by AB, and +distinguish it from BA, which is supposed as being described by a +point moving from B to A, and hence in a direction or in a “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.</p> + +<p><span class="pagenum"><a name="page690" id="page690"></a>690</span></p> + +<p>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.</p> + +<p>We may also say, with Clifford, that AB denotes the “step” of +going from A to B.</p> + +<table class="flt" style="float: right; width: 230px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:184px; height:129px" src="images/img690a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 2.</span></td></tr></table> + +<p>§ 9. If we have three points A, B, C in a line (fig. 2), the step AB +will bring us from A to B, and the step +BC from B to C. Hence both steps are +equivalent to the one step AC. This is +expressed by saying that AC is the +“sum” of AB and BC; in symbols—</p> + +<p class="center">AB + BC = AC,</p> + +<p class="noind">where account is to be taken of the +sense.</p> + +<p>This equation is true whatever be the +position of the three points on the line. +As a special case we have</p> + +<p class="center">AB + BA = 0,</p> +<div class="aut">(1)</div> + +<p class="noind">and similarly</p> + +<p class="center">AB + BC + CA = 0,</p> +<div class="aut">(2)</div> + +<p class="noind">which again is true for any three points in a line.</p> + +<p>We further write</p> + +<p class="center">AB = −BA.</p> + +<p class="noind">where − denotes negative sense.</p> + +<p>We can then, just as in algebra, change subtraction of segments +into addition by changing the sense, so that AB − 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.</p> + +<p>§ 10. Of the many formulae which exist between points in a line +we shall have to use only one more, which connects the segments +between any four points A, B, C, D in a line. We have</p> + +<p class="center">BC = BD + DC, CA = CD + DA, AB = AD + DB;</p> + +<p class="noind">or multiplying these by AD, BD, CD respectively, we get</p> + +<table class="reg" summary="poem"><tr><td> <div class="poemr"> +<p>BC · AD = BD · AD + DC · AD = BD · AD − CD · AD</p> + +<p>CA · BD = CD · BD + DA · BD = CD · BD − AD · BD</p> + +<p>AB · CD = AD · CD + DB · CD = AD · CD − BD · CD.</p> +</div> </td></tr></table> + +<p class="noind">It will be seen that the sum of the right-hand sides vanishes, hence +that</p> + +<p class="center">BC · AD + CA · BD + AB · CD = 0</p> +<div class="aut">(3)</div> + +<p class="noind">for any four points on a line.</p> + +<table class="flt" style="float: right; width: 390px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:335px; height:30px" src="images/img690b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 3.</span></td></tr></table> + +<p>§ 11. If C is any point in the line AB, then we say that C divides +the segment AB in the ratio AC/CB, account being taken of the +sense of the two segments AC and CB. If C lies between A and B +the ratio is positive, as AC and CB have the same sense. But if +C lies without the segment AB, <i>i.e.</i> if C divides AB externally, then +the ratio is negative. +To see how the value of +this ratio changes with +C, we will move C along +the whole line (fig. 3), +whilst A and B remain fixed. If C lies at the point A, then AC = 0, +hence the ratio AC : CB vanishes. As C moves towards B, AC +increases and CB decreases, so that our ratio increases. At the +middle point M of AB it assumes the value +1, and then increases +till it reaches an infinitely large value, when C arrives at B. On +passing beyond B the ratio becomes negative. If C is at P we have +AC = AP = AB + BP, hence</p> + +<table class="math0" summary="math"> +<tr><td>AC</td> +<td rowspan="2">=</td> <td>AB</td> +<td rowspan="2">+</td> <td>BP</td> +<td rowspan="2">= −</td> <td>AB</td> +<td rowspan="2">− 1.</td></tr> +<tr><td class="denom">CB</td> <td class="denom">PB</td> +<td class="denom">PB</td> <td class="denom">BP</td></tr></table> + +<p class="noind">In the last expression the ratio AB : BP is positive, has its greatest +value ∞ 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 −∞ to −1.</p> + +<p>If, on the other hand, C is to the left of A, say at Q, we have +AC = AQ = AB + BQ = AB − QB, hence AC/CB = AB/QB − 1.</p> + +<p>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—</p> + +<p>“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, ∞ at B, now changes +sign to −∞, and increases till at an infinite distance it reaches +again the value −1. <i>It assumes therefore all possible values from +-∞ to +∞, and each value only once, so that not only does every +position of</i> C <i>determine a definite value of the ratio</i> AC : CB, <i>but also, +conversely, to every positive or negative value of this ratio belongs one +single point in the line</i> AB.</p> + +<p>[Relations between segments of lines are interesting as showing an +application of algebra to geometry. The genesis of such relations +from algebraic identities is very simple. For example, if a, b, c, x +be any four quantities, then</p> + +<table class="math0" summary="math"> +<tr><td>a</td> +<td rowspan="2">+</td> <td>b</td> +<td rowspan="2">+</td> <td>c</td> +<td rowspan="2">=</td> <td>x</td> +<td rowspan="2">;</td></tr> +<tr><td class="denom">(a − b)(a − c)(x − a)</td> <td class="denom">(b − c)(b − a)(x − b)</td> +<td class="denom">(c − a)(c − b)(x − c)</td> <td class="denom">(x − a)(x − b)(x − c)</td></tr></table> + +<p class="noind">this may be proved, cumbrously, by multiplying up, or, simply, by +decomposing the right-hand member of the identity into partial +fractions. Now take a line ABCDX, and let AB = a, AC = b, AD = c, +AX = x. Then obviously (a − 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:—</p> + +<table class="math0" summary="math"> +<tr><td>AB</td> +<td rowspan="2">+</td> <td>AC</td> +<td rowspan="2">+</td> <td>AD</td> +<td rowspan="2">=</td> <td>AX</td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">BC · BD · BX</td> <td class="denom">CD · CB · CX</td> +<td class="denom">DB · DC · DX</td> <td class="denom">BX · CX · DX</td></tr></table> + +<p>Conversely, if a metrical relation be given, its validity may be +tested by reducing to an algebraic equation, which is an identity +if the relation be true. For example, if ABCDX be five collinear +points, prove</p> + +<table class="math0" summary="math"> +<tr><td>AD · AX</td> +<td rowspan="2">+</td> <td>BD · BX</td> +<td rowspan="2">+</td> <td>CD · CX</td> +<td rowspan="2">= 1.</td></tr> +<tr><td class="denom">AB · AC</td> <td class="denom">BC · BA</td> +<td class="denom">CA · CB</td></tr></table> + +<p class="noind">Clearing of fractions by multiplying throughout by AB · BC · CA, +we have to prove</p> + +<p class="center">−AD · AX · BC − BD · BX · CA − CD · CX · AB = AB · BC · CA.</p> + +<p class="noind">Take A as origin and let AB = a, AC = b, AD = c, AX = x. Substituting +for the segments in terms of a, b, c, x, we obtain on simplification</p> + +<p class="center">a²b − ab² = −ab² + a²b, an obvious identity.</p> + +<p>An alternative method of testing a relation is illustrated in the +<span class="correction" title="amended from example: following">following example:—</span> If A, B, C, D, E, F be six collinear points, +then</p> + +<table class="math0" summary="math"> +<tr><td>AE · AF</td> +<td rowspan="2">+</td> <td>BE · BF</td> +<td rowspan="2">+</td> <td>CE · CF</td> +<td rowspan="2">+</td> <td>DE · DF</td> +<td rowspan="2">= 0.</td></tr> +<tr><td class="denom">AB · AC · AD</td> <td class="denom">BC · BD · BA</td> +<td class="denom">CD · CA · CB</td> <td class="denom">DA · DB · DC</td></tr></table> + +<p class="noind">Clearing of fractions by multiplying throughout by AB · BC · CD · DA, +and reducing to a common origin O (calling OA = a, OB = b, &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.</p> + +<p>The relations connecting five points which we have instanced above +may be readily deduced from the six-point relation; the first by +taking D at infinity, and the second by taking F at infinity, and then +making the obvious permutations of the points.]</p> + +<p class="pt2 center sc">Projection and Cross-ratios</p> + +<p>§ 12. If we join a point A to a point S, then the point where the +line SA cuts a fixed plane π is called the projection of A on the +plane π from S as centre of projection. If we have two planes π +and π′ and a point S, we may project every point A in π 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 π we get as its projection a corresponding figure in π′.</p> + +<p>We shall determine such properties of figures as remain true for +the projection, and which are called projective properties. For this +purpose it will be sufficient to consider at first only constructions in +one plane.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:472px; height:377px" src="images/img690c.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 4.</span></td> +<td class="caption"><span class="sc">Fig. 5.</span></td></tr></table> + +<p>Let us suppose we have given in a plane two lines p and p′ 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 +<span class="pagenum"><a name="page691" id="page691"></a>691</span> +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′.</p> + +<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.</p> + +<p>§ 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</p> + +<table class="math0" summary="math"> +<tr><td>AC</td> +<td rowspan="2">:</td> <td>AD</td> +<td rowspan="2">=</td> <td>A′C′</td> +<td rowspan="2">:</td> <td>A′D′</td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">CB</td> <td class="denom">DB</td> +<td class="denom">C′B′</td> <td class="denom">D′B′</td></tr></table> + +<p>This is easily proved by aid of similar triangles.</p> + +<table class="flt" style="float: right; width: 350px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:304px; height:237px" src="images/img691.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 6.</span></td></tr></table> + +<p>Through the points A and B on p draw parallels to p′, which cut +the projecting rays in +C<span class="su">2</span>, D<span class="su">2</span>, B<span class="su">2</span> and A<span class="su">1</span>, C<span class="su">1</span>, +D<span class="su">1</span>, as indicated in +fig. 6. The two triangles +ACC<span class="su">2</span> and BCC<span class="su">1</span> will be +similar, as will also be +the triangles ADD<span class="su">2</span> and +BDD<span class="su">1</span>.</p> + +<p>The proof is left to +the reader.</p> + +<p>This result is of fundamental +importance.</p> + +<p>The expression +AC/CB : AD/DB has been +called by Chasles the +“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</p> + +<p><span class="sc">Fundamental Theorem.</span>—<i>The cross-ratio of four points in a +line is equal to the cross-ratio of their projections on any other line +which lies in the same plane with it.</i></p> + +<p>§ 14. Before we draw conclusions from this result, we must investigate +the meaning of a cross-ratio somewhat more fully.</p> + +<p>If four points A, B, C, D are given, and we wish to form their +cross-ratio, we have first to divide them into two groups of two, +the points in each group being taken in a definite order. Thus, +let A, B be the first, C, D the second pair, A and C being the first +points in each pair. The cross-ratio is then the ratio AC : CB +divided by AD : DB. This will be denoted by (AB, CD), so that</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">(AB, CD) =</td> <td>AC</td> +<td rowspan="2">:</td> <td>AD</td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">CB</td> <td class="denom">DB</td></tr></table> + +<p>This is easily remembered. In order to write it out, make first +the two lines for the fractions, and put above and below these +the letters A and B in their places, thus, A/*B : A/*B; and then fill +up, crosswise, the first by C and the other by D.</p> + +<p>§ 15. If we take the points in a different order, the value of the +cross-ratio will change. We can do this in twenty-four different +ways by forming all permutations of the letters. But of these +twenty-four cross-ratios groups of four are equal, so that there are +really only six different ones, and these six are reciprocals in pairs.</p> + +<p>We have the following rules:—</p> + +<p>I. If in a cross-ratio the two groups be interchanged, its value +remains unaltered, <i>i.e.</i></p> + +<p class="center">(AB, CD) = (CD, AB) = (BA, DC) = (DC, BA).</p> + +<p>II. If in a cross-ratio the two points belonging to one of the two +groups be interchanged, the cross-ratio changes into its reciprocal, <i>i.e.</i></p> + +<p class="center">(AB, CD) = 1/(AB, DC) = 1/(BA, CD) = 1/(CD, BA) = 1/(DC, AB).</p> + +<p>From I. and II. we see that eight cross-ratios are associated with +(AB, CD).</p> + +<p>III. If in a cross-ratio the two middle letters be interchanged, +the cross-ratio α changes into its complement 1 − α, <i>i.e.</i> (AB, CD) = +1 − (AC, BD).</p> + +<p>[§ 16. If λ = (AB, CD), μ = (AC, DB), ν = (AD, BC), then λ, μ, ν +and their reciprocals 1/λ, 1/μ, 1/ν are the values of the total number +of twenty-four cross-ratios. Moreover, λ, μ, ν are connected by the +relations</p> + +<p class="center">λ + 1/μ = μ + 1/ν = ν + 1/λ = +−λμν = 1;</p> + +<p class="noind">this proposition may be proved by substituting for λ, μ, ν 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 λ, μ, ν are positive, and the remaining one negative.</p> + +<p>The following scheme shows the twenty-four cross-ratios expressed +in terms of λ, μ, ν.]</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl lb rb cl tb">(AB, CD)<br />(BA, DC)<br />(CD, AB)<br />(DC, BA)</td> <td class="tccm rb cl tb">λ</td> <td class="tccm rb cl tb">1 − μ</td> <td class="tccm rb2 cl tb">1/(1 − ν)</td> + <td class="tcl rb cl tb">(AD, BC)<br />(BC, AD)<br />(CB, DA)<br />(DA, CB)</td> <td class="tccm rb cl tb">(λ − 1)/λ</td> <td class="tccm rb cl tb">μ/(μ − 1)</td> <td class="tccm rb cl tb">ν</td></tr> + + +<tr><td class="tcl lb rb">(AC, DB)<br />(BD, CA)<br />(CA, BD)<br />(DB, AC)</td> <td class="tccm rb">1/(1 − λ)</td> <td class="tccm rb">1/μ</td> <td class="tccm rb2">(ν − 1)/ν</td> + <td class="tcl rb">(AC, BD)<br />(BD, AC)<br />(CA, DB)<br />(DB, CA)</td> <td class="tccm rb">1 − λ</td> <td class="tccm rb">μ</td> <td class="tccm rb">ν/(ν − 1)</td></tr> + +<tr><td class="tcl lb rb cl bb">(AB, DC)<br />(BA, CD)<br />(CD, BA)<br />(DC, AB)</td> <td class="tccm rb cl bb">1/λ</td> <td class="tccm rb cl bb">1/(1 − μ)</td> <td class="tccm rb2 cl bb">1 − ν</td> + <td class="tcl rb cl bb">(AD, CB)<br />(BC, DA)<br />(CB, AD)<br />(DA, BC)</td> <td class="tccm rb cl bb">λ/(λ − 1)</td> <td class="tccm rb cl bb">(μ − 1)/μ</td> <td class="tccm rb cl bb">1/ν</td></tr> +</table> + +<p>§ 17. If one of the points of which a cross-ratio is formed is the +point at infinity in the line, the cross-ratio changes into a simple +ratio. It is convenient to let the point at infinity occupy the last +place in the symbolic expression for the cross-ratio. Thus if I is a +point at infinity, we have (AB, CI) = −AC/CB, because AI : IB = −1.</p> + +<p>Every common ratio of three points in a line may thus be expressed +as a cross-ratio, by adding the point at infinity to the group +of points.</p> + +<p class="pt2 center sc">Harmonic Ranges</p> + +<p>§ 18. If the points have special positions, the cross-ratios may +have such a value that, of the six different ones, two and two become +equal. If the first two shall be equal, we get λ = 1/λ, or λ² = 1, +λ = ±1.</p> + +<p>If we take λ = +1, we have (AB, CD) = 1, or AC/CB = AD/DB; +that is, the points C and D coincide, provided that A and B are +different.</p> + +<p>If we take λ = −1, so that (AB, CD) = −1, we have AC/CB = +−AD/DB. <i>Hence C and D divide AB internally and externally in the +same ratio.</i></p> + +<p>The four points are in this case said to be <i>harmonic points</i>, and +C <i>and</i> D <i>are said to be harmonic conjugates with regard to</i> A <i>and</i> B.</p> + +<p>But we have also (CD, AB) = −1, so that A and B are harmonic +conjugates with regard to C and D.</p> + +<p>The principal property of harmonic points is that their cross-ratio +remains unaltered if we interchange the two points belonging to one +pair, viz.</p> + +<p class="center">(AB, CD) = (AB, DC) = (BA, CD).</p> + +<p>For four harmonic points the six cross-ratios become equal two +and two:</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">λ = −1, 1 − λ = 2,</td> <td>λ</td> +<td rowspan="2">= ½,</td> <td>1</td> +<td rowspan="2">= −1,</td> <td>1</td> +<td rowspan="2">= ½,</td> <td>λ − 1</td> +<td rowspan="2">= 2.</td></tr> +<tr><td class="denom">λ − 1</td> <td class="denom">λ</td> +<td class="denom">1 − λ</td> <td class="denom">λ</td></tr></table> + +<p>Hence if we get four points whose cross-ratio is 2 or ½, then they +are harmonic, but not arranged so that conjugates are paired. If +this is the case the cross-ratio = −1.</p> + +<p>§ 19. If we equate any two of the above six values of the cross-ratios, +we get either λ = 1, 0, ∞, or λ = −1, 2, ½, or else λ becomes +a root of the equation λ² − λ + 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.</p> + +<p>§ 20. From the definition of harmonic points, and by aid of § 11, +the following properties are easily deduced.</p> + +<p>If C and D are harmonic conjugates with regard to A and B, +then one of them lies in, the other without AB; it is impossible +to move from A to B without passing either through C or through +D; the one blocks the finite way, the other the way through infinity. +This is expressed by saying A and B are “separated” by +C and D.</p> + +<p>For every position of C there will be one and only one point +D which is its harmonic conjugate with regard to any point pair +A, B.</p> + +<p>If A and B are different points, and if C coincides with A or B, +D does. But if A and B coincide, one of the points C or D, lying +between them, coincides with them, and the other may be anywhere +in the line. It follows that, “<i>if of four harmonic conjugates two +coincide, then a third coincides with them, and the fourth may be any +point in the line</i>.”</p> + +<p>If C is the middle point between A and B, then D is the point at +infinity; for AC : CB = +1, hence AD : DB must be equal to −1. +<i>The harmonic conjugate of the point at infinity in a line with regard +to two points</i> A, B <i>is the middle point of</i> AB.</p> + +<p>This important property gives a first example how metric properties +are connected with projective ones.</p> + +<p>[§ 21. <i>Harmonic properties of the complete quadrilateral and quadrangle.</i></p> + +<p><span class="pagenum"><a name="page692" id="page692"></a>692</span></p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:500px; height:212px" src="images/img692.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 7.</span></td> +<td class="caption"><span class="sc">Fig. 8.</span></td></tr></table> + +<p>A figure formed by four lines in a plane is called a <i>complete quadrilateral</i>, +or, shorter, a <i>four-side</i>. The four sides meet in six points, +named the “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 αβγ the harmonic +triangle of the quadrilateral. A figure formed by four coplanar +points is named a <i>complete quadrangle</i>, or, shorter, a <i>four-point</i>. +The four points may be joined by six lines, named the “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.</p> + +<p>The harmonic property of the complete quadrilateral is: Any +diagonal or harmonic line is harmonically divided by the other +two; and of a complete quadrangle: The angle at any harmonic +point is divided harmonically by the joins to the other harmonic +points. To prove the first theorem, we have to prove (AA′, βγ), +(BB′, γα), (CC′, βα) are harmonic. Consider the cross-ratio (CC′, αβ). +Then projecting from A on BB′ we have A(CC′, αβ) = A(B′B, αγ). +Projecting from A′ on BB′, A′(CC′, αβ) = A′(BB′, αγ). Hence +(B′B, αγ) = (BB′, αγ), <i>i.e.</i> the cross-ratio (BB′, αγ) equals that of its +reciprocal; hence the range is harmonic.</p> + +<p>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.]</p> + +<p>This property of the complete quadrilateral allows the solution +of the problem:</p> + +<p><i>To construct the harmonic conjugate</i> D <i>to a point</i> C <i>with regard to two +given points</i> A <i>and</i> B.</p> + +<p>Through A draw any two lines, and through C one cutting the +former two in G and H. Join these points to B, cutting the former +two lines in E and F. The point D where EF cuts AB will be the +harmonic conjugate required.</p> + +<p>This remarkable construction requires nothing but the drawing +of lines, and is therefore independent of measurement. In a similar +manner the harmonic conjugate of the line VA for two lines VC, +VD is constructed with the aid of the property of the complete +quadrangle.</p> + +<p>§ 22. <i>Harmonic Pencils.</i>—The theory of cross-ratios may be extended +from points in a row to lines in a flat pencil and to planes in +an axial pencil. We have seen (§ 13) that if the lines which join four +points A, B, C, D to any point S be cut by any other line in A′, B′, C′, +D′, then (AB, CD) = (A′B′, C′D′). In other words, four lines in a +flat pencil are cut by every other line in four points whose cross-ratio +is constant.</p> + +<p><i>Definition.</i>—By the cross-ratio of four rays in a flat pencil is +meant the cross-ratio of the four points in which the rays are cut +by any line. If a, b, c, d be the lines, then this cross-ratio is denoted +by (ab, cd).</p> + +<p><i>Definition.</i>—By the cross-ratio of four planes in an axial pencil +is understood the cross-ratio of the four points in which any line +cuts the planes, or, what is the same thing, the cross-ratio of the +four rays in which any plane cuts the four planes.</p> + +<p>In order that this definition may have a meaning, it has to be +proved that all lines cut the pencil in points which have the same +cross-ratio. This is seen at once for two intersecting lines, as their +plane cuts the axial pencil in a flat pencil, which is itself cut by +the two lines. The cross-ratio of the four points on one line is +therefore equal to that on the other, and equal to that of the four +rays in the flat pencil.</p> + +<p>If two non-intersecting lines p and q cut the four planes in A, B, +C, D and A′, 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″).</p> + +<p>§ 23. We may now also extend the notion of harmonic elements, +viz.</p> + +<p><i>Definition.</i>—Four rays in a flat pencil and four planes in an axial +pencil are said to be harmonic if their cross-ratio equals -1, that is, +if they are cut by a line in four harmonic points.</p> + +<p>If we understand by a “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:</p> + +<p><i>The diagonals and median lines of a parallelogram form an harmonic +pencil</i>; and</p> + +<p><i>At a vertex of any triangle, the two sides, the median line, and the +line parallel to the base form an harmonic pencil.</i></p> + +<p>Taking the parallelogram a rectangle, or the triangle isosceles, +we get:</p> + +<p><i>Any two lines and the bisections of their angles form an harmonic +pencil.</i> Or:</p> + +<p><i>In an harmonic pencil, if two conjugate rays are perpendicular, +then the other two are equally inclined to them</i>; and, conversely, <i>if +one ray bisects the angle between conjugate rays, it is perpendicular to +its conjugate</i>.</p> + +<p>This connects perpendicularity and bisection of angles with +projective properties.</p> + +<p>§ 24. We add a few theorems and problems which are easily proved +or solved by aid of harmonics.</p> + +<p>An harmonic pencil is cut by a line parallel to one of its rays in +three equidistant points.</p> + +<p>Through a given point to draw a line such that the segment +determined on it by a given angle is bisected at that point.</p> + +<p>Having given two parallel lines, to bisect on either any given +segment without using a pair of compasses.</p> + +<p>Having given in a line a segment and its middle point, to draw +through any given point in the plane a line parallel to the given line.</p> + +<p>To draw a line which joins a given point to the intersection of two +given lines which meet off the drawing paper (by aid of § 21).</p> + +<p class="pt2 center sc">Correspondence. Homographic and Perspective Ranges</p> + +<p>§ 25. Two rows, p and p′, which are one the projection of the +other (as in fig. 5), stand in a definite relation to each other, characterized +by the following properties.</p> + +<p>1. <i>To each point in either corresponds one point in the other</i>; that +is, those points are said to correspond which are projections of one +another.</p> + +<p>2. <i>The cross-ratio of any four points in one equals that of the corresponding +points in the other.</i></p> + +<p>3. <i>The lines joining corresponding points all pass through the same +point.</i></p> + +<p>If we suppose corresponding points marked, and the rows brought +into any other position, then the lines joining corresponding points +will no longer meet in a common point, and hence the third of the +above properties will not hold any longer; but we have still a +correspondence between the points in the two rows possessing the first +two properties. Such a correspondence has been called a <i>one-one +correspondence</i>, whilst the two rows between which such correspondence +has been established are said to be <i>projective</i> or <i>homographic</i>. +Two rows which are each the projection of the other are therefore +<i>projective</i>. We shall presently see, also, that any two projective +rows may always be placed in such a position that one appears as +the projection of the other. If they are in such a position the rows +are said to be in <i>perspective position</i>, or simply to be in <i>perspective</i>.</p> + +<p>§ 26. The notion of a one-one correspondence between rows may +be extended to flat and axial pencils, viz. a flat pencil will be said +to be projective to a flat pencil if to each ray in the first corresponds +one ray in the second, and if the cross-ratio of four rays in one equals +that of the corresponding rays in the second.</p> + +<p>Similarly an axial pencil may be projective to an axial pencil. +But a flat pencil may also be projective to an axial pencil, or either +pencil may be projective to a row. The definition is the same in each +case: there is a one-one correspondence between the elements, and +four elements have the same cross-ratio as the corresponding ones.</p> + +<p>§ 27. There is also in each case a special position which is called +<i>perspective</i>, viz.</p> + +<p>1. Two projective rows are perspective if they lie in the same +plane, and if the one row is a projection of the other.</p> + +<p>2. Two projective flat pencils are perspective—(1) if they lie in +the same plane, and have a row as a common section; (2) if they +lie in the same pencil (in space), and are both sections of the same +axial pencil; (3) if they are in space and have a row as common +section, or are both sections of the same axial pencil, one of the +conditions involving the other.</p> + +<p>3. Two projective axial pencils, if their axes meet, and if they +have a flat pencil as a common section.</p> + +<p>4. A row and a projective flat pencil, if the row is a section of the +pencil, each point lying in its corresponding line.</p> + +<p>5. A row and a projective axial pencil, if the row is a section of the +pencil, each point lying in its corresponding line.</p> + +<p>6. A flat and a projective axial pencil, if the former is a section +of the other, each ray lying in its corresponding plane.</p> + +<p>That in each case the correspondence established by the position +indicated is such as has been called projective follows at once from +the definition. It is not so evident that the perspective position may +always be obtained. We shall show in § 30 this for the first three +<span class="pagenum"><a name="page693" id="page693"></a>693</span> +cases. First, however, we shall give a few theorems which relate to +the general correspondence, not to the perspective position.</p> + +<p>§ 28. <i>Two rows or pencils, flat or axial, which are projective to a +third are projective to each other</i>; this follows at once from the +definitions.</p> + +<p>§ 29. <i>If two rows, or two pencils, either flat or axial, or a row and a +pencil, be projective, we may assume to any three elements in the one +the three corresponding elements in the other, and then the correspondence +is uniquely determined.</i></p> + +<p>For if in two projective rows we assume that the points A, B, C +in the first correspond to the given points A′, B′, C′ in the second, +then to any fourth point D in the first will correspond a point D′ +in the second, so that</p> + +<p class="center">(AB, CD) = (A′B′, C′D′).</p> + +<p class="noind">But there is only one point, D′, which makes the cross-ratio +(A′B′, C′D′) equal to the given number (AB, CD).</p> + +<p>The same reasoning holds in the other cases.</p> + +<p>§ 30. If two rows are perspective, then the lines joining corresponding +points all meet in a point, the centre of projection; and +the point in which the two bases of the rows intersect as a point +in the first row coincides with its corresponding point in the +second.</p> + +<p>This follows from the definition. The converse also holds, +viz.</p> + +<p><i>If two projective rows have such a position that one point in the one +coincides with its corresponding point in the other, then they are perspective, +that is, the lines joining corresponding points all pass through +a common point, and form a flat pencil.</i></p> + +<p>For let A, B, C, D ... be points in the one, and A′, 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′.</p> + +<p>§ 31. If two projected flat pencils in the same plane are in perspective, +then the intersections of corresponding lines form a row, +and the line joining the two centres as a line in the first pencil +corresponds to the same line as a line in the second. And conversely,</p> + +<p><i>If two projective pencils in the same plane, but with different centres, +have one line in the one coincident with its corresponding line in the +other, then the two pencils are perspective, that is, the intersection of +corresponding lines lie in a line.</i></p> + +<p>The proof is the same as in § 30.</p> + +<p>§ 32. If two projective flat pencils in the same point (pencil in +space), but not in the same plane, are perspective, then the planes +joining corresponding rays all pass through a line (they form an +axial pencil), and the line common to the two pencils (in which +their planes intersect) corresponds to itself. And conversely:—</p> + +<p>If two flat pencils which have a common centre, but do not lie +in a common plane, are placed so that one ray in the one coincides +with its corresponding ray in the other, then they are perspective, +that is, the planes joining corresponding lines all pass through a +line.</p> + +<p>§ 33. If two projective axial pencils are perspective, then the intersection +of corresponding planes lie in a plane, and the plane common +to the two pencils (in which the two axes lie) corresponds to itself. +And conversely:—</p> + +<p>If two projective axial pencils are placed in such a position that a +plane in the one coincides with its corresponding plane, then the two +pencils are perspective, that is, corresponding planes meet in lines +which lie in a plane.</p> + +<p>The proof again is the same as in § 30.</p> + +<p>§ 34. These theorems relating to perspective position become +illusory if the projective rows of pencils have a common base. We +then have:—</p> + +<p>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.</p> + +<p><i>Proof</i> (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.</p> + +<p>The last theorem may also be stated thus:—</p> + +<p>In two projective rows or pencils, which have a common base +but are not identical, not more than two elements in the one can +coincide with their corresponding elements in the other.</p> + +<p>Thus two projective rows on the same line cannot have more +than two pairs of coincident points unless every point coincides +with its corresponding point.</p> + +<table class="flt" style="float: right; width: 370px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:309px; height:275px" src="images/img693a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 9.</span></td></tr> +<tr><td class="figright1"><img style="width:301px; height:266px" src="images/img693b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 10.</span></td></tr> +<tr><td class="figright1"><img style="width:316px; height:361px" src="images/img693c.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 11.</span></td></tr></table> + +<p>It is easy to construct two projective rows on the same line, +which have two pairs of corresponding points coincident. Let the +points A, B, C as points belonging to the one row correspond to A, +B, and C′ as points in the second. Then A and B coincide with their +corresponding points, but C does not. It is, however, not necessary +that two such rows +have twice a point +coincident with its corresponding +point; it is +possible that this happens +only once or not +at all. Of this we shall +see examples later.</p> + +<p>§ 35. If two projective +rows or pencils are in +perspective position, we +know at once which +element in one corresponds +to any given +element in the other. +If p and q (fig. 9) are +two projective rows, so +that K corresponds to +itself, and if we know +that to A and B in p +correspond A′ 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.</p> + +<p>If two flat pencils, S<span class="su">1</span> and S<span class="su">2</span>, in a plane are perspective (fig. 10), +we need only to know two pairs, a, a′ and b, b′, of corresponding +rays in order to find the +axis s of projection. This +being known, a ray c′ in +S<span class="su">2</span>, corresponding to a given +ray c in S<span class="su">1</span>, is found by +joining S<span class="su">2</span> to the point +where c cuts the axis s.</p> + +<p>A similar construction +holds in the other cases +of perspective figures.</p> + +<p>On this depends the +solution of the following +general problem.</p> + +<p>§ 36. Three pairs of corresponding +elements in two +projective rows or pencils +being given, to determine +for any element in one +the corresponding element +in the other.</p> + +<p>We solve this in the two cases of two projective rows and of two +projective flat pencils in a plane.</p> + +<p><i>Problem</i> I.—Let A, B, C be +three points in a row s, A′, B′, C′ +the corresponding points in a +projective row s′, both being in a +plane; it is required to find for +any point D in s the corresponding +point D′ in s′.</p> + +<p><i>Problem</i> II.—Let a, b, c be +three rays in a pencil S, a′, b′, c′ +the corresponding rays in a projective +pencil S′, both being in +the same plane; it is required to +find for any ray d in S the corresponding +ray d′ in S′.</p> + +<p>The solution is made to depend on the construction of an auxiliary +row or pencil which is perspective to both the given ones. This is +found as follows:—</p> + +<p><i>Solution of Problem</i> 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 B<span class="su">1</span> +of SB and S′B′ to the +intersection C<span class="su">1</span> of SC +and S′C′ by the line s<span class="su">1</span>. +Then a row on s<span class="su">1</span> will +be perspective to s with +S as centre of projection, +and to s′ 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 D<span class="su">1</span>, +where the line SD cuts +s<span class="su">1</span>, and to draw S′D<span class="su">1</span>; +the point where this line +cuts s′ will be the required +point D′.</p> + +<p><i>Proof.</i>—The rows s +and s′ are both perspective +to the row s<span class="su">1</span>, hence +they are projective to +one another. To A, B, +C, D on s correspond +A<span class="su">1</span>, B<span class="su">1</span>, C<span class="su">1</span>, D<span class="su">1</span> on s<span class="su">1</span>, and +to these correspond A′, B′, C′, D′ on s′; so that D and D′ are +corresponding points as required.</p> + +<p><span class="pagenum"><a name="page694" id="page694"></a>694</span></p> + +<table class="flt" style="float: left; width: 315px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:251px; height:310px" src="images/img694a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 12.</span></td></tr> +<tr><td class="figleft1"><img style="width:265px; height:233px" src="images/img694b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 13.</span></td></tr></table> + +<p><i>Solution of Problem</i> II.—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 S<span class="su">1</span> +be the point where the line b<span class="su">1</span>, +which joins B and B′, cuts the +line c<span class="su">1</span>, which joins C and C′. +Then a pencil S<span class="su">1</span> will be perspective +to S with s as axis of +projection. To find the ray d′ in +S′ corresponding to a given ray d +in S, cut d by s at D; project +this point from S<span class="su">1</span> to D′ on s′ +and join D′ to S′. This will be +the required ray.</p> + +<p><i>Proof.</i>—That the pencil S<span class="su">1</span> is +perspective to S and also to S′ +follows from construction. To +the lines a<span class="su">1</span>, b<span class="su">1</span>, c<span class="su">1</span>, d<span class="su">1</span> in S<span class="su">1</span> correspond +the lines a, b, c, d in S and +the lines a′, b′, c′, d′ in S′, so that d +and d′ are corresponding rays.</p> + +<p>In the first solution the two +centres, S, S′, are <i>any</i> two points +on a line joining any two corresponding +points, so that the solution +of the problem allows of a great many different constructions. +<i>But whatever construction be used, the point</i> D′, <i>corresponding to</i> D, +<i>must be always the same</i>, according to the theorem in § 29. This +gives rise to a number of theorems, into which, however, we shall +not enter. The same remarks hold for the second problem.</p> + +<p>§ 37. <i>Homological Triangles.</i>—As a further application of the +theorems about perspective rows and pencils we shall prove the +following important theorem.</p> + +<p><i>Theorem.</i>—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.</p> + +<p><i>Proof.</i>—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.</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc">If</td> <td class="tcl">S<span class="su">1</span> be the centre of projection of rows</td> <td class="tcl">b and c,</td></tr> + +<tr><td class="tcc"> </td> <td class="tcl">S<span class="su">2</span>    ”     ”     ”</td> <td class="tcl">c and a,</td></tr> + +<tr><td class="tcc"> </td> <td class="tcl">S<span class="su">3</span>    ”     ”     ”</td> <td class="tcl">a and b,</td></tr> +</table> + +<p class="noind">and if the line S<span class="su">1</span>S<span class="su">2</span> cuts a in A<span class="su">1</span>, and b in B<span class="su">1</span>, and c in C<span class="su">1</span>, then A<span class="su">1</span>, B<span class="su">1</span> +will be corresponding points +in a and b, both corresponding +to C<span class="su">1</span> in c. But a and b are +perspective, therefore the line +A<span class="su">1</span>B<span class="su">1</span>, that is S<span class="su">1</span>S<span class="su">2</span>, joining +corresponding points must +pass through the centre of +projection S<span class="su">3</span> of a and b. In +other words, S<span class="su">1</span>, S<span class="su">2</span>, S<span class="su">3</span> lie in a +line. This is Desargues’ celebrated +theorem if we state it +thus:—</p> + +<p><i>Theorem of Desargues.</i>—If +each of two triangles has one +vertex on each of three concurrent +lines, then the intersections +of corresponding sides +lie in a line, those sides +being called corresponding which are opposite to vertices on the +same line.</p> + +<p>The converse theorem holds also, viz.</p> + +<p><i>Theorem.</i>—If the sides of one triangle meet those of another in +three points which lie in a line, then the vertices lie on three lines +which meet in a point.</p> + +<p>The proof is almost the same as before.</p> + +<p>§ 38. <i>Metrical Relations between Projective Rows.</i>—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</p> + +<p class="center">(AB, JI) = (A′B′, J′I′),</p> + +<p class="noind">or</p> + +<p class="center">AJ/JB : AI/IB = A′J′/J′B′ : A′I′/I′B′.</p> + +<p>But, by § 17,</p> + +<p class="center">AI/IB = A′J′/J′B′ = −1;</p> + +<p class="noind">therefore the last equation changes into</p> + +<p class="center">AJ · A′I′ = BJ · B′I′,</p> + +<p class="noind">that is to say—</p> + +<p><i>Theorem.</i>—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 <i>Power of the correspondence</i>.</p> + +<p>[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 αxx′ + βx + γx′ ++ δ = 0. Conversely, if a relation of this nature holds, then points +corresponding to solutions in x, x′ form homographic ranges.]</p> + +<p>§ 39. <i>Similar Rows.</i>—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</p> + +<p class="center">(AB, CI) = (A′B′, C′I′),</p> + +<p class="noind">which reduces to</p> + +<p class="center">AC/CB = A′C′/C′B′ or AC/A′C′ = BC/B′C′,</p> + +<p class="noind">that is, corresponding segments are proportional. Conversely, if +corresponding segments are proportional, then to the point at +infinity in one corresponds the point at infinity in the other. If we call +such rows <i>similar</i>, we may state the result thus—</p> + +<p><i>Theorem.</i>—Two projective rows are similar if to the point at +infinity in one corresponds the point at infinity in the other, and +conversely, if two rows are similar then they are projective, and the +points at infinity are corresponding points.</p> + +<p>From this the well-known propositions follow:—</p> + +<p>Two lines are cut proportionally (in similar rows) by a series of +parallels. The rows are perspective, with centre of projection at +infinity.</p> + +<p>If two similar rows are placed parallel, then the lines joining +homologous points pass through a common point.</p> + +<p>§ 40. If two flat pencils be projective, then there exists in either, +one single pair of lines at right angles to one another, such that the +corresponding lines in the other pencil are again at right angles.</p> + +<table class="flt" style="float: right; width: 300px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:252px; height:248px" src="images/img694c.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 14.</span></td></tr></table> + +<p>To prove this, we place the pencils in perspective position (fig. 14) +by making one ray coincident +with its corresponding +ray. Corresponding rays +meet then on a line p. And +now we draw the circle which +has its centre O on p, and +which passes through the +centres S and S′ 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 <i>to every +right angle in the one pencil corresponds a right angle in the +other</i>.</p> + +<p class="pt2 center sc" style="clear: both;">Principle of Duality</p> + +<p>§ 41. It has been stated in § 1 that not only points, but also planes +and lines, are taken as elements out of which figures are built up. +We shall now see that the construction of one figure which possesses +certain properties gives rise in many cases to the construction of +another figure, by replacing, according to definite rules, elements +of one kind by those of another. The new figure thus obtained will +then possess properties which may be stated as soon as those of the +original figure are known.</p> + +<p>We obtain thus a principle, known as the <i>principle of duality</i> +or of <i>reciprocity</i>, which enables us to construct to any figure not +containing any measurement in its construction a <i>reciprocal</i> figure, +as it is called, and to deduce from any theorem a <i>reciprocal</i> theorem, +for which no further proof is needed.</p> + +<p>It is convenient to print reciprocal propositions on opposite sides +of a page broken into two columns, and this plan will occasionally +be adopted.</p> + +<p>We begin by repeating in this form a few of our former statements:—</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p>Two points determine a line.</p></td> +<td class="tcl" style="width: 50%;"><p>Two planes determine a line.</p></td></tr> + +<tr><td class="tcl rb3"><p>Three points which are not in a line determine a plane.</p></td> +<td class="tcl"><p>Three planes which do not pass through a line determine a point.</p></td></tr> + +<tr><td class="tcl rb3"><p>A line and a point without it determine a plane.</p></td> +<td class="tcl"><p>A line and a plane not through it determine a point.</p></td></tr> + +<tr><td class="tcl rb3"><p>Two lines in a plane determine a point.</p></td> +<td class="tcl"><p>Two lines through a point determine a plane.</p></td></tr></table> + +<p>These propositions show that it will be possible, when any figure +is given, to construct a second figure by taking planes instead of +points, and points instead of planes, but lines where we had lines.</p> + +<p><span class="pagenum"><a name="page695" id="page695"></a>695</span></p> + +<p>For instance, if in the first figure we take a plane and three points +in it, we have to take in the second figure a point and three planes +through it. The three points in the first, together with the three +lines joining them two and two, form a triangle; the three planes +in the second and their three lines of intersection form a trihedral +angle. A triangle and a trihedral angle are therefore reciprocal +figures.</p> + +<p>Similarly, to any figure in a plane consisting of points and lines +will correspond a figure consisting of planes and lines passing through +a point S, and hence belonging to the pencil which has S as centre.</p> + +<p>The figure reciprocal to four points in space which do not lie +in a plane will consist of four planes which do not meet in a point. +In this case each figure forms a tetrahedron.</p> + +<p>§ 42. As other examples we have the following:—</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">To a row</td> <td class="tcc">is reciprocal</td> <td class="tcl">an axial pencil,</td></tr> + +<tr><td class="tcl">to a flat pencil</td> <td class="tcc">”</td> <td class="tcl">a flat pencil,</td></tr> + +<tr><td class="tcl">to a field of points and lines</td> <td class="tcc">”</td> <td class="tcl">a pencil of planes and lines,</td></tr> + +<tr><td class="tcl">to the space of points</td> <td class="tcc">”</td> <td class="tcl">the space of planes.</td></tr> +</table> + +<p class="noind">For the row consists of a line and all the points in it, reciprocal to +it therefore will be a line with all planes through it, that is, an axial +pencil; and so for the other cases.</p> + +<p>This correspondence of reciprocity breaks down, however, if we +take figures which contain measurement in their construction. For +instance, there is no figure reciprocal to two planes at <i>right angles</i>, +because there is no segment in a row which has a magnitude as +definite as a right angle.</p> + +<p>We add a few examples of reciprocal propositions which are easily +proved.</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p><i>Theorem.</i>—If A, B, C, D are any four points in space, and if + the lines AB and CD meet, then all four points lie in a plane, + hence also AC and BD, as well as AD and BC, meet.</p></td> + +<td class="tcl" style="width: 50%;"><p><i>Theorem.</i>—If α, β, γ, δ are four planes in space, and if the + lines αβ and γδ meet, then all four planes lie in a point (pencil), + hence also αγ and βδ, as well as αδ and βγ, meet.</p></td></tr></table> + +<p>Theorem.—<i>If of any number of lines every one meets every other, +whilst all do not</i></p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p><i>lie in a point, then all lie in a plane</i>.</p></td> + +<td class="tcl" style="width: 50%;"><p><i>lie in a plane, then all lie in a point</i> (<i>pencil</i>).</p></td></tr></table> + +<p>§ 43. Reciprocal figures as explained lie both in space of three +dimensions. If the one is confined to a plane (is formed of elements +which lie in a plane), then the reciprocal figure is confined to a pencil +(is formed of elements which pass through a point).</p> + +<p>But there is also a more special principle of duality, according to +which figures are reciprocal which lie both in a plane or both in a +pencil. In the plane we take points and lines as reciprocal elements, +for they have this fundamental property in common, that two +elements of one kind determine one of the other. In the pencil, +on the other hand, lines and planes have to be taken as reciprocal, +and here it holds again that two lines or planes determine one plane +or line.</p> + +<p>Thus, to one plane figure we can construct one reciprocal figure +in the plane, and to each one reciprocal figure in a pencil. We +mention a few of these. At first we explain a few names:—</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p>A figure consisting of n points in a plane will be called an n-point.</p></td> +<td class="tcl" style="width: 50%;"><p>A figure consisting of n lines in a plane will be called an n-side.</p></td></tr> + +<tr><td class="tcl rb3"><p>A figure consisting of n planes in a pencil will be called an n-flat.</p></td> +<td class="tcl"><p>A figure consisting of n lines in a pencil will be called an n-edge.</p></td></tr></table> + +<p>It will be understood that an n-side is different from a polygon +of n sides. The latter has sides of finite length and n vertices, the +former has sides all of infinite extension, and every point where +two of the sides meet will be a vertex. A similar difference exists +between a solid angle and an n-edge or an n-flat. We notice particularly—</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p>A four-point has six sides, of which two and two are opposite, + and three diagonal points, which are intersections of opposite sides.</p></td> + +<td class="tcl" style="width: 50%;"><p>A four-side has six vertices, of which two and two are opposite, + and three diagonals, which join opposite vertices.</p></td></tr> + +<tr><td class="tcl rb3"><p>A four-flat has six edges, of which two and two are opposite, + and three diagonal planes, which pass through opposite edges.</p></td> + +<td class="tcl"><p>A four-edge has six faces, of which two and two are opposite, + and three diagonal edges, which are intersections of opposite faces.</p></td></tr></table> + +<p>A four-side is usually called a complete quadrilateral, and a four-point +a complete quadrangle. The above notation, however, seems +better adapted for the statement of reciprocal propositions.</p> + +<p>§ 44.</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p>If a point moves in a plane it describes a plane curve.</p></td> + +<td class="tcl" style="width: 50%;"><p>If a line moves in a plane it envelopes a plane curve (fig. 15).</p></td></tr> + +<tr><td class="tcl rb3"><p>If a plane moves in a pencil it envelopes a cone.</p></td> + +<td class="tcl"><p>If a line moves in a pencil it describes a cone.</p></td></tr></table> + +<p>A curve thus appears as generated either by points, and then we +call it a “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.</p> + +<table class="flt" style="float: right; width: 240px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:192px; height:126px" src="images/img695.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 15.</span></td></tr></table> + +<p>To a surface as locus of points corresponds, in the same manner, +a surface as envelope of planes; and to +a curve in space as locus of points corresponds +a developable surface as envelope +of planes.</p> + +<p>It will be seen from the above that +we may, by aid of the principle of +duality, construct for every figure a +reciprocal figure, and that to any +property of the one a reciprocal property +of the other will exist, as long +as we consider only properties which +depend upon nothing but the positions and intersections of the +different elements and not upon measurement.</p> + +<p>For such propositions it will therefore be unnecessary to prove +more than one of two reciprocal theorems.</p> + +<p class="pt2 center sc" style="clear: both;">Generation of Curves and Cones of Second Order +or Second Class</p> + +<p>§ 45. <i>Conics.</i>—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:—</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p><i>Theorem.</i>—If two projective flat pencils lie in a plane, but + are neither in perspective nor concentric, then the locus of + intersections of corresponding rays is a curve of the second + order, that is, no line contains more than two points of the locus.</p></td> + +<td class="tcl" style="width: 50%;"><p><i>Theorem.</i>—If two projective rows lie in a plane, but are + neither in perspective nor on a common base, then the envelope + of lines joining corresponding points is a curve of the second + class, that is, through no point pass more than two of the enveloping lines.</p></td></tr> + +<tr><td class="tcl rb3"><p>Proof.—We draw any line t. This cuts each of the pencils in a + row, so that we have on t two rows, and these are projective + because the pencils are projective. If corresponding rays + of the two pencils meet on the line t, their intersection will be a + point in the one row which coincides with its corresponding + point in the other. But two projective rows on the same base + cannot have more than two points of one coincident with + their corresponding points in the other (§ 34).</p></td> + +<td class="tcl"><p><i>Proof.</i>—We take any point T and join it to all points in each + row. This gives two concentric pencils, which are projective + because the rows are projective. If a line joining corresponding + points in the two rows passes through T, it will be a line in the + one pencil which coincides with its corresponding line in the + other. But two projective concentric flat pencils in the same + plane cannot have more than two lines of one coincident with their + corresponding line in the other (§ 34).</p></td></tr></table> + +<p>It will be seen that the proofs are reciprocal, so that the one may +be copied from the other by simply interchanging the words point +and line, locus and envelope, row and pencil, and so on. We shall +therefore in future prove seldom more than one of two reciprocal +theorems, and often state one theorem only, the reader being recommended +to go through the reciprocal proof by himself, and to supply +the reciprocal theorems when not given.</p> + +<p>§ 46. We state the theorems in the pencil reciprocal to the last, +without proving them:—</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p><i>Theorem.</i>—If two projective flat pencils are concentric, but + are neither perspective nor coplanar, then the envelope of the + planes joining corresponding rays is a cone of the second class; + that is, no line through the common centre contains more + than two of the enveloping planes.</p></td> + +<td class="tcl" style="width: 50%;"><p><i>Theorem.</i>—If two projective axial pencils lie in the same + pencil (their axes meet in a point), but are neither perspective + nor co-axial, then the locus of lines joining corresponding + planes is a cone of the second order; that is, no plane in the + pencil contains more than two of these lines.</p></td></tr></table> + +<p>§ 47. Of theorems about cones of second order and cones of second +class we shall state only very few. We point out, however, the +following connexion between the curves and cones under consideration:</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p>The lines which join any point in space to the points on a curve + of the second order form a cone of the second order.</p></td> + +<td class="tcl" style="width: 50%;"><p>Every plane section of a cone of the second order is a curve of + the second order.</p></td></tr> + +<tr><td class="tcl rb3"><p>The planes which join any point in space to the lines enveloping + a curve of the second class envelope themselves a cone of the second class.</p></td> + +<td class="tcl"><p>Every plane section of a cone of the second class is a curve of + the second class.</p></td></tr></table> + +<p>By its aid, or by the principle of duality, it will be easy to obtain +theorems about them from the theorems about the curves.</p> + +<p>We prove the first. A curve of the second order is generated by +two projective pencils. These pencils, when joined to the point in +space, give rise to two projective axial pencils, which generate the +cone in question as the locus of the lines where corresponding planes +meet.</p> + +<p><span class="pagenum"><a name="page696" id="page696"></a>696</span></p> + +<p>§48.</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p><i>Theorem.</i>—The curve of second order which is generated by two + projective flat pencils passes through the centres of the two pencils.</p></td> + +<td class="tcl" style="width: 50%;"><p><i>Theorem.</i>—The envelope of second class which is generated + by two projective rows contains the bases of these rows as enveloping + lines or tangents.</p></td></tr> + +<tr><td class="tcl rb3"><p><i>Proof.</i>—If S and S′ are the two pencils, then to the ray SS′ or p′ + in the pencil S′ corresponds in the pencil S a ray p, which is + different from p′, for the pencils are not perspective. But p and + p′ meet at S, so that S is a point on the curve, and similarly S′.</p></td> + +<td class="tcl"><p><i>Proof.</i>—If s and s′ are the two rows, then to the point ss′ or P′ + as a point in s′ corresponds in s a point P, which is not coincident + with P′, for the rows are not perspective. But P and P′ are + joined by s, so that s is one of the enveloping lines, and similarly s′.</p></td></tr></table> + +<p>It follows that every line in one of the two pencils cuts the curve +in two points, viz. once at the centre S of the pencil, and once +where it cuts its corresponding ray in the other pencil. These two +points, however, coincide, if the line is cut by its corresponding +line at S itself. The line p in S, which corresponds to the line +SS′ 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 <i>tangent</i> to the curve, +touching the latter at the point S, which is called the “point of +contact.”</p> + +<p>In the same manner we get in the reciprocal investigation the +result that through every point in one of the rows, say in s, two +tangents may be drawn to the curve, the one being s, the other the +line joining the point to its corresponding point in s′. 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—</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p><i>Theorem.</i>—To the line joining the centres of the projective + pencils as a line in one pencil corresponds in the other the + tangent at its centre.</p></td> + +<td class="tcl" style="width: 50%;"><p><i>Theorem.</i>—To the point of intersection of the bases of two + projective rows as a point in one row corresponds in the other the + <i>point of contact</i> of its base.</p></td></tr></table> + +<p>§ 49. Two projective pencils are determined if three pairs of +corresponding lines are given. Hence if a<span class="su">1</span>, b<span class="su">1</span>, c<span class="su">1</span> are three lines in a +pencil S<span class="su">1</span>, and a<span class="su">2</span>, b<span class="su">2</span>, c<span class="su">2</span> the corresponding lines in a projective pencil +S<span class="su">2</span>, the correspondence and therefore the curve of the second order +generated by the points of intersection of corresponding rays is +determined. Of this curve we know the two centres S<span class="su">1</span> and S<span class="su">2</span>, +and the three points a<span class="su">1</span>a<span class="su">2</span>, b<span class="su">1</span>b<span class="su">2</span>, c<span class="su">1</span>c<span class="su">2</span>, hence five points in all. This +and the reciprocal considerations enable us to solve the following +two problems:</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p><i>Problem.</i>—To construct a curve of the second order, of which five + points S<span class="su">1</span>, S<span class="su">2</span>, A, B, C are given.</p></td> + +<td class="tcl" style="width: 50%;"><p><i>Problem.</i>—To construct a curve of the second class, of which five + tangents u<span class="su">1</span>, u<span class="su">2</span>, a, b, c are given.</p></td></tr></table> + +<p>In order to solve the left-hand problem, we take two of the given +points, say S<span class="su">1</span> and S<span class="su">2</span>, as centres of pencils. These we make projective +by taking the rays a<span class="su">1</span>, b<span class="su">1</span>, c<span class="su">1</span>, which join S<span class="su">1</span> to A, B, C respectively, +as corresponding to the rays a<span class="su">2</span>, b<span class="su">2</span>, c<span class="su">2</span>, which join S<span class="su">2</span> to A, B, C +respectively, so that three rays meet their corresponding rays at +the given points A, B, C. This determines the correspondence of +the pencils which will generate a curve of the second order passing +through A, B, C and through the centres S<span class="su">1</span> and S<span class="su">2</span>, hence through +the five given points. To find more points on the curve we have to +construct for any ray in S<span class="su">1</span> the corresponding ray in S<span class="su">2</span>. This has +been done in § 36. But we repeat the construction in order to deduce +further properties from it. We also solve the right-hand problem. +Here we select two, viz. u<span class="su">1</span>, u<span class="su">2</span> of the five given lines, u<span class="su">1</span>, u<span class="su">2</span>, a, b, c, +as bases of two rows, and the points A<span class="su">1</span>, B<span class="su">1</span>, C<span class="su">1</span> where a, b, c cut u<span class="su">1</span> +as corresponding to the points A<span class="su">2</span>, B<span class="su">2</span>, C<span class="su">2</span> where a, b, c cut u<span class="su">2</span>.</p> + +<p>We get then the following solutions of the two problems:</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p><i>Solution.</i>—Through the point + A draw any two lines, u<span class="su">1</span> and u<span class="su">2</span> + (fig. 16), the first u<span class="su">1</span> to cut the + pencil S<span class="su">1</span> in a row AB<span class="su">1</span>C<span class="su">1</span>, the + other u<span class="su">2</span> to cut the pencil S<span class="su">2</span> in a + row AB<span class="su">2</span>C<span class="su">2</span>. These two rows will + be perspective, as the point A + corresponds to itself, and the + centre of projection will be the + point S, where the lines B<span class="su">1</span>B<span class="su">2</span> + and C<span class="su">1</span>C<span class="su">2</span> meet. To find now for + any ray d<span class="su">1</span> in S<span class="su">1</span> its corresponding + ray d<span class="su">2</span> in S<span class="su">2</span>, we determine the + point D<span class="su">1</span> where d<span class="su">1</span> cuts u<span class="su">1</span>, project + this point from S to D<span class="su">2</span> on u<span class="su">2</span> and + join S<span class="su">2</span> to D<span class="su">2</span>. This will be the + required ray d<span class="su">2</span> which cuts d<span class="su">1</span> at + some point D on the curve.</p></td> + +<td class="tcl" style="width: 50%;"><p><i>Solution.</i>—In the line a take + any two points S<span class="su">1</span> and S<span class="su">2</span> as + centres of pencils (fig. 17), the + first S<span class="su">1</span> (A<span class="su">1</span>B<span class="su">1</span>C<span class="su">1</span>) to project the + row u<span class="su">1</span>, the other S<span class="su">2</span> (A<span class="su">2</span>B<span class="su">2</span>C<span class="su">2</span>) to + project the row u<span class="su">2</span>. These two + pencils will be perspective, the + line S<span class="su">1</span>A<span class="su">1</span> being the same as the + corresponding line S<span class="su">2</span>A<span class="su">2</span>, and the + axis of projection will be the line + u, which joins the intersection B + of S<span class="su">1</span>B<span class="su">1</span> and S<span class="su">2</span>B<span class="su">2</span> to the intersection + C of S<span class="su">1</span>C<span class="su">1</span> and S<span class="su">2</span>C<span class="su">2</span>. To find + now for any point D<span class="su">1</span> in u<span class="su">1</span> the + corresponding point D<span class="su">2</span> in u<span class="su">2</span>, we + draw S<span class="su">1</span>D<span class="su">1</span> and project the point + D where this line cuts u from S<span class="su">2</span> + to u<span class="su">2</span>. This will give the required + point D<span class="su">2</span>, and the line d joining D<span class="su">1</span> + to D<span class="su">2</span> will be a new tangent to the + curve.</p></td></tr></table> + +<p>§ 50. These constructions prove, when rightly interpreted, very +important properties of the curves in question.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:348px; height:319px" src="images/img696a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 16.</span></td></tr></table> + +<p>If in fig. 16 we draw in the pencil S<span class="su">1</span> the ray k<span class="su">1</span> which passes +through the auxiliary centre S, it will be found that the corresponding +ray k<span class="su">2</span> cuts it on u<span class="su">2</span>. Hence—</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p><i>Theorem.</i>—In the above construction the bases of the auxiliary + rows u<span class="su">1</span> and u<span class="su">2</span> cut the curve + where they cut the rays S<span class="su">2</span>S and + S<span class="su">1</span>S respectively.</p></td> + +<td class="tcl" style="width: 50%;"><p><i>Theorem.</i>—In the above construction (fig. 17) the tangents to + the curve from the centres of the auxiliary pencils S<span class="su">1</span> and S<span class="su">2</span> are the + lines which pass through u<span class="su">2</span>u and + u<span class="su">1</span>u respectively.</p></td></tr></table> + +<p>As A is any given point on the curve, and u<span class="su">1</span> any line through +it, we have solved the problems:</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p><i>Problem.</i>—To find the second point in which any line through a +known point on the curve cuts the curve.</p></td> + +<td class="tcl" style="width: 50%;"><p><i>Problem.</i>—To find the second tangent which can be drawn +from any point in a given tangent to the curve.</p></td></tr></table> + +<p>If we determine in S<span class="su">1</span> (fig. 16) the ray corresponding to the ray +S<span class="su">2</span>S<span class="su">1</span> in S<span class="su">2</span>, we get the tangent at S<span class="su">1</span>. Similarly, we can determine +the point of contact of the tangents u<span class="su">1</span> or u<span class="su">2</span> in fig. 17.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:386px; height:266px" src="images/img696b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 17.</span></td></tr></table> + +<table class="flt" style="float: right; width: 300px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:248px; height:183px" src="images/img696c.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 18.</span></td></tr></table> + +<p>§ 51. If five points are given, of which not three are in a line, +then we can, as has just been shown, always draw a curve of the +second order through them; we select two of the points as centres of +projective pencils, and then one such curve is determined. It will +be presently shown that we get always the same curve if two other +points are taken as centres of pencils, that therefore five points +<i>determine</i> one curve of the second order, and reciprocally, that five +tangents determine one curve of the second class. Six points taken +at random will therefore not lie on a curve of the second order. In +order that this may be the case a certain condition has to be satisfied, +and this condition is easily obtained +from the construction in +§ 49, fig. 16. If we consider the +conic determined by the five +points A, S<span class="su">1</span>, S<span class="su">2</span>, K, L, then the +point D will be on the curve if, +and only if, the points on D<span class="su">1</span>, S, +D<span class="su">2</span> be in a line.</p> + +<p>This may be stated differently +if we take AKS<span class="su">1</span>DS<span class="su">2</span>L (figs. 16 +and 18) as a hexagon inscribed +in the conic, then AK and DS<span class="su">2</span> +will be opposite sides, so will be +KS<span class="su">1</span> and S<span class="su">2</span>L, as well as S<span class="su">1</span>D and +LA. The first two meet in D<span class="su">2</span>, +the others in S and D<span class="su">1</span> respectively. We may therefore state the +required condition, together with the reciprocal one, as follows:—</p> + +<p><span class="pagenum"><a name="page697" id="page697"></a>697</span></p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p><i>Pascal’s Theorem.</i>—If a hexagon be inscribed in a curve of the + second order, then the intersections of opposite sides are three points in a line.</p></td> + +<td class="tcl" style="width: 50%;"><p><i>Brianchon’s Theorem.</i>—If a hexagon be circumscribed about + a curve of the second class, then the lines joining opposite vertices + are three lines meeting in a point.</p></td></tr></table> + +<p>These celebrated theorems, which are known by the names of +their discoverers, are perhaps the most fruitful in the whole theory +of conics. Before we go over to their applications we have to show +that we obtain the same curve if we take, instead of S<span class="su">1</span>, S<span class="su">2</span>, any two +other points on the curve as centres of projective pencils.</p> + +<p>§ 52. We know that the curve depends only upon the correspondence +between the pencils S<span class="su">1</span> and S<span class="su">2</span>, and not upon the special construction +used for finding new points on the curve. The point A +(fig. 16 or 18), through which the two auxiliary rows u<span class="su">1</span>, u<span class="su">2</span> were +drawn, may therefore be changed to any other point on the curve. +Let us now suppose the curve drawn, and keep the points S<span class="su">1</span>, S<span class="su">2</span>, +K, L and D, and hence also the point S fixed, whilst we move A +along the curve. Then the line AL will describe a pencil about +L as centre, and the point D<span class="su">1</span> a row on S<span class="su">1</span>D perspective to the +pencil L. At the same time AK describes a pencil about K and D<span class="su">2</span> +a row perspective to it on S<span class="su">2</span>D. But by Pascal’s theorem D<span class="su">1</span> and +D<span class="su">2</span> will always lie in a line with S, so that the rows described by D<span class="su">1</span> +and D<span class="su">2</span> are perspective. It follows that the pencils K and L will +themselves be projective, corresponding rays meeting on the curve. +This proves that we get the same curve whatever pair of the five +given points we take as centres of projective pencils. Hence—</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p>Only one curve of the second order can be drawn which passes through five given points.</p></td> + +<td class="tcl" style="width: 50%;"><p>Only one curve of the second class can be drawn which touches five given lines.</p></td></tr></table> + +<p>We have seen that if on a curve of the second order two points +coincide at A, the line joining them becomes the tangent at A. +If, therefore, a point on the curve and its tangent are given, this +will be equivalent to having given two points on the curve. Similarly, +if on the curve of second class a tangent and its point of +contact are given, this will be equivalent to two given tangents.</p> + +<p>We may therefore extend the last theorem:</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p>Only one curve of the second order can be drawn, of which + four points and the tangent at one of them, or three points and the + tangents at two of them, are given.</p></td> + +<td class="tcl" style="width: 50%;"><p>Only one curve of the second class can be drawn, of which four + tangents and the point of contact at one of them, or three tangents + and the points of contact at two of them, are given.</p></td></tr></table> + +<p>§ 53. At the same time it has been proved:</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p>If all points on a curve of the second order be joined to any + two of them, then the two pencils thus formed are projective, those + rays being corresponding which meet on the curve. Hence—</p></td> + +<td class="tcl" style="width: 50%;"><p>All tangents to a curve of second class are cut by any two of + them in projective rows, those being corresponding points which + lie on the same tangent. Hence—</p></td></tr> + +<tr><td class="tcl rb3"><p>The cross-ratio of four rays joining a point S on a curve of + second order to four fixed points A, B, C, D in the curve is independent + of the position of S, and is called the cross-ratio of the + four points A, B, C, D.</p></td> + +<td class="tcl"><p>The cross-ratio of the four points in which any tangent u is + cut by four fixed tangents a, b, c, d is independent of the position of + u, and is called the cross-ratio of the four tangents a, b, c, d.</p></td></tr> + +<tr><td class="tcl rb3"><p>If this cross-ratio equals −1 the four points are said to be + four harmonic points.</p></td> + +<td class="tcl"><p>If this cross-ratio equals −1 the four tangents are said to be + four harmonic tangents.</p></td></tr></table> + +<p>We have seen that a curve of second order, as generated by +projective pencils, has at the centre of each pencil one tangent; +and further, that any point on the curve may be taken as centre of +such pencil. Hence—</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p>A curve of second order has at every point one tangent.</p></td> + +<td class="tcl" style="width: 50%;"><p>A curve of second class has on every tangent a point of contact.</p></td></tr></table> + +<p>§ 54. We return to Pascal’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.</p> + +<p>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.</p> + +<p>If the points 1 2 3 4 5 are given, and we want a 6th point on a +line drawn through 1, we know all the sides of the hexagon with +the exception of 5 6, and this is found by Pascal’s theorem.</p> + +<p>If this line should happen to pass through 1, then 6 and 1 coincide, +or the line 6 1 is the tangent at 1. And always if two consecutive +vertices of the hexagon approach nearer and nearer, then the side +joining them will ultimately become a tangent.</p> + +<p>We may therefore consider a pentagon inscribed in a curve of +second order and the tangent at one of its vertices as a hexagon, +and thus get the theorem:</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p>Every pentagon inscribed in a curve of second order has the + property that the intersections of two pairs of non-consecutive + sides lie in a line with the point where the fifth side cuts the tangent + at the opposite vertex.</p></td> + +<td class="tcl" style="width: 50%;"><p>Every pentagon circumscribed about a curve of the second class + has the property that the lines which join two pairs of non-consecutive + vertices meet on that line which joins the fifth vertex + to the point of contact of the opposite side.</p></td></tr></table> + +<p>This enables us also to solve the following problems.</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p>Given five points on a curve of second order to construct the + tangent at any one of them.</p></td> + +<td class="tcl" style="width: 50%;"><p>Given five tangents to a curve of second class to construct the + point of contact of any one of them.</p></td></tr></table> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:390px; height:354px" src="images/img697a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 19.</span></td></tr></table> + +<p>If two pairs of adjacent vertices coincide, the hexagon becomes a +quadrilateral, with tangents at two vertices. These we take to be +opposite, and get the following theorems:</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p>If a quadrilateral be inscribed in a curve of second order, the + intersections of opposite sides, and also the intersections of the + tangents at opposite vertices, lie in a line (fig. 19).</p></td> + +<td class="tcl" style="width: 50%;"><p>If a quadrilateral be circumscribed about a curve of second + class, the lines joining opposite vertices, and also the lines joining + points of contact of opposite sides, meet in a point.</p></td></tr></table> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:365px; height:294px" src="images/img697b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 20.</span></td></tr></table> + +<p>If we consider the hexagon made up of a triangle and the tangents +at its vertices, we get—</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p>If a triangle is inscribed in a curve of the second order, the + points in which the sides are cut by the tangents at the opposite + vertices meet in a point.</p></td> + +<td class="tcl" style="width: 50%;"><p>If a triangle be circumscribed about a curve of second class, + the lines which join the vertices to the points of contact of the + opposite sides meet in a point (fig. 20).</p></td></tr></table> + +<p>§ 55. Of these theorems, those about the quadrilateral give rise to +a number of others. Four points A, B, C, D may in three different +ways be formed into a quadrilateral, for we may take them in the +order ABCD, or ACBD, or ACDB, so that either of the points +B, C, D may be taken as the vertex opposite to A. Accordingly we +may apply the theorem in three different ways.</p> + +<p>Let A, B, C, D be four points on a curve of second order (fig. 21), +and let us take them as forming a quadrilateral by taking the points +in the order ABCD, so that A, C and also B, D are pairs of opposite +vertices. Then P, Q will be the points where opposite sides meet, +<span class="pagenum"><a name="page698" id="page698"></a>698</span> +and E, F the intersections of tangents at opposite vertices. The +four points P, Q, E, F lie therefore in a line. The quadrilateral +ACBD gives us in the same way the four points Q, R, G, H in a line, +and the quadrilateral ABDC a line containing the four points R, P, +I, K. These three lines form a triangle PQR.</p> + +<p>The relation between the points and lines in this figure may be +expressed more clearly if we consider ABCD as a four-point inscribed +in a conic, and the tangents at these points as a four-side circumscribed +about it,—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—</p> + +<p><i>Any four-point on a curve of the second order and the four-side +formed by the tangents at these points stand in this relation that the +diagonal points of the four-point lie in the diagonals of the four-side.</i> +And conversely,</p> + +<p><i>If a four-point and a circumscribed four-side stand in the above +relation, then a curve of the second order may be described which passes +through the four points and touches there the four sides of these figures.</i></p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:396px; height:707px" src="images/img698a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 21.</span></td></tr></table> + +<p>That the last part of the theorem is true follows from the fact +that the four points A, B, C, D and the line a, as tangent at A, determine +a curve of the second order, and the tangents to this curve at +the other points B, C, D are given by the construction which leads +to fig. 21.</p> + +<p>The theorem reciprocal to the last is—</p> + +<p><i>Any four-side circumscribed about a curve of second class and the +four-point formed by the points of contact stand in this relation that the +diagonals of the four-side pass through the diagonal points of the +four-point.</i> And conversely,</p> + +<p><i>If a four-side and an inscribed four-point stand in the above relation, +then a curve of the second class may be described which touches the sides +of the four-side at the points of the four-point.</i></p> + +<p>§ 56. The four-point and the four-side in the two reciprocal +theorems are alike. Hence if we have a four-point ABCD and a +four-side abcd related in the manner described, then not only may +a curve of the second order be drawn, but also a curve of the second +class, which both touch the lines a, b, c, d at the points A, B, C, D.</p> + +<p>The curve of second order is already more than determined by the +points A, B, C and the tangents a, b, c at A, B and C. The point D +may therefore be <i>any</i> point on this curve, and d any tangent to the +curve. On the other hand the curve of the second class is more +than determined by the three tangents a, b, c and their points of +contact A, B, C, so that d is any tangent to this curve. It follows +that every tangent to the curve of second order is a tangent of a +curve of the second class having the same point of contact. In +other words, the curve of second order is a curve of second class, +and <i>vice versa</i>. Hence the important theorems—</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p><i>Every curve of second order is a curve of second class.</i></p></td> + +<td class="tcl" style="width: 50%;"><p><i>Every curve of second class is a curve of second order.</i></p></td></tr></table> + +<p>The curves of second order and of second class, having thus been +proved to be identical, shall henceforth be called by the common +name of <i>Conics</i>.</p> + +<p>For these curves hold, therefore, all properties which have been +proved for curves of second order or of second class. We may +therefore now state Pascal’s and Brianchon’s theorem thus—</p> + +<p><i>Pascal’s Theorem.</i>—If a hexagon be inscribed in a conic, then +the intersections of opposite sides lie in a line.</p> + +<p><i>Brianchon’s Theorem.</i>—If a hexagon be circumscribed about a +conic, then the diagonals forming opposite centres meet in a point.</p> + +<p>§ 57. If we suppose in fig. 21 that the point D together with the +tangent d moves along the curve, whilst A, B, C and their tangents +a, b, c remain fixed, then the ray DA will describe a pencil about +A, the point Q a projective row on the fixed line BC, the point F +the row b, and the ray EF a pencil about E. But EF passes always +through Q. Hence the pencil described by AD is projective to the +pencil described by EF, and therefore to the row described by F on +b. At the same time the line BD describes a pencil about B projective +to that described by AD (§ 53). Therefore the pencil BD +and the row F on b are projective. Hence—</p> + +<p><i>If on a conic a point</i> A <i>be taken and the tangent a at this point, then +the cross-ratio of the four rays which join</i> A <i>to any four points on the +curve is equal to the cross-ratio of the points in which the tangents at +these points cut the tangent at</i> A.</p> + +<p>§ 58. There are theorems about cones of second order and second +class in a pencil which are reciprocal to the above, according to § 43. +We mention only a few of the more important ones.</p> + +<p>The locus of intersections of corresponding planes in two projective +axial pencils whose axes meet is a cone of the second order.</p> + +<p>The envelope of planes which join corresponding lines in two +projective flat pencils, not in the same plane, is a cone of the second +class.</p> + +<p>Cones of second order and cones of second class are identical.</p> + +<p>Every plane cuts a cone of the second order in a conic.</p> + +<p><i>A cone of second order is uniquely determined by five of its edges +or by five of its tangent planes, or by four edges and the tangent plane +at one of them, &c. &c.</i></p> + +<p><i>Pascal’s Theorem.</i>—If a solid angle of six faces be inscribed in a +cone of the second order, then the intersections of opposite faces +are three lines in a plane.</p> + +<p><i>Brianchon’s Theorem.</i>—If a solid angle of six edges be circumscribed +about a cone of the second order, then the planes through +opposite edges meet in a line.</p> + +<p>Each of the other theorems about conics may be stated for cones +of the second order.</p> + +<table class="flt" style="float: right; width: 360px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:310px; height:314px" src="images/img698b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 22.</span></td></tr></table> + +<p>§ 59. <i>Projective Definitions of the Conics.</i>—We now consider the +shape of the conics. We know that any line in the plane of the conic, +and hence that the line at infinity, either has no point in common +with the curve, or one (counting for two coincident points) or two +distinct points. If the line at infinity has no point on the curve the +latter is altogether finite, and is called an <i>Ellipse</i> (fig. 21). If the line +at infinity has only one point in common with the conic, the latter +extends to infinity, and has the line at infinity a tangent. It is +called a <i>Parabola</i> (fig. 22). If, lastly, the line at infinity cuts the +curve in two points, it +consists of two separate +parts which each extend +in two branches to the +points at infinity where +they meet. The curve is +in this case called an +<i>Hyperbola</i> (see fig. 20). +The tangents at the +two points at infinity +are finite because the +line at infinity is not +a tangent. They are +called <i>Asymptotes</i>. The +branches of the hyperbola +approach these lines +indefinitely as a point on +the curves moves to infinity.</p> + +<p>§ 60. That the circle +belongs to the curves of +the second order is seen +at once if we state in +a slightly different form the theorem that in a circle all angles at +the circumference standing upon the same arc are equal. If two +points S<span class="su">1</span>, S<span class="su">2</span> on a circle be joined to any other two points A and B +on the circle, then the angle included by the rays S<span class="su">1</span>A and S<span class="su">1</span>B is +equal to that between the rays S<span class="su">2</span>A and S<span class="su">2</span>B, so that as A moves +along the circumference the rays S<span class="su">1</span>A and S<span class="su">2</span>A describe equal and +therefore projective pencils. The circle can thus be generated by +two projective pencils, and is a curve of the second order.</p> + +<p><span class="pagenum"><a name="page699" id="page699"></a>699</span></p> + +<p>If we join a point in space to all points on a circle, we get a (circular) +cone of the second order (§ 43). Every plane section of this cone is a +conic. This conic will be an ellipse, a parabola, or an hyperbola, +according as the line at infinity in the plane has no, one or two points +in common with the conic in which the plane at infinity cuts the +cone. It follows that our curves of second order may be obtained +as sections of a circular cone, and that they are identical with the +“Conic Sections” of the Greek mathematicians.</p> + +<p>§ 61. Any two tangents to a parabola are cut by all others in +projective rows; but the line at infinity being one of the tangents, +the points at infinity on the rows are corresponding points, and the +rows therefore similar. Hence the theorem—</p> + +<p><i>The tangents to a parabola cut each other proportionally.</i></p> + +<p class="pt2 center sc">Pole and Polar</p> + +<p>§ 62. We return once again to fig. 21, which we obtained in § 55.</p> + +<p>If a four-side be circumscribed about and a four-point inscribed +in a conic, so that the vertices of the second are the points of contact +of the sides of the first, then the triangle formed by the diagonals +of the first is the same as that formed by the diagonal points of the +other.</p> + +<p>Such a triangle will be called a <i>polar-triangle</i> of the conic, so that +PQR in fig. 21 is a polar-triangle. It has the property that on the +side p opposite P meet the tangents at A and B, and also those at C +and D. From the harmonic properties of four-points and four-sides +it follows further that the points L, M, where it cuts the lines AB +and CD, are harmonic conjugates with regard to AB and CD +respectively.</p> + +<p>If the point P is given, and we draw a line through it, cutting +the conic in A and B, then the point Q harmonic conjugate to P +with regard to AB, and the point H where the tangents at A and B +meet, are determined. But they lie both on p, and therefore this +line is determined. If we now draw a second line through P, cutting +the conic in C and D, then the point M harmonic conjugate to P +with regard to CD, and the point G where the tangents at C and D +meet, must also lie on p. As the first line through P already determines +p, the second may be any line through P. Now every two +lines through P determine a four-point ABCD on the conic, and +therefore a polar-triangle which has one vertex at P and its opposite +side at p. This result, together with its reciprocal, gives the +theorems—</p> + +<p><i>All polar-triangles which have one vertex in common have also the +opposite side in common.</i></p> + +<p><i>All polar-triangles which have one side in common have also the +opposite vertex in common.</i></p> + +<p>§ 63. To any point P in the plane of, but not on, a conic corresponds +thus one line p as the side opposite to P in all polar-triangles which +have one vertex at P, and reciprocally to every line p corresponds +one point P as the vertex opposite to p in all triangles which have p +as one side.</p> + +<p>We call the line p the <i>polar</i> of P, and the point P the <i>pole</i> of the +line p with regard to the conic.</p> + +<p>If a point lies on the conic, we call the tangent at that point its +polar; and reciprocally we call the point of contact the pole of +tangent.</p> + +<p>§ 64. From these definitions and former results follow—</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;"><p>The polar of any point P not on the conic is a line p, which has + the following properties:—</p></td> + +<td class="tcl" style="width: 50%;"><p>The pole of any line p not a tangent to the conic is a point + P, which has the following properties:—</p></td></tr> + +<tr><td class="tcl rb3">1. On every line through P which cuts the conic, the polar + of P contains the harmonic conjugate of P with regard to those + points on the conic.</td> + +<td class="tcl">1. Of all lines through a point on p from which two tangents + may be drawn to the conic, the pole P contains the line which is + harmonic conjugate to p, with regard to the two tangents.</td></tr> + +<tr><td class="tcl rb3">2. If tangents can be drawn from P, their points of contact lie + on p.</td> + +<td class="tcl">2. If p cuts the conic, the tangents at the intersections + meet at P.</td></tr> + +<tr><td class="tcl rb3">3. Tangents drawn at the points where any line through P + cuts the conic meet on p; and conversely,</td> + +<td class="tcl">3. The point of contact of tangents drawn from any point + on p to the conic lie in a line with P; and conversely,</td></tr> + +<tr><td class="tcl rb3">4. If from any point on p, tangents be drawn, their points + of contact will lie in a line with P.</td> + +<td class="tcl">4. Tangents drawn at points where any line through P cuts the + conic meet on p.</td></tr> + +<tr><td class="tcl rb3">5. Any four-point on the conic which has one diagonal point at + P has the other two lying on p.</td> + +<td class="tcl">5. Any four-side circumscribed about a conic which has one + diagonal on p has the other two meeting at P.</td></tr></table> + +<p>The truth of 2 follows from 1. If T be a point where p cuts the +conic, then one of the points where PT cuts the conic, and which +are harmonic conjugates with regard to PT, coincides with T; hence +the other does—that is, PT touches the curve at T.</p> + +<p>That 4 is true follows thus: If we draw from a point H on the +polar one tangent a to the conic, join its point of contact A to the +pole P, determine the second point of intersection B of this line with +the conic, and draw the tangent at B, it will pass through H, and +will therefore be the second tangent which may be drawn from H to +the curve.</p> + +<p>§ 65. The second property of the polar or pole gives rise to the +theorem—</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;">From a point in the plane of a conic, two, one or no tangents + may be drawn to the conic, as its polar has two, + one, or no points in common with the curve.</td> + +<td class="tcl" style="width: 50%;">A line in the plane of a conic has two, one or no points in + common with the conic, according as two, one or no tangents + can be drawn from its pole to the conic.</td></tr></table> + +<p>Of any point in the plane of a conic we say that it was <i>without</i>, +on or <i>within</i> the curve according as two, one or no tangents to the +curve pass through it. The points on the conic separate those within +the conic from those without. That this is true for a circle is known +from elementary geometry. That it also holds for other conics +follows from the fact that every conic may be considered as the +projection of a circle, which will be proved later on.</p> + +<p>The fifth property of pole and polar stated in § 64 shows how +to find the polar of any point and the pole of any line by aid of the +straight-edge only. Practically it is often convenient to draw three +secants through the pole, and to determine only one of the diagonal +points for two of the four-points formed by pairs of these lines and +the conic (fig. 22).</p> + +<p>These constructions also solve the problem—</p> + +<p>From a point without a conic, to draw the two tangents to the +conic by aid of the straight-edge only.</p> + +<p>For we need only draw the polar of the point in order to find the +points of contact.</p> + +<p>§ 66. The property of a polar-triangle may now be stated thus—</p> + +<p>In a polar-triangle each side is the polar of the opposite vertex, +and each vertex is the pole of the opposite side.</p> + +<table class="flt" style="float: right; width: 340px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:286px; height:303px" src="images/img699.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 23.</span></td></tr></table> + +<p>If P is one vertex of a polar-triangle, then the other vertices, Q +and R, lie on the polar p of P. One of these vertices we may choose +arbitrarily. For if from +any point Q on the polar +a secant be drawn cutting +the conic in A and D (fig. +23), and if the lines joining +these points to P cut the +conic again at B and C, +then the line BC will pass +through Q. Hence P and +Q are two of the vertices +on the polar-triangle which +is determined by the four-point +ABCD. The third +vertex R lies also on the +line p. It follows, therefore, +also—</p> + +<p><i>If</i> Q <i>is a point on the polar +of</i> P, <i>then</i> P <i>is a point on the +polar of</i> Q; and reciprocally,</p> + +<p><i>If</i> q <i>is a line through the +pole of</i> p, <i>then</i> p <i>is a line +through the pole of</i> q.</p> + +<p>This is a very important theorem. It may also be stated +thus—</p> + +<p><i>If a point moves along a line describing a row, its polar turns about +the pole of the line describing a pencil.</i></p> + +<p><i>This pencil is projective to the row, so that the cross-ratio of four +poles in a row equals the cross-ratio of its four polars, which pass +through the pole of the row.</i></p> + +<p>To prove the last part, let us suppose that P, A and B in fig. 23 +remain fixed, whilst Q moves along the polar p of P. This will +make CD turn about P and move R along p, whilst QD and RD +describe projective pencils about A and B. Hence Q and R describe +projective rows, and hence PR, which is the polar of Q, describes a +pencil projective to either.</p> + +<p>§ 67. Two points, of which one, and therefore each, lies on the +polar of the other, are said to be <i>conjugate with regard to the conic</i>; +and two lines, of which one, and therefore each, passes through the +pole of the other, are said to be <i>conjugate with regard to the conic</i>. +Hence all points conjugate to a point P lie on the polar of P; all lines +conjugate to a line p pass through the pole of p.</p> + +<p>If the line joining two conjugate poles cuts the conic, then the +poles are harmonic conjugates with regard to the points of intersection; +hence one lies within the other without the conic, and all +points conjugate to a point within a conic lie without it.</p> + +<p>Of a polar-triangle any two vertices are conjugate poles, any two +sides conjugate lines. If, therefore, one side cuts a conic, then +one of the two vertices which lie on this side is within and the other +without the conic. The vertex opposite this side lies also without, +for it is the pole of a line which cuts the curve. In this case therefore +one vertex lies within, the other two without. If, on the +other hand, we begin with a side which does not cut the conic, +then its pole lies within and the other vertices without. Hence—</p> + +<p>Every polar-triangle has one and only one vertex within the conic.</p> + +<p>We add, without a proof, the theorem—</p> + +<p>The four points in which a conic is cut by two conjugate polars +are four harmonic points in the conic.</p> + +<p>§ 68. If two conics intersect in four points (they cannot have +more points in common, § 52), there exists one and only one +<span class="pagenum"><a name="page700" id="page700"></a>700</span> +four-point which is inscribed in both, and therefore one polar-triangle +common to both.</p> + +<p><i>Theorem.</i>—Two conics which intersect in four points have always +one and only one common polar-triangle; and reciprocally,</p> + +<p>Two conics which have four common tangents have always one +and only one common polar-triangle.</p> + +<p class="pt2 center sc">Diameters and Axes of Conics</p> + +<p>§ 69. <i>Diameters.</i>—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:—</p> + +<p><i>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.</i></p> + +<p>This line is called a <i>diameter</i>.</p> + +<p><i>The polar of every point at infinity is a diameter.</i></p> + +<p><i>The tangents at the end points of a diameter are parallel, and are +parallel to the chords bisected by the diameter.</i></p> + +<p><i>All diameters pass through a common point, the pole of the line at +infinity.</i></p> + +<p><i>All diameters of a parabola are parallel</i>, the pole to the line at +infinity being the point where the curve touches the line at infinity.</p> + +<p>In case of the ellipse and hyperbola, the pole to the line at infinity +is a finite point called the <i>centre</i> of the curve.</p> + +<p><i>A centre of a conic bisects every chord through it.</i></p> + +<p><i>The centre of an ellipse is within the curve</i>, for the line at infinity +does not cut the ellipse.</p> + +<p><i>The centre of an hyperbola is without the curve</i>, because the line at +infinity cuts the curve. Hence also—</p> + +<p><i>From the centre of an hyperbola two tangents can be drawn to the +curve which have their point of contact at infinity.</i> These are called +<i>Asymptotes</i> (§ 59).</p> + +<p><i>To construct a diameter</i> of a conic, draw two parallel chords and +join their middle points.</p> + +<p><i>To find the centre</i> of a conic, draw two diameters; their intersection +will be the centre.</p> + +<p>§ 70. <i>Conjugate Diameters.</i>—A polar-triangle with one vertex at +the centre will have the opposite side at infinity. The other two +sides pass through the centre, and are called <i>conjugate diameters</i>, +each being the polar of the point at infinity on the other.</p> + +<p><i>Of two conjugate diameters each bisects the chords parallel to the +other, and if one cuts the curve, the tangents at its ends are parallel to +the other diameter.</i></p> + +<p>Further—</p> + +<p><i>Every parallelogram inscribed in a conic has its sides parallel to +two conjugate diameters</i>; and</p> + +<p><i>Every parallelogram circumscribed about a conic has as diagonals two +conjugate diameters.</i></p> + +<p>This will be seen by considering the parallelogram in the first +case as an inscribed four-point, in the other as a circumscribed +four-side, and determining in each case the corresponding polar-triangle. +The first may also be enunciated thus—</p> + +<p><i>The lines which join any point on an ellipse or an hyperbola to the +ends of a diameter are parallel to two conjugate diameters.</i></p> + +<p>§ 71. <i>If every diameter is perpendicular to its conjugate the conic is +a circle.</i></p> + +<p>For the lines which join the ends of a diameter to any point on +the curve include a right angle.</p> + +<p><i>A conic which has more than one pair of conjugate diameters at right +angles to each other is a circle.</i></p> + +<table class="flt" style="float: right; width: 260px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:209px; height:221px" src="images/img700a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 24.</span></td></tr></table> + +<p>Let AA′ and BB′ (fig. 24) be one pair of conjugate diameters at +right angles to each other, CC and DD′ a second pair. If we draw +through the end point A of one +diameter a chord AP parallel to +DD′, and join P to A′, then PA and +PA′ are, according to § 70, parallel to +two conjugate diameters. But PA is +parallel to DD′, hence PA′ is parallel +to CC, and therefore PA and PA′ +are perpendicular. If we further +draw the tangents to the conic at A +and A′, these will be perpendicular +to AA′, they being parallel to the +conjugate diameter BB′. We know +thus five points on the conic, viz. the +points A and A′ with their tangents, +and the point P. Through these a +circle may be drawn having AA′ as +diameter; and as through five points +one conic only can be drawn, this circle must coincide with the +given conic.</p> + +<p>§ 72. <i>Axes.</i>—Conjugate diameters perpendicular to each other +are called <i>axes</i>, and the points where they cut the curve <i>vertices</i> +of the conic.</p> + +<p>In a circle every diameter is an axis, every point on it is a vertex; +and any two lines at right angles to each other may be taken as a +pair of axes of any circle which has its centre at their intersection.</p> + +<table class="flt" style="float: left; width: 340px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:289px; height:261px" src="images/img700b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 25.</span></td></tr></table> + +<p>If we describe on a diameter AB of an ellipse or hyperbola a circle +concentric to the conic, it will cut the latter in A and B (fig. 25). +Each of the semicircles in which it is divided by AB will be partly +within, partly without the curve, and must cut the latter therefore +again in a point. The circle and the conic have thus four points +A, B, C, D, and therefore +one polar-triangle, in common +(§ 68). Of this the +centre is one vertex, for +the line at infinity is the +polar to this point, both +with regard to the circle +and the other conic. The +other two sides are conjugate +diameters of both, +hence perpendicular to +each other. This gives—</p> + +<p>An ellipse as well as an +hyperbola has one pair of +axes.</p> + +<p>This reasoning shows at +the same time <i>how to construct +the axis of an ellipse +or of an hyperbola</i>.</p> + +<p><i>A parabola has one axis</i>, +if we define an axis as a diameter perpendicular to the chords +which it bisects. It is easily constructed. The line which bisects +any two parallel chords is a diameter. Chords perpendicular to it +will be bisected by a parallel diameter, and this is the axis.</p> + +<p>§ 73. The first part of the right-hand theorem in § 64 may be +stated thus: any two conjugate lines through a point P without a +conic are harmonic conjugates with regard to the two tangents +that may be drawn from P to the conic.</p> + +<p>If we take instead of P the centre C of an hyperbola, then the +conjugate lines become conjugate diameters, and the tangents +asymptotes. Hence—</p> + +<p><i>Any two conjugate diameters of an hyperbola are harmonic conjugates +with regard to the asymptotes.</i></p> + +<p>As the axes are conjugate diameters at right angles to one another, +it follows (§ 23)—</p> + +<p><i>The axes of an hyperbola bisect the angles between the asymptotes.</i></p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:352px; height:337px" src="images/img700c.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 26.</span></td></tr></table> + +<p>Let O be the centre of the hyperbola (fig. 26), t any secant which +cuts the hyperbola in C, D and the asymptotes in E, F, then the +line OM which bisects the chord CD is a diameter conjugate to the +diameter OK which is parallel to the secant t, so that OK and OM +are harmonic with regard to the asymptotes. The point M therefore +bisects EF. But by construction M bisects CD. It follows +that DF = EC, and ED = CF; or</p> + +<p><i>On any secant of an hyperbola the segments between the curve and the +asymptotes are equal.</i></p> + +<p>If the chord is changed into a tangent, this gives—</p> + +<p><i>The segment between the asymptotes on any tangent to an hyperbola +is bisected by the point of contact.</i></p> + +<p>The first part allows a simple solution of the problem to find any +number of points on an hyperbola, of which the asymptotes and one +point are given. This is equivalent to three points and the tangents +at two of them. This construction requires measurement.</p> + +<p>§ 74. For the parabola, too, follow some metrical properties. A +diameter PM (fig. 27) bisects every chord conjugate to it, and the +pole P of such a chord BC lies on the diameter. But a diameter cuts +the parabola once at infinity. Hence—</p> + +<p><i>The segment</i> PM <i>which joins the middle point</i> M <i>of a chord of a parabola +to the pole</i> P <i>of the chord is bisected by the parabola at</i> A.</p> + +<table class="flt" style="float: right; width: 340px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:293px; height:282px" src="images/img701a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 27.</span></td></tr></table> + +<p>§ 75. Two asymptotes and any two tangents to an hyperbola +may be considered as a quadrilateral circumscribed about the +<span class="pagenum"><a name="page701" id="page701"></a>701</span> +hyperbola. But in such a quadrilateral the intersections of the +diagonals and the points of contact of opposite sides lie in a line +(§ 54). If therefore DEFG +(fig. 28) is such a quadrilateral, +then the diagonals +DF and GE will meet on +the line which joins the +points of contact of the +asymptotes, that is, on the +line at infinity; hence they +are parallel. From this +the following theorem is +a simple deduction:</p> + +<p><i>All triangles formed by a +tangent and the asymptotes +of an hyperbola are equal in +area.</i></p> + +<p>If we draw at a point P +(fig. 28) on an hyperbola +a tangent, the part HK +between the asymptotes +is bisected at P. The +parallelogram PQOQ′ +formed by the asymptotes and lines parallel to them through +P will be half the triangle OHK, and will therefore be constant. +If we now take the asymptotes OX and OY as oblique +axes of co-ordinates, the lines OQ and QP will be the co-ordinates of +P, and will satisfy the equation xy = const. = a².</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:323px; height:333px" src="images/img701b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 28.</span></td></tr></table> + +<p><i>For the asymptotes as axes of co-ordinates the equation of the hyperbola +is</i> xy = const.</p> + +<p class="pt2 center sc">Involution</p> + +<table class="flt" style="float: right; width: 270px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:221px; height:49px" src="images/img701c.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 29.</span></td></tr></table> + +<p>§ 76. If we have two projective rows, ABC on u and A′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:</p> + +<p><i>If two projective rows lie on the same base, and if it happens that to one +point in the base the same point corresponds, whether we consider the +point as belonging to the first or to the second row, then the same will +happen for every point in the base—that is to say, to every point in the +line corresponds the same point in the first as in the second row.</i></p> + +<table class="flt" style="float: right; width: 280px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:226px; height:45px" src="images/img701d.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 30.</span></td></tr></table> + +<p>In order to determine the correspondence, we may assume three +pairs of corresponding points in two projective rows. Let then +A′, 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</p> + +<p class="center">(AB, CD) = (A′B′, C′D′)</p> + +<p>but replacing the dashed letters by those undashed ones which +denote the same points, the second cross-ratio equals (BA, DD′), +which, according to § 15, equals (AB, D′D); so that the equation +becomes</p> + +<p class="center">(AB, CD) = (AB, D′D).</p> + +<p>This requires that C and D′ coincide.</p> + +<p>§ 77. Two projective rows on the same base, which have the above +property, that to every point, whether it be considered as a point in +the one or in the other row, corresponds the same point, are said +to be in <i>involution</i>, or to form an <i>involution</i> of points on the line.</p> + +<p>We mention, but without proving it, that any two projective +rows may be placed so as to form an involution.</p> + +<p>An involution may be said to consist of a row of pairs of points, +to every point A corresponding a point A′, 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.</p> + +<p>From the definition, according to which an involution may be +considered as made up of two projective rows, follow at once the +following important properties:</p> + +<p>1. The cross-ratio of four points equals that of the four conjugate +points.</p> + +<p>2. If we call a point which coincides with its mate a “focus” +or “double point” of the involution, we may say: An involution +has either two foci, or one, or none, and is called respectively a +hyperbolic, parabolic or elliptic involution (§ 34).</p> + +<p>3. In <span class="correction" title="amended from a">an</span> hyperbolic involution any two conjugate points are +harmonic conjugates with regard to the two foci.</p> + +<p>For if A, A′ be two conjugate points, F<span class="su">1</span>, F<span class="su">2</span> the two foci, then to the +points F<span class="su">1</span>, F<span class="su">2</span>, A, A′ in the one row correspond the points F<span class="su">1</span>, F<span class="su">2</span>, A′, A +in the other, each focus corresponding to itself. Hence (F<span class="su">1</span>F<span class="su">2</span>, AA′) = +(F<span class="su">1</span>F<span class="su">2</span>, A′A)—that is, we may interchange the two points AA′ without +altering the value of the cross-ratio, which is the characteristic +property of harmonic conjugates (§ 18).</p> + +<p>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.</p> + +<p>In an hyperbolic involution the centre is the middle point between +the foci.</p> + +<p>5. The product of the distances of two conjugate points A, A′ +from the centre O is constant: OA · OA′ = c.</p> + +<p>For let A, A′ and B, B′ be two pairs of conjugate points, the +centre, I the point at infinity, then</p> + +<p class="center">(AB, OI) = (A′B′, IO),</p> + +<p class="noind">or</p> + +<p class="center">OA · OA′ = OB · OB′.</p> + +<p>In order to determine the distances of the foci from the centre, +we write F for A and A′ and get</p> + +<p class="center">OF² = c; OF = ±√c.</p> + +<p class="noind">Hence if c is positive OF is real, and has two values, equal and +opposite. The involution is hyperbolic.</p> + +<p>If c = 0, OF = 0, and the two foci both coincide with the centre. +If c is negative, √c becomes imaginary, and there are no foci. +Hence we may write—</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">In an hyperbolic involution,</td> <td class="tcl">OA · OA′ = k²,</td></tr> + +<tr><td class="tcl">In a parabolic involution,</td> <td class="tcl">OA · OA′ = 0,</td></tr> + +<tr><td class="tcl">In an elliptic involution,</td> <td class="tcl">OA · OA′ = −k².</td></tr> +</table> + +<p>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.</p> + +<p>In the first case, for instance, OA · OA′ is positive; hence OA +and OA′ have the same sign.</p> + +<p>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.</p> + +<p><i>Proof.</i>—We have OA . OA′ = OB · OB′ = k² in case of an hyperbolic +involution. Let A and B be the points in each pair which are +nearer to the centre O. If now A, A′ and B, B′ lie on the same side of +O, and if B is nearer to O than A, so that OB < OA, then OB′ > OA′; +hence B′ lies farther away from O than A′, or the segment AA′ lies +within BB′. And so on for the other cases.</p> + +<p>6. An involution is determined—</p> + +<table class="reg" summary="poem"><tr><td> <div class="poemr"> +<p>(α) By two pairs of conjugate points. Hence also</p> +<p>(β) By one pair of conjugate points and the centre;</p> +<p>(γ) By the two foci;</p> +<p>(δ) By one focus and one pair of conjugate points;</p> +<p>(ε) By one focus and the centre.</p> +</div> </td></tr></table> + +<p>7. The condition that A, B, C and A′, B′, C′ may form an involution +may be written in one of the forms—</p> + +<p class="center">(AB, CC′) = (A′B′, C′C),</p> + +<p class="noind">or</p> + +<p class="center">(AB, CA′) = (A′B′, C′A),</p> + +<p class="noind">or</p> + +<p class="center">(AB, C′A′) = (A′B′, CA),</p> + +<p class="noind">for each expresses that in the two projective rows in which A, B, C +<span class="pagenum"><a name="page702" id="page702"></a>702</span> +and A′, B′, C′ are conjugate points two conjugate elements may be +interchanged.</p> + +<p>8. Any three pairs. A, A′, B, B′, C, C′, of conjugate points are +connected by the relations:</p> + +<table class="math0" summary="math"> +<tr><td>AB′ · BC′ · CA′</td> +<td rowspan="2">=</td> <td>AB′ · BC · C′A′</td> +<td rowspan="2">=</td> <td>AB · B′C′ · CA′</td> +<td rowspan="2">=</td> <td>AB · B′C · C′A′</td> +<td rowspan="2">= −1.</td></tr> +<tr><td class="denom">A′B · B′C · C′A</td> <td class="denom">A′B · B′C′ · CA</td> +<td class="denom">A′B′ · BC · C′A</td> <td class="denom">A′B′ · BC′ · CA</td></tr></table> + +<p>These relations readily follow by working out the relations in (7) +(above).</p> + +<p>§ 78. <i>Involution of a quadrangle.—The sides of any four-point are +cut by any line in six points in involution, opposite sides being cut in +conjugate points.</i></p> + +<p>Let A<span class="su">1</span>B<span class="su">1</span>C<span class="su">1</span>D<span class="su">1</span> (fig. 31) be the four-point. If its sides be cut by +the line p in the points A, A′, B, B′, C, C′, if further, C<span class="su">1</span>D<span class="su">1</span> cuts the +line A<span class="su">1</span>B<span class="su">1</span> in C<span class="su">2</span>, and if we project the row A<span class="su">1</span>B<span class="su">1</span>C<span class="su">2</span>C to p once from +D<span class="su">1</span> and once from C<span class="su">1</span>, we get (A′B′, C′C) = (BA, C′C).</p> + +<p>Interchanging in the last cross-ratio the letters in each pair we get +(A′B′, C′C) = (AB, CC′). Hence by § 77 (7) the points are in involution.</p> + +<p>The theorem may also be stated thus:</p> + +<p><i>The three points in which any line cuts the sides of a triangle and the +projections, from any point in the plane, of the vertices of the triangle +on to the same line are six points in involution.</i></p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:421px; height:325px" src="images/img702a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 31.</span></td></tr></table> + +<p>Or again—</p> + +<p>The projections from any point on to any line of the six vertices +of a four-side are six points in involution, the projections of opposite +vertices being conjugate points.</p> + +<p>This property gives a simple means to construct, by aid of the +straight edge only, in an involution of which two pairs of conjugate +points are given, to any point its conjugate.</p> + +<p>§ 79. <i>Pencils in Involution.</i>—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 +<i>focal rays</i> (double lines) or <i>planes</i>, but nothing corresponding to a +centre.</p> + +<p>An involution in a flat pencil contains always one, and in general +only one, pair of conjugate rays which are perpendicular to one +another. For in two projective flat pencils exist always two corresponding +right angles (§ 40).</p> + +<p>Each involution in an axial pencil contains in the same manner +one pair of conjugate planes at right angles to one another.</p> + +<p>As a rule, there exists but one pair of conjugate lines or planes +at right angles to each other. But it is possible that there are +more, and then there is an infinite number of such pairs. An involution +in a flat pencil, in which every ray is perpendicular to its +conjugate ray, is said to be <i>circular</i>. That such involution is +possible is easily seen thus: if in two concentric flat pencils each +ray on one is made to correspond to that ray on the other which +is perpendicular to it, then the two pencils are projective, for if +we turn the one pencil through a right angle each ray in one coincides +with its corresponding ray in the other. But these two projective +pencils are in involution.</p> + +<p>A circular involution has no focal rays, because no ray in a pencil +coincides with the ray perpendicular to it.</p> + +<p>§ 80. <i>Every elliptical involution in a row may be considered as a +section of a circular involution.</i></p> + +<p>In an elliptical involution any two segments AA′ 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:</p> + +<table class="flt" style="float: right; width: 310px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:257px; height:158px" src="images/img702b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 32.</span></td></tr></table> + +<p><i>Every elliptical involution has +the property that there are two +definite points in the plane from +which any two conjugate points +are seen under a right angle.</i></p> + +<p>At the same time the following +problem has been solved:</p> + +<p>To determine the centre and +also the point corresponding +to any given point in an elliptical involution of which two pairs of +conjugate points are given.</p> + +<p>§ 81. <i>Involution Range on a Conic.</i>—By the aid of § 53, the points +on a conic may be made to correspond to those on a line, so that the +row of points on the conic is projective to a row of points on a line. +We may also have two projective rows on the same conic, and these +will be in involution as soon as one point on the conic has the same +point corresponding to it all the same to whatever row it belongs. +An involution of points on a conic will have the property (as follows +from its definition, and from § 53) that the lines which join conjugate +points of the involution to any point on the conic are conjugate lines +of an involution in a pencil, and that a fixed tangent is cut by the +tangents at conjugate points on the conic in points which are again +conjugate points of an involution on the fixed tangent. For such +involution on a conic the following theorem holds:</p> + +<p><i>The lines which join corresponding points in an involution on a conic +all pass through a fixed point; and reciprocally, the points of intersection +of conjugate lines in an involution among tangents to a conic +lie on a line.</i></p> + +<table class="flt" style="float: left; width: 400px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:350px; height:288px" src="images/img702c.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 33</span></td></tr></table> + +<p>We prove the first part only. The involution is determined by +two pairs of conjugate points, say by A, A′ 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 <i>axis</i> 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.”</p> + +<p class="pt2 center sc" style="clear: both;">Involution Determined by a Conic on a Line.—Foci</p> + +<p>§ 82. The polars, with regard to a conic, of points in a row p form +a pencil P projective to the row (§ 66). This pencil cuts the base of +the row p in a projective row.</p> + +<p>If A is a point in the given row, A′ 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—</p> + +<p><i>A conic determines on every line in its plane an involution, in which +those points are conjugate which are also conjugate with regard to the +conic.</i></p> + +<p><i>If the line cuts the conic the involution is hyperbolic, the points of +intersection being the foci.</i></p> + +<p><i>If the line touches the conic the involution is parabolic, the two foci +coinciding at the point of contact.</i></p> + +<p><i>If the line does not cut the conic the involution is elliptic, having no +foci.</i></p> + +<p><span class="pagenum"><a name="page703" id="page703"></a>703</span></p> + +<p>If, on the other hand, we take a point P in the plane of a conic, +we get to each line a through P one conjugate line which joins P +to the pole of a. These pairs of conjugate lines through P form an +involution in the pencil at P. The focal rays of this involution are +the tangents drawn from P to the conic. This gives the theorem +reciprocal to the last, viz:—</p> + +<p><i>A conic determines in every pencil in its plane an involution, corresponding +lines being conjugate lines with regard to the conic.</i></p> + +<p><i>If the point is without the conic the involution is hyperbolic, the +tangents from the points being the focal rays.</i></p> + +<p><i>If the point lies on the conic the involution is parabolic, the tangent +at the point counting for coincident focal rays.</i></p> + +<p><i>If the point is within the conic the involution is elliptic, having no +focal rays.</i></p> + +<p>It will further be seen that the involution determined by a conic +on any line p is a section of the involution, which is determined by +the conic at the pole P of p.</p> + +<p>§ 83. <i>Foci.</i>—The centre of a pencil in which the conic determines +a circular involution is called a “focus” of the conic.</p> + +<p>In other words, a focus is such a point that every line through it is +perpendicular to its conjugate line. The polar to a focus is called a +<i>directrix</i> of the conic.</p> + +<p>From the definition it follows that <i>every focus lies on an axis</i>, for +the line joining a focus to the centre of the conic is a diameter to +which the conjugate lines are perpendicular; and <i>every line joining +two foci is an axis</i>, for the perpendiculars to this line through the foci +are conjugate to it. These conjugate lines pass through the pole of +the line, the pole lies therefore at infinity, and the line is a diameter, +hence by the last property an axis.</p> + +<p>It follows that all <i>foci lie on one axis</i>, for no line joining a point +in one axis to a point in the other can be an axis.</p> + +<p>As the conic determines in the pencil which has its centre at a focus +a circular involution, no tangents can be drawn from the focus to +the conic. Hence <i>each focus lies within a conic</i>; and <i>a directrix does +not cut the conic</i>.</p> + +<p>Further properties are found by the following considerations:</p> + +<p>§ 84. Through a point P one line p can be drawn, which is with +regard to a given conic conjugate to a given line q, viz. that line +which joins the point P to the pole of the line q. If the line q is made +to describe a pencil about a point Q, then the line p will describe a +pencil about P. These two pencils will be projective, for the line +p passes through the pole of q, and whilst q describes the pencil Q, +its pole describes a projective row, and this row is perspective to +the pencil P.</p> + +<p>We now take the point P on an axis of the conic, draw any line +p through it, and from the pole of p draw a perpendicular q to p. +Let q cut the axis in Q. Then, in the pencils of conjugate lines, +which have their centres at P and Q, the lines p and q are conjugate +lines at right angles to one another. Besides, to the axis as a ray +in either pencil will correspond in the other the perpendicular to the +axis (§ 72). The conic generated by the intersection of corresponding +lines in the two pencils is therefore the circle on PQ as diameter, +<i>so that every line in P is perpendicular to its corresponding line +in Q</i>.</p> + +<p>To every point P on an axis of a conic corresponds thus a point +Q, such that conjugate lines through P and Q are perpendicular.</p> + +<p>We shall show that these <i>point-pairs</i> P, Q <i>form an involution</i>. +To do this let us move P along the axis, and with it the line p, +keeping the latter parallel to itself. Then P describes a row, p a +perspective pencil (of parallels), and the pole of p a projective row. +At the same time the line q describes a pencil of parallels perpendicular +to p, and perspective to the row formed by the pole of p. The point +Q, therefore, where q cuts the axis, describes a row projective to the +row of points P. The two points P and Q describe thus two projective +rows on the axis; and not only does P as a point in the first +row correspond to Q, but also Q as a point in the first corresponds +to P. The two rows therefore form an involution. <i>The centre of +this involution, it is easily seen, is the centre of the conic.</i></p> + +<p><i>A focus of this involution has the property that any two conjugate +lines through it are perpendicular; hence, it is a focus to the conic.</i></p> + +<p>Such involution exists on each axis. But only one of these can +have foci, because all foci lie on the same axis. The involution on +one of the axes is elliptic, and appears (§ 80) therefore as the section +of two circular involutions in two pencils whose centres lie in the +other axis. These centres are foci, hence the one axis contains two +foci, the other axis none; <i>or every central conic has two foci which lie +on one axis equidistant from the centre</i>.</p> + +<p>The axis which contains the foci is called the <i>principal axis</i>; in +case of an hyperbola it is the axis which cuts the curve, because the +foci lie within the conic.</p> + +<p>In case of the parabola there is but one axis. The involution +on this axis has its centre at infinity. One focus is therefore at +infinity, the one focus only is finite. <i>A parabola has only one +focus.</i></p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:356px; height:210px" src="images/img703a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 34.</span></td></tr></table> + +<p>§ 85. If through any point P (fig. 34) on a conic the tangent PT +and the normal PN (<i>i.e.</i> the perpendicular to the tangent through +the point of contact) be drawn, these will be conjugate lines with +regard to the conic, and at right angles to each other. They will +therefore cut the principal axis in two points, which are conjugate +in the involution considered in § 84; hence they are harmonic +conjugates with regard to the foci. If therefore the two foci F<span class="su">1</span> and +F<span class="su">2</span> be joined to P, these lines will be harmonic with regard to the +tangent and normal. As the latter are perpendicular, they will +bisect the angles between the other pair. Hence—</p> + +<p><i>The lines joining any point on a conic to the two foci are equally +inclined to the tangent and normal at that point.</i></p> + +<p>In case of the parabola this becomes—</p> + +<p><i>The line joining any point on a parabola to the focus and the diameter +through the point, are equally inclined to the tangent and normal at +that point.</i></p> + +<p>From the definition of a focus it follows that—</p> + +<p><i>The segment of a tangent between the directrix and the point of +contact is seen from the focus belonging to the directrix under a right +angle</i>, because the lines joining the focus to the ends of this +segment are conjugate with regard to the conic, and therefore +perpendicular.</p> + +<p>With equal ease the following theorem is proved:</p> + +<p><i>The two lines which join the points of contact of two tangents each +to one focus, but not both to the same, are seen from the intersection of +the tangents under equal angles.</i></p> + +<p>§ 86. Other focal properties of a conic are obtained by the following +considerations:</p> + +<table class="flt" style="float: right; width: 370px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:317px; height:550px" src="images/img703b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 35.</span></td></tr></table> + +<p>Let F (fig. 35) be a focus to a conic, f the corresponding directrix, +A and B the points of contact of two tangents meeting at T, and P +the point where the +line AB cuts the directrix. +Then TF will be +the polar of P (because +polars of F and T meet +at P). Hence TF and +PF are conjugate lines +through a focus, and +therefore perpendicular. +They are further harmonic +conjugates with +regard to FA and FB +(§§ 64 and 13), so that +they bisect the angles +formed by these lines. +This by the way +proves—</p> + +<p><i>The segments between +the point of intersection +of two tangents to a conic +and their points of contact +are seen from a focus +under equal angles.</i></p> + +<p>If we next draw +through A and B lines +parallel to TF, then the +points A<span class="su">1</span>, B<span class="su">1</span> where +these cut the directrix +will be harmonic conjugates +with regard to P +and the point where FT +cuts the directrix. The +lines FT and FP bisect +therefore also the angles +between FA<span class="su">1</span> and FB<span class="su">1</span>. +From this it follows +easily that the triangles +FAA<span class="su">1</span> and FBB<span class="su">1</span> are +equiangular, and therefore similar, so that FA : AA<span class="su">1</span> = FB : BB<span class="su">1</span>.</p> + +<p>The triangles AA<span class="su">1</span>A<span class="su">2</span> and BB<span class="su">1</span>B<span class="su">2</span> formed by drawing perpendiculars +from A and B to the directrix are also similar, so that AA<span class="su">1</span> : AA<span class="su">2</span> = += BB<span class="su">1</span> : BB<span class="su">2</span>. This, combined with the above proportion, gives +FA : AA<span class="su">2</span> = FB : BB<span class="su">2</span>. Hence the theorem:</p> + +<p><i>The ratio of the distances of any point on a conic from a focus and +the corresponding directrix is constant.</i></p> + +<p>To determine this ratio we consider its value for a vertex on the +principal axis. In an ellipse the focus lies between the two vertices +on this axis, hence the focus is nearer to a vertex than to the corresponding +directrix. Similarly, in an hyperbola a vertex is nearer +<span class="pagenum"><a name="page704" id="page704"></a>704</span> +to the directrix than to the focus. In a parabola the vertex lies +halfway between directrix and focus.</p> + +<p>It follows in an ellipse the ratio between the distance of a point +from the focus to that from the directrix is less than unity, in the +parabola it equals unity, and in the hyperbola it is greater than +unity.</p> + +<p>It is here the same which focus we take, because the two foci +lie symmetrical to the axis of the conic. If now P is any point on +the conic having the distances r<span class="su">1</span> and r<span class="su">2</span> from the foci and the distances +d<span class="su">1</span> and d<span class="su">2</span> from the corresponding directrices, then r<span class="su">1</span>/d<span class="su">1</span> = r<span class="su">2</span>/d<span class="su">2</span> = e, +where e is constant. Hence also (r<span class="su">1</span> ± r<span class="su">2</span>) / (d<span class="su">1</span> ± d<span class="su">2</span>) = e.</p> + +<p>In the ellipse, which lies between the directrices, d<span class="su">1</span> + d<span class="su">2</span> is constant, +therefore also r<span class="su">1</span> + r<span class="su">2</span>. In the hyperbola on the other hand d<span class="su">1</span> − d<span class="su">2</span> is +constant, equal to the distance between the directrices, therefore +in this case r<span class="su">1</span> − r<span class="su">2</span> is constant.</p> + +<p>If we call the distances of a point on a conic from the focus its +focal distances we have the theorem:</p> + +<p><i>In an ellipse the sum of the focal distances is constant; and in an +hyperbola the difference of the focal distances is constant.</i></p> + +<p><i>This constant sum or difference equals in both cases the length of +the principal axis.</i></p> + +<p class="pt2 center sc">Pencil of Conics</p> + +<p>§ 87. Through four points A, B, C, D in a plane, of which no three +lie in a line, an infinite number of conics may be drawn, viz. through +these four points and any fifth one single conic. This system of +conics is called a pencil of conics. Similarly, all conics touching four +fixed lines form a system such that any fifth tangent determines one +and only one conic. We have here the theorems:</p> + +<table class="nobctr" summary="Contents"> +<tr><td class="tcl rb3" style="width: 50%;">The pairs of points in which any line is cut by a system of + conics through four fixed points are in involution.</td> + +<td class="tcl" style="width: 50%;">The pairs of tangents which can be drawn from a point to + a system of conics touching four fixed lines are in involution.</td></tr></table> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:371px; height:298px" src="images/img704a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 36.</span></td></tr></table> + +<p>We prove the first theorem only. Let ABCD (fig. 36) be the +four-point, then any line t will cut two opposite sides AC, BD in +the points E, E′, 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).</p> + +<p>If we cut these pencils by t we get</p> + +<p class="center">(EF, MN) = (F′E′, MN)</p> + +<p class="noind">or</p> + +<p class="center">(EF, MN) = (E′F′, NM).</p> + +<p>But this is, according to § 77 (7), the condition that M, N are +corresponding points in the involution determined by the point pairs +E, E′, 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.</p> + +<p>§ 88. There follow several important theorems:</p> + +<p><i>Through four points two, one, or no conics may be drawn which touch +any given line, according as the involution determined by the given +four-point on the line has real, coincident or imaginary foci.</i></p> + +<p><i>Two, one, or no conics may be drawn which touch four given lines +and pass through a given point, according as the involution determined +by the given four-side at the point has real, coincident or imaginary +focal rays.</i></p> + +<p>For the conic through four points which touches a given line has +its point of contact at a focus of the involution determined by the +four-point on the line.</p> + +<p>As a special case we get, by taking the line at infinity:</p> + +<p><i>Through four points of which none is at infinity either two or no +parabolas may be drawn.</i></p> + +<p>The problem of drawing a conic through four points and touching +a given line is solved by determining the points of contact on the +line, that is, by determining the foci of the involution in which the +line cuts the sides of the four-point. The corresponding remark +holds for the problem of drawing the conics which touch four lines +and pass through a given point.</p> + +<p class="pt2 center sc">Ruled Quadric Surfaces</p> + +<p>§ 89. We have considered hitherto projective rows which lie in +the same plane, in which case lines joining corresponding points +envelop a conic. We shall now consider projective rows whose +bases do not meet. In this case, corresponding points will be joined +by lines which do not lie in a plane, but on some surface, which +like every surface generated by lines is called a <i>ruled</i> surface. This +surface clearly contains the bases of the two rows.</p> + +<p>If the points in either row be joined to the base of the other, we +obtain two axial pencils which are also projective, those planes +being corresponding which pass through corresponding points in the +given rows. If A′, A be two corresponding points, α, α′ the planes in +the axial pencils passing through them, then AA′ will be the line +of intersection of the corresponding planes α, α′ and also the line +joining corresponding points in the rows.</p> + +<p>If we cut the whole figure by a plane this will cut the axial pencils +in two projective flat pencils, and the curve of the second order +generated by these will be the curve in which the plane cuts the +surface. Hence</p> + +<p><i>The locus of lines joining corresponding points in two projective +rows which do not lie in the same plane is a surface which contains the +bases of the rows, and which can also be generated by the lines of intersection +of corresponding planes in two projective axial pencils. This +surface is cut by every plane in a curve of the second order, hence either +in a conic or in a line-pair. No line which does not lie altogether on +the surface can have more than two points in common with the surface, +which is therefore said to be of the second order or is called a ruled +quadric surface.</i></p> + +<p>That no line which does not lie on the surface can cut the surface +in more than two points is seen at once if a plane be drawn through +the line, for this will cut the surface in a conic. It follows also that +a line which contains more than two points of the surface lies altogether +on the surface.</p> + +<p>§ 90. Through any point in space one line can always be drawn +cutting two given lines which do not themselves meet.</p> + +<p>If therefore three lines in space be given of which no two meet, +then through every point in either one line may be drawn cutting +the other two.</p> + +<p><i>If a line moves so that it always cuts three given lines of which no +two meet, then it generates a ruled quadric surface.</i></p> + +<p>Let a, b, c be the given lines, and p, q, r ... lines cutting them in the +points A, A′, 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.</p> + +<p>Of the lines p, q, r ... no two can meet, for otherwise the lines +a, b, c which cut them would also lie in their plane. There is a single +infinite number of them, for one passes through each point of a. +These lines are said to form a set of lines on the surface.</p> + +<p>If now three of the lines p, q, r be taken, then every line d cutting +them will have three points in common with the surface, and will +therefore lie altogether on it. This gives rise to a second set of lines +on the surface. From what has been said the theorem follows:</p> + +<p><i>A ruled quadric surface contains two sets of straight lines. Every +line of one set cuts every line of the other, but no two lines of the same +set meet.</i></p> + +<p><i>Any two lines of the same set may be taken as bases of two projective +rows, or of two projective pencils which generate the surface. They are +cut by the lines of the other set in two projective rows.</i></p> + +<p>The plane at infinity like every other plane cuts the surface either +in a conic proper or in a line-pair. In the first case the surface is +called an <i>Hyperboloid of one sheet</i>, in the second an <i>Hyperbolic +Paraboloid</i>.</p> + +<p>The latter may be generated by a line cutting three lines of which +one lies at infinity, that is, cutting two lines and remaining parallel +to a given plane.</p> + +<p class="pt2 center sc">Quadric Surfaces</p> + +<p>§ 91. The conics, the cones of the second order, and the ruled +quadric surfaces complete the figures which can be generated by +projective rows or flat and axial pencils, that is, by those aggregates +of elements which are of one dimension (§§ 5, 6). We shall +now consider the simpler figures which are generated by aggregates of +two dimensions. The space at our disposal will not, however, allow +us to do more than indicate a few of the results.</p> + +<p>§ 92. We establish a correspondence between the lines and planes +in pencils in space, or reciprocally between the points and lines in +two or more planes, but consider principally pencils.</p> + +<p>In two pencils we may either make planes correspond to planes +and lines to lines, or else planes to lines and lines to planes. If +hereby the condition be satisfied that to a flat, or axial, pencil +corresponds in the first case a projective flat, or axial, pencil, and in +the second a projective axial, or flat, pencil, the pencils are said to be +<i>projective</i> in the first case and <i>reciprocal</i> in the second.</p> + +<p>For instance, two pencils which join two points S<span class="su">1</span> and S<span class="su">2</span> to the +different points and lines in a given plane π are projective (and +in perspective position), if those lines and planes be taken as +<span class="pagenum"><a name="page705" id="page705"></a>705</span> +corresponding which meet the plane π in the same point or in the +same line. In this case every plane through both centres S<span class="su">1</span> and S<span class="su">2</span> +of the two pencils will correspond to itself. If these pencils are +brought into any other position they will be projective (but not +perspective).</p> + +<p><i>The correspondence between two projective pencils is uniquely +determined, if to four rays (or planes) in the one the corresponding +rays (or planes) in the other are given, provided that no three rays of +either set lie in a plane.</i></p> + +<p>Let a, b, c, d be four rays in the one, a′, 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′.</p> + +<p>Similarly the correspondence between two reciprocal pencils is +determined if for four rays in the one the corresponding planes in +the other are given.</p> + +<p>§ 93. We may now combine—</p> + +<div class="list"> +<p>1. Two reciprocal pencils.</p> +</div> + +<div class="list1"> +<p>Each ray cuts its corresponding plane in a point, the locus +of these points is a quadric surface.</p> +</div> + +<div class="list"> +<p>2. Two projective pencils.</p> +</div> + +<div class="list1"> +<p>Each plane cuts its corresponding plane in a line, but a +ray as a rule does not cut its corresponding ray. The +locus of points where a ray cuts its corresponding ray +is a twisted cubic. The lines where a plane cuts its +corresponding plane are secants.</p> +</div> + +<div class="list"> +<p>3. Three projective pencils.</p> +</div> + +<div class="list1"> +<p>The locus of intersection of corresponding planes is a +cubic surface.</p> +</div> + +<p>Of these we consider only the first two cases.</p> + +<p>§ 94. If two pencils are reciprocal, then to a plane in either corresponds +a line in the other, to a flat pencil an axial pencil, and so on. +Every line cuts its corresponding plane in a point. If S<span class="su">1</span> and S<span class="su">2</span> be +the centres of the two pencils, and P be a point where a line a<span class="su">1</span> in the +first cuts its corresponding plane α<span class="su">2</span>, <i>then the line</i> b<span class="su">2</span> <i>in the pencil</i> S<span class="su">2</span> +<i>which passes through</i> P <i>will meet its corresponding plane β<span class="su">1</span> in</i> P. For +b<span class="su">2</span> is a line in the plane α<span class="su">2</span>. The corresponding plane β<span class="su">1</span> must therefore +pass through the line a<span class="su">1</span>, hence through P.</p> + +<p>The points in which the lines in S<span class="su">1</span> cut the planes corresponding +to them in S<span class="su">2</span> are therefore the same as the points in which the lines +in S<span class="su">2</span> cut the planes corresponding to them in S<span class="su">1</span>.</p> + +<p><i>The locus of these points is a surface which is cut by a plane in a +conic or in a line-pair and by a line in not more than two points unless +it lies altogether on the surface. The surface itself is therefore called a +quadric surface, or a surface of the second order.</i></p> + +<p>To prove this we consider any line p in space.</p> + +<p>The flat pencil in S<span class="su">1</span> which lies in the plane drawn through p +and the corresponding axial pencil in S<span class="su">2</span> determine on p two projective +rows, and those points in these which coincide with their +corresponding points lie on the surface. But there exist only two, +or one, or no such points, unless every point coincides with its +corresponding point. In the latter case the line lies altogether on +the surface.</p> + +<p>This proves also that a plane cuts the surface in a curve of the +second order, as no line can have more than two points in common +with it. To show that this is a curve of the same kind as those +considered before, we have to show that it can be generated by +projective flat pencils. We prove first that this is true for any +plane through the centre of one of the pencils, and afterwards that +every point on the surface may be taken as the centre of such pencil. +Let then α<span class="su">1</span> be a plane through S<span class="su">1</span>. To the flat pencil in S<span class="su">1</span> which +it contains corresponds in S<span class="su">2</span> a projective axial pencil with axis +a<span class="su">2</span> and this cuts α<span class="su">1</span> in a second flat pencil. These two flat pencils +in α<span class="su">1</span> are projective, and, in general, neither concentric nor perspective. +They generate therefore a conic. But if the line a<span class="su">2</span> passes +through S<span class="su">1</span> the pencils will have S<span class="su">1</span> as common centre, and may +therefore have two, or one, or no lines united with their corresponding +lines. The section of the surface by the plane α<span class="su">1</span> will be accordingly +a line-pair or a single line, or else the plane α<span class="su">1</span> will have only the +point S<span class="su">1</span> in common with the surface.</p> + +<p>Every line l<span class="su">1</span> through S<span class="su">1</span> cuts the surface in two points, viz. first +in S<span class="su">1</span> and then at the point where it cuts its corresponding plane. +If now the corresponding plane passes through S<span class="su">1</span>, as in the case +just considered, then the two points where l<span class="su">1</span> cuts the surface coincide +at S<span class="su">1</span>, and the line is called a tangent to the surface with S<span class="su">1</span> as point +of contact. Hence if l<span class="su">1</span> be a tangent, it lies in that plane τ<span class="su">1</span> which +corresponds to the line S<span class="su">2</span>S<span class="su">1</span> as a line in the pencil S<span class="su">2</span>. The section +of this plane has just been considered. It follows that—</p> + +<p><i>All tangents to quadric surface at the centre of one of the reciprocal +pencils lie in a plane which is called the tangent plane to the surface +at that point as point of contact.</i></p> + +<p><i>To the line joining the centres of the two pencils as a line in one +corresponds in the other the tangent plane at its centre.</i></p> + +<p><i>The tangent plane to a quadric surface either cuts the surface in +two lines, or it has only a single line, or else only a single point in +common with the surface.</i></p> + +<p><i>In the first case the point of contact is said to be hyperbolic, in the +second parabolic, in the third elliptic.</i></p> + +<p>§ 95. It remains to be proved that every point S on the surface +may be taken as centre of one of the pencils which generate the +surface. Let S be any point on the surface Φ′ generated by the +reciprocal pencils S<span class="su">1</span> and S<span class="su">2</span>. We have to establish a reciprocal +correspondence between the pencils S and S<span class="su">1</span>, so that the surface +generated by them is identical with Φ. To do this we draw two +planes α<span class="su">1</span> and β<span class="su">1</span> through S<span class="su">1</span>, cutting the surface Φ in two conics +which we also denote by α<span class="su">1</span> and β<span class="su">1</span>. These conics meet at S<span class="su">1</span>, and +at some other point T where the line of intersection of α<span class="su">1</span> and β<span class="su">1</span> +cuts the surface.</p> + +<p>In the pencil S we draw some plane σ which passes through T, +but not through S<span class="su">1</span> or S<span class="su">2</span>. It will cut the two conics first at T, and +therefore each at some other point which we call A and B respectively. +These we join to S by lines a and b, and now establish the +required correspondence between the pencils S<span class="su">1</span> and S as follows:—To +S<span class="su">1</span>T shall correspond the plane σ, to the plane α<span class="su">1</span> the line a, and +to β<span class="su">1</span> the line b, hence to the flat pencil in α<span class="su">1</span> the axial pencil a. +These pencils are made projective by aid of the conic in α<span class="su">1</span>.</p> + +<p>In the same manner the flat pencil in β<span class="su">1</span> is made projective to the +axial pencil b by aid of the conic in β<span class="su">1</span>, corresponding elements being +those which meet on the conic. This determines the correspondence, +for we know for more than four rays in S<span class="su">1</span> the corresponding planes +in S. The two pencils S and S<span class="su">1</span> thus made reciprocal generate a +quadric surface Φ′, which passes through the point S and through +the two conics α<span class="su">1</span> and β<span class="su">1</span>.</p> + +<p>The two surfaces Φ and Φ′ have therefore the points S and S<span class="su">1</span> and +the conics α<span class="su">1</span> and β<span class="su">1</span> in common. To show that they are identical, +we draw a plane through S and S<span class="su">2</span>, cutting each of the conics α<span class="su">1</span> and +β<span class="su">1</span> in two points, which will always be possible. This plane cuts +Φ and Φ′ in two conics which have the point S and the points where +it cuts α<span class="su">1</span> and β<span class="su">1</span> in common, that is five points in all. The conics +therefore coincide.</p> + +<p>This proves that all those points P on Φ′ lie on Φ which have the +property that the plane SS<span class="su">2</span>P cuts the conics α<span class="su">1</span>, β<span class="su">1</span> in two points +each. If the plane SS<span class="su">2</span>P has not this property, then we draw a plane +SS<span class="su">1</span>P. This cuts each surface in a conic, and these conics have in +common the points S, S<span class="su">1</span>, one point on each of the conics α<span class="su">1</span>, β<span class="su">1</span>, and +one point on one of the conics through S and S<span class="su">2</span> which lie on both +surfaces, hence five points. They are therefore coincident, and our +theorem is proved.</p> + +<p>§ 96. The following propositions follow:—</p> + +<p><i>A quadric surface has at every point a tangent plane.</i></p> + +<p><i>Every plane section of a quadric surface is a conic or a line-pair.</i></p> + +<p><i>Every line which has three points in common with a quadric surface +lies on the surface.</i></p> + +<p><i>Every conic which has five points in common with a quadric surface +lies on the surface.</i></p> + +<p><i>Through two conics which lie in different planes, but have two points +in common, and through one external point always one quadric surface +may be drawn.</i></p> + +<p>§ 97. <i>Every plane which cuts a quadric surface in a line-pair is a +tangent plane.</i> For every line in this plane through the centre of +the line-pair (the point of intersection of the two lines) cuts the +surface in two coincident points and is therefore a tangent to the +surface, <i>the centre of the line-pair being the point of contact</i>.</p> + +<p><i>If a quadric surface contains a line, then every plane through this +line cuts the surface in a line-pair (or in two coincident lines).</i> For +this plane cannot cut the surface in a conic. Hence:—</p> + +<p><i>If a quadric surface contains one line p then it contains an infinite +number of lines, and through every point</i> Q <i>on the surface, one line</i> +q <i>can be drawn which cuts</i> p. For the plane through the point Q +and the line p cuts the surface in a line-pair which must pass through +Q and of which p is one line.</p> + +<p><i>No two such lines</i> q <i>on the surface can meet</i>. For as both meet p +their plane would contain p and therefore cut the surface in a +triangle.</p> + +<p><i>Every line which cuts three lines</i> q <i>will be on the surface</i>; for it +has three points in common with it.</p> + +<p><i>Hence the quadric surfaces which contain lines are the same as the +ruled quadric surfaces considered in</i> §§ 89-93, but with one important +exception. In the last investigation we have left out of consideration +the possibility of a plane having only one line (two coincident +lines) in common with a quadric surface.</p> + +<p>§ 98. To investigate this case we suppose first that there is one +point A on the surface through which two different lines a, b can be +drawn, which lie altogether on the surface.</p> + +<p>If P is any other point on the surface which lies neither on a nor +b, then the plane through P and a will cut the surface in a second +line a′ 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′ <i>may</i> +coincide, but then they must coincide with PA.</p> + +<p>If this happens for one point P, it happens for every other point +Q. For if two different lines could be drawn through Q, then by the +same reasoning the line PQ would be altogether on the surface, +hence two lines would be drawn through P against the assumption. +From this follows:—</p> + +<p><i>If there is one point on a quadric surface through which one, but only +one, line can be drawn on the surface, then through every point one line</i> +<span class="pagenum"><a name="page706" id="page706"></a>706</span> +<i>can be drawn, and all these lines meet in a point. The surface is a cone +of the second order</i>.</p> + +<p><i>If through one point on a quadric surface, two, and only two, lines +can be drawn on the surface, then through every point two lines may +be drawn, and the surface is ruled quadric surface.</i></p> + +<p><i>If through one point on a quadric surface no line on the surface can +be drawn, then the surface contains no lines.</i></p> + +<p>Using the definitions at the end of § 95, we may also say:—</p> + +<p><i>On a quadric surface the points are all hyperbolic, or all parabolic, +or all elliptic.</i></p> + +<p>As an example of a quadric surface with elliptical points, we +mention the sphere which may be generated by two reciprocal +pencils, where to each line in one corresponds the plane perpendicular +to it in the other.</p> + +<p>§ 99. <i>Poles and Polar Planes.</i>—The theory of poles and polars +with regard to a conic is easily extended to quadric surfaces.</p> + +<p>Let P be a point in space not on the surface, which we suppose +not to be a cone. On every line through P which cuts the surface +in two points we determine the harmonic conjugate Q of P with +regard to the points of intersection. Through one of these lines we +draw two planes α and β. The locus of the points Q in α is a line a, +the polar of P with regard to the conic in which α cuts the surface. +Similarly the locus of points Q in β is a line b. This cuts a, because +the line of intersection of α and β contains but one point Q. The +locus of all points Q therefore is a plane. <i>This plane is called the +polar plane of the point</i> P, <i>with regard to the quadric surface. If</i> P +<i>lies on the surface we take the tangent plane of P as its polar.</i></p> + +<p>The following propositions hold:—</p> + +<p>1. <i>Every point has a polar plane</i>, which is constructed by drawing +the polars of the point with regard to the conics in which two planes +through the point cut the surface.</p> + +<p>2. <i>If</i> Q <i>is a point in the polar of</i> P, <i>then</i> P <i>is a point in the polar +of</i> Q, because this is true with regard to the conic in which a plane +through PQ cuts the surface.</p> + +<p>3. <i>Every plane is the polar plane of one point, which is called the +Pole of the plane.</i></p> + +<p>The pole to a plane is found by constructing the polar planes of +three points in the plane. Their intersection will be the pole.</p> + +<p>4. <i>The points in which the polar plane of P cuts the surface are +points of contact of tangents drawn from P to the surface</i>, as is easily +seen. Hence:—</p> + +<p>5. <i>The tangents drawn from a point P to a quadric surface form a +cone of the second order</i>, for the polar plane of P cuts it in a conic.</p> + +<p>6. <i>If the pole describes a line a, its polar plane will turn about +another line</i> a′, as follows from 2. <i>These lines a and a′ are said to be +conjugate with regard to the surface.</i></p> + +<p>§ 100. The pole of the line at infinity is called the <i>centre</i> of the +surface. If it lies at the infinity, the plane at infinity is a tangent +plane, and the surface is called a <i>paraboloid</i>.</p> + +<p><i>The polar plane to any point at infinity passes through the centre, +and is called a diametrical plane.</i></p> + +<p><i>A line through the centre is called a diameter. It is bisected at the +centre. The line conjugate to it lies at infinity.</i></p> + +<p><i>If a point moves along a diameter its polar plane turns about the +conjugate line at infinity</i>; that is, <i>it moves parallel to itself, its centre +moving on the first line.</i></p> + +<p><i>The middle points of parallel chords lie in a plane</i>, viz. in the polar +plane of the point at infinity through which the chords are drawn.</p> + +<p><i>The centres of parallel sections lie in a diameter which is a line +conjugate to the line at infinity in which the planes meet.</i></p> + +<p class="pt2 center sc">Twisted Cubics</p> + +<p>§ 101. If two pencils with centres S<span class="su">1</span> and S<span class="su">2</span> are made projective, +then to a ray in one corresponds a ray in the other, to a plane a +plane, to a flat or axial pencil a projective flat or axial pencil, and +so on.</p> + +<p>There is a double infinite number of lines in a pencil. We shall +see that a single infinite number of lines in one pencil meets its +corresponding ray, and that the points of intersection form a curve +in space.</p> + +<p>Of the double infinite number of planes in the pencils each will +meet its corresponding plane. This gives a system of a double +infinite number of lines in space. We know (§ 5) that there is a +quadruple infinite number of lines in space. From among these we +may select those which satisfy one or more given conditions. The +systems of lines thus obtained were first systematically investigated +and classified by Plücker, in his <i>Geometrie des Raumes</i>. He uses the +following names:—</p> + +<p>A <i>treble infinite</i> number of lines, that is, all lines which satisfy one +condition, are said to form a <i>complex of lines</i>; <i>e.g.</i> all lines cutting +a given line, or all lines touching a surface.</p> + +<p>A <i>double infinite</i> number of lines, that is, all lines which satisfy +two conditions, or which are common to two complexes, are said to +form a <i>congruence of lines</i>; <i>e.g.</i> all lines in a plane, or all lines +cutting two curves, or all lines cutting a given curve twice.</p> + +<p>A <i>single infinite</i> number of lines, that is, all lines which satisfy +three conditions, or which belong to three complexes, form a <i>ruled +surface</i>; <i>e.g.</i> one set of lines on a ruled quadric surface, or developable +surfaces which are formed by the tangents to a curve.</p> + +<p>It follows that all lines in which corresponding planes in two +projective pencils meet form a congruence. We shall see this congruence +consists of all lines which cut a twisted cubic twice, or of +all <i>secants</i> to a twisted cubic.</p> + +<p>§ 102. Let l<span class="su">1</span> be the line S<span class="su">1</span>S<span class="su">2</span> as a line in the pencil S<span class="su">1</span>. To it +corresponds a line l<span class="su">2</span> in S<span class="su">2</span>. <i>At each of the centres two corresponding +lines meet.</i> The two axial pencils with l<span class="su">1</span> and l<span class="su">2</span> as axes are projective, +and, as, their axes meet at S<span class="su">2</span>, the intersections of corresponding +planes form a cone of the second order (§ 58), with S<span class="su">2</span> as +centre. If π<span class="su">1</span> and π<span class="su">2</span> be corresponding planes, then their intersection +will be a line p<span class="su">2</span> which passes through S<span class="su">2</span>. Corresponding to it in +S<span class="su">1</span> will be a line p<span class="su">1</span> which lies in the plane π<span class="su">1</span>, and which therefore +meets p<span class="su">2</span> at some point P. Conversely, if p<span class="su">2</span> be any line in S<span class="su">2</span> which +meets its corresponding line p<span class="su">1</span> at a point P, then to the plane l<span class="su">2</span>p<span class="su">2</span> +will correspond the plane l<span class="su">1</span>p<span class="su">1</span>, that is, the plane S<span class="su">1</span>S<span class="su">2</span>P. These +planes intersect in p<span class="su">2</span>, so that p<span class="su">2</span> is a line on the quadric cone generated +by the axial pencils l<span class="su">1</span> and l<span class="su">2</span>. Hence:—</p> + +<p><i>All lines in one pencil which meet their corresponding lines in the +other form a cone of the second order which has its centre at the centre +of the first pencil, and passes through the centre of the second.</i></p> + +<p>From this follows that the points in which corresponding rays +meet lie on two cones of the second order which have the ray joining +their centres in common, and form therefore, together with the line +S<span class="su">1</span>S<span class="su">2</span> or l<span class="su">1</span>, the intersection of these cones. Any plane cuts each of the +cones in a conic. These two conics have necessarily that point in +common in which it cuts the line l<span class="su">1</span>, and therefore besides either +one or three other points. It follows that the curve is of the third +order as a plane may cut it in three, but not in more than three, +points. Hence:—</p> + +<p><i>The locus of points in which corresponding lines on two projective +pencils meet is a curve of the third order or a “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.</i></p> + +<p><i>A line belonging to the congruence determined by the pencils is a +secant of the cubic; it has two, or one, or no points in common with +this cubic, and is called accordingly a secant proper, a tangent, or a +secant improper of the cubic.</i> A secant improper may be considered, +to use the language of coordinate geometry, as a secant with +imaginary points of intersection.</p> + +<p>§ 103. If a<span class="su">1</span> and a<span class="su">2</span> be any two corresponding lines in the two +pencils, then corresponding planes in the axial pencils having a<span class="su">1</span> and +a<span class="su">2</span> as axes generate a ruled quadric surface. If P be any point on +the cubic k, and if p<span class="su">1</span>, p<span class="su">2</span> be the corresponding rays in S<span class="su">1</span> and S<span class="su">2</span> which +meet at P, then to the plane a<span class="su">1</span>p<span class="su">1</span> in S<span class="su">1</span> corresponds a<span class="su">2</span>p<span class="su">2</span> in S<span class="su">2</span>. These +therefore meet in a line through P.</p> + +<p>This may be stated thus:—</p> + +<p><i>Those secants of the cubic which cut a ray</i> a<span class="su">1</span>, <i>drawn through the +centre</i> S<span class="su">1</span> <i>of one pencil, form a ruled quadric surface which passes through +both centres, and which contains the twisted cubic</i> k. <i>Of such surfaces +an infinite number exists. Every ray through</i> S<span class="su">1</span> <i>or</i> S<span class="su">2</span> <i>which is not a +secant determines one of them.</i></p> + +<p>If, however, the rays a<span class="su">1</span> and a<span class="su">2</span> are secants meeting at A, then the +ruled quadric surface becomes a cone of the second order, having +A as centre. Or <i>all lines of the congruence which pass through a point +on the twisted cubic k form a cone of the second order</i>. In other words, +the projection of a twisted cubic from any point in the curve on to +any plane is a conic.</p> + +<p>If a<span class="su">1</span> is not a secant, but made to pass through any point Q in +space, the ruled quadric surface determined by a<span class="su">1</span> will pass through +Q. <i>There will therefore be one line of the congruence passing through</i> +Q, <i>and only one.</i> For if two such lines pass through Q, then the lines +S<span class="su">1</span>Q and S<span class="su">2</span>Q will be corresponding lines; hence Q will be a point on +the cubic k, and an infinite number of secants will pass through it. +Hence:—</p> + +<p><i>Through every point in space not on the twisted cubic one and only +one secant to the cubic can be drawn.</i></p> + +<p>§ 104. The fact that all the secants through a point on the cubic +form a quadric cone shows that the centres of the projective pencils +generating the cubic are not distinguished from any other points on +the cubic. If we take any two points S, S′ 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:—</p> + +<p><i>On a twisted cubic any two points may be taken as centres of projective +pencils which generate the cubic, corresponding planes being +those which meet on the same secant.</i></p> + +<p>Of the two projective pencils at S and S′ 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 α in S will be cut by its +corresponding plane α′ always in the same secant a. Whilst S′ +moves along the curve the plane α′ will turn about a, describing an +axial pencil.</p> + +<p><span class="pagenum"><a name="page707" id="page707"></a>707</span></p> + +<p><span class="sc">Authorities.</span>—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 <i>Geometrie der +Lage</i>, and to this excellent work those readers are referred who wish +for a more exhaustive treatment of the subject. Other works +especially valuable as showing the development of the subject are: +Monge, <i>Géométrie descriptive</i>: Carnot, <i>Géométrie de position</i> +(1803), containing a theory of transversals; Poncelet’s great work +<i>Traité des propriétés projectives des figures</i> (1822); Möbins, <i>Barycentrischer +Calcul</i> (1826); Steiner, <i>Abhängigkeit geometrischer +Gestalten</i> (1832), containing the first full discussion of the projective +relations between rows, pencils, &c.; Von Staudt, <i>Geometrie der +Lage</i> (1847) and <i>Beiträge zur Geometrie der Lage</i> (1856-1860), in +which a system of geometry is built up from the beginning without +any reference to number, so that ultimately a number itself gets +a geometrical definition, and in which imaginary elements are +systematically introduced into pure geometry; Chasles, <i>Aperçu +historique</i> (1837), in which the author gives a brilliant account of +the progress of modern geometrical methods, pointing out the +advantages of the different purely geometrical methods as compared +with the analytical ones, but without taking as much account of +the German as of the French authors; Id., <i>Rapport sur les progrès +de la géométrie</i> (1870), a continuation of the <i>Aperçu</i>; Id., <i>Traité de +géométrie supérieure</i> (1852); Cremona, <i>Introduzione ad una teoria +geometrica delle curve piane</i> (1862) and its continuation <i>Preliminari +di una teoria geometrica delle superficie</i> (German translations by +Curtze). As more elementary books, we mention: Cremona, +<i>Elements of Projective Geometry</i>, translated from the Italian by +C. Leudesdorf (2nd ed., 1894); J.W. Russell, <i>Pure Geometry</i> (2nd ed., +1905).</p> +</div> +<div class="author">(O. H.)</div> + +<p class="pt2 center sc">III. Descriptive Geometry</p> + +<p>This branch of geometry is concerned with the methods for +representing solids and other figures in three dimensions by +drawings in one plane. The most important method is that +which was invented by Monge towards the end of the 18th +century. It is based on parallel projections to a plane by rays +perpendicular to the plane. Such a projection is called orthographic +(see <span class="sc"><a href="#artlinks">Projection</a></span>, § 18). If the plane is horizontal the +projection is called the plan of the figure, and if the plane is +vertical the elevation. In Monge’s method a figure is represented +by its plan and elevation. It is therefore often called drawing +in plan and elevation, and sometimes simply orthographic +projection.</p> + +<div class="condensed"> +<p>§ 1. We suppose then that we have two planes, one horizontal, +the other vertical, and these we call the planes of plan and of elevation +respectively, or the horizontal and the vertical plane, and +denote them by the letters π<span class="su">1</span> and π<span class="su">2</span>. Their line of intersection is +called the axis, and will be denoted by xy.</p> + +<p>If the surface of the drawing paper is taken as the plane of the +plan, then the vertical plane will be the plane perpendicular to it +through the axis xy. To bring this also into the plane of the drawing +paper we turn it about the axis till it coincides with the horizontal +plane. This process of turning one plane down till it coincides with +another is called <i>rabatting</i> one to the other. Of course there is no +necessity to have one of the two planes horizontal, but even when +this is not the case it is convenient to retain the above names.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:466px; height:205px" src="images/img707a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 37.</span></td> +<td class="caption"><span class="sc">Fig. 38.</span></td></tr></table> + +<p>The whole arrangement will be better understood by referring to +fig. 37. A point A in space is there projected by the perpendicular +AA<span class="su">1</span> and AA<span class="su">2</span> to the planes π<span class="su">1</span> and π<span class="su">2</span> so that A<span class="su">1</span> and A<span class="su">2</span> are the +horizontal and vertical projections of A.</p> + +<p>If we remember that a line is perpendicular to a plane that is +perpendicular to every line in the plane if only it is perpendicular +to any two intersecting lines in the plane, we see that the axis which +is perpendicular both to AA<span class="su">1</span> and to AA<span class="su">2</span> is also perpendicular to +A<span class="su">1</span>A<span class="su">0</span> and to A<span class="su">2</span>A<span class="su">0</span> because these four lines are all in the same plane. +Hence, if the plane π<span class="su">2</span> be turned about the axis till it coincides with +the plane π<span class="su">1</span>, then A<span class="su">2</span>A<span class="su">0</span> will be the continuation of A<span class="su">1</span>A<span class="su">0</span>. This +position of the planes is represented in fig. 38, in which the line A<span class="su">1</span>A<span class="su">2</span> +is perpendicular to the axis x.</p> + +<p>Conversely any two points A<span class="su">1</span>, A<span class="su">2</span> in a line perpendicular to the +axis will be the projections of some point in space when the plane +π<span class="su">2</span> is turned about the axis till it is perpendicular to the plane π<span class="su">1</span>, +because in this position the two perpendiculars to the planes π<span class="su">1</span> +and π<span class="su">2</span> through the points A<span class="su">1</span> and A<span class="su">2</span> will be in a plane and therefore +meet at some point A.</p> + +<p><i>Representation of Points.</i>—We have thus the following method +of representing in a single plane the position of points in space:—<i>we +take in the plane a line xy as the axis, and then any pair of points +A<span class="su">1</span>, A<span class="su">2</span> in the plane on a line perpendicular to the axis represent a +point A in space</i>. If the line A<span class="su">1</span>A<span class="su">2</span> cuts the axis at A<span class="su">0</span>, and if at A<span class="su">1</span> +a perpendicular be erected to the plane, then the point A will be in +it at a height A<span class="su">1</span>A = A<span class="su">0</span>A<span class="su">2</span> above the plane. This gives the position +of the point A relative to the plane π<span class="su">1</span>. In the same way, if in a +perpendicular to π<span class="su">2</span> through A<span class="su">2</span> a point A be taken such that A<span class="su">2</span>A = +A<span class="su">0</span>A<span class="su">1</span>, then this will give the point A relative to the plane π<span class="su">2</span>.</p> + +<table class="flt" style="float: right; width: 230px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:175px; height:182px" src="images/img707b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 39.</span></td></tr></table> + +<p>§ 2. The two planes π<span class="su">1</span>, π<span class="su">2</span> in their original position divide space +into four parts. These are called the four quadrants. We suppose +that the plane π<span class="su">2</span> is turned as indicated in +fig. 37, so that the point P comes to Q and +R to S, then the quadrant in which the +point A lies is called the first, and we say +that in the first quadrant a point lies above +the horizontal and in front of the vertical +plane. Now we go round the axis in the +sense in which the plane π<span class="su">2</span> is turned and +come in succession to the second, third +and fourth quadrant. In the second a +point lies above the plane of the plan and +behind the plane of elevation, and so on. +In fig. 39, which represents a side view of +the planes in fig. 37 the quadrants are +marked, and in each a point with its projection +is taken. Fig. 38 shows how these are represented when +the plane π<span class="su">2</span> is turned down. We see that</p> + +<p><i>A point lies in the first quadrant if the plan lies below, the elevation +above the axis; in the second if plan and elevation both lie above; in +the third if the plan lies above, the elevation below; in the fourth if plan +and elevation both lie below the axis.</i></p> + +<p><i>If a point lies in the horizontal plane</i>, its elevation lies in the axis +and the plan coincides with the point itself. <i>If a point lies in the +vertical plane</i>, its plan lies in the axis and the elevation coincides +with the point itself. <i>If a point lies in the axis</i>, both its plan and +elevation lie in the axis and coincide with it.</p> + +<p>Of each of these propositions, which will easily be seen to be true, +the converse holds also.</p> + +<p>§ 3. <i>Representation of a Plane.</i>—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 π<span class="su">1</span> and π<span class="su">2</span>, and no plane would appear different from any other. +But any plane α cuts each of the planes π<span class="su">1</span>, π<span class="su">2</span> in a line. These are +called the traces of the plane. They cut each other in the axis at the +point where the latter cuts the plane α.</p> + +<p><i>A plane is determined by its two traces, which are two lines that meet +on the axis</i>, and, conversely, <i>any two lines which meet on the axis +determine a plane</i>.</p> + +<p><i>If the plane is parallel to the axis its traces are parallel to the axis.</i> +Of these one may be at infinity; then the plane will cut one of the +planes of projection at infinity and will be parallel to it. Thus a +plane parallel to the horizontal plane of the plan has only one finite +trace, viz. that with the plane of elevation.</p> + +<table class="flt" style="float: right; width: 300px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:246px; height:207px" src="images/img707c.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 40.</span></td></tr></table> + +<p><i>If the plane passes through the axis both its traces coincide with the +axis.</i> This is the only case in which the representation of the plane +by its two traces fails. A third plane of projection is therefore +introduced, which is best taken perpendicular to the other two. +We call it simply the third plane and denote it by π<span class="su">3</span>. As it is +perpendicular to π<span class="su">1</span>, it may be +taken as the plane of elevation, +its line of intersection γ with π<span class="su">1</span> +being the axis, and be turned +down to coincide with π<span class="su">1</span>. This +is represented in fig. 40. OC is +the axis xy whilst OA and OB +are the traces of the third plane. +They lie in one line γ. The plane +is rabatted about γ to the horizontal +plane. A plane α through +the axis xy will then show in it +a trace α<span class="su">3</span>. In fig. 40 the lines OC +and OP will thus be the traces +of a plane through the axis xy, +which makes an angle POQ with +the horizontal plane.</p> + +<p>We can also find the trace +which any other plane makes +with π<span class="su">3</span>. In rabatting the plane +π<span class="su">3</span> its trace OB with the plane π<span class="su">2</span> will come to the position OD. +Hence a plane β having the traces CA and CB will have with the +third plane the trace β<span class="su">3</span>, or AD if OD = OB.</p> + +<p><span class="pagenum"><a name="page708" id="page708"></a>708</span></p> + +<p>It also follows immediately that—</p> + +<p><i>If a plane α is perpendicular to the horizontal plane, then every point +in it has its horizontal projection in the horizontal trace of the plane</i>, +as all the rays projecting these points lie in the plane itself.</p> + +<p><i>Any plane which is perpendicular to the horizontal plane has its +vertical trace perpendicular to the axis.</i></p> + +<p><i>Any plane which is perpendicular to the vertical plane has its horizontal +trace perpendicular to the axis and the vertical projections of all +points in the plane lie in this trace.</i></p> + +<p>§ 4. <i>Representation of a Line.</i>—A line is determined either by two +points in it or by two planes through it. We get accordingly two +representations of it either by projections or by traces.</p> + +<p>First.—<i>A line a is represented by its projections</i> a<span class="su">1</span> <i>and</i> a<span class="su">2</span> <i>on the +two planes</i> π<span class="su">1</span> <i>and</i> π<span class="su">2</span>. These may be any two lines, for, bringing +the planes π<span class="su">1</span>, π<span class="su">2</span> into their original position, the planes through these +lines perpendicular to π<span class="su">1</span> and π<span class="su">2</span> respectively will intersect in some line +a which has a<span class="su">1</span>, a<span class="su">2</span> as its projections.</p> + +<p>Secondly.—<i>A line a is represented by its traces—that is, by the points +in which it cuts the two planes</i> π<span class="su">1</span>, π<span class="su">2</span>. Any two points may be taken +as the traces of a line in space, for it is determined when the planes +are in their original position as the line joining the two traces. This +representation becomes undetermined if the two traces coincide in +the axis. In this case we again use a third plane, or else the projections +of the line.</p> + +<p>The fact that there are different methods of representing points +and planes, and hence two methods of representing lines, suggests +the principle of duality (section ii., <i>Projective Geometry</i>, § 41). It +is worth while to keep this in mind. It is also worth remembering +that traces of planes or lines always lie in the planes or lines which +they represent. Projections do not as a rule do this excepting when +the point or line projected lies in one of the planes of projection.</p> + +<p>Having now shown how to represent points, planes and lines, +we have to state the conditions which must hold in order that these +elements may lie one in the other, or else that the figure formed by +them may possess certain metrical properties. It will be found that +the former are very much simpler than the latter.</p> + +<p>Before we do this, however, we shall explain the notation used; +for it is of great importance to have a systematic notation. We +shall denote points in space by capitals A, B, C; planes in space +by Greek letters α, β, γ; lines in space by small letters a, b, c; +horizontal projections by suffixes 1, like A<span class="su">1</span>, a<span class="su">1</span>; vertical projections +by suffixes 2, like A<span class="su">2</span>, a<span class="su">2</span>; traces by single and double dashes α′ α″, +a′, a″. Hence P<span class="su">1</span> will be the horizontal projection of a point P in +space; a line a will have the projections a<span class="su">1</span>, a<span class="su">2</span> and the traces a′ and +a″; a plane α has the traces α′ and α″.</p> + +<p>§ 5. <i>If a point lies in a line, the projections of the point lie in the +projections of the line.</i></p> + +<p><i>If a line lies in a plane, the traces of the line lie in the traces of the +plane.</i></p> + +<p>These propositions follow at once from the definitions of the +projections and of the traces.</p> + +<p>If a point lies in two lines its projections must lie in the projections +of both. Hence</p> + +<p><i>If two lines, given by their projections, intersect, the intersection of +their <span class="correction" title="amended from plans">planes</span> and the intersection of their elevations must lie in a line +perpendicular to the axis</i>, because they must be the projections of +the point common to the two lines.</p> + +<p>Similarly—<i>If two lines given by their traces lie in the same plane +or intersect, then the lines joining their horizontal and vertical traces +respectively must meet on the axis</i>, because they must be the traces +of the plane through them.</p> + +<p>§ 6. <i>To find the projections of a line which joins two points A, B +given by their projections</i> A<span class="su">1</span>, A<span class="su">2</span> <i>and</i> B<span class="su">1</span>, B<span class="su">2</span>, we join A<span class="su">1</span>, B<span class="su">1</span> and A<span class="su">2</span>, +B<span class="su">2</span>; these will be the projections required. For example, the +traces of a line are two points in the line whose projections are +known or at all events easily found. They are the traces themselves +and the feet of the perpendiculars from them to the axis.</p> + +<table class="flt" style="float: right; width: 300px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:246px; height:207px" src="images/img708a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 41.</span></td></tr></table> + +<p>Hence <i>if</i> a′ a″ (fig. 41) <i>are the traces of a line a, and if the perpendiculars +from them cut the axis in</i> P <i>and</i> Q <i>respectively, then the +line</i> a′Q <i>will be the horizontal and</i> +a″P <i>the vertical projection of the +line</i>.</p> + +<p>Conversely, if the projections +a<span class="su">1</span>, a<span class="su">2</span> of a line are given, and if +these cut the axis in Q and P +respectively, then <i>the perpendiculars</i> +Pa′ <i>and</i> Qa″ <i>to the axis +drawn through these points cut the +projections</i> a<span class="su">1</span> <i>and</i> a<span class="su">2</span> <i>in the traces</i> +a′ <i>and</i> a″.</p> + +<p><i>To find the line of intersection of +two planes</i>, we observe that this +line lies in both planes; its traces +must therefore lie in the traces +of both. Hence the points where the horizontal traces of the given +planes meet will be the horizontal, and the point where the vertical +traces meet the vertical trace of the line required.</p> + +<p>§ 7. <i>To decide whether a point</i> A, <i>given by its projections, lies in +a plane α, given by its traces</i>, we draw a line p by joining A to some +point in the plane α and determine its traces. If these lie in the +traces of the plane, then the line, and therefore the point A, lies +in the plane; otherwise not. This is conveniently done by joining +A<span class="su">1</span> to some point p′ in the trace α′; this gives p<span class="su">1</span>; and the point +where the perpendicular from p′ to the axis cuts the latter we join +to A<span class="su">2</span>; this gives p<span class="su">2</span>. If the vertical trace of this line lies in the +vertical trace of the plane, then, and then only, does the line p, and +with it the point A, lie in the plane α.</p> + +<p>§ 8. <i>Parallel planes have parallel traces</i>, because parallel planes are +cut by any plane, hence also by π<span class="su">1</span> and by π<span class="su">2</span>, in parallel lines.</p> + +<p><i>Parallel lines have parallel projections</i>, because points at infinity +are projected to infinity.</p> + +<p><i>If a line is parallel to a plane, then lines through the traces of the +line and parallel to the traces of the plane must meet on the axis</i>, because +these lines are the traces of a plane parallel to the given plane.</p> + +<p>§ 9. <i>To draw a plane through two intersecting lines or through two +parallel lines</i>, we determine the traces of the lines; the lines joining +their horizontal and vertical traces respectively will be the horizontal +and vertical traces of the plane. They will meet, at a finite point +or at infinity, on the axis if the lines do intersect.</p> + +<p><i>To draw a plane through a line and a point without the line</i>, we +join the given point to any point in the line and determine the plane +through this and the given line.</p> + +<p><i>To draw a plane through three points which are not in a line</i>, we +draw two of the lines which each join two of the given points and +draw the plane through them. If the traces of all three lines AB, +BC, CA be found, these must lie in two lines which meet on the +axis.</p> + +<p>§ 10. We have in the last example got more points, or can easily +get more points, than are necessary for the determination of the +figure required—in this case the traces of the plane. This will +happen in a great many constructions and is of considerable importance. +It may happen that some of the points or lines obtained +are not convenient in the actual construction. The horizontal +traces of the lines AB and AC may, for instance, fall very near +together, in which case the line joining them is not well defined. +Or, one or both of them may fall beyond the drawing paper, so that +they are practically non-existent for the construction. In this case +the traces of the line BC may be used. Or, if the vertical traces of +AB and AC are both in convenient position, so that the vertical +trace of the required plane is found and one of the horizontal traces +is got, then we may join the latter to the point where the vertical +trace cuts the axis.</p> + +<p>The draughtsman must remember that the lines which he draws +are not mathematical lines without thickness, and therefore every +drawing is affected by some errors. It is therefore very desirable +to be able constantly to check the latter. Such checks always +present themselves when the same result can be obtained by different +constructions, or when, as in the above case, some lines must meet +on the axis, or if three points must lie in a line. A careful draughtsman +will always avail himself of these checks.</p> + +<p>§ 11. <i>To draw a plane through a given point parallel to a given +plane α</i>, we draw through the point two lines which are parallel to +the plane α, and determine the plane through them; or, as we +know that the traces of the required plane are parallel to those of +the given one (§ 8), we need only draw one line l through the point +parallel to the plane and find one of its traces, say the vertical trace +l″; a line through this parallel to the vertical trace of α will be the +vertical trace β″ of the required plane β, and a line parallel to the +horizontal trace of α meeting β″ on the axis will be the horizontal +trace β′.</p> + +<table class="flt" style="float: right; width: 340px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:286px; height:183px" src="images/img708b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 42.</span></td></tr></table> + +<p>Let A<span class="su">1</span> A<span class="su">2</span> (fig. 42) be the given point, α′ α″ the given plane, a +line l<span class="su">1</span> through A<span class="su">1</span>, parallel to α′ and a horizontal line l<span class="su">2</span> through +A<span class="su">2</span> will be the projections of +a line l through A parallel +to the plane, because the +horizontal plane through +this line will cut the plane +α in a line c which has its +horizontal projection c<span class="su">1</span> +parallel to α′.</p> + +<p>§ 12. We now come to +the metrical properties of +figures.</p> + +<p><i>A line is perpendicular +to a plane if the projections +of the line are perpendicular +to the traces of the plane.</i> We prove it for the horizontal +projection. If a line p is perpendicular to a plane α, every plane +through p is perpendicular to α; hence also the vertical plane which +projects the line p to p<span class="su">1</span>. As this plane is perpendicular both to the +horizontal plane and to the plane α, it is also perpendicular to their +intersection—that is, to the horizontal trace of α. It follows that +every line in this projecting plane, therefore also p<span class="su">1</span>, the plan of p, is +perpendicular to the horizontal trace of α.</p> + +<p><i>To draw a plane through a given point A perpendicular to a given +line p</i>, we first draw through some point O in the axis lines γ′, γ″ +perpendicular respectively to the projections p<span class="su">1</span> and p<span class="su">2</span> of the given +line. These will be the traces of a plane γ which is perpendicular +to the given line. We next draw through the given point A a plane +parallel to the plane γ; this will be the plane required.</p> + +<p><span class="pagenum"><a name="page709" id="page709"></a>709</span></p> + +<p>Other metrical properties depend on the determination of the real +size or shape of a figure.</p> + +<p>In general the projection of a figure differs both in size and shape +from the figure itself. But figures in a plane parallel to a plane +of projection will be identical with their projections, and will thus +be given in their true dimensions. In other cases there is the +problem, constantly recurring, either to find the true shape and +size of a plane figure when plan and elevation are given, or, conversely, +to find the latter from the known true shape of the figure +itself. To do this, the plane is turned about one of its traces till it +is laid down into that plane of projection to which the trace belongs. +This is technically called rabatting the plane respectively into the +plane of the plan or the elevation. As there is no difference in the +treatment of the two cases, we shall consider only the case of rabatting +a plane α into the plane of the plan. The plan of the figure is +a parallel (orthographic) projection of the figure itself. The results +of parallel projection (see <span class="sc"><a href="#artlinks">Projection</a></span>, §§ 17 and 18) may therefore +now be used. The trace α′ 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 α′ and that corresponding +lines meet on this trace. We also see that the correspondence is +completely determined if we know for one point or one line in the +plan the corresponding point or line in the rabatted plane.</p> + +<p>Before, however, we treat of this we consider some special cases.</p> + +<p>§ 13. <i>To determine the distance between two points A, B given by their +projections</i> A<span class="su">1</span>, B<span class="su">1</span> <i>and</i> A<span class="su">2</span>, B<span class="su">2</span>, <i>or, in other words, to determine the true +length of a line the plan and elevation of which are given.</i></p> + +<table class="flt" style="float: right; width: 310px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:242px; height:225px" src="images/img709a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 43.</span></td></tr> +<tr><td class="figright1"><img style="width:257px; height:216px" src="images/img709b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 44.</span></td></tr></table> + +<p><i>Solution.</i>—The two points A, B in space lie vertically above their +plans A<span class="su">1</span>, B<span class="su">1</span> (fig. 43) and A<span class="su">1</span>A = A<span class="su">0</span>A<span class="su">2</span>, B<span class="su">1</span>B = +B<span class="su">0</span>B<span class="su">2</span>. The four points +A, B, A<span class="su">1</span>, B<span class="su">1</span> therefore form a plane +quadrilateral on the base A<span class="su">1</span>B<span class="su">1</span> and +having right angles at the base. +This plane we rabatt about A<span class="su">1</span>B<span class="su">1</span> +by drawing A<span class="su">1</span>A and B<span class="su">1</span>B perpendicular +to A<span class="su">1</span>B<span class="su">1</span> and making +A<span class="su">1</span>A = A<span class="su">0</span>A<span class="su">2</span>, B<span class="su">1</span>B = B<span class="su">0</span>B<span class="su">2</span>. Then +AB will give the length required.</p> + +<p>The construction might have +been performed in the elevation +by making A<span class="su">2</span>A = A<span class="su">0</span>A<span class="su">1</span> and +B<span class="su">2</span>B = B<span class="su">0</span>B<span class="su">1</span> on lines perpendicular +to A<span class="su">2</span>B<span class="su">2</span>. Of course AB must have +the same length in both cases.</p> + +<p>This figure may be turned into +a model. Cut the paper along +A<span class="su">1</span>A, AB and BB<span class="su">1</span>, and fold the +piece A<span class="su">1</span>ABB<span class="su">1</span> over along A<span class="su">1</span>B<span class="su">1</span> till +it stands upright at right angles to the horizontal plane. The points +A, B will then be in their true position in space relative to +π<span class="su">1</span>. Similarly +if B<span class="su">2</span>BAA<span class="su">2</span> be cut out and turned along A<span class="su">2</span>B<span class="su">2</span> through a right +angle we shall get AB in its true position relative to the plane +π<span class="su">2</span>. Lastly we fold the whole plane of the paper along the axis x +till the plane π<span class="su">2</span> is at right angles to π<span class="su">1</span>. In this position the two +sets of points AB will coincide if the drawing has been accurate.</p> + +<p>Models of this kind can be made in many cases and their construction +cannot be too highly recommended in order to realize +orthographic projection.</p> + +<p>§ 14. <i>To find the angle between two given lines</i> a, b <i>of which the +projections</i> a<span class="su">1</span>, b<span class="su">1</span> <i>and</i> a<span class="su">2</span>, b<span class="su">2</span> <i>are given.</i></p> + +<p><i>Solution.</i>—Let a<span class="su">1</span>, b<span class="su">1</span> (fig. 44) meet in P<span class="su">1</span>, a<span class="su">2</span>, b<span class="su">2</span> in T, then if the line +P<span class="su">1</span>T is not perpendicular to the axis the two lines will not meet. In +this case we draw a line parallel +to b to meet the line a. This is +easiest done by drawing first the +line P<span class="su">1</span>P<span class="su">2</span> perpendicular to the +axis to meet a<span class="su">2</span> in P<span class="su">2</span>, and then +drawing through P<span class="su">2</span> a line c<span class="su">2</span> +parallel to b<span class="su">2</span>; then b<span class="su">1</span>, c<span class="su">2</span> will be +the projections of a line c which +is parallel to b and meets a in P. +The plane α 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 α′ of their +plane. On rabatting the point +P comes to a point S on the line +P<span class="su">1</span>Q perpendicular to a′c′, so +that QS = QP. But QP is the hypotenuse of a triangle PP<span class="su">1</span>Q with +a right angle P<span class="su">1</span>. This we construct by making QR = P<span class="su">0</span>P<span class="su">2</span>; then +P<span class="su">1</span>R = PQ. The lines a′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.</p> + +<p><i>To determine the angle between a line and a plane</i>, we draw through +any point in the line a perpendicular to the plane (§ 12) and determine +the angle between it and the given line. The complement of this +angle is the required one.</p> + +<p><i>To determine the angle between two planes</i>, we draw through any +point two lines perpendicular to the two planes and determine the +angle between the latter as above.</p> + +<p>In special cases it is simpler to determine at once the angle between +the two planes by taking a plane section perpendicular to the intersection +of the two planes and rabatt this. This is especially the +case if one of the planes is the horizontal or vertical plane of projection.</p> + +<p>Thus in fig. 45 the angle P<span class="su">1</span>QR is the angle which the plane α +makes with the horizontal plane.</p> + +<p>§ 15. We return to the general case of rabatting a plane α of +which the traces α′ α″ are given.</p> + +<table class="flt" style="float: left; width: 350px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:300px; height:238px" src="images/img709c.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 45.</span></td></tr></table> + +<p>Here it will be convenient to determine first the position which +the trace α″—which is a line in α—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 α″ +originally coincident with +its elevation P<span class="su">2</span>, and if O +is the point where α″ cuts +the axis xy, so that O is +also in α′, then the point P +will after rabatting the +plane assume such a position +that OP = OP<span class="su">2</span>. At +the same time the plan is +an orthographic projection +of the plane α. Hence the +line joining P to the plan +P<span class="su">1</span> will after rabatting be +perpendicular to α′. But +P<span class="su">1</span> is known; it is the foot +of the perpendicular from +P<span class="su">2</span> to the axis xy. We +draw therefore, to find P, +from P<span class="su">1</span> a perpendicular P<span class="su">1</span>Q to α′ and find on it a point P such that +OP = OP<span class="su">2</span>. Then the line OP will be the position of α″ when +rabatted. This line corresponds therefore to the plan of +α″—that +is, to the axis xy, corresponding points on these lines being those +which lie on a perpendicular to α′.</p> + +<p>We have thus one pair of corresponding lines and can now find +for any point B<span class="su">1</span> in the plan the corresponding point B in the rabatted +plane. We draw a line through B<span class="su">1</span>, say B<span class="su">1</span>P<span class="su">1</span>, cutting α′ in C. To it +corresponds the line CP, and the point where this is cut by the projecting +ray through B<span class="su">1</span>, perpendicular to α′, is the required point B.</p> + +<p>Similarly any figure in the rabatted plane can be found when the +plan is known; but this is usually found in a different manner +without any reference to the general theory of parallel projection. +As this method and the reasoning employed for it have their peculiar +advantages, we give it also.</p> + +<p>Supposing the planes π<span class="su">1</span> and π<span class="su">2</span> to be in their positions in space +perpendicular to each other, we take a section of the whole figure +by a plane perpendicular to the trace α′ about which we are going +to rabatt the plane α. Let this section pass through the point Q in +α′. Its traces will then be the lines QP<span class="su">1</span> and P<span class="su">1</span>P<span class="su">2</span> (fig. 9). These +will be at right angles, and will therefore, together with the section +QP<span class="su">2</span> of the plane α, form a right-angled triangle QP<span class="su">1</span>P<span class="su">2</span> with the +right angle at P<span class="su">1</span>, and having the sides P<span class="su">1</span>Q and P<span class="su">1</span>P<span class="su">2</span> which both +are given in their true lengths. This triangle we rabatt about its +base P<span class="su">1</span>Q, making P<span class="su">1</span>R = P<span class="su">1</span>P<span class="su">2</span>. The line QR will then give the true +length of the line QP in space. If now the plane α be turned about +α′ the point P will describe a circle about Q as centre with radius +QP = QR, in a plane perpendicular to the trace α′. Hence when the +plane α has been rabatted into the horizontal plane the point P will +lie in the perpendicular P<span class="su">1</span>Q to α′, so that QP = QR.</p> + +<p>If A<span class="su">1</span> is the plan of a point A in the plane α, and if A<span class="su">1</span> lies in QP<span class="su">1</span>, +then the point A will lie vertically above A<span class="su">1</span> in the line QP. On +turning down the triangle QP<span class="su">1</span>P<span class="su">2</span>, the point A will come to A<span class="su">0</span>, the +line A<span class="su">1</span>A<span class="su">0</span> being perpendicular to QP<span class="su">1</span>. Hence A will be a point in +QP such that QA = QA<span class="su">0</span>.</p> + +<p>If B<span class="su">1</span> is the plan of another point, but such that A<span class="su">1</span>B<span class="su">1</span> is parallel +to α′, then the corresponding line AB will also be parallel to α′. +Hence, if through A a line AB be drawn parallel to α′, and B<span class="su">1</span>B +perpendicular to α′, then their intersection gives the point B. Thus +of any point given in plan the real position in the plane α, when +rabatted, can be found by this second method. This is the one +most generally given in books on geometrical drawing. The first +method explained is, however, in most cases preferable as it gives +the draughtsman a greater variety of constructions. It requires a +somewhat greater amount of theoretical knowledge.</p> + +<p>If instead of our knowing the plan of a figure the latter is itself +given, then the process of finding the plan is the reverse of the +above and needs little explanation. We give an example.</p> + +<p>§ 16. <i>It is required to draw the plan and elevation of a polygon of +which the real shape and position in a given plane α are known.</i></p> + +<table class="flt" style="float: right; width: 410px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:358px; height:511px" src="images/img710a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 46.</span></td></tr></table> + +<p>We first rabatt the plane α (fig. 46) as before so that P<span class="su">1</span> comes to +P, hence OP<span class="su">1</span> to OP. Let the given polygon in α be the figure +ABCDE. We project, not the vertices, but the sides. To project +the line AB, we produce it to cut α′ in F and OP in G, and draw GG<span class="su">1</span> +perpendicular to α′; then G<span class="su">1</span> corresponds to G, therefore FG<span class="su">1</span> to FG. +In the same manner we might project all the other sides, at least +<span class="pagenum"><a name="page710" id="page710"></a>710</span> +those which cut OF and OP in convenient points. It will be best, +however, first to produce all the sides to cut OP and α′ and then to +draw all the projecting rays through A, B, C ... perpendicular to +α′, and in the same +direction the lines +G, G<span class="su">1</span>, &c. By +drawing FG we +get the points A<span class="su">1</span>, +B<span class="su">1</span> on the projecting +ray through A +and B. We then +join B to the point +M where BC produced +meets the +trace α′. This +gives C<span class="su">1</span>. So we +go on till we have +found E<span class="su">1</span>. The +line A<span class="su">1</span> E<span class="su">1</span> must +then meet AE in +α′, and this gives +a check. If one +of the sides cuts +α′ 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 α′.</p> + +<p>Having got the plan we easily find the elevation. The elevation +of G is above G<span class="su">1</span> in α″, and that of F is at F<span class="su">2</span> in the axis. This +gives the elevation F<span class="su">2</span>G<span class="su">2</span> of FG and in it we get A<span class="su">2</span>B<span class="su">2</span> in the verticals +through A<span class="su">1</span> and B<span class="su">1</span>. As a check we have OG = OG<span class="su">2</span>. Similarly the +elevation of the other sides and vertices are found.</p> + +<p>§ 17. We proceed to give some applications of the above principles +to the representation of solids and of the solution of problems +connected with them.</p> + +<p><i>Of a pyramid are given its base, the length of the perpendicular from +the vertex to the base, and the point where this perpendicular cuts the +base; it is required first to develop the whole surface of the pyramid +into one plane, and second to determine its section by a plane which +cuts the plane of the base in a given line and makes a given angle +with it.</i></p> + +<p>1. As the planes of projection are not given we can take them as we +like, and we select them in such a manner that the solution becomes +as simple as possible. We take the plane of the base as the horizontal +plane and the vertical plane perpendicular to the plane of the section. +Let then (fig. 47) ABCD be the base of the pyramid, V<span class="su">1</span> the plan of +the vertex, then the elevations of A, B, C, D will be in the axis at +A<span class="su">2</span>, B<span class="su">2</span>, C<span class="su">2</span>, D<span class="su">2</span>, and the vertex at some point V<span class="su">2</span> above V<span class="su">1</span> at a known +distance from the axis. The lines V<span class="su">1</span>A, V<span class="su">1</span>B, &c., will be the plans +and the lines V<span class="su">2</span>A<span class="su">2</span>, V<span class="su">2</span>B<span class="su">2</span>, &c., the elevations of the edges of the +pyramid, of which thus plan and elevation are known.</p> + +<p>We develop the surface into the plane of the base by turning +each lateral face about its lower edge into the horizontal plane by +the method used in § 14. If one face has been turned down, say +ABV to ABP, then the point Q to which the vertex of the next +face BCV comes can be got more simply by finding on the line +V<span class="su">1</span>Q perpendicular to BC the point Q such that BQ = BP, for these +lines represent the same edge BV of the pyramid. Next R is +found by making CR = CQ, and so on till we have got the last vertex—in +this case S. The fact that AS must equal AP gives a convenient +check.</p> + +<p>2. The plane α whose section we have to determine has its horizontal +trace given perpendicular to the axis, and its vertical trace +makes the given angle with the axis. This determines it. To find +the section of the pyramid by this plane there are two methods +applicable: we find the sections of the plane either with the faces +or with the edges of the pyramid. We use the latter.</p> + +<p>As the plane α is perpendicular to the vertical plane, the trace +α″ contains the projection of every figure in it; the points E<span class="su">2</span>, F<span class="su">2</span>, +G<span class="su">2</span>, H<span class="su">2</span> where this trace cuts the elevations of the edges will therefore +be the elevations of the points where the edges cut α. From these +we find the plans E<span class="su">1</span>, F<span class="su">1</span>, G<span class="su">1</span>, H<span class="su">1</span>, and by joining them the plan +of the section. If from E<span class="su">1</span>, F<span class="su">1</span> lines be drawn perpendicular to AB, +these will determine the points E, F on the developed face in which +the plane α 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 α be rabatted to the plan, we get the real shape of the +section as shown in the figure in EFGH. This is done easily by +making F<span class="su">0</span>F = OF<span class="su">2</span>, &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 +α with a hole cut out representing the true section be bent along the +trace α′ till its edge coincides with α″, the edges of the hole ought to +coincide with the lines EF, FG, &c., on the faces.</p> + +<p>§ 18. Polyhedra like the pyramid in § 17 are represented by the +projections of their edges and vertices. But solids bounded by +curved surfaces, or surfaces themselves, cannot be thus represented.</p> + +<p>For a surface we may use, as in case of the plane, its traces—that +is, the curves in which it cuts the planes of projection. We may +also project points and curves on the surface. A ray cuts the +surface generally in more than one point; hence it will happen +that some of the rays touch the surface, if two of these points coincide. +The points of contact of these rays will form some curve on the surface, +and this will appear from the centre of projection as the boundary +of the surface or of part of the surface. The outlines of all surfaces +of solids which we see about us are formed by the points at which +rays through our eye touch the surface. The projections of these +contours are therefore best adapted to give an idea of the shape of a +surface.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:481px; height:566px" src="images/img710b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 47.</span></td></tr></table> + +<p>Thus the tangents drawn from any finite centre to a sphere form +a right circular cone, and this will be cut by any plane in a conic. +It is often called the projection of a sphere, but it is better called +the contour-line of the sphere, as it is the boundary of the projections +of all points on the sphere.</p> + +<p>If the centre is at infinity the tangent cone becomes a right +circular cylinder touching the sphere along a great circle, and if +the projection is, as in our case, orthographic, then the section of +this cone by a plane of projection will be a circle equal to the great +circle of the sphere. We get such a circle in the plan and another in +the elevation, their centres being plan and elevation of the centre of +the sphere.</p> + +<p>Similarly the rays touching a cone of the second order will lie +in two planes which pass through the vertex of the cone, the contour-line +of the projection of the cone consists therefore of two lines +meeting in the projection of the vertex. These may, however, +be invisible if no real tangent rays can be drawn from the centre of +projection; and this happens when the ray projecting the centre +of the vertex lies within the cone. In this case the traces of the +cone are of importance. Thus in representing a cone of revolution +with a vertical axis we get in the plan a circular trace of the surface +whose centre is the plan of the vertex of the cone, and in the elevation +the contour, consisting of a pair of lines intersecting in the elevation +of the vertex of the cone. The circle in the plan and the pair of lines +in the elevation do not determine the surface, for an infinite number +of surfaces might be conceived which pass through the circular trace +and touch two planes through the contour lines in the vertical plane. +The surface becomes only completely defined if we write down to +the figure that it shall represent a cone. The same holds for all +<span class="pagenum"><a name="page711" id="page711"></a>711</span> +surfaces. Even a plane is fully represented by its traces only under +the silent understanding that the traces are those of a plane.</p> + +<p>§ 19. Some of the simpler problems connected with the representation +of surfaces are the determination of plane sections and of +the curves of intersection of two such surfaces. The former is +constantly used in nearly all problems concerning surfaces. Its +solution depends of course on the nature of the surface.</p> + +<p>To determine the curve of intersection of two surfaces, we take a +plane and determine its section with each of the two surfaces, +rabatting this plane if necessary. This gives two curves which lie +in the same plane and whose intersections will give us points on +both surfaces. It must here be remembered that two curves in +space do not necessarily intersect, hence that the points in which +their projections intersect are not necessarily the projections of +points common to the two curves. This will, however, be the case +if the two curves lie in a common plane. By taking then a number +of plane sections of the surfaces we can get as many points on their +curve of intersection as we like. These planes have, of course, to +be selected in such a way that the sections are curves as simple as +the case permits of, and such that they can be easily and accurately +drawn. Thus when possible the sections should be straight lines +or circles. This not only saves time in drawing but determines all +points on the sections, and therefore also the points where the two +curves meet, with equal accuracy.</p> + +<p>§ 20. We give a few examples how these sections have to be +selected. A cone is cut by every plane through the vertex in lines, +and if it is a cone of revolution by planes perpendicular to the +axis in circles.</p> + +<p>A cylinder is cut by every plane parallel to the axis in lines, and +if it is a cylinder of revolution by planes perpendicular to the axis +in circles.</p> + +<p>A sphere is cut by every plane in a circle.</p> + +<p>Hence in case of two cones situated anywhere in space we take +sections through both vertices. These will cut both cones in lines. +Similarly in case of two cylinders we may take sections parallel to +the axis of both. In case of a sphere and a cone of revolution with +vertical axis, horizontal sections will cut both surfaces in circles +whose plans are circles and whose elevations are lines, whilst vertical +sections through the vertex of the cone cut the latter in lines and +the sphere in circles. To avoid drawing the projections of these +circles, which would in general be ellipses, we rabatt the plane and +then draw the circles in their real shape. And so on in other cases.</p> + +<p>Special attention should in all cases be paid to those points in +which the tangents to the projection of the curve of intersection are +parallel or perpendicular to the axis x, or where these projections +touch the contour of one of the surfaces.</p> +</div> +<div class="author">(O. H.)</div> + +<p class="pt2 center sc">IV. Analytical Geometry</p> + +<p>1. In the name <i>geometry</i> there is a lasting record that the +science had its origin in the knowledge that two distances may +be compared by measurement, and in the idea that measurement +must be effectual in the dissociation of different directions as well +as in the comparison of distances in the same direction. The +distance from an observer’s eye of an object seen would be +specified as soon as it was ascertained that a rod, straight to the +eye and of length taken as known, could be given the direction +of the line of vision, and had to be moved along it a certain +number of times through lengths equal to its own in order to +reach the object from the eye. Moreover, if a field had for two +of its boundaries lines straight to the eye, one running from south +to north and the other from west to east, the position of a point +in the field would be specified if the rod, when directed west, +had to be shifted from the point one observed number of times +westward to meet the former boundary, and also, when directed +south, had to be shifted another observed number of times +southward to meet the latter. Comparison by measurement, +the beginning of geometry, involved counting, the basis of arithmetic; +and the science of number was marked out from the +first as of geometrical importance.</p> + +<p>But the arithmetic of the ancients was inadequate as a science +of number. Though a length might be recognized as known +when measurement certified that it was so many times a standard +length, it was not every length which could be thus specified +in terms of the same standard length, even by an arithmetic +enriched with the notion of fractional number. The idea of +possible incommensurability of lengths was introduced into +Europe by Pythagoras; and the corresponding idea of irrationality +of number was absent from a crude arithmetic, while there +were great practical difficulties in the way of its introduction. +Hence perhaps it arose that, till comparatively modern times, +appeal to arithmetical aid in geometrical reasoning was in all +possible ways restrained. Geometry figured rather as the helper +of the more difficult science of arithmetic.</p> + +<p>2. It was reserved for algebra to remove the disabilities of +arithmetic, and to restore the earliest ideas of the land-measurer +to the position of controlling ideas in geometrical investigation. +This unified science of pure number made comparatively little +headway in the hands of the ancients, but began to receive +due attention shortly after the revival of learning. It expresses +whole classes of arithmetical facts in single statements, gives +to arithmetical laws the form of equations involving symbols +which may mean any known or sought numbers, and provides +processes which enable us to analyse the information given by an +equation and derive from that equation other equations, which +express laws that are in effect consequences or causes of a law +started from, but differ greatly from it in form. Above all, for +present purposes, it deals not only with integral and fractional +number, but with number regarded as capable of continuous +growth, just as distance is capable of continuous growth. The +difficulty of the arithmetical expression of irrational number, +a difficulty considered by the modern school of analysts to have +been at length surmounted (see <span class="sc"><a href="#artlinks">Function</a></span>), is not vital to it. +It can call the ratio of the diagonal of a square to a side, for +instance, or that of the circumference of a circle to a diameter, +a number, and let a or x denote that number, just as properly +as it may allow either letter to denote any rational number +which may be greater or less than the ratio in question by a +difference less than any minute one we choose to assign.</p> + +<p>Counting only, and not the counting of objects, is of the essence +of arithmetic, and of algebra. But it is lawful to count objects, +and in particular to count equal lengths by measure. The +widened idea is that even when a or x is an irrational number +we may speak of a or x unit lengths by measure. We may give +concrete interpretation to an algebraical equation by allowing +its terms all to mean numbers of times the same unit length, +or the same unit area, or &c. and in any equation lawfully +derived from the first by algebraical processes we may do the +same. Descartes in his <i>Géométrie</i> (1637) was the first to systematize +the application of this principle to the inherent first +notions of geometry; and the methods which he instituted have +become the most potent methods of all in geometrical research. +It is hardly too much to say that, when known facts as to a +geometrical figure have once been expressed in algebraical +terms, all strictly consequential facts as to the figure can be +deduced by almost mechanical processes. Some may well be +unexpected consequences; and in obtaining those of which +there has been suggestion beforehand the often bewildering +labour of constant attention to the figure is obviated. These +are the methods of what is now called <i>analytical</i>, or sometimes +<i>algebraical</i>, <i>geometry</i>.</p> + +<p>3. The modern use of the term “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 <i>analysis</i>. Or again, being asked to prove a +theorem A, we ascertain that it must be true if another theorem +B is, that B must be if C is, and so on, thus eventually finding +that the theorem A is the consequence, through a chain of intermediaries, +of a theorem Z of which the establishment is easy. +This geometrical analysis is not the subject of the present article; +but in the reasoning from form to form of an equation or system +<span class="pagenum"><a name="page712" id="page712"></a>712</span> +of equations, with the object of basing the algebraical proof +of a geometrical fact on other facts of a more obvious character, +the same logic is utilized, and the name “analytical geometry” +is thus in part explained.</p> + +<p>4. In algebra real positive number was alone at first dealt +with, and in geometry actual signless distance. But in algebra +it became of importance to say that every equation of the first +degree has a root, and the notion of negative number was introduced. +The negative unit had to be defined as what can be +added to the positive unit and produce the sum zero. The +corresponding notion was readily at hand in geometry, where +it was clear that a unit distance can be measured to the left +or down from the farther end of a unit distance already measured +to the right or up from a point O, with the result of reaching O +again. Thus, to give full interpretation in geometry to the +algebraically negative, it was only necessary to associate distinctness +of sign with oppositeness of direction. Later it was discovered +that algebraical reasoning would be much facilitated, and that +conclusions as to the real would retain all their soundness, if a pair +of imaginary units ±√−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 √−1, with a, b, +real. The corresponding enrichment could be given to geometry, +with corresponding advantages and the same absence of danger, +and this was done. On a line of measurement of distance we +contemplate as existing, not only an infinite continuum of points +at real distances from an origin of measurement O, but a doubly +infinite continuum of points, all but the singly infinite continuum +of real ones imaginary, and imaginary in conjugate pairs, a +conjugate pair being at imaginary distances from O, which have +a real arithmetic and a real geometric mean. To geometry +enriched with this conception all algebra has its application.</p> + +<p>5. Actual geometry is one, two or three-dimensional, <i>i.e.</i> +lineal, plane or solid. In one-dimensional geometry positions +and measurements in a single line only are admitted. Now +descriptive constructions for points in a line are impossible +without going out of the line. It has therefore been held that +there is a sense in which no science of geometry strictly confined +to one dimension exists. But an algebra of one variable can be +applied to the study of distances along a line measured from a +chosen point on it, so that the idea of construction as distinct +from measurement is not essential to a one-dimensional geometry +aided by algebra. In geometry of two dimensions, the +flat of the land-measurer, the passage from one point O to any +other point, can be effected by two successive marches, one east +or west and one north or south, and, as will be seen, an algebra +of two variables suffices for geometrical exploitation. In +geometry of three dimensions, that of space, any point can be +reached from a chosen one by three marches, one east or west, +one north or south, and one up or down; and we shall see that +an algebra of three variables is all that is necessary. With +three dimensions actual geometry stops; but algebra can supply +any number of variables. Four or more variables have been +used in ways analogous to those in which one, two and three +variables are used for the purposes of one, two and three-dimensional +geometry, and the results have been expressed in +quasi-geometrical language on the supposition that a higher +space can be conceived of, though not realized, in which four +independent directions exist, such that no succession of marches +along three of them can effect the same displacement of a point +as a march along the fourth; and similarly for higher numbers +than four. Thus analytical, though not actual, geometries exist +for four and more dimensions. They are in fact algebras furnished +with nomenclature of a geometrical cast, suggested by convenient +forms of expression which actual geometry has, in return for +benefits received, conferred on algebras of one, two and three +variables.</p> + +<p>We will confine ourselves to the dimensions of actual geometry, +and will devote no space to the one-dimensional, except incidentally +as existing within the two-dimensional. The analytical +method will now be explained for the cases of two and three +dimensions in succession. The form of it originated by Descartes, +and thence known as Cartesian, will alone be considered in much +detail.</p> + +<div class="condensed"> +<p class="pt2 center">I. <i>Plane Analytical Geometry.</i></p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:520px; height:209px" src="images/img712.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 48.</span></td> +<td class="caption"><span class="sc">Fig. 49.</span></td></tr></table> + +<p>6. <i>Coordinates.</i>—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.</p> + +<p>It is necessary to attend to signs; x has one sign or the other +according as the point P is on one side or the other of the axis of y, +and y one sign or the other according as P is on one side or the other +of the axis of x. Using the letters N, E, S, W, as in a map, and +considering the plane as divided into four quadrants by the axes, +the signs are usually taken to be:</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc">x</td> <td class="tcc">y</td> <td class="tcc">For quadrant</td></tr> + +<tr><td class="tcc">+</td> <td class="tcc">+</td> <td class="tcc">N   E</td></tr> +<tr><td class="tcc">+</td> <td class="tcc">−</td> <td class="tcc">S   E</td></tr> +<tr><td class="tcc">−</td> <td class="tcc">+</td> <td class="tcc">N   W</td></tr> +<tr><td class="tcc">−</td> <td class="tcc">−</td> <td class="tcc">S   W</td></tr> +</table> + +<p class="noind">A point is referred to as the point (a, b), when its coordinates are +x = a, y = b. A point may be fixed, or it may be variable, <i>i.e.</i> be +regarded for the time being as free to move in the plane. The +coordinates (x, y) of a variable point are algebraic variables, and are +said to be “current coordinates.”</p> + +<p>The axes of x and y are usually (as in fig. 48) taken at right angles +to one another, and we then speak of them as rectangular axes, +and of x and y as “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 ω +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.</p> + +<p>7. <i>Equations and loci.</i> If (x, y) is the point P, and if we are +given that x = 0, we are told that, in fig. 48 or fig. 49, the point M lies +at O, whatever value y may have, <i>i.e.</i> we are told the one fact that +P lies on the axis of y. Conversely, if P lies anywhere on the axis +of y, we have always OM = 0, <i>i.e.</i> x = 0. Thus the equation x = 0 is +one satisfied by the coordinates (x, y) of every point in the axis of y, +and not by those of any other point. We say that x = 0 is the +equation of the axis of y, and that the axis of y is the locus represented +by the equation x = 0. Similarly y = 0 is the equation of the +axis of x. An equation x = a, where a is a constant, expresses that +P lies on a parallel to the axis of y through a point M on the axis +of x such that OM = a. Every line parallel to the axis of y has an +equation of this form. Similarly, every line parallel to the axis of x +has an equation of the form y = b, where b is some definite constant.</p> + +<p>These are simple cases of the fact that a single equation in the +current coordinates of a variable point (x, y) imposes one limitation +on the freedom of that point to vary. The coordinates of a point +<span class="pagenum"><a name="page713" id="page713"></a>713</span> +taken at random in the plane will, as a rule, not satisfy the equation, +but infinitely many points, and in most cases infinitely many real +ones, have coordinates which do satisfy it, and these points are +exactly those which lie upon some locus of one dimension, a straight +line or more frequently a curve, which is said to be represented by +the equation. Take, for instance, the equation y = mx, where m +is a given constant. It is satisfied by the coordinates of every point +P, which is such that, in fig. 48, the distance MP, with its proper sign, +is m times the distance OM, with its proper sign, <i>i.e.</i> by the coordinates +of every point in the straight line through O which we +arrive at by making a line, originally coincident with x′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 = φ(x), where φ(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>i.e.</i> let x take all real values +from −∞ to ∞. 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 = φ(x) +represents it. More generally still, take an equation f(x, y) = 0 +which involves both x and y under a functional form. Any particular +value given to x in it produces from it an equation for the determination +of a value or values of y, which go with that value of x in specifying +a point or points (x, y), of which the coordinates satisfy the +equation f(x, y) = 0. Here again, as x takes all values, the point or +points describe a path or paths, which constitute a locus represented +by the equation. Except when y enters to the first degree only in +f(x, y), it is not to be expected that all the values of y, determined +as going with a chosen value of x, will be necessarily real; indeed +it is not uncommon for all to be imaginary for some ranges of values +of x. The locus may largely consist of continua of imaginary +points; but the real parts of it constitute a real curve or real curves. +Note that we have to allow x to admit of all imaginary, as well as +of all real, values, in order to obtain all imaginary parts of the +locus.</p> + +<p>A locus or curve may be algebraically specified in another way; +viz. we may be given two equations x = f(θ), y = F(θ), which express +the coordinates of any point of it as two functions of the same +variable parameter θ to which all values are open. As θ takes all +values in turn, the point (x, y) traverses the curve.</p> + +<p>It is a good exercise to trace a number of curves, taken as defined +by the equations which represent them. This, in simple cases, can +be done approximately by plotting the values of y given by the +equation of a curve as going with a considerable number of values +of x, and connecting the various points (x, y) thus obtained. But +methods exist for diminishing the labour of this tentative process.</p> + +<p>Another problem, which will be more attended to here, is that of +determining the equations of curves of known interest, taken as +defined by geometrical properties. It is not a matter for surprise +that the curves which have been most and longest studied geometrically +are among those represented by equations of the simplest +character.</p> + +<table class="flt" style="float: right; width: 300px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:250px; height:245px" src="images/img713.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 50.</span></td></tr></table> + +<p>8. <i>The Straight Line.</i>—This is the simplest type of locus. Also +the simplest type of equation in x and y is Ax + By + C = 0, one of +the first degree. Here the coefficients A, B, C are constants. They +are, like the current coordinates, x, y, numerical. But, in giving +interpretation to such an equation, we must of course refer to +numbers Ax, By, C of unit magnitudes of the same kind, of units +of counting for instance, or unit lengths or unit squares. It will +now be seen that every straight +line has an equation of the first +degree, and that every equation +of the first degree represents a +straight line.</p> + +<p>It has been seen (§ 7) that lines +parallel to the axes have equations +of the first degree, free +from one of the variables. Take +now a straight line ABC inclined +to both axes. Let it make a +given angle α with the positive +direction of the axis of x, <i>i.e.</i> in +fig. 50 let this be the angle +through which Ax must be revolved +counter-clockwise about +A in order to be made coincident +with the line. Let C, of +coordinates (h, k), be a fixed point +on the line, and P(x, y) any other point upon it. Draw the ordinates +CD, PM of C and P, and let the parallel to the axis of x through C +meet PM, produced if necessary, in R. The right-angled triangle +CRP tells us that, with the signs appropriate to their directions +attached to CR and RP,</p> + +<p class="center">RP = CR tan α, <i>i.e.</i> MP − DC = (OM − OD) tan α,</p> + +<p class="noind">and this gives that</p> + +<p class="center">y − k = tan α (x − h),</p> + +<p class="noind">an equation of the first degree satisfied by x and y. No point not +on the line satisfies the same equation; for the line from C to any +point off the line would make with CR some angle β different from α, +and the point in question would satisfy an equation y − k = tan β(x − h), +which is inconsistent with the above equation.</p> + +<p>The equation of the line may also be written y = mx + b, where +m = tan α, and b = k − h tan α. Here b is the value obtained for y +from the equation when 0 is put for x, <i>i.e.</i> it is the numerical measure, +with proper sign, of OB, the intercept made by the line on the axis +of y, measured from the origin. For different straight lines, m and b +may have any constant values we like.</p> + +<p>Now the general equation of the first degree Ax + By + C = 0 may +be written y = −(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</p> + +<table class="math0" summary="math"> +<tr><td>x</td> +<td rowspan="2">+</td> <td>y</td> +<td rowspan="2">= 1</td></tr> +<tr><td class="denom">a</td> <td class="denom">b</td></tr></table> + +<p class="noind">is the equation of a straight line determined by the two conditions +that it makes intercepts OA, OB on the two axes, of which a and b +are the numerical measures with proper signs: note that in fig. 50 a +is negative. Again,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">y − y<span class="su">1</span> =</td> <td>y<span class="su">2</span> − y<span class="su">1</span></td> +<td rowspan="2">(x − x<span class="su">1</span>),</td></tr> +<tr><td class="denom">x<span class="su">2</span> − x<span class="su">1</span></td></tr></table> + +<p class="noind"><i>i.e.</i></p> + +<p class="center">(y<span class="su">1</span> − y<span class="su">2</span>) x − (x<span class="su">1</span> − x<span class="su">2</span>) y + x<span class="su">1</span>y<span class="su">2</span> − x<span class="su">2</span>y<span class="su">1</span> = 0,</p> + +<p class="noind">represents the line determined by the data that it passes through +two given points (x<span class="su">1</span>, y<span class="su">1</span>) and (x<span class="su">2</span>, y<span class="su">2</span>). To prove this find m in the +equation y − y<span class="su">1</span> = m(x − x<span class="su">1</span>) of a line through (x<span class="su">1</span>, y<span class="su">1</span>), from the condition +that (x<span class="su">2</span>, y<span class="su">2</span>) lies on the line.</p> + +<p>In this paragraph the coordinates have been assumed rectangular. +Had they been oblique, the doctrine of similar triangles would have +given the same results, except that in the forms of equation y − k = m(x − h), +y = mx + b, we should not have had m = tan α.</p> + +<p>9. <i>The Circle.</i>—It is easy to write down the equation of a given +circle. Let (h, k) be its given centre C, and ρ the numerical measure +of its given radius. Take P (x, y) any point on its circumference, +and construct the triangle CRP, in fig. 50 as above. The fact that +this is right-angled tells us that</p> + +<p class="center">CR² + RP² = CP²,</p> + +<p class="noind">and this at once gives the equation</p> + +<p class="center">(x − h)² + (y − k)² = ρ².</p> + +<p class="noind">A point not upon the circumference of the particular circle is at some +distance from (h, k) different from ρ, and satisfies an equation +inconsistent with this one; which accordingly represents the circumference, +or, as we say, the circle.</p> + +<p>The equation is of the form</p> + +<p class="center">x² + y² + 2Ax + 2By + C = 0.</p> + +<p class="noind">Conversely every equation of this form represents a circle: we have +only to take −A, −B, A² + B² − C for h, k, ρ² respectively, to obtain +its centre and radius. But this statement must appear too unrestricted. +Ought we not to require A² + B² − C to be positive? +Certainly, if by circle we are only to mean the visible round circumference +of the geometrical definition. Yet, analytically, we +contemplate altogether imaginary circles, for which ρ² is negative, +and circles, for which ρ = 0, with all their reality condensed into +their centres. Even when ρ² 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 = ρ², +we obtain two values of y, real, coincident or imaginary, +each of which goes with the abscissa x as the ordinate of a point, +real or imaginary, on what is represented by the equation of the +circle.</p> + +<p>The doctrine of the imaginary on a circle, and in geometry generally, +is of purely algebraical inception; but it has been in its entirety +accepted by modern pure geometers, and signal success has attended +the efforts of those who, like K.G.C. von Staudt, have striven to +base its conclusions on principles not at all algebraical in form, +though of course cognate to those adopted in introducing the +imaginary into algebra.</p> + +<p>A circle with its centre at the origin has an equation x² + y² = ρ².</p> + +<p>In oblique coordinates the general equation of a circle is +x² + 2xy cos ω + y² + 2Ax + 2By + C = 0.</p> + +<p>10. The conic sections are the next simplest loci; and it will be +seen later that they are the loci represented by equations of the +second degree. Circles are particular cases of conic sections; and +<span class="pagenum"><a name="page714" id="page714"></a>714</span> +they have just been seen to have for their equations a particular +class of equations of the second degree. Another particular class +of such equations is that included in the form (Ax + By + C)(A′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>i.e.</i> +if, and only if, (x, y) lies on one or other of two straight lines. The +condition that ax² + 2hxy + by² + 2gx + 2fy + c = 0, which is often +written (a, b, c, f, g, h)(x, y, I)² = 0, takes this form is abc + 2fgh − af² − +bg² − ch² = 0. Note that the two lines may, in particular cases, be +parallel or coincident.</p> + +<p>Any equation like F<span class="su">1</span>(x, y) F<span class="su">2</span>(x, y) ... F<span class="su">n</span>(x, y) = 0, of which +the left-hand side breaks up into factors, represents all the loci +separately represented by F<span class="su">1</span>(x, y) = 0, F<span class="su">2</span>(x, y) = 0, ... F<span class="su">n</span>(x, y) = 0. +In particular an equation of degree n which is free from x represents +n straight lines parallel to the axis of x, and one of degree n which +is homogeneous in x and y, <i>i.e.</i> one which upon division by x<span class="sp">n</span>, becomes +an equation in the ratio y/x, represents n straight lines through +the origin.</p> + +<p>Curves represented by equations of the third degree are called +cubic curves. The general equation of this degree will be written +(*)(x, y, I)³ = 0.</p> + +<table class="flt" style="float: right; width: 330px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:281px; height:263px" src="images/img714a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 51.</span></td></tr></table> + +<p>11. <i>Descriptive Geometry.</i>—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.</p> + +<p>12. It is worth while to +illustrate this by the instance +of the well-known +theorem of the radical centre +of three circles. The theorem +is that, given any three circles +A, B, C (fig. 51), the common +chords αα′, ββ′, γγ′ of the +three pairs of circles meet in +a point.</p> + +<p>The geometrical proof is +metrical throughout:—</p> + +<p>Take O the point of intersection +of αα′, ββ′, and joining +this with γ′, suppose that γ′O does not pass through γ, but that it +meets the circles A, B in two distinct points γ<span class="su">2</span>, γ<span class="su">1</span> respectively. We +have then the known metrical property of intersecting chords of a +circle; viz. in circle C, where αα′, ββ′, are chords meeting at a point O,</p> + +<p class="center">Oα·Oα′ = Oβ·Oβ′,</p> + +<p class="noind">where, as well as in what immediately follows, Oα, &c. denote, of +course, <i>lengths</i> or <i>distances</i>.</p> + +<p>Similarly in circle A,</p> + +<p class="center">Oβ·Oβ′ = Oγ<span class="su">2</span>·Oγ′,</p> + +<p class="noind">and in circle B,</p> + +<p class="center">Oα·Oα′ = Oγ<span class="su">1</span>·Oγ′.</p> + +<p class="noind">Consequently Oγ<span class="su">1</span>·Oγ′ = Oγ<span class="su">2</span>·Oγ′, that is, Oγ<span class="su">1</span> = Oγ<span class="su">2</span>, or the points +γ<span class="su">1</span> and γ<span class="su">2</span> coincide; that is, they each coincide with γ.</p> + +<p>We contrast this with the analytical method:—</p> + +<p>Here it only requires to be known that an equation Ax + By + C = 0 +represents a line, and an equation x² + y² + Ax + By + C = 0 represents +a circle. A, B, C have, in the two cases respectively, metrical +significations; but these we are not concerned with. Using S to +denote the function x² + y² + Ax + By + C, the equation of a circle is +S = o. Let the equation of any other circle be S′, = x² + y² + A′x + B′y + C′ = 0; +the equation S-S′ = 0 is a linear equation (S − S′ is in +fact = (A − A′)x + (B − B′)y + C-C), and it thus represents a line; +this equation is satisfied by the coordinates of each of the points of +intersection of the two circles (for at each of these points S = 0 and +S′ = 0, therefore also S − S′ = 0); hence the equation S − S′ = 0 is +that of the line joining the two points of intersection of the two circles, +or say it is the equation of the common chord of the two circles. +Considering then a third circle S″, = x² + y² + A″x + B″y + C″ = 0, the +equations of the common chords are S − S′ = 0, S − S″ = 0, S′ − S″ = 0 +(each of these a linear equation); at the intersection of the first and +second of these lines S = S′ and S = S″, therefore also S′ = S″, or the +equation of the third line is satisfied by the coordinates of the point +in question; that is, the three chords intersect in a point O, the coordinates +of which are determined by the equations S = S′ = S″.</p> + +<p>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 <i>any</i> three circles the common +chords intersect in a point (of course real) which is the radical centre. +And by this very theorem, given two circles with imaginary intersections, +we can, by drawing circles which meet each of them in +real points, construct the radical axis of the first-mentioned two +circles.</p> + +<p>13. The principle employed in showing that the equation of the +common chord of two circles is S − S′ = 0 is one of very extensive +application, and some more illustrations of it may be given.</p> + +<p>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 (x<span class="su">0</span>, y<span class="su">0</span>); if S<span class="su">0</span>, S′<span class="su">0</span> are +what S, S′ respectively become on writing for (x, y) the values (x<span class="su">0</span>, y<span class="su">0</span>), +then the value of k is k = S<span class="su">0</span> ÷ S′<span class="su">0</span>. The equation in fact is SS′<span class="su">0</span> − S<span class="su">0</span>S′ = 0; +and starting from this equation we at once verify it <i>a posteriori</i>; +the equation is a linear equation satisfied by the values of (x, y) +which make S = 0, S′ = 0; and satisfied also by the values (x<span class="su">0</span>, y<span class="su">0</span>); +and it is thus the equation of the line in question.</p> + +<p>If, as before, S = 0, S′ = 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 (x<span class="su">0</span>, y<span class="su">0</span>) we have again k = S<span class="su">0</span> ÷ S′<span class="su">0</span>. In the +particular case k = 1, the circle becomes the common chord (more +accurately it becomes the common chord together with the line +infinity; see § 23 below).</p> + +<p>If S denote the general quadric function,</p> + +<p class="center">S = ax<span class="sp">2</span> + 2hxy + by<span class="sp">2</span> + 2fy + 2gx + c,</p> + +<p class="noind">then the equation S = 0 represents a conic; assuming this, then, if +S′ = 0 represents another conic, the equation S − kS′ = 0 represents +<i>any</i> conic through the four points of intersection of the two conics.</p> + +<table class="flt" style="float: right; width: 300px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:246px; height:143px" src="images/img714b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 52.</span></td></tr></table> + +<p>14. The object still being to illustrate the mode of working with +coordinates for descriptive purposes, +we consider the theorem +of the polar of a point in regard +to a circle. Given a circle and +a point O (fig. 52), we draw +through O any two lines meeting +the circle in the points A, A′ 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′.</p> + +<p>Taking O as the origin, and for the axes any two lines through O +at right angles to each other, the equation of the circle will be</p> + +<p class="center">x<span class="sp">2</span> + y<span class="sp">2</span> + 2Ax + 2By + C = 0;</p> + +<p class="noind">and if the equation of the line OAA′ 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</p> + +<p class="center">x<span class="sp">2</span> (1 + m<span class="sp">2</span>) + 2x (A + Bm) + C = 0.</p> + +<p class="noind">If(x<span class="su">1</span>, y<span class="su">1</span>) are the coordinates of A, and (x<span class="su">2</span>, y<span class="su">2</span>) of A′, then the roots +of this equation are x<span class="su">1</span>, x<span class="su">2</span>, whence easily</p> + +<table class="math0" summary="math"> +<tr><td>1</td> +<td rowspan="2">+</td> <td>1</td> +<td rowspan="2">= −2</td> <td>A + Bm</td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">x<span class="su">1</span></td> <td class="denom">x<span class="su">2</span></td> +<td class="denom">C</td></tr></table> + +<p class="noind">And similarly, if the equation of the line OBB′ is taken to be y = m′x<span class="su">1</span> +and the coordinates of B, B′ to be (x<span class="su">3</span>, y<span class="su">3</span>) and (x<span class="su">4</span>, y<span class="su">4</span>) respectively, +then</p> + +<table class="math0" summary="math"> +<tr><td>1</td> +<td rowspan="2">+</td> <td>1</td> +<td rowspan="2">= −2</td> <td>A + Bm′</td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">x<span class="su">3</span></td> <td class="denom">x<span class="su">4</span></td> +<td class="denom">C′</td></tr></table> + +<p>We have then by § 8</p> + +<table class="reg" summary="poem"><tr><td> <div class="poemr"> +<p>x (y<span class="su">1</span> − y<span class="su">4</span>) − y (x<span class="su">1</span> − x<span class="su">4</span>) + x<span class="su">1</span>y<span class="su">4</span> − x<span class="su">4</span>y<span class="su">1</span> = 0,</p> + +<p>x (y<span class="su">2</span> − y<span class="su">3</span>) − y (x<span class="su">2</span> − x<span class="su">3</span>) + x<span class="su">2</span>y<span class="su">3</span> − x<span class="su">3</span>y<span class="su">2</span> = 0,</p> +</div> </td></tr></table> + +<p class="noind">as the equations of the lines AB′ and A′B respectively. Reducing +by means of the relations y<span class="su">1</span> − mx<span class="su">1</span> = 0, y<span class="su">2</span> − mx<span class="su">2</span> = 0, y<span class="su">3</span> − m′x<span class="su">3</span> = 0, +y<span class="su">4</span> − m′x<span class="su">4</span> = 0, the two equations become</p> + +<table class="reg" summary="poem"><tr><td> <div class="poemr"> +<p>x (mx<span class="su">1</span> − m′x<span class="su">4</span>) − y (x<span class="su">1</span> − x<span class="su">4</span>) + (m′ − m) x<span class="su">1</span>x<span class="su">4</span> = 0,</p> + +<p>x (mx<span class="su">2</span> − m′x<span class="su">3</span>) − y (x<span class="su">2</span> − x<span class="su">3</span>) + (m′ − m) x<span class="su">2</span>x<span class="su">3</span> = 0,</p> +</div> </td></tr></table> + +<p class="noind">and if we divide the first of these equations by x<span class="su">1</span>x<span class="su">4</span>, and the second +by x<span class="su">2</span>x<span class="su">3</span> and then add, we obtain</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">x <span class="f150">{</span> m <span class="f150">(</span></td> <td>1</td> +<td rowspan="2">+</td> <td>1</td> +<td rowspan="2"><span class="f150">)</span> − m′ <span class="f150">(</span></td> <td>1</td> +<td rowspan="2">+</td> <td>1</td> +<td rowspan="2"><span class="f150">) }</span> − y <span class="f150">{</span></td> <td>1</td> +<td rowspan="2">+</td> <td>1</td> +<td rowspan="2">− <span class="f150">(</span></td> <td>1</td> +<td rowspan="2">+</td> <td>1</td> +<td rowspan="2"><span class="f150">) }</span> + 2m′ − 2m = 0,</td></tr> +<tr><td class="denom">x<span class="su">3</span></td> <td class="denom">x<span class="su">4</span></td> +<td class="denom">x<span class="su">1</span></td> <td class="denom">x<span class="su">2</span></td> +<td class="denom">x<span class="su">3</span></td> <td class="denom">x<span class="su">4</span></td> +<td class="denom">x<span class="su">1</span></td> <td class="denom">x<span class="su">2</span></td></tr></table> + +<p class="noind">or, what is the same thing,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2"><span class="f150">(</span></td> <td>1</td> +<td rowspan="2">+</td> <td>1</td> +<td rowspan="2"><span class="f150">)</span> (y − m′x) − <span class="f150">(</span></td> <td>1</td> +<td rowspan="2">+</td> <td>1</td> +<td rowspan="2"><span class="f150">)</span> (y − mx) + 2m′ − 2m = 0,</td></tr> +<tr><td class="denom">x<span class="su">1</span></td> <td class="denom">x<span class="su">2</span></td> +<td class="denom">x<span class="su">3</span></td> <td class="denom">x<span class="su">4</span></td></tr></table> + +<p class="noind">which by what precedes is the equation of a line through the point Q. +Substituting herein for 1/x<span class="su">1</span> + 1/x<span class="su">2</span>, 1/x<span class="su">3</span> + 1/x<span class="su">4</span> their foregoing values, the +equation becomes</p> + +<p class="center">−(A + Bm) (y − m′x) + (A + Bm′) (y − mx) + C (m′ − m) = 0;</p> + +<p class="noind">that is,</p> + +<p class="center">(m − m′) (Ax + By + C) = 0;</p> + +<p><span class="pagenum"><a name="page715" id="page715"></a>715</span></p> + +<p class="noind">or finally it is Ax + By + C = 0, showing that the point Q lies in a line +the position of which is independent of the particular lines OAA′, +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′.</p> + +<p>15. In § 8 it has been seen that two conditions determine the +equation of a straight line, because in Ax + By + C = 0 one of the +coefficients may be divided out, leaving only two parameters to be +determined. Similarly five conditions instead of six determine an +equation of the second degree (a, b, c, f, g, h)(x, y, 1)² = 0, and nine +instead of ten determine a cubic (*)(x, y, 1)³ = 0. It thus appears +that a cubic can be made to pass through 9 given points, and that +the cubic so passing through 9 given points is completely determined. +There is, however, a remarkable exception. Considering two given +cubic curves S = 0, S′ = 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 = S<span class="su">0</span> ÷ S′<span class="su">0</span>, if the coordinates +are (x<span class="su">0</span>, y<span class="su">0</span>), and S<span class="su">0</span>, S′<span class="su">0</span> denote the corresponding values of S, S′). +The resulting curve SS′<span class="su">0</span> − S′S<span class="su">0</span> = 0 may be regarded as the cubic +determined by the conditions of passing through 8 of the 9 points +and through the given point (x<span class="su">0</span>, y<span class="su">0</span>); and from the equation it +thence appears that the curve passes through the remaining one of +the 9 points. In other words, we thus have the theorem, any cubic +curve which passes through 8 of the 9 intersections of two given +cubic curves passes through the 9th intersection.</p> + +<p>The applications of this theorem are very numerous; for instance, +we derive from it Pascal’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.</p> + +<p>16. <i>Metrical Theory resumed.</i> <i>Projections and Perpendiculars.</i>—It +is a metrical fact of fundamental importance, already used in § 8, +that, if a finite line PQ be projected on any other line OO′ by perpendiculars +PP′, QQ′ to OO′, the length of the projection P′Q′ is +equal to that of PQ multiplied by the cosine of the acute angle +between the two lines. Also the algebraical sum of the projections +of the sides of any closed polygon upon any line is zero, because as a +point goes round the polygon, from any vertex A to A again, the +point which is its projection on the line passes from A′ the projection +of A to A′ again, <i>i.e.</i> traverses equal distances along the line in +positive and negative senses. If we consider the polygon as consisting +of two broken lines, each extending from the same initial +to the same terminal point, the sum of the projections of the lines +which compose the one is equal, in sign and magnitude, to the sum +of the projections of the lines composing the other. Observe that +the projection on a line of a length perpendicular to the line is +zero.</p> + +<p>Let us hence find the equation of a straight line such that the +perpendicular OD on it from the origin is of length ρ taken as +positive, and is inclined to the axis of x at an angle xOD = α, +measured counter-clockwise from Ox. Take any point P(x, y) on +the line, and construct OM and MP as in fig. 48. The sum of the +projections of OM and MP on OD is OD itself; and this gives the +equation of the line</p> + +<p class="center">x cos α + y sin α = ρ.</p> + +<p class="noind">Observe that cos α and sin α here are the sin α and −cos α, or the +−sin α and cos α of § 8 according to circumstances.</p> + +<p>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 α + y′ sin α = OE, +and the perpendicular distance required is OD − OE, +<i>i.e.</i> ρ − x′ cos α − y′ sin α; 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.</p> + +<p>The general equation Ax + By + C = 0 may be given the form +x cos α + y sin α − ρ = 0 by dividing it by √(A² + B³). Thus (Ax′ + +By′ + C) ÷ √(A² + B²) is in absolute value the perpendicular distance +of (x′, y′) from the line Ax + By + C = 0. Remember, however, that +there is an essential ambiguity of sign attached to a square root. +The expression found gives the distance taken positively when +(x′, y′) is on the origin side of the line, if the sign of C is given to +√(A² + B²).</p> + +<p>17. <i>Transformation of Coordinates.</i>—We often need to adopt new +axes of reference in place of old ones; and the above principle of +projections readily expresses the old coordinates of any point in +terms of the new.</p> + +<table class="flt" style="float: right; width: 310px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:264px; height:216px" src="images/img715.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 53.</span></td></tr></table> + +<p>Suppose, for instance, that we want to take for new origin the +point O′ 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 θ 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:</p> + +<p class="center">x = h + X cos θ + Y cos (θ + ½π) = +h + X cos θ − Y sin θ,</p> + +<p class="noind">and</p> + +<p class="center">y = k + X cos (½π − θ) + Y cos θ = +k + X sin θ + Y cos θ.</p> + +<p>Be careful to observe that these +formulae do not apply to every +conceivable change of reference from one set of rectangular axes to +another. It might have been required to take O′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.</p> + +<p>Were the new axes oblique, making angles α, β respectively with +the old axis of x, and so inclined at the angle β − α, the same method +would give the formulae</p> + +<p class="center">x = h + X cos α + Y cos β, y = k + X sin α + Y sin β.</p> + +<p>18. <i>The Conic Sections.</i>—The conics, as they are now called, were +at first defined as curves of intersection of planes and a cone; but +Apollonius substituted a definition free from reference to space of +three dimensions. This, in effect, is that a conic is the locus of a +point the distance of which from a given point, called the focus, has +a given ratio to its distance from a given line, called the directrix +(see <span class="sc"><a href="#artlinks">Conic Section</a></span>). If e : 1 is the ratio, e is called the eccentricity. +The distances are considered signless.</p> + +<p>Take (h, k) for the focus, and x cos α + y sin α − p = 0 for the +directrix. The absolute values of √{(x − h)² + (y − k)²} and p − x cos α − +y sin α are to have the ratio e : 1; and this gives</p> + +<p class="center">(x − h)² + (y − k)² = e² (p − x cos α − y sin α)²</p> + +<p class="noind">as the general equation, in rectangular coordinates, of a conic.</p> + +<p>It is of the second degree, and is the general equation of that +degree. If, in fact, we multiply it by an unknown λ, we can, by +solving six simultaneous equations in the six unknowns λ, h, k, e, p, α, +so choose values for these as to make the coefficients in the equation +equal to those in any equation of the second degree which may be +given. There is no failure of this statement in the special case +when the given equation represents two straight lines, as in § 10, +but there is speciality: if the two lines intersect, the intersection +and either bisector of the angle between them are a focus and +directrix; if they are united in one line, any point on the line and a +perpendicular to it through the point are: if they are parallel, +the case is a limiting one in which e and h² + k² have become infinite +while e<span class="sp">−2</span>(h² + k²) remains finite. In the case (§ 9) of an equation +such as represents a circle there is another instance of proceeding +to a limit: e has to become 0, while ep remains finite: moreover α +is indeterminate. The centre of a circle is its focus, and its directrix +has gone to infinity, having no special direction. This last fact +illustrates the necessity, which is also forced on plane geometry by +three-dimensional considerations, of treating all points at infinity +in a plane as lying on a single straight line.</p> + +<p>Sometimes, in reducing an equation to the above focus and directrix +form, we find for h, k, e, p, tan α, or some of them, only imaginary +values, as quadratic equations have to be solved; and we have in +fact to contemplate the existence of entirely imaginary conics. +For instance, no real values of x and y satisfy x² + 2y² + 3 = 0. Even +when the locus represented is real, we obtain, as a rule, four sets of +values of h, k, e, p, of which two sets are imaginary; a real conic +has, besides two real foci and corresponding directrices, two others +that are imaginary.</p> + +<p>In oblique as well as rectangular coordinates equations of the +second degree represent conics.</p> + +<p>19. <i>The three Species of Conics.</i>—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.</p> + +<p>In the case of a parabola, let 2c be the distance between the given +focus and directrix, and take axes referred to which these are the +point (c, 0) and the line x = − c. The equation becomes (x − c)² + y² = +(x + c)², <i>i.e.</i> y² = 4cx.</p> + +<p>In the other cases, take a such that a(e ~ e<span class="sp">−1</span>) is the distance of focus +from directrix, and so choose axes that these are (ae, 0) and x = ae<span class="sp">-1</span>, +thus getting the equation(x − ae)² + y² = e²(x − ae<span class="sp">-1</span>)², <i>i.e.</i> (1 − e²)x² + y² = +a²(1 − e²). When e < 1, <i>i.e.</i> in the case of an ellipse, this may be +written x²/a² + y²/b² = 1, where b² = a²(1 − e²); and when e > 1, <i>i.e.</i> +in the case of an hyperbola, x²/a² − y²/b² = 1, where b² = a²(e² − 1). +<span class="pagenum"><a name="page716" id="page716"></a>716</span> +The axes thus chosen for the ellipse and hyperbola are called the +principal axes.</p> + +<p>In figs. 54, 55, 56 in order, conics of the three species, thus referred, +are depicted.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:511px; height:227px" src="images/img716a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 54</span></td> +<td class="caption"><span class="sc">Fig. 55</span></td></tr></table> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter"><img style="width:331px; height:189px" src="images/img716b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 56.</span></td></tr></table> + +<p>The oblique straight lines in fig. 56 are the <i>asymptotes</i> x/a = ±y/b +of the hyperbola, lines to which the curve tends with unlimited +closeness as it goes to infinity. The hyperbola would have an equation +of the form xy = c if referred to its asymptotes as axes, the coordinates +being then oblique, unless a = b, in which case the hyperbola +is called rectangular. An ellipse has two imaginary asymptotes. +In particular a circle x² + y² = a², a particular ellipse, has for asymptotes +the imaginary lines x = ±y √−1. These run from the centre +to the so-called circular points at infinity.</p> + +<p>20. <i>Tangents and Curvature.</i>—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.</p> + +<p>The <i>normal</i> 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.</p> + +<p>In the case of the conic (a, b, c, f, g, h) (x, y, 1)² = 0 we find that +A/B = (ax′ + hy′ + g)/(hx′ + by′ + f).</p> + +<p>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 <i>centre of curvature</i> 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 <i>radius +of curvature</i>.</p> + +<p>21. <i>Differential Plane Geometry.</i>—The language and notation of the +differential calculus are very useful in the study of tangents and +curvature. Denoting by (ξ, η) the current coordinates, we find, +as above, that the tangent at a point (x, y) of a curve is η − y = +(ξ − x)dy/dx, where dy/dx is found from the equation of the curve. If +this be f(x, y) = 0 the tangent is (ξ − x) (∂f/∂x) + (η − y) (∂f/∂y) = 0. If ρ +and (α, β) are the radius and centre of curvature at (x, y), we find that +q(α − x) = −p(1 + p²), q(β − y) = 1 + p², q²ρ² = (1 + p²)³, where p, q denote +dy/dx, d²y/dx² respectively. (See <span class="sc"><a href="#artlinks">Infinitesimal Calculus</a></span>.)</p> + +<p>In any given case we can, at all events in theory, eliminate x, y +between the above equations for α − x and β − y, and the equation +of the curve. The resulting equation in (α, β) represents the locus +of the centre of curvature. This is the <i>evolute</i> of the curve.</p> + +<p>22. <i>Polar Coordinates.</i>—In plane geometry the distance of any +point P from a fixed origin (or pole) O, and the inclination xOP of OP +to a fixed line Ox, determine the point: r, the numerical measure +of OP, the <i>radius vector</i>, and θ, the circular measure of xOP, the +<i>inclination</i>, are called polar coordinates of P. The formulae x = +r cos θ, y = r sin θ 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 (θ − α), r{1 − e cos (θ − α)} = 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 θ, with m integral, and also the +curves called spirals, which have equations giving r as functions of +θ itself, and not merely of sin θ and cos θ. In the geometry of +motion under central forces the advantage of working with polar +coordinates is great.</p> + +<p>23. <i>Trilinear and Areal Coordinates.</i>—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 Δ the area, of the triangle. We might, as in § 6, take CA, +CB as axes of x and y, inclined at an angle C. Any point P (x, y) +in the plane is at perpendicular distances y sin C and x sin C from +CA and CB. Call these β and α respectively. The signs of β and α +are those of y and x, <i>i.e.</i> β is positive or negative according as P lies +on the same side of CA as B does or the opposite, and similarly for α. +An equation in (x, y) of any degree may, upon replacing in it x and y +by α cosec C and β cosec C, be written as one of the same degree in +(α, β). Now let γ 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 aα + bβ + cγ = 2Δ. +By means of this relation in α, β, γ we can give an equation considered +countless other forms, involving two or all of α, β, γ. In +particular we may make it <i>homogeneous</i> in α, β, γ: 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 (aα + bβ + cγ)/2Δ just sufficient +to raise them, in each case, to the highest degree.</p> + +<p>We call (α, β, γ) <i>trilinear coordinates</i>, and an equation in them +the trilinear equation of the locus represented. Trilinear equations +are, as a rule, dealt with in their homogeneous forms. An advantage +thus gained is that we need not mean by (α, β, γ) the actual measures +of the perpendicular distances, but any properly signed numbers +which have the same ratio two and two as these distances.</p> + +<p>In place of α, β, γ 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 = aα/2Δ, +y = bβ/2Δ, z = cγ/2Δ, the properly signed ratios of the triangular +areas PBC, PCA, PAB to the triangular area ABC. These are called +the <i>areal</i> coordinates of P. In areal coordinates the relation which +enables us to make any equation homogeneous takes the simple +form x + y + z = 1; and, as before, we need mean by x, y, z, in a +homogeneous equation, only signed numbers in the right ratios.</p> + +<p>Straight lines and conics are represented in trilinear and in areal, +because in Cartesian, coordinates by equations of the first and +second degrees respectively, and these degrees are preserved when +the equations are made homogeneous. What must be said about +points infinitely far off in order to make universal the statement, +to which there is no exception as long as finite distances alone are +considered, that <i>every</i> homogeneous equation of the first degree +represents a straight line? Let the point of areal coordinates +(x′, 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 <i>line at infinity</i> has for equation aα + bβ + +cγ = 0.</p> + +<p>On the one special line at infinity parallel lines are treated as +meeting. There are on it two special (imaginary) points, the circular +points at infinity of § 19, through which all circles pass in the same +sense. In fact if S = O be one circle, in areal coordinates, +S + (x + y + z)(lx + my + nz) = 0 may, by proper choice of l, m, n, be +made any other; since the added terms are once lx + my + nz, and +have the generality of any expression like a′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.</p> + +<p>24. Let us consider the perpendicular distance of a point (α′, β′, γ′) +from a line lα + mβ + nγ. 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 θ − y sin θ, &c., for α &c.; we can then apply § 16 to +get the perpendicular distance; and finally revert to the trilinear +notation. The result is to find that the required distance is</p> + +<p class="center">(lα′ + mβ′ + nγ′) / {l, m, n},</p> + +<p class="noind">where {l, m, n}² = l² + m² + n² − 2mn cos A − 2nl cos B − 2lm cos C.</p> + +<p>In areal coordinates the perpendicular distance from (x′, y′, z′) +<span class="pagenum"><a name="page717" id="page717"></a>717</span> +to lx + my + nz = 0 is 2Δ(lx′ + my′ + nz′)/{al, bm, cn}. In both cases +the coordinates are of course actual values.</p> + +<p>Now let ξ, η, ζ be the perpendiculars on the line from the vertices +A, B, C, <i>i.e.</i> the points (1, 0, 0), (0, 1, 0), (0, 0, 1), with signs in +accord with a convention that oppositeness of sign implies distinction +between one side of the line and the other. Three applications +of the result above give</p> + +<p class="center">ξ/l = 2Δ / {al, bm, cn} = η/m = ζ/n;</p> + +<p class="noind">and we thus have the important fact that ξx′ + ηy′ + ζz′ 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 ξ, η, ζ +respectively. We have also that ξx + ηy + ζz = 0 is the areal equation +of the line on which the perpendiculars are ξ, η, ζ; and, by equating +the two expressions for the perpendiculars from (x′, y′, z′) on the +line, that in all cases {aξ, bη, cζ}² = 4Δ².</p> + +<p>25. <i>Line-coordinates.</i> <i>Duality.</i>—A quite different order of ideas +may be followed in applying analysis to geometry. The notion of a +straight line specified may precede that of a point, and points may +be dealt with as the intersections of lines. The specification of +a line may be by means of coordinates, and that of a point by an +equation, satisfied by the coordinates of lines which pass through it. +Systems of <i>line-coordinates</i> will here be only briefly considered. +Every such system is allied to some system of point-coordinates; +and space will be saved by giving prominence to this fact, and not +recommencing <i>ab initio</i>.</p> + +<p>Suppose that any particular system of point-coordinates, in which +lx + my + nz = 0 may represent any straight line, is before us: notice +that not only are trilinear and areal coordinates such systems, but +Cartesian coordinates also, since we may write x/z, y/z for the +Cartesian x, y, and multiply through by z. The line is exactly +assigned if l, m, n, or their mutual ratios, are known. Call (l, m, n) +the <i>coordinates</i> of the line. Now keep x, y, z constant, and let the +coordinates of the line vary, but always so as to satisfy the equation. +This equation, which we now write xl + ym + zn = 0, is satisfied by +the coordinates of every line through a certain fixed point, and by +those of no other line; it is the equation of that point in the line-coordinates +l, m, n.</p> + +<p>Line-coordinates are also called <i>tangential</i> coordinates. A curve +is the envelope of lines which touch it, as well as the locus of points +which lie on it. A homogeneous equation of degree above the first +in l, m, n is a relation connecting the coordinates of every line which +touches some curve, and represents that curve, regarded as an +envelope. For instance, the condition that the line of coordinates +(l, m, n), <i>i.e.</i> the line of which the allied point-coordinate equation +is lx + my + nz = 0, may touch a conic (a, b, c, f, g, h) (x, y, z)² = 0, +is readily found to be of the form (A, B, C, F, G, H) (l, m, n)² = 0, +<i>i.e.</i> to be of the second degree in the line-coordinates. It is not hard +to show that the <i>general</i> equation of the second degree in l, m, n +thus represents a conic; but the degenerate conics of line-coordinates +are not line-pairs, as in point-coordinates, but point-pairs.</p> + +<p>The degree of the point-coordinate equation of a curve is the +<i>order</i> of the curve, the number of points in which it cuts a straight +line. That of the line-coordinate equation is its <i>class</i>, the number +of tangents to it from a point. The order and class of a curve are +generally different when either exceeds two.</p> + +<p>26. The system of line-coordinates allied to the areal system of +point-coordinates has special interest.</p> + +<p>The l, m, n of this system are the perpendiculars ξ, η, ζ of § 24; +and x′ξ + y′η + z′ζ = 0 is the equation of the point of areal coordinates +(x′, y′, z′), <i>i.e.</i> is a relation which the perpendiculars from the vertices +of the triangle of reference on every line through the point, but no +other line, satisfy. Notice that a non-homogeneous equation of the +first degree in ξ, η, ζ does not, as a homogeneous one does, represent +a point, but a circle. In fact x′ξ + y′η + z′ζ = R expresses the constancy +of the perpendicular distance of the fixed point x′ξ + y′η + +z′ζ = 0 from the variable line (ξ, η, ζ), <i>i.e.</i> the fact that (ξ, η, ζ) touches +a circle with the fixed point for centre. The relation in any ξ, η, ζ +which enables us to make an equation homogeneous is not linear, +as in point-coordinates, but quadratic, viz. it is the relation {aξ, bη, +cζ}² = 4Δ² of § 24. Accordingly the homogeneous equation of the +above circle is</p> + +<p class="center">4Δ² (x′ξ + y′η + z′ζ)² = R² {aξ, bη, cζ}².</p> + +<p>Every circle has an equation of this form in the present system of +line-coordinates. Notice that the equation of any circle is satisfied +by those coordinates of lines which satisfy both x′ξ + y′η + z′ζ = 0, +the equation of its centre, and {aξ, bη, cζ}² = 0. This last equation, +of which the left-hand side satisfies the condition for breaking up +into two factors, represents the two imaginary circular points at +infinity, through which all circles and their asymptotes pass.</p> + +<p>There is strict duality in descriptive geometry between point-line-locus +and line-point-envelope theorems. But in metrical geometry +duality is encumbered by the fact that there is in a plane one special +line only, associated with distance, while of special points, associated +with direction, there are two: moreover the line is real, and the +points both imaginary.</p> + +<p class="pt2 center">II. <i>Solid Analytical Geometry.</i></p> + +<p>27. Any point in space may be specified by three coordinates. +We consider three fixed planes of reference, and generally, as in all +that follows, three which are at right angles two and two. They +intersect, two and two, in lines x′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.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:513px; height:254px" src="images/img717a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 57.</span></td> +<td class="caption"><span class="sc">Fig. 58.</span></td></tr></table> + +<p>In fig. 57 the delineation is on a plane of the paper taken parallel +to the plane zOx, the points of a solid figure being projected on that +plane by parallels to some chosen line through O in the positive +octant. Sometimes it is clearer to delineate, as in fig. 58, by projection +parallel to that line in the octant which is equally inclined to +Ox, Oy, Oz upon a plane of the paper perpendicular to it. It is +possible by parallel projection to delineate equal scales along Ox, +Oy, Oz by scales having any ratios we like along lines in a plane +having any mutual inclinations we like.</p> + +<table class="flt" style="float: right; width: 375px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:325px; height:293px" src="images/img717b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 59.</span></td></tr></table> + +<p>For the delineation of a surface of simple form it frequently +suffices to delineate the sections by the coordinate planes; and, in +particular, when the surface has symmetry about each coordinate +plane, to delineate the +quarter-sections belonging +to a single +octant. Thus fig. 59 +conveniently represents +an octant of the +wave surface, which +cuts each coordinate +plane in a circle and +an ellipse. Or we may +delineate a series of +contour lines, <i>i.e.</i> sections +by planes parallel +to xOy, or some other +chosen plane; of course +other sections may be +indicated too for +greater clearness. For +the delineation of a +curve a good method +is to represent, as +above, a series of points +P thereof, each accompanied by its ordinate PN, which serves to +refer it to the plane of xy. The employment of stereographic +projection is also interesting.</p> + +<p>28. In plane geometry, reckoning the line as a curve of the first +order, we have only the point and the curve. In solid geometry, +reckoning a line as a curve of the first order, and the plane as a surface +of the first order, we have the point, the curve and the surface; +but the increase of complexity is far greater than would hence at +first sight appear. In plane geometry a curve is considered in +connexion with lines (its tangents); but in solid geometry the curve +is considered in connexion with lines and planes (its tangents and +osculating planes), and the surface also in connexion with lines and +planes (its tangent lines and tangent planes); there are surfaces +arising out of the line—cones, skew surfaces, developables, doubly +and triply infinite systems of lines, and whole classes of theories +which have nothing analogous to them in plane geometry: it is thus +a very small part indeed of the subject which can be even referred +to in the present article.</p> + +<p>In the case of a surface we have between the coordinates (x, y, z) +a single, or say a onefold relation, which can be represented by a +single relation ƒ(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 = ƒ(x, y) is a particular case of each of these +modes of representation; in other words, we have in the first mode +ƒ(x, y, z) = z − ƒ(x, y), and in the second mode x = p, y = q for the +expression of two of the coordinates in terms of the parameters.</p> + +<p><span class="pagenum"><a name="page718" id="page718"></a>718</span></p> + +<p>In the case of a curve we have between the coordinates (x, y, z) a +twofold relation: two equations ƒ(x, y, z) = 0, φ(x, y, z) = 0 give +such a relation; <i>i.e.</i> the curve is here considered as the intersection +of two surfaces (but the curve is not always the complete intersection +of two surfaces, and there are hence difficulties); or, again, the coordinates +may be given each of them as a function of a single variable +parameter. The form y = φ(x), z = ψ(x), where two of the coordinates +are given in terms of the third, is a particular case of each of these +modes of representation.</p> + +<p>29. The remarks under plane geometry as to descriptive and +metrical propositions, and as to the non-metrical character of the +method of coordinates when used for the proof of a descriptive +proposition, apply also to solid geometry; and they might be +illustrated in like manner by the instance of the theorem of the radical +centre of four spheres. The proof is obtained from the consideration +that S and S′ being each of them a function of the form x² + y² + z² + +ax + by + cz + d, the difference S-S′ is a mere linear function of the +coordinates, and consequently that S-S′ = 0 is the equation of the +plane containing the circle of intersection of the two spheres S = 0 +and S′ = 0.</p> + +<table class="flt" style="float: left; width: 200px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:155px; height:298px" src="images/img718.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 60.</span></td></tr></table> + +<p>30. <i>Metrical Theory.</i>—The foundation in solid geometry of the +metrical theory is in fact the before-mentioned theorem that if a +finite right line PQ be projected upon any other line OO′ by lines +perpendicular to OO′, then the length of the +projection P′Q′ is equal to the length of PQ +into the cosine of its inclination to P′Q′—or +(in the form in which it is now convenient +to state the theorem) the perpendicular +distance P′Q′ of two parallel planes is equal +to the inclined distance PQ into the cosine +of the inclination. The principle of § 16, +that the algebraical sum of the projections of +the sides of any closed polygon on any line is +zero, or that the two sets of sides of the +polygon which connect a vertex A and a +vertex B have the same sum of projections +on the line, in sign and magnitude, as we pass +from A to B, is applicable when the sides do +not all lie in one plane.</p> + +<p>31. Consider the skew quadrilateral QMNP, +the sides QM, MN, NP being respectively +parallel to the three rectangular axes Ox, +Oy, Oz; let the lengths of these sides be +ξ, η, ζ, and that of the side QP be = ρ; and +let the cosines of the inclinations (or say the cosine-inclinations) of +ρ to the three axes be α, β, γ; then projecting successively on +the three sides and on QP we have</p> + +<p class="center">ξ, η, ζ = ρα, ρβ, ργ,</p> + +<p class="noind">and</p> + +<p class="center">ρ = αξ + βη + γζ,</p> + +<p class="noind">whence ρ² = ξ² + η² + ζ², which is the relation between a distance ρ +and its projections ξ, η, ζ upon three rectangular axes. And from +the same equations we obtain α² + β² + γ² = 1, which is a relation connecting +the cosine-inclinations of a line to three rectangular axes.</p> + +<p>Suppose we have through Q any other line QT, and let the cosine-inclinations +of this to the axes be α′, β′, γ′, and δ be its cosine-inclination +to QP; also let ρ be the length of the projection of QP +upon QT; then projecting on QT we have</p> + +<p class="center">ρ = α′ξ + β′η + γ′ζ = ρδ.</p> + +<p>And in the last equation substituting for ξ, η, ζ their values ρα, +ρβ, ργ we find</p> + +<p class="center">δ = αα′ + ββ′ + γγ′,</p> + +<p class="noind">which is an expression for the mutual cosine-inclination of two +lines, the cosine-inclinations of which to the axes are α, β, γ and +α′, β′, γ′ respectively. We have of course α² + β² + γ² = 1 and +α′² + β′² + γ′² = 1; and hence also</p> + +<p class="center">1 − δ² = (α² + β² + γ²)(α′² + β′² + γ′²) − (αα′ + ββ′ + γγ′)²,<br /> + += (βγ′ − β′γ)² + (γα′ − γ′α)² + (αβ′ − α′β)²;</p> + +<p class="noind">so that the sine of the inclination can only be expressed as a square +root. These formulae are the foundation of spherical trigonometry.</p> + +<p>32. <i>Straight Lines, Planes and Spheres.</i>—The foregoing formulae +give at once the equations of these loci.</p> + +<p>For first, taking Q to be a fixed point, coordinates (a, b, c), and +the cosine-inclinations (α, β, γ) 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 ξ, η, ζ then are +x − a, y − b, z − c, and we thus have</p> + +<table class="math0" summary="math"> +<tr><td>x − a</td> +<td rowspan="2">=</td> <td>y − b</td> +<td rowspan="2">=</td> <td>z − c</td> +<td rowspan="2">(= ρ),</td></tr> +<tr><td class="denom">α</td> <td class="denom">β</td> +<td class="denom">γ</td></tr></table> + +<p class="noind">which (omitting the last equation, = ρ) are the equations of the line +through the point (a, b, c), the cosine-inclinations to the axes being +α, β, γ, and these quantities being connected by the relation +α² + β² + γ² = 1. This equation may be omitted, and then α, β, γ, +instead of being equal, will only be proportional, to the cosine-inclinations.</p> + +<p>Using the last equation, and writing</p> + +<p class="center">x, y, z = a + αρ, b + βρ, c + γρ,</p> + +<p class="noind">these are expressions for the current coordinates in terms of a +parameter ρ, which is in fact the distance from the fixed point +(a, b, c).</p> + +<p>It is easy to see that, if the coordinates (x, y, z) are connected by +any two linear equations, these equations can always be brought +into the foregoing form, and hence that the two linear equations +represent a line.</p> + +<p>Secondly, taking for greater simplicity the point Q to be coincident +with the origin, and α′, β′, γ′, p to be constant, then p is the perpendicular +distance of a plane from the origin, and α′, β′, γ′ are the cosine-inclinations +of this distance to the axes (α′² + β′² + γ′² = 1). P is +any point in this plane, and taking its coordinates to be (x, y, z) then +(ξ, η, ζ) are = (x, y, z), and the foregoing equation p = α′ξ + β′η + γ′ζ +becomes</p> + +<p class="center">α′x + β′y + γ′z = p,</p> + +<p class="noind">which is the equation of the plane in question.</p> + +<p>If, more generally, Q is not coincident with the origin, then, +taking its coordinates to be (a, b, c), and writing p<span class="su">1</span> instead of p, the +equation is</p> + +<p class="center">α′ (x − a) + β′ (y − b) + γ′ (z − c) = p<span class="su">1</span>;</p> + +<p class="noind">and we thence have p<span class="su">1</span> = p − (aα′ + bβ′ + cγ′), which is an expression +for the perpendicular distance of the point (a, b, c) from the plane +in question.</p> + +<p>It is obvious that any linear equation Ax + By + Cz + D = O between +the coordinates can always be brought into the foregoing form, +and hence that such an equation represents a plane.</p> + +<p>Thirdly, supposing Q to be a fixed point, coordinates (a, b, c), +and the distance QP = ρ, to be constant, say this is = d, then, as +before, the values of ξ, η, ζ are x − a, y − b, z − c, and the equation +ξ² + η² + ζ² = ρ² becomes</p> + +<p class="center">(x − a)² + (y − b)² + (z − c)² = d²,</p> + +<p class="noind">which is the equation of the sphere, coordinates of the centre = (a, b, c), +and radius = d.</p> + +<p>A quadric equation wherein the terms of the second order are +x² + y² + z², viz. an equation</p> + +<p class="center">x² + y² + z² + Ax + By + Cz + D = 0,</p> + +<p class="noind">can always, it is clear, be brought into the foregoing form; and it +thus appears that this is the equation of a sphere, coordinates of +the centre −½A, −½B, −½C, and squared radius = ¼(A² + B² + C²) − D.</p> + +<p>33. <i>Cylinders, Cones, ruled Surfaces.</i>—If the two equations of a +straight line involve a parameter to which any value may be given, +we have a singly infinite system of lines. They cover a surface, and +the equation of the surface is obtained by eliminating the parameter +between the two equations.</p> + +<p>If the lines all pass through a given point, then the surface is a +cone; and, in particular, if the lines are all parallel to a given line, +then the surface is a cylinder.</p> + +<p>Beginning with this last case, suppose the lines are parallel to +the line x = mz, y = nz, the equations of a line of the system are +x = mz + a, y = nz + b,—where a, b are supposed to be functions of +the variable parameter, or, what is the same thing, there is between +them a relation ƒ(a, b) = 0: we have a = x − mz, b = y − nz, and the +result of the elimination of the parameter therefore is ƒ(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 ƒ(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 +ƒ(x, y) = 0, the equation of the cylinder is ƒ(x − mz, y − nz) = 0. Thus, +if the curve of intersection be the circle (x − α)² + (y − β)² = γ², we +have (x − mz − α)² + (y − nz − β)² = γ² as the equation of an oblique +cylinder on this base, and thus also (x − α)² + (y − β)² = γ² as the +equation of the right cylinder.</p> + +<p>If the lines all pass through a given point (a, b, c), then the equations +of a line are x − a = α(z − c), y − b = β(z − c), where α, β are +functions of the variable parameter, or, what is the same thing, +there exists between them an equation ƒ(α, β) = 0; the elimination +of the parameter gives, therefore, ƒ[(x − a)/(x − c′), (y − b)/(z − c)] = 0; and this +equation, or, what is the same thing, any homogeneous equation +ƒ(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)<span class="sp">n</span> = 0, is the general +equation of the cone having the point (a, b, c) for its vertex. Taking +the vertex to be at the origin, the equation is (*)(x, y, z)<span class="sp">n</span> = 0; and, +in particular, (*)(x, y, z)² = 0 is the equation of a cone of the second +order, or quadricone, having the origin for its vertex.</p> + +<p>34. In the general case of a singly infinite system of lines, the +locus is a ruled surface (or <i>regulus</i>). Now, when a line is changing +its position in space, it may be looked upon as in a state of turning +about some point in itself, while that point is, as a rule, in a state of +moving out of the plane in which the turning takes place. If instantaneously +it is only in a state of turning, it is usual, though not +strictly accurate, to say that it intersects its consecutive position. +A regulus such that consecutive lines on it do not intersect, in this +sense, is called a skew surface, or <i>scroll</i>; one on which they do is +called a developable surface or <i>torse</i>.</p> + +<p>Suppose, for instance, that the equations of a line (depending on +<span class="pagenum"><a name="page719" id="page719"></a>719</span> +the variable parameter θ) are x/a + y/c = θ +(1 + y/b), x/a − z/c = (1/θ)(1 − y/b); +then, eliminating θ we have x²/a² − z²/c² = 1 − y²/b², or say, x²/a² + y²/b² − z²/c² = 1, +the equation of a quadric surface, afterwards called the hyperboloid +of one sheet; this surface is consequently a scroll. It is to be remarked +that we have upon the surface a second singly infinite +series of lines; the equations of a line of this second system (depending +on the variable parameter φ) are</p> + +<table class="math0" summary="math"> +<tr><td>x</td> +<td rowspan="2">+</td> <td>z</td> +<td rowspan="2">= φ <span class="f150">(</span> 1 −</td> <td>y</td> +<td rowspan="2"><span class="f150">)</span>,  </td> <td>x</td> +<td rowspan="2">−</td> <td>z</td> +<td rowspan="2">=</td> <td>1</td> +<td rowspan="2"><span class="f150">(</span> 1 +</td> <td>y</td> +<td rowspan="2"><span class="f150">)</span>.</td></tr> +<tr><td class="denom">a</td> <td class="denom">c</td> +<td class="denom">b</td> <td class="denom">a</td> +<td class="denom">c</td> <td class="denom">φ</td> +<td class="denom">b</td></tr></table> + +<p class="noind">It is easily shown that any line of the one system intersects every +line of the other system.</p> + +<p>Considering any curve (of double curvature) whatever, the tangent +lines of the curve form a singly infinite system of lines, each line +intersecting the consecutive line of the system,—that is, they form +a developable, or torse; the curve and torse are thus inseparably +connected together, forming a single geometrical figure. An osculating +plane of the curve (see § 38 below) is a tangent plane of the torse +all along a generating line.</p> + +<p>35. <i>Transformation of Coordinates.</i>—There is no difficulty in +changing the origin, and it is for brevity assumed that the origin +remains unaltered. We have, then, two sets of rectangular axes, +Ox, Oy, Oz, and Ox<span class="su">1</span>, Oy<span class="su">1</span>, Ozx<span class="su">1</span>, the mutual cosine-inclinations being +shown by the diagram—</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc bb"> </td> <td class="tcc lb rb">x</td> <td class="tcc rb">y</td> <td class="tcc rb">z</td></tr> + +<tr><td class="tcc bb">x<span class="su">1</span></td> <td class="tcc allb">α</td> <td class="tcc allb">β</td> <td class="tcc allb">γ</td></tr> + +<tr><td class="tcc bb">y<span class="su">1</span></td> <td class="tcc allb">α</td> <td class="tcc allb">β′</td> <td class="tcc allb">γ′</td></tr> + +<tr><td class="tcc bb">z<span class="su">1</span></td> <td class="tcc allb">α″</td> <td class="tcc allb">β″</td> <td class="tcc allb">γ″</td></tr> + +</table> + +<p class="noind">that is, α, β, γ are the cosine-inclinations of +Ox<span class="su">1</span> to Ox, Oy, Oz; +α′, β′, γ′ those of Oy<span class="su">1</span>, &c.</p> + +<p>And this diagram gives also the linear expressions of the coordinates +(x<span class="su">1</span>, y<span class="su">1</span>, z<span class="su">1</span>) or (x, y, z) of either set in terms of those of the +other set; we thus have</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">x<span class="su">1</span> = α x + β y + γ z,</td> <td class="tcl">x = αx<span class="su">1</span> + α′y<span class="su">1</span> + α″z<span class="su">1</span>,</td></tr> + +<tr><td class="tcl">y<span class="su">1</span> = α′x + β′y + γ′z,</td> <td class="tcl">y = βx<span class="su">1</span> + β′y<span class="su">1</span> + β″z<span class="su">1</span>,</td></tr> + +<tr><td class="tcl">z<span class="su">1</span> = α″x + β″y + γ″z,</td> <td class="tcl">z = γx<span class="su">1</span> + γ′y<span class="su">1</span> + γ″z<span class="su">1</span>,</td></tr> +</table> + +<p class="noind">which are obtained by projection, as above explained. Each of +these equations is, in fact, nothing else than the before-mentioned +equation p = α′ξ + β′η + γ′ζ, adapted to the problem in hand.</p> + +<p>But we have to consider the relations between the nine coefficients. +By what precedes, or by the consideration that we must have +identically x² + y² + z² = x<span class="su">1</span>² + y<span class="su">1</span>² + z<span class="su">1</span>², it appears that these satisfy +the relations—</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">α²</td> <td class="tcl">+ β²</td> <td class="tcl">+ γ²</td> <td class="tcl">= 1,  </td> <td class="tcl">α² +</td> <td class="tcl">α′²</td> <td class="tcl">+ α″²</td> <td class="tcl">= 1,</td></tr> +<tr><td class="tcl">α′²</td> <td class="tcl">+ β′²</td> <td class="tcl">+ γ′²</td> <td class="tcl">= 1,  </td> <td class="tcl">β²</td> <td class="tcl">+ β′²</td> <td class="tcl">+ β″²</td> <td class="tcl">= 1,</td></tr> +<tr><td class="tcl">α″²</td> <td class="tcl">+ β″²</td> <td class="tcl">+ γ″²</td> <td class="tcl">= 1,  </td> <td class="tcl">γ²</td> <td class="tcl">+ γ′²</td> <td class="tcl">+ γ″²</td> <td class="tcl">= 1,</td></tr> +<tr><td class="tcl">α′a″</td> <td class="tcl">+ β′β″</td> <td class="tcl">+ γ′γ″</td> <td class="tcl">= 0,  </td> <td class="tcl">βγ</td> <td class="tcl">+β′γ′</td> <td class="tcl">+ β″γ″</td> <td class="tcl">= 0,</td></tr> +<tr><td class="tcl">α″α</td> <td class="tcl">+ β″β</td> <td class="tcl">+ γ″γ</td> <td class="tcl">= 0,  </td> <td class="tcl">γα</td> <td class="tcl">+ γ′α′</td> <td class="tcl">+ γ″α″</td> <td class="tcl">= 0,</td></tr> +<tr><td class="tcl">αα′</td> <td class="tcl">+ ββ′</td> <td class="tcl">+ γγ′</td> <td class="tcl">= 0,  </td> <td class="tcl">αβ</td> <td class="tcl">+α′β′</td> <td class="tcl">+ α″β″</td> <td class="tcl">= 0,</td></tr> +</table> + +<p class="noind">either set of six equations being implied in the other set.</p> + +<p>It follows that the square of the determinant</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc lb">α,</td> <td class="tcc">β,</td> <td class="tcc rb">γ</td></tr> + +<tr><td class="tcc lb">α′,</td> <td class="tcc">β′,</td> <td class="tcc rb">γ′</td></tr> + +<tr><td class="tcc lb">α″,</td> <td class="tcc">β″,</td> <td class="tcc rb">γ″</td></tr> +</table> + +<p class="noind">is = 1; and hence that the determinant itself is = ±1. The distinction +of the two cases is an important one: if the determinant is += + 1, then the axes Ox<span class="su">1</span>, Oy<span class="su">1</span>, Oz<span class="su">1</span> are such that they can by a +rotation about O be brought to coincide with Ox, Oy, Oz respectively; +if it is = −1, then they cannot. But in the latter case, by +measuring x<span class="su">1</span>, y<span class="su">1</span>, z<span class="su">1</span> in the opposite directions we change the signs of +all the coefficients and so make the determinant to be = + 1; hence +the former case need alone be considered, and it is accordingly +assumed that the determinant is = +1. This being so, it is found +that we have the equality α = β′γ″ − β″γ′, and eight like ones, +obtained from this by cyclical interchanges of the letters α, β, γ, +and of unaccented, singly and doubly accented letters.</p> + +<p>36. The nine cosine-inclinations above are, as has been seen, +connected by six equations. It ought then to be possible to express +them all in terms of three parameters. An elegant means of doing +this has been given by Rodrigues, who has shown that the tabular +expression of the formulae of transformation may be written</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc bb"> </td> <td class="tcc lb rb">x</td> <td class="tcc rb">y</td> <td class="tcc rb">z</td></tr> + +<tr><td class="tcc bb">x<span class="su">1</span></td> <td class="tcc allb">1 + λ² − μ² − ν²</td> <td class="tcc allb">2(λμ − ν)</td> <td class="tcc allb">2(νλ + μ)</td></tr> + +<tr><td class="tcc bb">y<span class="su">1</span></td> <td class="tcc allb">2(λμ + ν)</td> <td class="tcc allb">1 − λ² + μ² − ν²</td> <td class="tcc allb">2(μν + λ)</td></tr> + +<tr><td class="tcc bb">z<span class="su">1</span></td> <td class="tcc allb">2(νλ − μ)</td> <td class="tcc allb"> 2(μν + λ)</td> <td class="tcc allb">1 − λ² − μ² + ν²</td></tr> + +<tr><td class="tcc" colspan="4">÷ (1 + λ² + μ² + ν²),</td></tr> +</table> + +<p class="noind">the meaning being that the coefficients in the transformation are +fractions, with numerators expressed as in the table, and the common +denominator.</p> + +<p>37. <i>The Species of Quadric Surfaces</i>.—Surfaces represented by +equations of the second degree are called <i>quadric</i> surfaces. Quadric +surfaces are either <i>proper</i> or <i>special</i>. The special ones arise when the +coefficients in the general equation are limited to satisfy certain +special equations; they comprise (1) plane-pairs, including in +particular one plane twice repeated, and (2) cones, including in +particular cylinders; there is but one form of cone, but cylinders +may be elliptic, parabolic or hyperbolic.</p> + +<p>A discussion of the general equation of the second degree shows +that the <i>proper</i> quadric surfaces are of five kinds, represented +respectively, when referred to the most convenient axes of reference, +by equations of the five types (a and b positive):</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">(1)   </td> <td class="tcl">z = x²/2a + y²/2b, elliptic paraboloid.</td></tr> + +<tr><td class="tcl">(2)   </td> <td class="tcl">z = x²/2a − y²/2b, hyperbolic paraboloid.</td></tr> + +<tr><td class="tcl">(3)   </td> <td class="tcl">x²/a² + y²/b² + z²/c² = 1, ellipsoid.</td></tr> + +<tr><td class="tcl">(4)   </td> <td class="tcl">x²/a² + y²/b² − z²/c² = 1, hyperboloid of one sheet.</td></tr> + +<tr><td class="tcl">(5)   </td> <td class="tcl">x²/a² + y²/b² − z²/c² = −1, hyperboloid of two sheets.</td></tr> +</table> + +<table class="flt" style="float: right; width: 270px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:221px; height:217px" src="images/img719a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig</span>. 61.</td></tr></table> + +<p>It is at once seen that these are distinct surfaces; and the equations +also show very readily the +general form and mode of generation +of the several surfaces.</p> + +<p>In the elliptic paraboloid (fig. 61) +the sections by the planes of zx and +zy are the parabolas</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">z =</td> <td>x²</td> +<td rowspan="2">,  z =</td> <td>y²</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">2a</td> <td class="denom">2b</td></tr></table> + +<p class="noind">having the common axes Oz; and +the section by any plane z = γ +parallel to that of xy is the ellipse</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">γ =</td> <td>x²</td> +<td rowspan="2">+</td> <td>y²</td> +<td rowspan="2">;</td></tr> +<tr><td class="denom">2a</td> <td class="denom">2b</td></tr></table> + +<p class="noind">so that the surface is generated by +a variable ellipse moving parallel to itself along the parabolas as +directrices.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:515px; height:232px" src="images/img719b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig</span>. 62.</td> +<td class="caption"><span class="sc">Fig</span>. 63.</td></tr></table> + +<table class="flt" style="float: right; width: 350px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:300px; height:249px" src="images/img719c.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig</span>. 64.</td></tr></table> + +<p>In the hyperbolic paraboloid (figs. 62 and 63) the sections by the +planes of zx, zy are the parabolas z = x²/2a, z = − y²/2b, having the opposite +axes Oz, Oz′, and the section by a plane z = γ parallel to that of +xy is the hyperbola γ = x²/2a − y²/2b, which has its transverse axis parallel +to Ox or Oy according as γ is positive or negative. The surface is thus +generated by a variable hyperbola moving parallel to itself along +the parabolas as directrices. The form is best seen from fig. 63, +which represents the sections +by planes parallel to +the plane of xy, or say the +contour lines; the continuous +lines are the sections +above the plane of +xy, and the dotted lines +the sections below this +plane. The form is, in +fact, that of a saddle.</p> + +<p>In the ellipsoid (fig. 64) +the sections by the planes +of zx, zy, and xy are each +of them an ellipse, and the +section by any parallel +plane is also an ellipse. +The surface may be considered +as generated by +an ellipse moving parallel to itself along two ellipses as directrices.</p> + +<p><span class="pagenum"><a name="page720" id="page720"></a>720</span></p> + +<p>In the hyperboloid of one sheet (fig. 65), the sections by the planes +of zx, zy are the hyperbolas</p> + +<table class="math0" summary="math"> +<tr><td>x²</td> +<td rowspan="2">−</td> <td>z²</td> +<td rowspan="2">= 1, </td> <td>y²</td> +<td rowspan="2">−</td> <td>z²</td> +<td rowspan="2">= 1,</td></tr> +<tr><td class="denom">c²</td> <td class="denom">c²</td> +<td class="denom">b²</td> <td class="denom">c²</td></tr></table> + +<p class="noind">having a common conjugate axis zOz′; the section by the plane of +x, y, and that by any parallel plane, is an ellipse; and the surface +may be considered as generated by a variable ellipse moving parallel +to itself along the two hyperbolas as directrices. If we imagine two +equal and parallel circular disks, their points connected by strings +of equal lengths, so that these are the generators of a right circular +cylinder, and if we turn one of the disks about its centre through an +angle in its plane, the strings in their new positions will be one +system of generators of a hyperboloid of one sheet, for which a = b; +and if we turn it through the same angle in the opposite direction, +we get in like manner the generators of the other system; there will +be the same general configuration when a ≠ b. The hyperbolic +paraboloid is also covered by two systems of rectilinear generators +as a method like that used in § 34 establishes without difficulty. +The figures should be studied to see how they can lie.</p> + +<table class="nobctr" style="clear: both;" summary="Illustration"> +<tr><td class="figcenter" colspan="2"><img style="width:518px; height:328px" src="images/img720.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 65.</span></td> +<td class="caption"><span class="sc">Fig. 66.</span></td></tr></table> + +<p>In the hyperboloid of two sheets (fig. 66) the sections by the planes +of zx and zy are the hyperbolas</p> + +<table class="math0" summary="math"> +<tr><td>z²</td> +<td rowspan="2">−</td> <td>x²</td> +<td rowspan="2">= 1, </td> <td>z²</td> +<td rowspan="2">−</td> <td>y²</td> +<td rowspan="2">= 1,</td></tr> +<tr><td class="denom">c²</td> <td class="denom">a²</td> +<td class="denom">c²</td> <td class="denom">b²</td></tr></table> + +<p class="noind">having a common transverse axis along z′Oz; the section by any +plane z = ±γ parallel to that of xy is the ellipse</p> + +<table class="math0" summary="math"> +<tr><td>x²</td> +<td rowspan="2">+</td> <td>y²</td> +<td rowspan="2">=</td> <td>γ²</td> +<td rowspan="2">− 1,</td></tr> +<tr><td class="denom">a²</td> <td class="denom">b²</td> +<td class="denom">c²</td></tr></table> + +<p class="noind">provided γ² > c², and the surface, consisting of two distinct portions +or sheets, may be considered as generated by a variable ellipse +moving parallel to itself along the hyperbolas as directrices.</p> + +<p>38. <i>Differential Geometry of Curves.</i>—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 θ, which may in particular be one +of themselves. Use the notation x′, x″ for dx/dθ, d²x/dθ², and similarly +as to y and z. Only a few formulae will be given. Call the +current coordinates (ξ, η, ζ).</p> + +<p>The <i>tangent</i> at (x, y, z) is the line tended to as a limit by the +connector of (x, y, z) and a neighbouring point of the curve when the +latter moves up to the former: its equations are</p> + +<p class="center">(ξ − x)/x′ = (η − y)/y′ = (ζ − z)/z′.</p> + +<p>The <i>osculating plane</i> at (x, y, z) is the plane tended to as a limit by +that through (x, y, z) and two neighbouring points of the curve as +these, remaining distinct, both move up to (x, y, z): its one equation +is</p> + +<p class="center">(ξ − x) (y′z″ − y″z′) + (η − y) (z′x″ − z″x′) + (ζ − z) (x′y″ − x″y′) = 0.</p> + +<p>The <i>normal plane</i> is the plane through (x, y, z) at right angles to the +tangent line, <i>i.e.</i> the plane</p> + +<p class="center">x′(ξ − x) + y′ (η − y) + z′ (ζ − z) = 0.</p> + +<p class="noind">It cuts the osculating plane in a line called the <i>principal normal</i>. +Every line through (x, y, z) in the normal plane is a normal. The +normal perpendicular to the osculating plane is called the <i>binormal</i>. +A tangent, principal normal, and binormal are a convenient set of +rectangular axes to use as those of reference, when the nature of a +curve near a point on it is to be discussed.</p> + +<p>Through (x, y, z) and three neighbouring points, all on the curve, +passes a single sphere; and as the three points all move up to (x, y, z) +continuing distinct, the sphere tends to a limiting size and position. +The limit tended to is the sphere of closest contact with the curve at +(x, y, z); its centre and radius are called the centre and radius of +<i>spherical curvature</i>. It cuts the osculating plane in a circle, called the +<i>circle of absolute curvature</i>; and the centre and radius of this circle +are the centre and radius of absolute curvature. The centre of +absolute curvature is the limiting position of the point where the +principal normal at (x, y, z) is cut by the normal plane at a neighbouring +point, as that point moves up to (x, y, z).</p> + +<p>39. <i>Differential Geometry of Surfaces.</i>—Let (x, y, z) be any chosen +point on a surface ƒ(x, y, z) = 0. As a second point of the surface +moves up to (x, y, z), its connector with (x, y, z) tends to a limiting +position, a tangent line to the surface at (x, y, z). All these tangent +lines at (x, y, z), obtained by approaching (x, y, z) from different +directions on a surface, lie in one plane</p> + +<table class="math0" summary="math"> +<tr><td>∂ƒ</td> +<td rowspan="2">(ξ − x) +</td> <td>∂ƒ</td> +<td rowspan="2">(η − y) +</td> <td>∂ƒ</td> +<td rowspan="2">(ζ − z) = 0.</td></tr> +<tr><td class="denom">∂x</td> <td class="denom">∂y</td> +<td class="denom">∂z</td></tr></table> + +<p class="noind">This plane is called the <i>tangent plane</i> at (x, y, z). One line through +(x, y, z) is at right angles to the tangent plane. This is the normal</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">(ξ − x) <span class="f150">/</span></td> <td>∂ƒ</td> +<td rowspan="2">= (η − y) <span class="f150">/</span></td> <td>∂ƒ</td> +<td rowspan="2">= (ζ − z) <span class="f150">/</span></td> <td>∂ƒ</td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">∂x</td> <td class="denom">∂y</td> +<td class="denom">∂z</td></tr></table> + +<p>The tangent plane is cut by the surface in a curve, real or imaginary, +with a node or double point at (x, y, z). Two of the tangent lines +touch this curve at the node. They are called the “chief tangents” +(<i>Haupt-tangenten</i>) at (x, y, z); they have closer contact with the +surface than any other tangents.</p> + +<p>In the case of a quadric surface the curve of intersection of a +tangent and the surface is of the second order and has a node, +it must therefore consist of two straight lines. Consequently a +quadric surface is covered by two sets of straight lines, a pair through +every point on it; these are imaginary for the ellipsoid, hyperboloid +of two sheets, and elliptic paraboloid.</p> + +<p>A surface of any order is covered by two singly infinite systems +of curves, a pair through every point, the tangents to which are all +chief tangents at their respective points of contact. These are +called <i>chief-tangent curves</i>; on a quadric surface they are the above +straight lines.</p> + +<p>40. The tangents at a point of a surface which bisect the angles +between the chief tangents are called the <i>principal tangents</i> at the +point. They are at right angles, and together with the normal +constitute a convenient set of rectangular axes to which to refer the +surface when its properties near the point are under discussion. +At a special point which is such that the chief tangents there run +to the circular points at infinity in the tangent plane, the principal +tangents are indeterminate; such a special point is called an umbilic +of the surface.</p> + +<p>There are two singly infinite systems of curves on a surface, a +pair cutting one another at right angles through every point upon it, +all tangents to which are principal tangents of the surface at their +respective points of contact. These are called <i>lines of curvature</i>, +because of a property next to be mentioned.</p> + +<p>As a point Q moves in an arbitrary direction on a surface from +coincidence with a chosen point P, the normal at it, as a rule, at +once fails to meet the normal at P; but, if it takes the direction of a +line of curvature through P, this is instantaneously not the case. +We have thus on the normal two centres of curvature, and the +distances of these from the point on the surface are the two <i>principal +radii of curvature</i> of the surface at that point; these are also the radii +of curvature of the sections of the surface by planes through the +normal and the two principal tangents respectively; or say they are +the radii of curvature of the normal sections through the two principal +tangents respectively. Take at the point the axis of z in the direction +of the normal, and those of x and y in the directions of the principal +tangents respectively, then, if the radii of curvature be a, b (the signs +being such that the coordinates of the two centres of curvature are +z = a and z = b respectively), the surface has in the neighbourhood +of the point the form of the paraboloid</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">z =</td> <td>x²</td> +<td rowspan="2">+</td> <td>y²</td> +<td rowspan="2">,</td></tr> +<tr><td class="denom">2a</td> <td class="denom">2b</td></tr></table> + +<p class="noind">and the chief-tangents are determined by the equation 0 = x²/2a + y²/2b. +The two centres of curvature may be on the same side of the point +or on opposite sides; in the former case a and b have the same sign, +the paraboloid is elliptic, and the chief-tangents are imaginary; +in the latter case a and b have opposite signs, the paraboloid is +hyperbolic, and the chief-tangents are real.</p> + +<p>The normal sections of the surface and the paraboloid by the same +plane have the same radius of curvature; and it thence readily +follows that the radius of curvature of a normal section of the surface +by a plane inclined at an angle θ to that of zx is given by the equation</p> + +<table class="math0" summary="math"> +<tr><td>1</td> +<td rowspan="2">=</td> <td>cos² θ</td> +<td rowspan="2">+</td> <td>sin² θ</td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">ρ</td> <td class="denom">a</td> +<td class="denom">b</td></tr></table> + +<p>The section in question is that by a plane through the normal +and a line in the tangent plane inclined at an angle θ 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 φ; then it is shown without +difficulty (Meunier’s theorem) that the radius of curvature of this +inclined section of the surface is = ρ cos φ.</p> + +<p><span class="sc">Authorities.</span>—The above article is largely based on that by +Arthur Cayley in the 9th edition of this work. Of early and important +recent publications on analytical geometry, special mention +<span class="pagenum"><a name="page721" id="page721"></a>721</span> +is to be made of R. Descartes, <i>Géométrie</i> (Leyden, 1637); John +Wallis, <i>Tractatus de sectionibus conicis nova methodo expositis</i> (1655, +<i>Opera mathematica</i>, i., Oxford, 1695); de l’Hospital, <i>Traité analytique +des sections coniques</i> (Paris, 1720); Leonhard Euler, <i>Introductio in +analysin infinitorum</i>, ii. (Lausanne, 1748); Gaspard Monge, “Application +d’algèbre à la géométrie” (<i>Journ. École Polytech.</i>, 1801); +Julius Plücker, <i>Analytisch-geometrische Entwickelungen</i>, 3 Bde. +(Essen, 1828-1831); <i>System der analytischen Geometrie</i> (Berlin, +1835); G. Salmon, <i>A Treatise on Conic Sections</i> (Dublin, 1848; +6th ed., London, 1879); Ch. Briot and J. Bouquet, <i>Leçons de géométrie +analytique</i> (Paris, 1851; 16th ed., 1897); M. Chasles, <i>Traité +de géométrie supérieure</i> (Paris, 1852); Wilhelm Fiedler, <i>Analytische +Geometrie der Kegelschnitte</i> nach G. Salmon frei bearbeitet (Leipzig, +5te Aufl., 1887-1888); N.M. Ferrers, <i>An Elementary Treatise on +Trilinear Coordinates</i> (London, 1861); Otto Hesse, <i>Vorlesungen +aus der analytischen Geometrie</i> (Leipzig, 1865, 1881); W.A. Whitworth, +<i>Trilinear Coordinates and other Methods of Modern Analytical +Geometry</i> (Cambridge, 1866); J. Booth, <i>A Treatise on Some New +Geometrical Methods</i> (London, i., 1873; ii., 1877); A. Clebsch-F. +Lindemann, <i>Vorlesungen über Geometrie</i>, Bd. i. (Leipzig, 1876, +2te Aufl., 1891); R. Baltser, <i>Analytische Geometrie</i> (Leipzig, 1882); +Charlotte A. Scott, <i>Modern Methods of Analytical Geometry</i> (London, +1894); G. Salmon, <i>A Treatise on the Analytical Geometry of three +Dimensions</i> (Dublin, 1862; 4th ed., 1882); Salmon-Fiedler, <i>Analytische +Geometrie des Raumes</i> (Leipzig, 1863; 4te Aufl., 1898); P. +Frost, <i>Solid Geometry</i> (London, 3rd ed., 1886; 1st ed., Frost and +J. Wolstenholme). See also E. Pascal, <i>Repertorio di matematiche +superiori, II. Geometria</i> (Milan, 1900), and articles now appearing +in the <i>Encyklopädie der mathematischen Wissenschaften</i>, Bd. iii. 1, 2.</p> +</div> +<div class="author">(E. B. El.)</div> + +<p class="pt2 center sc">V. Line Geometry</p> + +<p>Line geometry is the name applied to those geometrical +investigations in which the straight line replaces the point as +element. Just as ordinary geometry deals primarily with points +and systems of points, this theory deals in the first instance +with straight lines and systems of straight lines. In two dimensions +there is no necessity for a special line geometry, inasmuch +as the straight line and the point are interchangeable by the +principle of duality; but in three dimensions the straight line +is its own reciprocal, and for the better discussion of systems +of lines we require some new apparatus, <i>e.g.</i>, a system of coordinates +applicable to straight lines rather than to points. +The essential features of the subject are most easily elucidated +by analytical methods: we shall therefore begin with the notion +of line coordinates, and in order to emphasize the merits of the +system of coordinates ultimately adopted, we first notice a +system without these advantages, but often useful in special +investigations.</p> + +<div class="condensed"> +<p>In ordinary Cartesian coordinates the two equations of a straight +line may be reduced to the form y = rx + s, z = tx + u, and r, s, t, u +may be regarded as the four coordinates of the line. These coordinates +lack symmetry: moreover, in changing from one base of +reference to another the transformation is not linear, so that the +degree of an equation is deprived of real significance. For purposes +of the general theory we employ homogeneous coordinates; if +x<span class="su">1</span>y<span class="su">1</span>z<span class="su">1</span>w<span class="su">1</span> and x<span class="su">2</span>y<span class="su">2</span>z<span class="su">2</span>w<span class="su">2</span> are two points on the line, it is easily verified +that the six determinants of the array</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc lb rb">x<span class="su">1</span>y<span class="su">1</span>z<span class="su">1</span>w<span class="su">1</span></td></tr> + +<tr><td class="tcc lb rb">x<span class="su">2</span>y<span class="su">2</span>z<span class="su">2</span>w<span class="su">2</span></td></tr> +</table> + +<p class="noind">are in the same ratios for all point-pairs on the line, and further, +that when the point coordinates undergo a linear transformation +so also do these six determinants. We therefore adopt these six +determinants for the coordinates of the line, and express them by the +symbols l, λ, m, μ, n, ν where l = x<span class="su">1</span>w<span class="su">2</span> − x<span class="su">2</span>w<span class="su">1</span>, λ = y<span class="su">1</span>z<span class="su">2</span> − y<span class="su">2</span>z<span class="su">1</span>, &c. +There is the further advantage that if a<span class="su">1</span>b<span class="su">1</span>c<span class="su">1</span>d<span class="su">1</span> and a<span class="su">2</span>b<span class="su">2</span>c<span class="su">2</span>d<span class="su">2</span> be two +planes through the line, the six determinants</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc lb rb">a<span class="su">1</span>b<span class="su">1</span>c<span class="su">1</span>d<span class="su">1</span></td></tr> + +<tr><td class="tcc lb rb">a<span class="su">2</span>b<span class="su">2</span>c<span class="su">2</span>d<span class="su">2</span></td></tr> +</table> + +<p class="noind">are in the same ratios as the foregoing, so that except as regards a +factor of proportionality we have λ = b<span class="su">1</span>c<span class="su">2</span> − b<span class="su">2</span>c<span class="su">1</span>, l = c<span class="su">1</span>d<span class="su">2</span> − c<span class="su">2</span>d<span class="su">1</span>, &c. +The identical relation lλ + mμ + nν = o reduces the number of independent +constants in the six coordinates to four, for we are only +concerned with their mutual ratios; and the quadratic character +of this relation marks an essential difference between point geometry +and line geometry. The condition of intersection of two lines is</p> + +<p class="center">lλ′ + l′λ + mμ′ + m′μ + nν′ + n′ν = 0</p> + +<p class="noind">where the accented letters refer to the second line. If the coordinates +are Cartesian and l, m, n are direction cosines, the quantity on the +left is the mutual moment of the two lines.</p> + +<p>Since a line depends on four constants, there are three distinct types +of configurations arising in line geometry—those containing a triply-infinite, +a doubly-infinite and a singly-infinite number of lines; they +are called Complexes, Congruences, and Ruled Surfaces or Skews +respectively. A <i>Complex</i> is thus a system of lines satisfying one +condition—that is, the coordinates are connected by a single relation; +and the degree of the complex is the degree of this equation supposing +it to be algebraic. The lines of a complex of the nth degree which +pass through any point lie on a cone of the nth degree, those which +lie in any plane envelop a curve of the nth class and there are n lines +of the complex in any plane pencil; the last statement combines +the former two, for it shows that the cone is of the nth degree and +the curve is of the nth class. To find the lines common to four +complexes of degrees n<span class="su">1</span>, n<span class="su">2</span>, n<span class="su">3</span>, n<span class="su">4</span>, we have to solve five equations, viz. +the four complex equations together with the quadratic equation +connecting the line coordinates, therefore the number of common +lines is 2n<span class="su">1</span>n<span class="su">2</span>n<span class="su">3</span>n<span class="su">4</span>. As an example of complexes we have the lines +meeting a twisted curve of the nth degree, which form a complex +of the nth degree.</p> + +<p>A <i>Congruence</i> is the set of lines satisfying two conditions: thus +a finite number m of the lines pass through any point, and a finite +number n lie in any plane; these numbers are called the degree +and class respectively, and the congruence is symbolically written +(m, n).</p> + +<p>The simplest example of a congruence is the system of lines +constituted by all those that pass through m points and those that +lie in n planes; through any other point there pass m of these lines, +and in any other plane there lie n, therefore the congruence is of +degree m and class n. It has been shown by G.H. Halphen that the +number of lines common to two congruences is mm′ + 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.</p> + +<p>A <i>Ruled Surface</i>, <i>Regulus</i> or <i>Skew</i> is a configuration of lines +which satisfy three conditions, and therefore depend on only one +parameter. Such lines all lie on a surface, for we cannot draw one +through an arbitrary point; only one line passes through a point of +the surface; the simplest example, that of a quadric surface, is +really two skews on the same surface.</p> + +<p>The degree of a ruled surface <i>qua</i> line geometry is the number of +its generating lines contained in a linear complex. Now the number +which meets a given line is the degree of the surface <i>qua</i> point geometry, +and as the lines meeting a given line form a particular case +of linear complex, it follows that the degree is the same from whichever +point of view we regard it. The lines common to three complexes +of degrees, n<span class="su">1</span>n<span class="su">2</span>n<span class="su">3</span>, form a ruled surface of degree 2n<span class="su">1</span>n<span class="su">2</span>n<span class="su">3</span>; +but not every ruled surface is the complete intersection of three +complexes.</p> + +<p>In the case of a complex of the first degree (or linear complex) +the lines through a fixed point lie in a plane called the polar plane +or nul-plane of that point, and those lying in a fixed plane +pass through a point called the nul-point or pole of the +<span class="sidenote">Linear complex.</span> +plane. If the nul-plane of A pass through B, then the +nul-plane of B will pass through A; the nul-planes of all points on +one line l<span class="su">1</span> pass through another line l<span class="su">2</span>. The relation between l<span class="su">1</span> and +l<span class="su">2</span> is reciprocal; any line of the complex that meets one will also +meet the other, and every line meeting both belongs to the complex. +They are called conjugate or polar lines with respect to the complex. +On these principles can be founded a theory of reciprocation with +respect to a linear complex.</p> + +<p>This may be aptly illustrated by an elegant example due to A. +Voss. Since a twisted cubic can be made to satisfy twelve conditions, +it might be supposed that a finite number could be drawn to touch +four given lines, but this is not the case. For, suppose one such can +be drawn, then its reciprocal with respect to any linear complex +containing the four lines is a curve of the third class, <i>i.e.</i> another +twisted cubic, touching the same four lines, which are unaltered +in the process of reciprocation; as there is an infinite number of +complexes containing the four lines, there is an infinite number of +cubics touching the four lines, and the problem is poristic.</p> + +<p>The following are some geometrical constructions relating to the +unique linear complex that can be drawn to contain five arbitrary +lines:</p> + +<p>To construct the nul-plane of any point O, we observe that the +two lines which meet any four of the given five are conjugate lines +of the complex, and the line drawn through O to meet them is +therefore a ray of the complex; similarly, by choosing another +four we can find another ray through O: these rays lie in the nul-plane, +and there is clearly a result involved that the five lines so +obtained all lie in one plane. A reciprocal construction will enable +us to find the nul-point of any plane. Proceeding now to the metrical +properties and the statical and dynamical applications, we remark +that there is just one line such that the nul-plane of any point on it +is perpendicular to it. This is called the central axis; if d be the +shortest distance, θ the angle between it and a ray of the complex, +then d tan θ = p, where p is a constant called the pitch or parameter. +Any system of forces can be reduced to a force R along a certain line, +and a couple G perpendicular to that line; the lines of nul-moment +<span class="pagenum"><a name="page722" id="page722"></a>722</span> +for the system form a linear complex of which the given line is the +central axis and the quotient G/R is the pitch. Any motion of a +rigid body can be reduced to a screw motion about a certain line, +<i>i.e.</i> to an angular velocity ω 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/ω for pitch (cf. Sir R.S. Ball, <i>Theory of Screws</i>).</p> + +<p>The following are some properties of a configuration of two linear +complexes:</p> + +<p>The lines common to the two-complexes also belong to an infinite +number of linear complexes, of which two reduce to single straight +lines. These two lines are conjugate lines with respect to each of +the complexes, but they may coincide, and then some simple modifications +are required. The locus of the central axis of this system +of complexes is a surface of the third degree called the cylindroid, +which plays a leading part in the theory of screws as developed +synthetically by Ball. Since a linear complex has an invariant of +the second degree in its coefficients, it follows that two linear complexes +have a lineo-linear invariant. This invariant is fundamental: +if the complexes be both straight lines, its vanishing is the condition +of their intersection as given above; if only one of them be a straight +line, its vanishing is the condition that this line should belong to the +other complex. When it vanishes for any two complexes they +are said to be in <i>involution</i> or <i>apolar</i>; the nul-points P, Q of any +plane then divide harmonically the points in which the plane meets +the common conjugate lines, and each complex is its own reciprocal +with respect to the other. As regards a configuration of these +linear complexes, the common lines from one system of generators +of a quadric, and the doubly infinite system of complexes containing +the common lines, include an infinite number of straight lines which +form the other system of generators of the same quadric.</p> + +<p>If the equation of a linear complex is Al + Bm + Cn + Dλ + Eμ + +Fν = 0, then for a line not belonging to the complex we may regard +the expression on the left-hand side as a multiple of the +moment of the line with respect to the complex, the word +<span class="sidenote">General line coordinates.</span> +moment being used in the statical sense; and we infer +that when the coordinates are replaced by linear functions +of themselves the new coordinates are multiples of the moments +of the line with respect to six fixed complexes. The essential features +of this coordinate system are the same as those of the original one, +viz. there are six coordinates connected by a quadratic equation, +but this relation has in general a different form. By suitable choice +of the six fundamental complexes, as they may be called, this connecting +relation may be brought into other simple forms of which +we mention two: (i.) When the six are mutually in involution it can +be reduced to x<span class="su">1</span>² + x<span class="su">2</span>² + x<span class="su">3</span>² + x<span class="su">4</span>² + x<span class="su">5</span>² + x<span class="su">6</span>² = 0; (ii.) When the first +four are in involution and the other two are the lines common to +the first four it is x<span class="su">1</span>² + x<span class="su">2</span>² + x<span class="su">3</span>² + x<span class="su">4</span>² − 2x<span class="su">5</span>x<span class="su">6</span> = 0. These generalized +coordinates might be explained without reference to actual magnitude, +just as homogeneous point coordinates can be; the essential +remark is that the equation of any coordinate to zero represents a +linear complex, a point of view which includes our original system, +for the equation of a coordinate to zero represents all the lines +meeting an edge of the fundamental tetrahedron.</p> + +<p>The system of coordinates referred to six complexes mutually +in involution was introduced by Felix Klein, and in many cases is +more useful than that derived directly from point coordinates; <i>e.g.</i> +in the discussion of quadratic complexes: by means of it Klein has +developed an analogy between line geometry and the geometry of +spheres as treated by G. Darboux and others. In fact, in that +geometry a point is represented by <i>five</i> coordinates, connected by a +relation of the same type as the one just mentioned when the five +fundamental spheres are mutually at right angles and the equation +of a sphere is of the first degree. Extending this to four dimensions +of space, we obtain an exact analogue of line geometry, in which +(i.) a point corresponds to a line; (ii.) a linear complex to a hypersphere; +(iii.) two linear complexes in involution to two orthogonal +hyperspheres; (iv.) a linear complex and two conjugate lines to +a hypersphere and two inverse points. Many results may be obtained +by this principle, and more still are suggested by trying to extend +the properties of circles to spheres in three and four dimensions. +Thus the elementary theorem, that, given four lines, the circles +circumscribed to the four triangles formed by them are concurrent, +may be extended to six hyperplanes in four dimensions; and then +we can derive a result in line geometry by translating the inverse +of this theorem. Again, just as there is an infinite number of spheres +touching a surface at a given point, two of them having contact of a +closer nature, so there is an infinite number of linear complexes +touching a non-linear complex at a given line, and <i>three</i> of these +have contact of a closer nature (cf. Klein, <i>Math. Ann.</i> v.).</p> + +<p>Sophus Lie has pointed out a different analogy with sphere +geometry. Suppose, in fact, that the equation of a sphere of radius +r is</p> + +<p class="center">x² + y² + z² + 2ax + 2by + 2cz + d = 0,</p> + +<p class="noind">so that r² = a² + b² + c² − d; then introducing the quantity e to make +this equation homogeneous, we may regard the sphere as given by +the six coordinates a, b, c, d, e, r connected by the equation a² + +b² + c² − r² − de = 0, and it is easy to see that two spheres touch, if +the polar form 2aa<span class="su">1</span> + 2bb<span class="su">1</span> + 2cc<span class="su">1</span> − 2rr<span class="su">1</span> − de<span class="su">1</span> − d<span class="su">1</span>e vanishes. Comparing +this with the equation x<span class="su">1</span>² + x<span class="su">2</span>² + x<span class="su">3</span>² + x<span class="su">4</span>² − 2x<span class="su">5</span>x<span class="su">6</span> = 0 given +above, it appears that this sphere geometry and line geometry are +identical, for we may write a = x<span class="su">1</span>, b = x<span class="su">2</span>, c = x<span class="su">3</span>, r = x<span class="su">4</span><span class="ov">δ − 1</span>, d = x<span class="su">5</span>, +e = ½x<span class="su">6</span>; but it is to be noticed that a sphere is really replaced by two +lines whose coordinates only differ in the sign of x<span class="su">4</span>, so that they are +polar lines with respect to the complex x<span class="su">4</span> = 0. Two spheres which +touch correspond to two lines which intersect, or more accurately +to two pairs of lines (p, p′) 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.</p> + +<p>Some of the principles of line geometry may be brought into +clearer light by admitting the ideas of space of four and five +dimensions.</p> + +<p>Thus, regarding the coordinates of a line as homogeneous coordinates +in five dimensions, we may say that line geometry is +equivalent to geometry on a quadric surface in five dimensions. +A linear complex is represented by a hyperplane section; and if +two such complexes are in involution, the corresponding hyperplanes +are conjugate with respect to the fundamental quadric. By projecting +this quadric stereographically into space of four dimensions +we obtain Klein’s analogy. In the same way geometry in a linear +complex is equivalent to geometry on a quadric in four dimensions; +when two lines intersect the representative points are on the same +generator of this quadric. Stereographic projection, therefore, +converts a curve in a linear complex, <i>i.e.</i> one whose tangents all +belong to the complex, into one whose tangents intersect a fixed +conic: when this conic is the imaginary circle at infinity the curve +is what Lie calls a minimal curve. Curves in a linear complex have +been extensively studied. The osculating plane at any point of such +a curve is the nul-plane of the point with respect to the complex, +and points of superosculation always coincide in pairs at the points +of contact of stationary tangents. When a point of such a curve is +given, the osculating plane is determined, hence all the curves through +a given point with the same tangent have the same torsion.</p> + +<p>The lines through a given point that belong to a complex of the +nth degree lie on a cone of the nth degree: if this cone has a double +line the point is said to be a singular point. Similarly, +<span class="sidenote">Non-linear complexes.</span> +a plane is said to be singular when the envelope of the +lines in it has a double tangent. It is very remarkable +that the same surface is the locus of the singular points +and the envelope of the singular planes: this surface is called the +singular surface, and both its degree and class are in general 2n(n − 1)², +which is equal to four for the quadratic complex.</p> + +<p>The singular lines of a complex F = 0 are the lines common to F +and the complex</p> + +<table class="math0" summary="math"> +<tr><td>δF</td> +<td rowspan="2"> </td> <td>δF</td> +<td rowspan="2">+</td> <td>δF</td> +<td rowspan="2"> </td> <td>δF</td> +<td rowspan="2">+</td> <td>δF</td> +<td rowspan="2"> </td> <td>δF</td> +<td rowspan="2">= 0.</td></tr> +<tr><td class="denom">δl</td> <td class="denom">δλ</td> +<td class="denom">δm</td> <td class="denom">δμ</td> +<td class="denom">δn</td> <td class="denom">δν</td></tr></table> + +<p class="noind">As already mentioned, at each line l of a complex there is an infinite +number of tangent linear complexes, and they all contain the lines +adjacent to l. If now l be a singular line, these complexes all reduce +to straight lines which form a plane pencil containing the line l. +Suppose the vertex of the pencil is A, its plane a, and one of its lines +ξ, then l′ being a complex line near l, meets ξ, 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 ξ 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 ξ′ any line of the pencil (A′, a′) so that ξ′ is +a tangent line at l′ to the complex: the mutual moments of the +pairs l′, ξ and l, ξ are each of the second order; hence the plane a′ +meets the lines l and ξ′ in two points very near A. This being true +for all singular planes, near a the point of contact of a with its +envelope is in A, <i>i.e.</i> the locus of singular points is the same as the +envelope of singular planes. Further, when a line touches a complex +it touches the singular surface, for it belongs to a plane pencil like +(Aa), and thus in Klein’s analogy the analogue of a focus of a hyper-surface +being a bitangent line of the complex is also a bitangent line +of the singular surface. The theory of cosingular complexes is thus +brought into line with that of confocal surfaces in four dimensions, +and guided by these principles the existence of cosingular quadratic +complexes can easily be established, the analysis required being +almost the same as that invented for confocal cyclides by Darboux +<span class="pagenum"><a name="page723" id="page723"></a>723</span> +and others. Of cosingular complexes of higher degree nothing is +known.</p> + +<p>Following J. Plücker, we give an account of the lines of a quadratic +complex that meet a given line.</p> + +<p>The cones whose vertices are on the given line all pass through +eight fixed points and envelop a surface of the fourth degree; the +conics whose planes contain the given line all lie on a surface of the +fourth class and touch eight fixed planes. It is easy to see by elementary +geometry that these two surfaces are identical. Further, +the given line contains four singular points A<span class="su">1</span>, A<span class="su">2</span>, A<span class="su">3</span>, A<span class="su">4</span>, and the +planes into which their cones degenerate are the eight common +tangent planes mentioned above; similarly, there are four singular +planes, a<span class="su">1</span>, a<span class="su">2</span>, a<span class="su">3</span>, a<span class="su">4</span>, through the line, and the eight points into +which their conics degenerate are the eight common points above. +The locus of the pole of the line with respect to all the conics in +planes through it is a straight line called the <i>polar line</i> of the given +one; and through this line passes the polar plane of the given line +with respect to each of the cones. The name polar is applied in the +ordinary analytical sense; any line has an infinite number of polar +complexes with respect to the given complex, for the equation of the +latter can be written in an infinite number of ways; one of these +polars is a straight line, and is the polar line already introduced. +The surface on which lie all the conics through a line l is called the +Plücker surface of that line: from the known properties of (2, 2) +correspondences it can be shown that the Plücker surface of l cuts l<span class="su">1</span> +in a range of the same cross ratio as that of the range in which the +Plücker surface of l<span class="su">1</span> cuts l. Applying this to the case in which l<span class="su">1</span> +is the polar of l, we find that the cross ratios of (A<span class="su">1</span>, A<span class="su">2</span>, +A<span class="su">3</span>, A<span class="su">4</span>) and (a<span class="su">1</span>, a<span class="su">2</span>, a<span class="su">3</span>, a<span class="su">4</span>) are equal. The identity of the locus of the A′s with the +envelope of the a′s follows at once; moreover, a line meets the +singular surface in four points having the same cross ratio as that +of the four tangent planes drawn through the line to touch the surface. +The Plücker surface has eight nodes, eight singular tangent +planes, and is a double line. The relation between a line and its +polar line is not a reciprocal one with respect to the complex; but +W. Stahl has pointed out that the relation is reciprocal as far as the +singular surface is concerned.</p> + +<p>To facilitate the discussion of the general quadratic complex we +<span class="sidenote">Quadratic complexes.</span> +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</p> + +<table class="ws" style="clear: both;" summary="Contents"> +<tr><td class="tcl">a<span class="su">1</span>x<span class="su">1</span><span class="sp">2</span></td> <td class="tcl">+ a<span class="su">2</span>x<span class="su">2</span><span class="sp">2</span></td> <td class="tcl">+ a<span class="su">3</span>x<span class="su">3</span><span class="sp">2</span></td> <td class="tcl">+ a<span class="su">4</span>x<span class="su">4</span><span class="sp">2</span></td> <td class="tcl">+ a<span class="su">5</span>x<span class="su">5</span><span class="sp">2</span></td> <td class="tcl">+ a<span class="su">6</span>x<span class="su">6</span><span class="sp">2</span></td> <td class="tcl">= 0</td></tr> +<tr><td class="tcl">x<span class="su">1</span><span class="sp">2</span></td> <td class="tcl">+ x<span class="su">2</span><span class="sp">2</span></td> <td class="tcl">+ x<span class="su">3</span><span class="sp">2</span></td> <td class="tcl">+ x<span class="su">4</span><span class="sp">2</span></td> <td class="tcl">+ x<span class="su">5</span><span class="sp">2</span></td> <td class="tcl">+ x<span class="su">6</span><span class="sp">2</span></td> <td class="tcl">= 0</td></tr> +</table> + +<p class="noind">subject to certain exceptions, which will be mentioned later.</p> + +<p>Taking the first equation to be that of the complex, we remark +that both equations are unaltered by changing the sign of any coordinate; +the geometrical meaning of this is, that the quadratic +complex is its own reciprocal with respect to each of the six fundamental +complexes, for changing the sign of a coordinate is equivalent +to taking the polar of a line with respect to the corresponding +fundamental complex. It is easy to establish the existence of +six systems of bitangent linear complexes, for the complex +l<span class="su">1</span>x<span class="su">1</span> + l<span class="su">2</span>x<span class="su">2</span> + l<span class="su">3</span>x<span class="su">3</span> + l<span class="su">4</span>x<span class="su">4</span> + l<span class="su">5</span>x<span class="su">5</span> + l<span class="su">6</span>x<span class="su">6</span> = 0 is a bitangent when</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">l<span class="su">1</span> = 0, and</td> <td>l<span class="su">2</span>²</td> +<td rowspan="2">+</td> <td>l<span class="su">3</span>²</td> +<td rowspan="2">+</td> <td>l<span class="su">4</span>²</td> +<td rowspan="2">+</td> <td>l<span class="su">5</span>²</td> +<td rowspan="2">+</td> <td>l<span class="su">6</span>²</td> +<td rowspan="2">= 0,</td></tr> +<tr><td class="denom">a<span class="su">2</span> − a<span class="su">1</span></td> <td class="denom">a<span class="su">3</span> − a<span class="su">1</span></td> +<td class="denom">a<span class="su">4</span> − a<span class="su">1</span></td> <td class="denom">a<span class="su">5</span> − a<span class="su">1</span></td> +<td class="denom">a<span class="su">6</span> − a<span class="su">1</span></td></tr></table> + +<p class="noind">and its lines of contact are conjugate lines with respect to the first +fundamental complex. We therefore infer the existence of six systems +of bitangent lines of the complex, of which the first is given by</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">x<span class="su">1</span> = 0,</td> <td>x<span class="su">2</span>²</td> +<td rowspan="2">+</td> <td>x<span class="su">3</span>²</td> +<td rowspan="2">+</td> <td>x<span class="su">4</span>²</td> +<td rowspan="2">+</td> <td>x<span class="su">5</span>²</td> +<td rowspan="2">+</td> <td>x<span class="su">6</span>²</td> +<td rowspan="2">= 0,</td></tr> +<tr><td class="denom">a<span class="su">2</span> − a<span class="su">1</span></td> <td class="denom">a<span class="su">3</span> − a<span class="su">1</span></td> +<td class="denom">a<span class="su">4</span> − a<span class="su">1</span></td> <td class="denom">a<span class="su">5</span> − a<span class="su">1</span></td> +<td class="denom">a<span class="su">6</span> − a<span class="su">1</span></td></tr></table> + +<p class="noind">Each of these lines is a bitangent of the singular surface, which is +therefore completely determined as being the focal surface of the +(2, 2) congruence above. It is thence easy to verify that the two +complexes Σax<span class="sp">2</span> = 0 and Σbx<span class="sp">2</span> = 0 are cosingular if b<span class="su">r</span> = a<span class="su">r</span>λ + μ/a<span class="su">r</span>ν + ρ.</p> + +<p>The singular surface of the general quadratic complex is the +famous quartic, with sixteen nodes and sixteen singular tangent +planes, first discovered by E.E. Kümmer.</p> + +<p>We cannot give a full account of its properties here, but we deduce +at once from the above that its bitangents break up into six (2, 2) +congruences, and the six linear complexes containing these are +mutually in involution. The nodes of the singular surface are points +whose complex cones are coincident planes, and the complex conic +in a singular tangent plane consists of two coincident points. This +configuration of sixteen points and planes has many interesting +properties; thus each plane contains six points which lie on a conic, +while through each point there pass six planes which touch a quadric +cone. In many respects the Kümmer quartic plays a part in three +dimensions analogous to the general quartic curve in two; it further +gives a natural representation of certain relations between hyperelliptic +functions (cf. R.W.H.T. Hudson, <i>Kümmer’s Quartic</i>, 1905).</p> + +<p>As might be expected from the magnitude of a form in six variables, +the number of projectivally distinct varieties of quadratic complexes +is very great; and in fact Adolf Weiler, by whom the +<span class="sidenote">Classification of quadratic complexes.</span> +question was first systematically studied on lines indicated +by Klein, enumerated no fewer than forty-nine different +types. But the principle of the classification is so important, +and withal so simple, that we give a brief sketch +which indicates its essential features.</p> + +<p>We have practically to study the intersection of two quadrics +F and F′ 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 + λF′) = 0 is</p> + +<p class="center">(λ − α)<span class="sp">s<span class="su">1</span> + s<span class="su">2</span> + s<span class="su">3</span> ...</span> (λ − β)<span class="sp">t<span class="su">1</span> + t<span class="su">2</span> + t<span class="su">3</span> + ...</span> ...</p> + +<p class="noind">where the root α occurs s<span class="su">1</span> + s<span class="su">2</span> + s<span class="su">3</span> ... times in the determinant, +s<span class="su">2</span> + s<span class="su">3</span> ... times in every first minor, s<span class="su">3</span> + ... times in every second +minor, and so on; the meaning of each exponent is then perfectly +definite. Every factor of the type (λ − α)<span class="sp">s</span> is called an <i>elementartheil</i> +(elementary divisor) of the determinant, and the condition of equivalence +of two pairs of quadratics is simply that their determinants have +the same elementary divisors. We write the pair of forms symbolically +thus [(s<span class="su">1</span>s<span class="su">2</span> ...), (t<span class="su">1</span>t<span class="su">2</span> ...), ...], letters in the inner brackets +referring to the same factor. Returning now to the two quadratics +representing the complex, the sum of the exponents will be six, +and two complexes are put in the same class if they have the same +symbolical expression; <i>i.e.</i> the actual values of the roots of the +determinantal equation need not be the same for both, but their +manner of occurrence, as far as here indicated, must be identical in +the two. The enumeration of all possible cases is thus reduced +to a simple question in combinatorial analysis, and the actual study +of any particular case is much facilitated by a useful rule of Klein’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, <i>Math. +Ann.</i> vol. vii.</p> + +<p>The detailed study of each variety of complex opens up a vast +subject; we only mention two special cases, the harmonic complex +and the tetrahedral complex.</p> + +<p>The harmonic complex, first studied by Battaglini, is generated +in an infinite number of ways by the lines cutting two quadrics +harmonically. Taking the most general case, and referring the +quadrics to their common self-conjugate tetrahedron, we can find its +equation in a simple form, and verify that this complex really +depends only on seventeen constants, so that it is not the most +general quadratic complex. It belongs to the general type in so far +as it is discussed above, but the roots of the determinant are in involution. +The singular surface is the “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>i.e.</i> 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 ∞<span class="sp">3</span> line integrals.</p> + +<p>As the coordinates of a line belonging to a congruence are functions +of two independent parameters, the theory of congruences is analogous +to that of surfaces, and we may regard it as a fundamental +inquiry to find the simplest form of surface into which +<span class="sidenote">Congruences.</span> +a given congruence can be transformed. Most of those +whose properties have been extensively discussed can be represented +on a plane by a birational transformation. But in addition to the +difficulties of the theory of algebraic surfaces, a subject still in its +infancy, the theory of congruences has other difficulties in that a +congruence is seldom completely represented, even by two equations.</p> + +<p>A fundamental theorem is that the lines of a congruence are in +general bitangents of a surface; in fact, since the condition of intersection +of two consecutive straight lines is ldλ + dmdμ + dndν = 0, a +line l of the congruence meets two adjacent lines, say l<span class="su">1</span> and l<span class="su">2</span>. +Suppose l, l<span class="su">1</span> lie in the plane pencil (A<span class="su">1</span>a<span class="su">1</span>) and l, l<span class="su">2</span> in the plane pencil +(A<span class="su">2</span>a<span class="su">2</span>), then the locus of the A′s is the same as the envelope of the +a′s, but a<span class="su">2</span> is the tangent plane at A<span class="su">1</span> and a<span class="su">1</span> at A<span class="su">2</span>. This surface is +called the focal surface of the congruence, and to it all the lines l +are bitangent. The distinctive property of the points A is that two +of the congruence lines through them coincide, and in like manner +the planes a each contain two coincident lines. The focal surface +consists of two sheets, but one or both may degenerate into curves; +<span class="pagenum"><a name="page724" id="page724"></a>724</span> +thus, for example, the normals to a surface are bitangents of the +surface of centres, and in the case of Dupin’s cyclide this surface +degenerates into two conics.</p> + +<p>In the discussion of congruences it soon becomes necessary to +introduce another number r, called the rank, which expresses the +number of plane pencils each of which contains an arbitrary line +and two lines of the congruence. The order of the focal surface is +2m(n − 1) − 2r, and its class is m(m − 1) − 2r. Our knowledge of +congruences is almost exclusively confined to those in which either +m or n does not exceed two. We give a brief account of those of +the second order without singular lines, those of order unity not +being especially interesting. A congruence generally has singular +points through which an infinite number of lines pass; a singular +point is said to be of order r when the lines through it lie on a cone +of the rth degree. By means of formulae connecting the number of +singular points and their orders with the class m of quadratic congruence +Kümmer proved that the class cannot exceed seven. The +focal surface is of degree four and class 2m; this kind of quartic +surface has been extensively studied by Kümmer, Cayley, Rohn and +others. The varieties (2, 2), (2, 3), (2, 4), (2, 5) all belong to at +least one Reye complex; and so also does the most important class +of (2, 6) congruences which includes all the above as special cases. +The congruence (2, 2) belongs to a linear complex and forty different +Reye complexes; as above remarked, the singular surface is +Kümmer’s sixteen-nodal quartic, and the same surface is focal for +six different congruences of this variety. The theory of (2, 2) +congruences is completely analogous to that of the surfaces called +cyclides in three dimensions. Further particulars regarding quadratic +congruences will be found in Kümmer’s memoir of 1866, and +the second volume of Sturm’s treatise. The properties of quadratic +congruences having singular lines, <i>i.e.</i> degenerate focal surfaces, are +not so interesting as those of the above class; they have been +discussed by Kümmer, Sturm and others.</p> + +<p>Since a ruled surface contains only ∞¹ elements, this theory is +practically the same as that of curves. If a linear complex contains +more than n generators of a ruled surface of the nth degree, +it contains all the generators, hence for n = 2 there are +<span class="sidenote">Ruled surfaces.</span> +three linearly independent complexes, containing all the +generators, and this is a well-known property of quadric surfaces. +In ruled cubics the generators all meet two lines which may or may +not coincide; these two cases correspond to the two main classes of +cubics discussed by Cayley and Cremona. As regards ruled quartics, +the generators must lie in one and may lie in two linear complexes. +The first class is equivalent to a quartic in four dimensions and is +always rational, but the latter class has to be subdivided into the +elliptic and the rational, just like twisted quartic curves. A quintic +skew may not lie in a linear complex, and then it is unicursal, while of +sextics we have two classes not in a linear complex, viz. the elliptic +variety, having thirty-six places where a linear complex contains +six consecutive generators, and the rational, having six such +places.</p> + +<p>The general theory of skews in two linear complexes is identical +with that of curves on a quadric in three dimensions and is known. +But for skews lying in only one linear complex there are difficulties; +the curve now lies in four dimensions, and we represent it in three by +stereographic projection as a curve meeting a given plane in n points +on a conic. To find the maximum deficiency for a given degree would +probably be difficult, but as far as degree eight the space-curve +theory of Halphen and Nöther can be translated into line geometry +at once. When the skew does not lie in a linear complex at all the +theory is more difficult still, and the general theory clearly cannot +advance until further progress is made in the study of twisted +curves.</p> + +<p><span class="sc">References</span>.—The earliest works of a general nature are Plücker, +<i>Neue Geometrie des Raumes</i> (Leipzig, 1868); and Kümmer, “Über +die algebraischen Strahlensysteme,” <i>Berlin Academy</i> (1866). Systematic +development on purely synthetic lines will be found in the +three volumes of Sturm, <i>Liniengeometrie</i> (Leipzig, 1892, 1893, 1896); +vol. i. deals with the linear and Reye complexes, vols. ii. and iii. +with quadratic congruences and complexes respectively. For a +highly suggestive review by Gino Loria see <i>Bulletin des sciences +mathématiques</i> (1893, 1897). A shorter treatise, giving a very +interesting account of Klein’s coordinates, is the work of Koenigs, +<i>La Géométrie réglée et ses applications</i> (Paris, 1898). English treatises +are C.M. Jessop, <i>Treatise on the Line Complex</i> (1903); R.W.H.T. +Hudson, <i>Kümmer’s Quartic</i> (1905). Many references to memoirs on +line geometry will be found in Hagen, <i>Synopsis der höheren Mathematik</i>, +ii. (Berlin, 1894); Loria, <i>Il passato ed il presente delle principali +teorie geometriche</i> (Milan, 1897); a clear résumé of the principal +results is contained in the very elegant volume of Pascal, <i>Repertorio +di mathematiche superiori</i>, ii. (Milan, 1900). Another treatise dealing +extensively with line geometry is Lie, <i>Geometrie der Berührungstransformationen</i> +(Leipzig, 1896). Many memoirs on the subject have +appeared in the <i>Mathematische Annalen</i>; a full list of these will be +found in the index to the first fifty volumes, p. 115. Perhaps the +two memoirs which have left most impression on the subsequent +development of the subject are Klein, “Zur Theorie der Liniencomplexe +des ersten und zweiten Grades,” <i>Math. Ann.</i> ii.; and Lie, +“Über Complexe, insbesondere Linien- und Kugelcomplexe,” +<i>Math. Ann.</i> v.</p> +</div> +<div class="author">(J. H. Gr.)</div> + +<p class="pt2 center sc">VI. Non-Euclidean Geometry</p> + +<p>The various metrical geometries are concerned with the +properties of the various types of congruence-groups, which are +defined in the study of the <i>axioms</i> of <i>geometry</i> and of their +immediate consequences. But this point of view of the subject +is the outcome of recent research, and historically the subject +has a different origin. Non-Euclidean geometry arose from the +discussion, extending from the Greek period to the present day, +of the various assumptions which are implicit in the traditional +Euclidean system of geometry. In the course of these investigations +it became evident that metrical geometries, each internally +consistent but inconsistent in many respects with each other +and with the Euclidean system, could be developed. A short +historical sketch will explain this origin of the subject, and +describe the famous and interesting progress of thought on the +subject. But previously a description of the chief characteristic +properties of elliptic and of hyperbolic geometries will be given, +assuming the standpoint arrived at below under VII. <i>Axioms +of Geometry</i>.</p> + +<p>First assume the equation to the absolute (cf. <i>loc. cit.</i>) to +be w² − x² − y² − z² = 0. The absolute is then real, and the +geometry is hyberbolic.</p> + +<div class="condensed"> +<p>The distance (d<span class="su">12</span>) between the two points (x<span class="su">1</span>, y<span class="su">1</span>, z<span class="su">1</span>, w<span class="su">1</span>) and (x<span class="su">2</span>, y<span class="su">2</span>, +z<span class="su">2</span>, w<span class="su">2</span>) is given by</p> + +<p class="center">cosh (d<span class="su">12</span>/γ) = (w<span class="su">1</span>w<span class="su">2</span> − x<span class="su">1</span>x<span class="su">2</span> − y<span class="su">1</span>y<span class="su">2</span> − z<span class="su">1</span>z<span class="su">2</span>) / {(w<span class="su">1</span>² − x<span class="su">1</span>² − y<span class="su">1</span>² − z<span class="su">1</span>²) +(w<span class="su">2</span>² − x<span class="su">2</span>² − y<span class="su">2</span>² − z<span class="su">2</span>²)}<span class="sp">1/2</span></p> +<div class="aut">(1)</div> + +<p class="noind">The only points to which the metrical geometry applies are those +within the region enclosed by the quadric; the other points are +“improper ideal points.” The angle (θ<span class="su">12</span>) between two planes, +l<span class="su">1</span>x + m<span class="su">1</span>y + n<span class="su">1</span>z + r<span class="su">1</span>w = 0 and l<span class="su">2</span>x + m<span class="su">2</span>y + n<span class="su">2</span>z + r<span class="su">2</span>w = 0, is given by</p> + +<p class="center">cos θ<span class="su">12</span> = (l<span class="su">1</span>l<span class="su">2</span> + m<span class="su">1</span>m<span class="su">2</span> + n<span class="su">1</span>n<span class="su">2</span> − r<span class="su">1</span>r<span class="su">2</span>) / {(l<span class="su">1</span>² + m<span class="su">1</span>² + n<span class="su">1</span>² − r<span class="su">1</span>²) +(l<span class="su">2</span>² + m<span class="su">2</span>² + n<span class="su">2</span>² − r<span class="su">2</span>²)}<span class="sp">1/2</span></p> +<div class="aut">(2)</div> + +<p class="noind">These planes only have a real angle of inclination if they possess a +line of intersection within the actual space, <i>i.e.</i> if they intersect. +Planes which do not intersect possess a shortest distance along a line +which is perpendicular to both of them. If this shortest distance is +δ<span class="su">12</span>, we have</p> + +<p class="center">cosh (δ<span class="su">12</span>/γ) = (l<span class="su">1</span>l<span class="su">2</span> + m<span class="su">1</span>m<span class="su">2</span> + n<span class="su">1</span>n<span class="su">2</span> − r<span class="su">1</span>r<span class="su">2</span>) / {(l<span class="su">1</span>² + m<span class="su">1</span>² + n<span class="su">1</span>² − r<span class="su">1</span>²) +(l<span class="su">2</span>² + m<span class="su">2</span>² + n<span class="su">2</span>² − r<span class="su">2</span>²)}<span class="sp">1/2</span></p> +<div class="aut">(3)</div> + +<table class="flt" style="float: right; width: 300px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:248px; height:220px" src="images/img724a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig</span>. 67.</td></tr></table> + +<p>Thus in the case of the two planes one and only one of the two, θ12 +and δ<span class="su">12</span>, is real. The same considerations hold for coplanar straight +lines (see VII. <i>Axioms of Geometry</i>). Let O (fig. 67) be the point +(0, 0, 0, 1), OX the line y = 0, +z = 0, OY the line z = 0, x = 0, and +OZ the line x = 0, y = 0. These are +the coordinate axes and are at +right angles to each other. Let +P be any point, and let ρ be the +distance OP, θ the angle POZ, and +φ the angle between the planes +ZOX and ZOP. Then the coordinates +of P can be taken to be</p> + +<p class="center">sinh (ρ/γ) sin θ cos φ, sinh (ρ/γ) sin θ +sin φ, sinh (ρ/γ) cos θ, cosh (ρ/γ).</p> + +<p>If ABC is a triangle, and the +sides and angles are named according +to the usual convention, we have</p> + +<p class="center">sinh (a/γ) / sin A = sinh (b/γ) / sin B = sinh (c/γ) / sin C,</p> +<div class="aut">(4)</div> + +<p class="noind">and also</p> + +<p class="center">cosh (a/γ) = cosh (b/γ) cosh (c/γ) − sinh (b/γ) sinh (c/γ) cos A,</p> +<div class="aut">(5)</div> + +<table class="flt" style="float: left; width: 260px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:211px; height:88px" src="images/img724b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig</span>. 68.</td></tr></table> + +<p class="noind">with two similar equations. The sum of the three angles of a triangle +is always less than two right angles. The area of the triangle ABC +is λ²(π − 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. <i>Axioms of +Geometry</i>) must be considered as making a zero angle with each +other. Also if B be a right angle, from the equation (5), remembering +that, in the limit,</p> + +<p class="center">cosh (a/γ) / cosh (b/γ) = cosh (a/γ) / sinh (b/γ) = 1,</p> + +<p><span class="pagenum"><a name="page725" id="page725"></a>725</span></p> + +<p class="noind">we have</p> + +<p class="center">cos A = tanh (c/2γ)</p> +<div class="aut">(6).</div> + +<p class="noind">The angle A is called by N.I. Lobatchewsky the “angle of parallelism.”</p> + +<p>The whole theory of lines and planes at right angles to each other +is simply the theory of conjugate elements with respect to the +absolute, where ideal lines and planes are introduced.</p> + +<p>Thus if l and l′ 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 (x<span class="su">1</span>, y<span class="su">1</span>, z<span class="su">1</span>, w<span class="su">1</span>) +and radius ρ, is</p> + +<p class="center">(w<span class="su">1</span>² − x<span class="su">1</span>² − y<span class="su">1</span>² − z<span class="su">1</span>²) (w² − x² − y² − z²) cosh² (ρ/γ) = +(w<span class="su">1</span>w − x<span class="su">1</span>x − y<span class="su">1</span>y − z<span class="su">1</span>z)²</p> +<div class="aut">(7).</div> + +<p class="noind">The equation of the surface of equal distance (σ) from the plane +lx + my + nz + rw = 0 is</p> + +<p class="center">(l² + m² + n² − r²) (w² − x² − y² − z²) sinh² (σ/γ) = +(rw + lx + my + nz)²</p> +<div class="aut">(8).</div> + +<p class="noind">A surface of equal distance is a sphere whose centre is improper; +and both types of surface are included in the family</p> + +<p class="center">k² (w² − x² − y² − z²) = (ax + by + cz + dw)²</p> +<div class="aut">(9).</div> + +<p>But this family also includes a third type of surfaces, which can +be looked on either as the limits of spheres whose centres have +approached the absolute, or as the limits of surfaces of equal distance +whose central planes have approached a position tangential to the +absolute. These surfaces are called limit-surfaces. Thus (9) denotes +a limit-surface, if d² − a² − b² − c² = 0. Two limit-surfaces only +differ in position. Thus the two limit-surfaces which touch the plane +YOZ at O, but have their concavities turned in opposite directions, +have as their equations</p> + +<p class="center">w² − x² − y² − z² = (w ± x)².</p> + +<p>The geodesic geometry of a sphere is elliptic, that of a surface of +equal distance is hyperbolic, and that of a limit-surface is parabolic +(<i>i.e.</i> <i>Euclidean</i>). The equation of the surface (cylinder) of equal +distance (δ) from the line OX is</p> + +<p class="center">(w² − x²) tanh² (δ/γ) − y² − z² = 0.</p> + +<p class="noind">This is not a ruled surface. Hence in this geometry it is not possible +for two straight lines to be at a constant distance from each other.</p> + +<p>Secondly, let the equation of the absolute be x² + y² + z² + +w² = 0. The absolute is now imaginary and the geometry is +elliptic.</p> + +<p>The distance (d<span class="su">12</span>) between the two points (x<span class="su">1</span>, y<span class="su">1</span>, z<span class="su">1</span>, w<span class="su">1</span>) and +(x<span class="su">2</span>, y<span class="su">2</span>, z<span class="su">2</span>, w<span class="su">2</span>) is given by</p> + +<p class="center">cos (d<span class="su">12</span>/γ) = ± (x<span class="su">1</span>x<span class="su">2</span> + y<span class="su">1</span>y<span class="su">2</span> + z<span class="su">1</span>z<span class="su">2</span> + w<span class="su">1</span>w<span class="su">2</span>) / +{(x<span class="su">1</span>² + y<span class="su">1</span>² + z<span class="su">1</span>² + w<span class="su">1</span>²) +(x<span class="su">2</span>² + y<span class="su">2</span>² + z<span class="su">2</span>² + w<span class="su">2</span>²)}<span class="sp">1/2</span></p> +<div class="aut">(10).</div> + +<p class="noind">Thus there are two distances between the points, and if one is d<span class="su">12</span>, +the other is πγ-d<span class="su">12</span>. Every straight line returns into itself, forming +a closed series. Thus there are two segments between any two +points, together forming the whole line which contains them; one +distance is associated with one segment, and the other distance with +the other segment. The complete length of every straight line is +πγ.</p> + +<p>The angle between the two planes l<span class="su">1</span>x + m<span class="su">1</span>y + n<span class="su">1</span>z + r + <span class="su">1</span>w = 0 and +l<span class="su">2</span>x + m<span class="su">2</span>y + n<span class="su">2</span>z + r<span class="su">2</span>w = 0 is</p> + +<p class="center">cos θ<span class="su">12</span> = (l<span class="su">1</span>l<span class="su">2</span> + m<span class="su">1</span>m<span class="su">2</span> + n<span class="su">1</span>n<span class="su">2</span> + r<span class="su">1</span>r<span class="su">2</span>) / +{(l<span class="su">1</span>² + m<span class="su">1</span>² + n<span class="su">1</span>² +r<span class="su">1</span>²) +(l<span class="su">2</span>² + m<span class="su">2</span>² + n<span class="su">2</span>² + r<span class="su">2</span>²)}<span class="sp">1/2</span></p> +<div class="aut">(11).</div> + +<p class="noind">The polar plane with respect to the absolute of the point (x<span class="su">1</span>, y<span class="su">1</span>, z<span class="su">1</span>, w<span class="su">1</span>) +is the real plane x<span class="su">1</span>x + y<span class="su">1</span>y + z<span class="su">1</span>z + w<span class="su">1</span>w = 0, and the pole of the plane +l<span class="su">1</span>x + m<span class="su">1</span>y + n<span class="su">1</span>z + r<span class="su">1</span>w = 0 is the point (l<span class="su">1</span>, m<span class="su">1</span>, n<span class="su">1</span>, r<span class="su">1</span>). Thus (from +equations 10 and 11) it follows that the angle between the polar +planes of the points (x<span class="su">1</span>, ...) and (x<span class="su">2</span>, ...) is d<span class="su">12</span>/γ, and that the +distance between the poles of the planes (l<span class="su">1</span>, ...) and (l<span class="su">2</span>, ...) is +γθ<span class="su">12</span>. Thus there is complete reciprocity between points and planes +in respect to all properties. This complete reign of the principle +of duality is one of the great beauties of this geometry. The theory +of lines and planes at right angles is simply the theory of conjugate +elements with respect to the absolute. A tetrahedron self-conjugate +with respect to the absolute has all its intersecting elements (edges +and planes) at right angles. If l and l′ 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 ½πγ. 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<a name="fa2d" id="fa2d" href="#ft2d"><span class="sp">2</span></a> to find three points P, Q, R such that P and Q are separated +by the plane p, but P and R are not separated by p, nor are Q +and R.</p> + +<p>Let A, B, C be any three non-collinear points, then four triangles +are defined by these points. Thus if a, b, c and A, B, C are the +elements of any one triangle, then the four triangles have as their +elements:</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcc">(1)</td> <td class="tcc">a,</td> <td class="tcc">b,</td> <td class="tcc">c,</td> <td class="tcc">A,</td> <td class="tcc">B,</td> <td class="tcc">C.</td></tr> + +<tr><td class="tcc">(2)</td> <td class="tcc">a,</td> <td class="tcc">πγ − b,</td> <td class="tcc">πγ − c,</td> <td class="tcc">A,</td> <td class="tcc">π − B,</td> <td class="tcc">π − C.</td></tr> + +<tr><td class="tcc">(3)</td> <td class="tcc">πγ − a,</td> <td class="tcc">b,</td> <td class="tcc">πγ − c,</td> <td class="tcc">π − A,</td> <td class="tcc">B,</td> <td class="tcc">π − C.</td></tr> + +<tr><td class="tcc">(4)</td> <td class="tcc">πγ − a,</td> <td class="tcc">πγ − b,</td> <td class="tcc">c,</td> <td class="tcc">π − A,</td> <td class="tcc">π − B,</td> <td class="tcc">C.</td></tr> +</table> + +<p class="noind">The formulae connecting the elements are</p> + +<p class="center">sin A/sin (a/γ) = sin B/sin (b/γ) = sin C/sin (c/γ),</p> +<div class="aut">(12)</div> + +<p class="noind">and</p> + +<p class="center">cos (a/γ) = cos (b/γ) cos (c/γ) + sin (b/γ) sin (c/γ) cos A,</p> +<div class="aut">(13)</div> + +<p class="noind">with two similar equations.</p> + +<p>Two cases arise, namely (I.) according as one of the four triangles +has as its sides the shortest segments between the angular points, +or (II.) according as this is not the case. When case I. holds there +is said to be a “principal triangle.”<a name="fa3d" id="fa3d" href="#ft3d"><span class="sp">3</span></a> If all the figures considered lie +within a sphere of radius ¼πγ 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 γ²(A + B + C − π). +Thus as in hyperbolic geometry the theory of similarity does not +hold, and the elements of a triangle are determined when its three +angles are given. The coordinates of a point can be written in the +form</p> + +<p class="center">sin (ρ/γ) sin Φ cos φ, sin (ρ/γ) sin Φ sin φ, sin (ρ/γ) cos Φ, cos (ρ/γ),</p> + +<p class="noind">where ρ, Φ and φ 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 (δ) from the line +XOX is</p> + +<p class="center">(x² + w²) tan² (δ/γ) − (y² + z²) = 0.</p> + +<p class="noind">This is also the surface of equal distance, ½πγ-δ, 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<a name="fa4d" id="fa4d" href="#ft4d"><span class="sp">4</span></a> of which the geodesic geometry is hyperbolic. +But it is to be noticed that the connectivity of these surfaces +is different to that of a Euclidean plane. For instance there are only +∞² congruence transformations of the cylindrical surfaces of equal +distance into themselves, instead of the ∞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.<a name="fa5d" id="fa5d" href="#ft5d"><span class="sp">5</span></a></p> +</div> + +<p>In both elliptic and hyperbolic geometry the spherical +geometry, <i>i.e.</i> the relations between the angles formed by lines +and planes passing through the same point, is the same as the +“spherical trigonometry” in Euclidean geometry. The constant +γ, 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 +γ in each case simply depends upon the magnitude of the arbitrary +unit of length in comparison with the natural unit of length +<span class="pagenum"><a name="page726" id="page726"></a>726</span> +which each particular instance of either geometry presents. +The existence of a natural unit of length is a peculiarity common +both to hyperbolic and elliptic geometries, and differentiates +them from Euclidean geometry. It is the reason for the failure +of the theory of similarity in them. If γ is very large, that is, +if the natural unit is very large compared to the arbitrary unit, +and if the lengths involved in the figures considered are not large +compared to the arbitrary unit, then both the elliptic and +hyperbolic geometries approximate to the Euclidean. For from +formulae (4) and (5) and also from (12) and (13) we find, after +retaining only the lowest powers of small quantities, as the +formulae for any triangle ABC,</p> + +<p class="center">a / sin A = b / sin B = c / sin C,</p> + +<p class="noind">and</p> + +<p class="center">a² = b² + c² − 2bc cos A,</p> + +<p class="noind">with two similar equations. Thus the geometries of small +figures are in both types Euclidean.</p> + +<p><i>History.</i>—“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.” +<span class="sidenote">Theory of parallels before Gauss.</span> +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 +<i>Elements</i>, 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 +<span class="grk" title="kai ean eis duo eutheias eutheia empiptousa tas entos kai epi ta +auta merê gônias duo orthôn elassonas poiê, ekballomenas tas +duo eutheias ep’ apeiron sympiptein, eph’ ha merê eisin hai tôn duo +orthôn elassones"> +καὶ ἐὰν εἰς δύο εὐθείας εὐθεῖα ἐμπίπτουσα τὰς ἐντὸς καὶ ἐπὶ τὰ +αὐτὰ μέρη γωνίας δύο ὀρθῶν ἐλάσσονας ποιῇ, ἐκβαλλομένας τὰς +δύο εὐθείας ἐπ᾽ ἄπειρον συμπίπτειν, ἐφ᾽ ἃ μέρη εἰσὶν αἱ τῶν δύο +ὀρθῶν ἐλάσσονες</span>.</p> + +<p>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.</p> + +<p>A new method, which, though it failed to lead to the desired +goal, proved in the end immensely fruitful, was invented by +Saccheri, in a work entitled <i>Euclides ab omni naevo +vindicatus</i> (Milan, 1733). If the postulate of parallels +<span class="sidenote">Saccheri.</span> +is involved in Euclid’s other assumptions, contradictions must +emerge when it is denied while the others are maintained. This +led Saccheri to attempt a <i>reductio ad absurdum</i>, in which he +mistakenly believed himself to have succeeded. What is interesting, +however, is not his fallacious conclusion, but the non-Euclidean +results which he obtains in the process. Saccheri +distinguishes three hypotheses (corresponding to what are now +known as Euclidean or parabolic, elliptic and hyperbolic geometry), +and proves that some one of the three must be universally +true. His three hypotheses are thus obtained: equal +perpendiculars AC, BD are drawn from a straight line AB, +and CD are joined. It is shown that the angles ACD, BDC are +equal. The first hypothesis is that these are both right angles; +the second, that they are both obtuse; and the third, that they +are both acute. Many of the results afterwards obtained by +Lobatchewsky and Bolyai are here developed. Saccheri fails +to be the founder of non-Euclidean geometry only because he +does not perceive the possible truth of his non-Euclidean hypotheses.</p> + +<p>Some advance is made by Johann Heinrich Lambert in his +<i>Theorie der Parallellinien</i> (written 1766; posthumously published +1786). Though he still believed in the necessary +truth of Euclidean geometry, he confessed that, in +<span class="sidenote">Lambert.</span> +all his attempted proofs, something remained undemonstrated. +He deals with the same three hypotheses as Saccheri, showing +that the second holds on a sphere, while the third would hold on +a sphere of purely imaginary radius. The second hypothesis +he succeeds in condemning, since, like all who preceded Bernhard +Riemann, he is unable to conceive of the straight line as finite +and closed. But the third hypothesis, which is the same as +Lobatchewsky’s, is not even professedly refuted.<a name="fa6d" id="fa6d" href="#ft6d"><span class="sp">6</span></a></p> + +<p>Non-Euclidean geometry proper begins with Karl Friedrich +Gauss. The advance which he made was rather philosophical +than mathematical: it was he (probably) who first +recognized that the postulate of parallels is possibly +<span class="sidenote">Three periods of non-Euclidean geometry.</span> +false, and should be empirically tested by measuring +the angles of large triangles. The history of non-Euclidean +geometry has been aptly divided by Felix +Klein into three very distinct periods. The first—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 <i>manifold</i>; but the characteristic +standpoint of the period is chiefly emphasized by Eugenio +Beltrami. The conception of measure of curvature is extended +by Riemann from surfaces to spaces, and a new kind of space, +finite but unbounded (corresponding to the second hypothesis +of Saccheri and Lambert), is shown to be possible. As opposed +to the second period, which is purely metrical, the third period +is essentially projective in its method. It begins with Arthur +Cayley, who showed that metrical properties are projective +properties relative to a certain fundamental quadric, and that +different geometries arise according as this quadric is real, +imaginary or degenerate. Klein, to whom the development of +Cayley’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 ∞<span class="sp">6</span> ways, no other +three-dimensional spaces than the above four are possible.</p> + +<p>Gauss published nothing on the theory of parallels, and it +was not generally known until after his death that he had +interested himself in that theory from a very early +date. In 1799 he announces that Euclidean geometry +<span class="sidenote">Gauss.</span> +would follow from the assumption that a triangle can be drawn +greater than any given triangle. Though unwilling to assume +this, we find him in 1804 still hoping to prove the postulate of +parallels. In 1830 he announces his conviction that geometry +is not an a priori science; in the following year he explains that +non-Euclidean geometry is free from contradictions, and that, +in this system, the angles of a triangle diminish without limit +when all the sides are increased. He also gives for the +<span class="pagenum"><a name="page727" id="page727"></a>727</span> +circumference of a circle of radius r the formula πk(e<span class="sp">r/k</span> − e<span class="sp">r −/k</span>), +where k is a constant depending upon the nature of the space. In +1832, in reply to the receipt of Bolyai’s <i>Appendix</i>, he gives an +elegant proof that the amount by which the sum of the angles of a +triangle falls short of two right angles is proportional to the area +of the triangle. From these and a few other remarks it appears +that Gauss possessed the foundations of hyperbolic geometry, +which he was probably the first to regard as perhaps true. It +is not known with certainty whether he influenced Lobatchewsky +and Bolyai, but the evidence we possess is against such a view.<a name="fa7d" id="fa7d" href="#ft7d"><span class="sp">7</span></a></p> + +<p>The first to publish a non-Euclidean geometry was Nicholas +Lobatchewsky, professor of mathematics in the new university +of Kazañ.<a name="fa8d" id="fa8d" href="#ft8d"><span class="sp">8</span></a> In the place of the disputed postulate +he puts the following: “All straight lines which, in +<span class="sidenote">Lobatchewsky.</span> +a plane, radiate from a given point, can, with respect +to any other straight line in the same plane, be divided into +two classes, the <i>intersecting</i> and the <i>non-intersecting</i>. The +<i>boundary line</i> of the one and the other class is called <i>parallel +to the given line</i>.” 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 Π(p). The determination of Π(p) is the chief problem (cf. +equation (6) above); it appears finally that, with a suitable +choice of the unit of length,</p> + +<p class="center">tan ½ Π(p) = e<span class="sp">−p</span>.</p> + +<p>Before obtaining this result it is shown that spherical trigonometry +is unchanged, and that the normals to a circle or a sphere +still pass through its centre. When the radius of the circle or +sphere becomes infinite all these normals become parallel, but the +circle or sphere does not become a straight line or plane. It +becomes what Lobatchewsky calls a limit-line or limit-surface. +The geometry on such a surface is shown to be Euclidean, limit-lines +replacing Euclidean straight lines. (It is, in fact, a surface +of zero measure of curvature.) By the help of these propositions +Lobatchewsky obtains the above value of Π(p), and thence the +solution of triangles. He points out that his formulae result +from those of spherical trigonometry by substituting ia, ib, ic, +for the sides a, b, c.</p> + +<p>John Bolyai, a Hungarian, obtained results closely corresponding +to those of Lobatchewsky. These he published in an appendix +to a work by his father, entitled <i>Appendix Scientiam +spatii absolute veram exhibens: a veritate aut falsitate</i> +<span class="sidenote">Bolyai.</span> +<i>Axiomatis XI. Euclidei</i> (<i>a priori haud unquam decidenda</i>) <i>independentem: +adjecta ad casum falsitatis, quadratura circuli +geometrica</i>.<a name="fa9d" id="fa9d" href="#ft9d"><span class="sp">9</span></a> This work was published in 1831, but its conception +dates from 1823. It reveals a profounder appreciation of the +importance of the new ideas, but otherwise differs little from +Lobatchewsky’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 πι², where i is what we should now call the +space-constant.</p> + +<p>The works of Lobatchewsky and Bolyai, though known and +valued by Gauss, remained obscure and ineffective until, in 1866, +they were translated into French by J. Hoüel. But +<span class="sidenote">Riemann.</span> +at this time Riemann’s dissertation, <i>Über die Hypothesen, +welche der Geometrie zu Grunde liegen</i>,<a name="fa10d" id="fa10d" href="#ft10d"><span class="sp">10</span></a> was already about to be +published. In this work Riemann, without any knowledge of +his predecessors in the same field, inaugurated a far more profound +discussion, based on a far more general standpoint; and by +its publication in 1867 the attention of mathematicians and +philosophers was at last secured. (The dissertation dates from +1854, but owing to changes which Riemann wished to make in it, +it remained unpublished until after his death.)</p> + +<p>Riemann’s work contains two fundamental conceptions, that +of a manifold and that of the <i>measure of curvature</i> of a continuous +manifold possessed of what he calls flatness in the smallest parts. +By means of these conceptions space is made to appear +<span class="sidenote">Definition of a manifold.</span> +at the end of a gradual series of more and more specialized +conceptions. Conceptions of magnitude, he explains, +are only possible where we have a general conception +capable of determination in various ways. The manifold consists +of all these various determinations, each of which is an element +of the manifold. The passage from one element to another may +be discrete or continuous; the manifold is called discrete or +continuous accordingly. Where it is discrete two portions of +it can be compared, as to magnitude, by counting; where +continuous, by measurement. But measurement demands +superposition, and consequently some magnitude independent +of its place in the manifold. In passing, in a continuous manifold, +from one element to another in a determinate way, we pass +through a series of intermediate terms, which form a one-dimensional +manifold. If this whole manifold be similarly +caused to pass over into another, each of its elements passes +through a one-dimensional manifold, and thus on the whole +a two-dimensional manifold is generated. In this way we can +proceed to n dimensions. Conversely, a manifold of n dimensions +can be analysed into one of one dimension and one of (n − 1) +dimensions. By repetitions of this process the position of an +element may be at last determined by n magnitudes. We may +here stop to observe that the above conception of a manifold +is akin to that due to Hermann Grassmann in the first edition +(1847) of his <i>Ausdehnungslehre</i>.<a name="fa11d" id="fa11d" href="#ft11d"><span class="sp">11</span></a></p> + +<p>Both concepts have been elaborated and superseded by the +modern procedure in respect to the axioms of geometry, and by +the conception of abstract geometry involved therein. +Riemann proceeds to specialize the manifold by considerations +<span class="sidenote">Measure of curvature.</span> +as to measurement. If measurement is to +be possible, some magnitude, we saw, must be independent of +position; let us consider manifolds in which lengths of lines are +such magnitudes, so that every line is measurable by every +other. The coordinates of a point being x<span class="su">1</span>, x<span class="su">2</span>, ... x<span class="su">n</span>, let us confine +ourselves to lines along which the ratios dx<span class="su">1</span> : dx<span class="su">2</span> : ... : dx<span class="su">n</span> +alter continuously. Let us also assume that the element of +length, ds, is unchanged (to the first order) when all its points +undergo the same infinitesimal motion. Then if all the increments +dx be altered in the same ratio, ds is also altered in this ratio. +Hence ds is a homogeneous function of the first degree of the +increments dx. Moreover, ds must be unchanged when all the +dx change sign. The simplest possible case is, therefore, that in +which ds is the square root of a quadratic function of the dx. +This case includes space, and is alone considered in what follows. +It is called the case of flatness in the smallest parts. Its further +discussion depends upon the measure of curvature, the second +of Riemann’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 δx say, determine a pencil, or one-dimensional +series, of shortest lines, any one of which is defined +<span class="pagenum"><a name="page728" id="page728"></a>728</span> +by λdx + μδx, where the parameter λ : μ may have any value. +This pencil generates a two-dimensional series of points, which +may be regarded as a surface, and for which we may apply +Gauss’s formula for the measure of curvature at any point. +Thus at every point of our manifold there is a measure of curvature +corresponding to every such pencil; but all these can be found +when n·<span class="ov">n − 1</span>/2 of them are known. If figures are to be freely +movable, it is necessary and sufficient that the measure of +curvature should be the same for all points and all directions +at each point. Where this is the case, <span class="correction" title="amended from it">if</span> α be the measure of +curvature, the linear element can be put into the form</p> + +<p class="center">ds = √(Σdx²) / (1 + ¼αΣx²).</p> + +<p class="noind">If α 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.</p> + +<p>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 dx<span class="su">1</span>, dx<span class="su">2</span>, ... is arbitrary; +(4) the idea of superposition, or congruence, is not adequately +analysed. The modern solution of these difficulties is properly +considered in connexion with the general subject of the axioms +of geometry.</p> + +<p>The publication of Riemann’s dissertation was closely followed +by two works of Hermann von Helmholtz,<a name="fa12d" id="fa12d" href="#ft12d"><span class="sp">12</span></a> again undertaken +in ignorance of the work of predecessors. In these a +<span class="sidenote">Helmholtz.</span> +proof is attempted that ds must be a rational integral +quadratic function of the increments of the coordinates. This +proof has since been shown by Lie to stand in need of correction +(see VII. <i>Axioms of Geometry</i>). Helmholtz’s remaining works +on the subject<a name="fa13d" id="fa13d" href="#ft13d"><span class="sp">13</span></a> are of almost exclusively philosophical interest. +We shall return to them later.</p> + +<p>The only other writer of importance in the second period is +Eugenio Beltrami, by whom Riemann’s work was brought into +connexion with that of Lobatchewsky and Bolyai. +As he gave, by an elegant method, a convenient +<span class="sidenote">Beltrami.</span> +Euclidean interpretation of hyperbolic plane geometry, his +results will be stated at some length<a name="fa14d" id="fa14d" href="#ft14d"><span class="sp">14</span></a>. The <i>Saggio</i> shows that +Lobatchewsky’s plane geometry holds in Euclidean geometry +on surfaces of constant negative curvature, straight lines being +replaced by geodesics. Such surfaces are capable of a conformal +representation on a plane, by which geodesics are represented +by straight lines. Hence if we take, as coordinates on the surface, +the Cartesian coordinates of corresponding points on the plane, +the geodesics must have linear equations.</p> + +<div class="condensed"> +<p>Hence it follows that</p> + +<p class="center">ds² = R²w<span class="sp">−4</span> {(α² − v²) du² + 2uvdudv + (α² − u²)dv²}</p> + +<p class="noind">where w² = α² − u² − v², and −1/R² is the measure of curvature +of our surface (note that k = γ as used above). The angle between +two geodesics u = const., v = const. is θ, where</p> + +<p class="center">cos θ = uv / √ {(α² − u²) (α² − v²)}, sin θ = aw / √ {(a² − u²) (a² − v²)}.</p> + +<p class="noind">Thus u = 0 is orthogonal to all geodesies v = const., and vice versa. +In order that sin θ may be real, w² must be positive; thus geodesics +have no real intersection when the corresponding straight +lines intersect outside the circle u² + v² = α². When they intersect on +this circle, θ = 0. Thus Lobatchewsky’s parallels are represented +by straight lines intersecting on the circle. Again, transforming +to polar coordinates u = r cos μ, v = r sin μ, and calling ρ the geodesic +distance of u, v from the origin, we have, for a geodesic through the +origin,</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">dρ = Radr / (a² − r²), ρ = ½R log</td> <td>a + r</td> +<td rowspan="2">, r = a tan h (ρ / R).</td></tr> +<tr><td class="denom">a − r</td></tr></table> + +<p class="noind">Thus points on the surface corresponding to points in the plane +on the limiting circle r = a, are all at an infinite distance from the +origin. Again, considering r constant, the arc of a geodesic circle +subtending an angle μ at the origin is</p> + +<p class="center">σ = Rrμ / √ (a² − r²) = μR sin h (ρ/R),</p> + +<p class="noind">whence the circumference of a circle of radius ρ is 2πR sin h (ρ/R). +Again, if α be the angle between any two geodesics</p> + +<p class="center">V − v = m (U − u), V − v = n (U − u),</p> + +<p class="noind">then</p> + +<p class="center">tan α = a (n − m)w / {(1 + mn)a² − (v − mu) (v − nu)}.</p> + +<p class="noind">Thus α 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 δ from the origin is u = a tan h (δ/R), and the +parallel to this through the origin is u = v sin h (δ/R). Hence Π (δ), +the angle which this parallel makes with v = 0, is given by</p> + +<p class="center">tan Π(δ) . sin h (δ/R) = 1, or tan ½Π(δ) = e<span class="sp">−δ/R</span></p> + +<p class="noind">which is Lobatchewsky’s formula. We also obtain easily for the +area of a triangle the formula R²(π − A − B − C).</p> + +<p>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 ρ and centre u<span class="su">0</span>v<span class="su">0</span> is</p> + +<p class="center">(a² − uu<span class="su">0</span> − vv<span class="su">0</span>)² = cos h² (ρ/R) w<span class="su">0</span>²w² = C²w²</p> +<div class="aut">(say).</div> + +<p class="noind">This equation remains real when ρ is a pure imaginary, and remains +finite when w<span class="su">0</span> = 0, provided ρ becomes infinite in such a way that +w<span class="su">0</span> cos h (ρ/R) remains finite. In the latter case the equation represents +a limit-line. In the former case, by giving different values to C, +we obtain concentric circles with the imaginary centre u<span class="su">0</span>v<span class="su">0</span>. One of +these, obtained by putting C = 0, is the straight line a² − uu<span class="su">0</span> − vv<span class="su">0</span> = 0. +Hence the others are each throughout at a constant distance from +this line. (It may be shown that all motions in a hyperbolic plane +consist, in a general sense, of rotations; but three types must +be distinguished according as the centre is real, imaginary or at +infinity. All points describe, accordingly, one of the three types of +circles.)</p> + +<p>The above Euclidean interpretation fails for three or more dimensions. +In the <i>Teoria fondamentale</i>, accordingly, where n dimensions +are considered, Beltrami treats hyperbolic space in a purely analytical +spirit. The paper shows that Lobatchewsky’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 <i>Saggio</i>)</p> + +<p class="center">ds² = R²x<span class="sp">−2</span> (dx² + dx<span class="su">1</span>² + dx<span class="su">2</span>² + ... + dx<span class="su">n</span>²)</p> + +<p class="noind">where</p> + +<p class="center">x² + x<span class="su">1</span>² + x<span class="su">2</span>² + ... + x<span class="su">n</span>² = a².</p> + +<p class="noind">He shows that geodesics are represented by linear equations between +x<span class="su">1</span>, x<span class="su">2</span>, ..., x<span class="su">n</span>, and that the geodesic distance ρ between two +points x and x′ is given by</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">cos h</td> <td>ρ</td> +<td rowspan="2">=</td> <td>a² − x<span class="su">1</span>x′<span class="su">1</span> − x<span class="su">2</span>x′<span class="su">2</span> − ... − x<span class="su">n</span>x′<span class="su">n</span></td></tr> +<tr><td class="denom">R</td> <td class="denom">{(a² − x<span class="su">1</span>² − x<span class="su">2</span>² − ... − x<span class="su">n</span>²) (a² − x′<span class="su">1</span>² − x′<span class="su">2</span>² − ... − x′<span class="su">n</span>²)}<span class="sp">1/2</span></td></tr></table> + +<p class="noind">(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</p> + +<p class="center">x<span class="su">1</span> = rλ<span class="su">1</span>, x<span class="su">2</span> = rλ<span class="su">2</span> ... x<span class="su">n</span> = rλ<span class="su">n</span>, where Σλ² = 1.</p> + +<p class="noind">Hence</p> + +<p class="center">ds² = (Radr / <span class="ov">a² − r²</span>)² + R²r²dΔ² / (a² − r²).</p> + +<p class="noind">where</p> + +<p class="center">dΔ² = Σdλ².</p> + +<p class="noind">Also calling ρ the geodesic distance from the origin, we have</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">cos h (ρ/R) =</td> <td>a</td> +<td rowspan="2">, sin h (ρ/R) =</td> <td>r</td> +<td rowspan="2">.</td></tr> +<tr><td class="denom">√(a² − r²)</td> <td class="denom">√(a² − r²)</td></tr></table> + +<p class="noind">Hence</p> + +<p class="center">ds² = dρ² + (R sin h (ρ/R))² dΔ².</p> + +<p class="noind">Putting</p> + +<p class="center">z<span class="su">1</span> = ρλ<span class="su">1</span>, z<span class="su">2</span> = ρλ<span class="su">2</span>, ... z<span class="su">n</span> = ρλ<span class="su">n</span>,</p> + +<p class="noind">we obtain</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ds² = Σdz² +</td> <td>1</td> +<td rowspan="2"><span class="f150">{ (</span></td> <td>R</td> +<td rowspan="2">sinh</td> <td>ρ</td> +<td rowspan="2"><span class="f150">)</span></td> <td>²</td> +<td rowspan="2">− 1 <span class="f150">}</span> Σ (z<span class="su">i</span>dz<span class="su">k</span> − z<span class="su">k</span>dz<span class="su">i</span>)².</td></tr> +<tr><td class="denom">ρ²</td> <td class="denom">ρ</td> +<td class="denom">R</td> <td> </td></tr></table> + +<p class="noind">Hence when ρ is small, we have approximately</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">ds² = Σdz² +</td> <td>1</td> +<td rowspan="2">Σ (z<span class="su">i</span>dz<span class="su">k</span> − z<span class="su">k</span>dz<span class="su">i</span>)²</td></tr> +<tr><td class="denom">3R²</td></tr></table> +<div class="aut">(1).</div> + +<p class="noind">Considering a surface element through the origin, we may choose +our axes so that, for this element,</p> + +<p class="center">z<span class="su">3</span> = z<span class="su">4</span> = ... = z<span class="su">n</span> = 0.</p> + +<p class="noind">Thus</p> + +<table class="math0" summary="math"> +<tr><td rowspan="2">dz<span class="su">1</span>² + dz<span class="su">2</span>² +</td> <td>1</td> +<td rowspan="2">(z<span class="su">1</span>dz<span class="su">2</span> − z<span class="su">2</span>dz<span class="su">1</span>)²</td></tr> +<tr><td class="denom">3R²</td></tr></table> +<div class="aut">(2).</div> + +<p class="noind">Now the area of the triangle whose vertices are (0, 0), (z<span class="su">1</span>, z<span class="su">2</span>), +(dz<span class="su">1</span>, dz<span class="su">2</span>) is ½(z<span class="su">1</span>, dz<span class="su">2</span> − z<span class="su">2</span>dz<span class="su">1</span>). Hence the quotient when the terms of +the fourth order in (2) are divided by the square of this triangle is +<span class="pagenum"><a name="page729" id="page729"></a>729</span> +4/3R²; hence, returning to general axes, the same is the quotient +when the terms of the fourth order in (1) are divided by the square +of the triangle whose vertices are (0, 0, ... 0), (z<span class="su">1</span>, z<span class="su">2</span>, z<span class="su">3</span>, ... z<span class="su">n</span>), +(dz<span class="su">1</span>, dz<span class="su">2</span>, dz<span class="su">3</span> ... dz<span class="su">n</span>). But −¾ of this quotient is defined by Riemann +as the measure of curvature.<a name="fa15d" id="fa15d" href="#ft15d"><span class="sp">15</span></a> Hence the measure of curvature is +−1/R², <i>i.e.</i> is constant and negative. The properties of parallels, +triangles, &c., are as in the <i>Saggio</i>. It is also shown that the analogues +of limit surfaces have zero curvature; and that spheres of +radius ρ have constant positive curvature 1/R² sinh² (ρ/R), so that +spherical geometry may be regarded as contained in the pseudo-spherical +(as Beltrami calls Lobatchewsky’s system).</p> +</div> + +<p>The <i>Saggio</i>, as we saw, gives a Euclidean interpretation +confined to two dimensions. But a consideration of the auxiliary +plane suggests a different interpretation, which may be +extended to any number of dimensions. If, instead +<span class="sidenote">Transition to the projective method.</span> +of referring to the pseudosphere, we merely <i>define</i> +distance and angle, in the Euclidean plane, as those +functions of the coordinates which gave us distance and angle +on the pseudosphere, we find that the geometry of our plane has +become Lobatchewsky’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 Σ<span class="su">xx</span> = 0, and if Σ<span class="su">xx</span>′ +be the polar form of Σ<span class="su">xx</span>, the distance ρ between x and x′ is +given by the projective formula</p> + +<p class="center">cos(ρ/k) = Σ<span class="su">xx</span>′ / {Σ<span class="su">xx</span>·Σ<span class="su">x</span>′<span class="su">x</span>′}<span class="sp">1/2</span>.</p> + +<p class="noind">That this formula is projective is rendered evident by observing +that e<span class="sp">−2iρ/k</span> is the anharmonic ratio of the range consisting of +the two points and the intersections of the line joining them with +the fundamental quadric. With this we are brought to the third +or projective period. The method of this period is due to Cayley; +its application to previous non-Euclidean geometry is due to +Klein. The projective method contains a generalization of discoveries +already made by Laguerre<a name="fa16d" id="fa16d" href="#ft16d"><span class="sp">16</span></a> in 1853 as regards Euclidean +geometry. The arbitrariness of this procedure of deriving +metrical geometry from the properties of conics is removed by +Lie’s theory of congruence. We then arrive at the stage of +thought which finds its expression in the modern treatment of +the axioms of geometry.</p> + +<p>The projective method leads to a discrimination, first made +by Klein,<a name="fa17d" id="fa17d" href="#ft17d"><span class="sp">17</span></a> of two varieties of Riemann’s space; Klein calls +these elliptic and spherical. They are also called the +polar and antipodal forms of elliptic space. The latter +<span class="sidenote">The two kinds of elliptic space.</span> +names will here be used. The difference is strictly +analogous to that between the diameters and the points +of a sphere. In the polar form two straight lines in a plane +always intersect in one and only one point; in the antipodal +form they intersect always in two points, which are antipodes. +According to the definition of geometry adopted in section VII. +(<i>Axioms of Geometry</i>), the antipodal form is not to be termed +“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.<a name="fa18d" id="fa18d" href="#ft18d"><span class="sp">18</span></a> But, in the antipodal form, a plane is, in these respects, +like a Euclidean plane.</p> + +<p>It is explained in section VII. in what sense the metrical +geometry of the material world can be considered to be determinate +and not a matter of arbitrary choice. The scientific +question as to the best available evidence concerning the nature +of this geometry is one beset with difficulties of a peculiar kind. +We are obstructed by the fact that all existing physical science +assumes the Euclidean hypothesis. This hypothesis has been +involved in all actual measurements of large distances, and in all +the laws of astronomy and physics. The principle of simplicity +would therefore lead us, in general, where an observation conflicted +with one or more of those laws, to ascribe this anomaly, +not to the falsity of Euclidean geometry, but to the falsity of the +laws in question. This applies especially to astronomy. On the +earth our means of measurement are many and direct, and so +long as no great accuracy is sought they involve few scientific +laws. Thus we acquire, from such direct measurements, a +very high degree of probability that the space-constant, if not +infinite, is yet large as compared with terrestrial distances. But +astronomical distances and triangles can only be measured by +means of the received laws of astronomy and optics, all of which +have been established by assuming the truth of the Euclidean +hypothesis. It therefore remains possible (until a detailed proof +of the contrary is forthcoming) that a large but finite space-constant, +with different laws of astronomy and optics, would +have equally explained the phenomena. We cannot, therefore, +accept the measurements of stellar parallaxes, &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.</p> + +<p>Finally, it is of interest to note that, though it is theoretically +possible to prove, by scientific methods, that our geometry is +non-Euclidean, it is wholly impossible to prove by such methods +that it is accurately Euclidean. For the unavoidable errors of +observation must always leave a slight margin in our measurements. +A triangle might be found whose angles were certainly +greater, or certainly less, than two right angles; but to prove +them <i>exactly</i> equal to two right angles must always be beyond our +powers. If, therefore, any man cherishes a hope of proving the +exact truth of Euclid, such a hope must be based, not upon +scientific, but upon philosophical considerations.</p> + +<div class="condensed"> +<p><span class="sc">Bibliography.</span>—The bibliography appended to section VII. should +be consulted in this connexion. Also, in addition to the citations +already made, the following works may be mentioned.</p> + +<p>For Lobatchewsky’s writings, cf. <i>Urkunden zur Geschichte der +nichteuklidischen Geometrie</i>, i., <i>Nikolaj Iwanowitsch Lobatschefsky</i>, +by F. Engel and P. Stäckel (Leipzig, 1898). For John Bolyai’s +<i>Appendix</i>, cf. <i>Absolute Geometrie nach Johann Bolyai</i>, by J. Frischauf +(Leipzig, 1872), and also the new edition of his father’s large work, +<i>Tentamen</i> ..., published by the Mathematical Society of Budapest; +the second volume contains the appendix. Cf. also J. Frischauf, +<i>Elemente der absoluten Geometrie</i> (Leipzig, 1876); M.L. Gérard, <i>Sur +la géométrie non-Euclidienne</i> (thesis for doctorate) (Paris, 1892); +de Tilly, <i>Essai sur les principes fondamentales de la géométrie et de la +mécanique</i> (Bordeaux, 1879); Sir R.S. Ball, “On the Theory of +Content,” <i>Trans. Roy. Irish Acad.</i> vol. xxix. (1889); F. Lindemann, +“Mechanik bei projectiver Maasbestimmung,” <i>Math. Annal.</i> vol. +vii.; W.K. Clifford, “Preliminary Sketch of Biquaternions,” <i>Proc. +of Lond. Math. Soc.</i> (1873), and <i>Coll. Works</i>; A. Buchheim, “On the +Theory of Screws in Elliptic Space,” <i>Proc. Lond. Math. Soc.</i> vols. xv., +xvi., xvii.; H. Cox, “On the Application of Quaternions and +Grassmann’s Algebra to different Kinds of Uniform Space,” <i>Trans. +Camb. Phil. Soc.</i> (1882); M. Dehn, “Die Legendarischen Sätze über +die Winkelsumme im Dreieck,” Math. Ann. vol. 53 (1900), and +“Über den Rauminhalt,” <i>Math. Annal.</i> vol. 55 (1902).</p> + +<p>For expositions of the whole subject, cf. F. Klein, <i>Nicht-Euklidische +Geometrie</i> (Göttingen, 1893); R. Bonola, <i>La Geometria non-Euclidea</i> +(Bologna, 1906); P. Barbarin, <i>La Géométrie non-Euclidienne</i> (Paris, +1902); W. Killing, <i>Die nicht-Euklidischen Raumformen in analytischer +Behandlung</i> (Leipzig, 1885). The last-named work also deals with +geometry of more than three dimensions; in this connexion cf. also +G. Veronese, <i>Fondamenti di geometria a più dimensioni ed a più specie</i> +<span class="pagenum"><a name="page730" id="page730"></a>730</span> +<i>di unità rettilinee</i> ... (Padua, 1891, German translation, Leipzig, +1894); G. Fontené, <i>L’Hyperespace à (n-1) dimensions</i> (Paris, 1892); +and A.N. Whitehead, <i>loc. cit.</i> Cf. also E. Study, “Über nicht-Euklidische +und Liniengeometrie,” <i>Jahr. d. Deutsch. Math. Ver.</i> +vol. xv. (1906); W. Burnside, “On the Kinematics of non-Euclidean +Space,” <i>Proc. Lond. Math. Soc.</i> vol. xxvi. (1894). A bibliography +on the subject up to 1878 has been published by G.B. Halsted, +<i>Amer. Journ. of Math.</i> vols. i. and ii.; and one up to 1900 by R. +Bonola, <i>Index operum ad geometriam absolutam spectantium</i> ... +(1902, and Leipzig, 1903).</p> +</div> +<div class="author">(B. A. W. R.; A. N. W.)</div> + +<p class="pt2 center sc">VII. Axioms of Geometry</p> + +<p>Until the discovery of the non-Euclidean geometries (Lobatchewsky, +1826 and 1829; J. Bolyai, 1832; B. Riemann, 1854), +geometry was universally considered as being exclusively +the science of existent space. (See section +<span class="sidenote">Theories of space.</span> +VI. <i>Non-Euclidean Geometry</i>.) In respect to the +science, as thus conceived, two controversies may be noticed. +First, there is the controversy respecting the absolute and +relational theories of space. According to the absolute theory, +which is the traditional view (held explicitly by Newton), space +has an existence, in some sense whatever it may be, independent +of the bodies which it contains. The bodies occupy space, and +it is not intrinsically unmeaning to say that any definite body +occupies <i>this</i> part of space, and not <i>that</i> part of space, without +reference to other bodies occupying space. According to the +relational theory of space, of which the chief exponent was +Leibnitz,<a name="fa19d" id="fa19d" href="#ft19d"><span class="sp">19</span></a> space is nothing but a certain assemblage of the relations +between the various particular bodies in space. The idea of +space with no bodies in it is absurd. Accordingly there can be +no meaning in saying that a body is <i>here</i> and not <i>there</i>, apart +from a reference to the other bodies in the universe. Thus, on +this theory, absolute motion is intrinsically unmeaning. It is +admitted on all hands that in practice only relative motion is +directly measurable. Newton, however, maintains in the +<i>Principia</i> (scholium to the 8th definition) that it is indirectly +measurable by means of the effects of “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.<a name="fa20d" id="fa20d" href="#ft20d"><span class="sp">20</span></a> 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 <i>Critique</i>) be classed +with the upholders of the absolute theory. The form of perception +has a type of existence proper to itself independently +of the particular bodies which it contains. For example he +writes:<a name="fa21d" id="fa21d" href="#ft21d"><span class="sp">21</span></a> “Space does not represent any quality of objects by +themselves, or objects in their relation to one another, <i>i.e.</i> space +does not represent any determination which is inherent in the +objects themselves, and would remain, even if all subjective +conditions of intuition were removed.”</p> + +<p>The second controversy is that between the view that the +axioms applicable to space are known only from experience, +and the view that in some sense these axioms are +given <i>a priori</i>. Both these views, thus broadly stated, +<span class="sidenote">Axioms.</span> +are capable of various subtle modifications, and a discussion +of them would merge into a general treatise on epistemology. +The cruder forms of the <i>a priori</i> view have been made quite +untenable by the modern mathematical discoveries. Geometers +now profess ignorance in many respects of the exact axioms +which apply to existent space, and it seems unlikely that a +profound study of the question should thus obliterate <i>a priori</i> +intuitions.</p> + +<p>Another question irrelevant to this article, but with some +relevance to the above controversy, is that of the derivation +of our perception of existent space from our various types of +sensation. This is a question for psychology.<a name="fa22d" id="fa22d" href="#ft22d"><span class="sp">22</span></a></p> + +<p><i>Definition of Abstract Geometry.</i>—Existent space is the subject +matter of only one of the applications of the modern science of +abstract geometry, viewed as a branch of pure mathematics. +Geometry has been defined<a name="fa23d" id="fa23d" href="#ft23d"><span class="sp">23</span></a> as “the study of series of two or more +dimensions.” It has also been defined<a name="fa24d" id="fa24d" href="#ft24d"><span class="sp">24</span></a> 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 (<i>e.g.</i> those of <i>points</i> and of +<i>straight lines</i>) are not ideas of determinate entities, but of any +entities for which the axioms are true. And a set of formal +geometrical axioms cannot in themselves be true or false, since +they are not determinate propositions, in that they do not refer +to a determinate subject matter. The axioms are propositional +functions.<a name="fa25d" id="fa25d" href="#ft25d"><span class="sp">25</span></a> When a set of axioms is given, we can ask (1) +whether they are consistent, (2) whether their “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<a name="fa26d" id="fa26d" href="#ft26d"><span class="sp">26</span></a> (with +respect to the undetermined subject matter) of which some at +least occur in each of the subsequent propositions.</p> + +<p>Any science is called a “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.</p> + +<p class="pt2 center"><i>Projective Geometry.</i></p> + +<p>Projective geometry may be developed from two undefined +fundamental ideas, namely, that of a “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 +<span class="pagenum"><a name="page731" id="page731"></a>731</span> +be presented<a name="fa27d" id="fa27d" href="#ft27d"><span class="sp">27</span></a> here as twelve in number, eight being “axioms +of classification,” and four being “axioms of order.”</p> + +<p><i>Axioms of Classification.</i>—The eight axioms of classification +are as follows:</p> + +<p>1. Points form a class of entities with at least two members.</p> + +<p>2. Any straight line is a class of points containing at least +three members.</p> + +<p>3. Any two distinct points lie in one and only one straight +line.</p> + +<p>4. There is at least one straight line which does not contain +all the points.</p> + +<p>5. If A, B, C are non-collinear points, and A′ 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.</p> + +<div class="condensed"> +<p><i>Definition.</i>—If A, B, C are any three non-collinear points, the +<i>plane</i> ABC is the class of points lying on the straight lines joining +A with the various points on the straight line BC.</p> +</div> + +<p>6. There is at least one plane which does not contain all the +points.</p> + +<p>7. There exists a plane α, and a point A not incident in α, +such that any point lies in some straight line which contains +both A and a point in α.</p> + +<div class="condensed"> +<p><i>Definition.</i>—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.</p> +</div> + +<p>8. Harm. (ABCD) implies that B and D are distinct points.</p> + +<p>In the above axioms 4 secures at least two dimensions, axiom +5 is the fundamental axiom of the plane, axiom 6 secures at +least three dimensions, and axiom 7 secures at most three +dimensions. From axioms 1-5 it can be proved that any two +distinct points in a straight line determine that line, that any +three non-collinear points in a plane determine that plane, that +the straight line containing any two points in a plane lies wholly +in that plane, and that any two straight lines in a plane intersect. +From axioms 1-6 Desargue’s well-known theorem on triangles +in perspective can be proved.</p> + +<div class="condensed"> +<p>The enunciation of this theorem is as follows: If ABC and +A′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.</p> + +<p>It has been proved<a name="fa28d" id="fa28d" href="#ft28d"><span class="sp">28</span></a> 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.</p> + +<p>It follows from axioms 1-5 that Harm. (ABCD) implies Harm. +(ADCB) and Harm. (CBAD), and that, if A, B, C be any three +distinct collinear points, there exists at least one point D such that +Harm. (ABCD). But it requires Desargues’s theorem, and hence +axiom 6, to prove that Harm. (ABCD) and Harm. (ABCD′) imply +the identity of D and D′.</p> +</div> + +<p>The necessity for axiom 8 has been proved by G. Fano,<a name="fa29d" id="fa29d" href="#ft29d"><span class="sp">29</span></a> who +has produced a three dimensional geometry of fifteen points, +<i>i.e.</i> a method of cross classification of fifteen entities, in which +each straight line contains three points, and each plane contains +seven straight lines. In this geometry axiom 8 does not hold. +Also from axioms 1-6 and 8 it follows that Harm. (ABCD) +implies Harm. (BCDA).</p> + +<div class="condensed"> +<p><i>Definitions.</i>—When two plane figures can be derived from one +another by a single projection, they are said to be in <i>perspective</i>. +When two plane figures can be derived one from the other by a finite +series of perspective relations between intermediate figures, they +are said to be <i>projectively</i> related. Any property of a plane figure +which necessarily also belongs to any projectively related figure, is +called a <i>projective</i> property.</p> + +<p>The following theorem, known from its importance as “the +fundamental theorem of projective geometry,” cannot be proved<a name="fa30d" id="fa30d" href="#ft30d"><span class="sp">30</span></a> +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<a name="fa31d" id="fa31d" href="#ft31d"><span class="sp">31</span></a> +(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:<a name="fa32d" id="fa32d" href="#ft32d"><span class="sp">32</span></a> 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>i.e.</i> 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.</p> + +<p>Thus, if we choose to take any one of these three theorems as an +axiom, all the theorems of projective geometry which do not require +ordinal or metrical ideas for their enunciation can be proved. Also +a conic can be defined as the locus of the points found by the usual +construction, based upon Pascal’s theorem, for points on the conic +through five given points. But it is unnecessary to assume here +any one of the suggested axioms; for the fundamental theorem can +be deduced from the axioms of order together with axioms 1-8.</p> +</div> + +<p><i>Axioms of Order.</i>—It is possible to define (cf. Pieri, <i>loc. cit.</i>) +the property upon which the order of points on a straight line +depends. But to secure that this property does in fact range +the points in a serial order, some axioms are required. A straight +line is to be a closed series; thus, when the points are in order, +it requires two points on the line to divide it into two distinct +complementary segments, which do not overlap, and together +form the whole line. Accordingly the problem of the definition +of order reduces itself to the definition of these two segments +formed by any two points on the line; and the axioms are +stated relatively to these segments.</p> + +<div class="condensed"> +<p><i>Definition.</i>—If A, B, C are three collinear points, the points on the +<i>segment</i> ABC are defined to be those points such as X, for which +there exist two points Y and Y′ with the property that Harm. +(AYCY′) and Harm. (BYXY′) both hold. The <i>supplementary +segment</i> ABC is defined to be the rest of the points on the line. +This definition is elucidated by noticing that with our ordinary +geometrical ideas, if B and X are any two points between A and C, +then the two pairs of points, A and C, B and X, define an involution +with real double points, namely, the Y and Y′ of the above definition. +The property of belonging to a segment ABC is projective, since +the harmonic relation is projective.</p> +</div> + +<p>The first three axioms of order (cf. Pieri, <i>loc. cit.</i>) are:</p> + +<p>9. If A, B, C are three distinct collinear points, the supplementary +segment ABC is contained within the segment BCA.</p> + +<p>10. If A, B, C are three distinct collinear points, the common +part of the segments BCA and CAB is contained in the supplementary +segment ABC.</p> + +<p>11. If A, B, C are three distinct collinear points, and D lies +In the segment ABC, then the segment ADC is contained +within the segment ABC.</p> + +<p>From these axioms all the usual properties of a closed order +follow. It will be noticed that, if A, B, C are any three collinear +points, C is necessarily traversed in passing from A to B by one +route along the line, and is not traversed in passing from A to B +along the other route. Thus there is no meaning, as referred +to closed straight lines, in the simple statement that C lies +between A and B. But there may be a relation of separation +between two pairs of collinear points, such as A and C, and +B and D. The couple B and D is said to separate A and C, if +<span class="pagenum"><a name="page732" id="page732"></a>732</span> +the four points are collinear and D lies in the segment complementary +to the segment ABC. The property of the separation +of pairs of points by pairs of points is projective. Also it can be +proved that Harm. (ABCD) implies that B and D separate +A and C.</p> + +<div class="condensed"> +<p><i>Definitions.</i>—A series of entities arranged in a serial order, open +or closed, is said to be <i>compact</i>, if the series contains no immediately +consecutive entities, so that in traversing the series from any one +entity to any other entity it is necessary to pass through entities +distinct from either. It was the merit of R. Dedekind and of +G. Cantor explicitly to formulate another fundamental property of +series. The Dedekind property<a name="fa33d" id="fa33d" href="#ft33d"><span class="sp">33</span></a> as applied to an open series can +be defined thus: An open series possesses the Dedekind property, +if, however, it be divided into two mutually exclusive classes u and +v, which (1) contain between them the whole series, and (2) are +such that every member of u precedes in the serial order every +member of v, there is always a member of the series, belonging to one +of the two, u or v, which precedes every member of v (other than +itself if it belong to v), and also succeeds every member of u (other +than itself if it belong to u). Accordingly in an open series with the +Dedekind property there is always a member of the series marking +the junction of two classes such as u and v. An open series is <i>continuous</i> +if it is compact and possesses the Dedekind property. A +closed series can always be transformed into an open series by taking +any arbitrary member as the first term and by taking one of the two +ways round as the ascending order of the series. Thus the definitions +of compactness and of the Dedekind property can be at once transferred +to a closed series.</p> +</div> + +<p>12. The last axiom of order is that there exists at least one +straight line for which the point order possesses the Dedekind +property.</p> + +<p>It follows from axioms 1-12 by projection that the Dedekind +property is true for all lines. Again the <i>harmonic system</i> ABC, +where A, B, C are collinear points, is defined<a name="fa34d" id="fa34d" href="#ft34d"><span class="sp">34</span></a> thus: take the +harmonic conjugates A′, 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.</p> + +<p>A system of numerical coordinates can now be introduced, +possessing the property that linear equations represent planes +and straight lines. The outline of the argument by which this +remarkable problem (in that “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 ∞.</p> + +<p>This association<a name="fa35d" id="fa35d" href="#ft35d"><span class="sp">35</span></a> is most easily effected by considering a +class of projective relations of the line with itself, called by +F. Schur (<i>loc. cit.</i>) <i>prospectivities</i>.</p> + +<table class="flt" style="float: right; width: 260px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:209px; height:150px" src="images/img732a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 69.</span></td></tr> +<tr><td class="figright1"><img style="width:230px; height:158px" src="images/img732b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 70.</span></td></tr> +<tr><td class="figright1"><img style="width:216px; height:122px" src="images/img732c.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 71.</span></td></tr> +<tr><td class="figright1"><img style="width:202px; height:156px" src="images/img732d.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 72.</span></td></tr></table> + +<div class="condensed"> +<p>Let l (fig. 69) be the given line, m and n any two lines intersecting +at U on l, S and S′ two points on n. Then a projective relation +between l and itself is formed by projecting l from S on to m, and +then by projecting m from S′ back on to l. All such projective +relations, however m, n, S and S′ be varied, are called “prospectivities,” +and U is the double point of the prospectivity. If a point +O on l is related to A by a prospectivity, then all prospectivities, +which (1) have the same double point +U, and (2) relate O to A, give the same +correspondent (Q, in figure) to any +point P on the line l; in fact they are +all the same prospectivity, however +m, n, S, and S′ may have been varied +subject to these conditions. Such +a prospectivity will be denoted by +(OAU²).</p> + +<p>The sum of two prospectivities, +written (OAU²) + (OBU²), is defined +to be that transformation of the line +l into itself which is obtained by first applying the prospectivity +(OAU²) and then applying the prospectivity (OBU²). Such a +transformation, when the two summands have the same double +point, is itself a prospectivity with that double point.</p> + +<p>With this definition of addition it can be proved that prospectivities +with the same double point satisfy all the axioms of magnitude. +Accordingly they can be associated in a one-one correspondence +with the positive and negative real numbers. Let E +(fig. 70) be any point on l, distinct from O and U. Then the +prospectivity (OEU²) is associated with unity, the prospectivity +(OOU²) is associated with zero, +and (OUU²) with ∞. The prospectivities +of the type (OPU²), +where P is any point on the segment +OEU, correspond to the positive +numbers; also if P′ is the +harmonic conjugate of P with +respect to O and U, the prospectivity +(OP′U²) is associated with +the corresponding negative number. +(The subjoined figure explains this +relation of the positive and negative +prospectivities.) Then any +point P on l is associated with the same number as is the prospectivity +(OPU²).</p> + +<p>It can be proved that the order of the numbers in algebraic order +of magnitude agrees with the order on the line of the associated +points. Let the numbers, assigned according to the preceding +specification, be said to be associated with the points according to +the “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 (OE<span class="su">1</span>U) and (OE<span class="su">2</span>V), where +E<span class="su">1</span> and E<span class="su">2</span> are arbitrarily chosen. +Then (cf. fig. 71) if M and N are +associated with the numbers x and +y according to these systems, the +coordinates of P are x and y. It then +follows that the equation of a straight +line is of the form ax + by + c = 0. Both coordinates of any point on +the line UV are infinite. This can be avoided by introducing +homogeneous coordinates X, Y, Z, where x = X/Z, and y = Y/Z, and +Z = 0 is the equation of UV.</p> + +<p>The procedure for three dimensions is similar. Let OUVW +(fig. 72) be any tetrahedron, and associate points on OU, OV, OW +with numbers according to the numeration +systems (OE<span class="su">1</span>U), (OE<span class="su">2</span>V), and +(OE<span class="su">3</span>W). Let the planes VWP, WUP, +UVP cut OU, OV, OW in L, M, N respectively; +and let x, y, z be the numbers +associated with L, M, N respectively. +Then P is the point (x, y, z). Also +homogeneous coordinates can be introduced +as before, thus avoiding the +infinities on the plane UVW.</p> + +<p>The cross ratio of a range of four +collinear points can now be defined +as a number characteristic of that range. Let the coordinates of any +point P<span class="su">r</span> of the range P<span class="su">1</span> P<span class="su">2</span> P<span class="su">3</span> P<span class="su">4</span> be</p> + +<table class="math0" summary="math"> +<tr><td>λ<span class="su">r</span>a + μ<span class="su">r</span> + a′</td> +<td rowspan="2">,  </td> <td>λ<span class="su">r</span>b + μ<span class="su">r</span>b′</td> +<td rowspan="2">,  </td> <td>λ<span class="su">r</span>c + μ<span class="su">r</span>c′</td> +<td rowspan="2">,   (r = 1, 2, 3, 4)</td></tr> +<tr><td class="denom">λ<span class="su">r</span> + μ<span class="su">r</span></td> <td class="denom">λ<span class="su">r</span> + μ<span class="su">r</span></td> +<td class="denom">λ<span class="su">r</span> + μ<span class="su">r</span></td></tr></table> + +<p class="noind">and let (λ<span class="su">r</span>μ<span class="su">s</span>) be written for λ<span class="su">r</span>μ<span class="su">s</span> -λ<span class="su">s</span>μ<span class="su">r</span>. Then the cross ratio +{P<span class="su">1</span> P<span class="su">2</span> P<span class="su">3</span> P<span class="su">4</span>} is defined to be the number +(λ<span class="su">1</span>μ<span class="su">2</span>)(λ<span class="su">3</span>μ<span class="su">4</span>) / (λ<span class="su">1</span>μ<span class="su">4</span>)(λ<span class="su">3</span>μ<span class="su">2</span>). +The equality of the cross ratios of the ranges (P<span class="su">1</span> P<span class="su">2</span> P<span class="su">3</span> P<span class="su">4</span>) and +(Q<span class="su">1</span> Q<span class="su">2</span> Q<span class="su">3</span> Q<span class="su">4</span>) is proved to be the necessary and sufficient condition +for their mutual projectivity. The cross ratios of all harmonic +ranges are then easily seen to be all equal to -1, by comparing with +the range (OE<span class="su">1</span>UE′<span class="su">1</span>) on the axis of x.</p> + +<p>Thus all the ordinary propositions of geometry in which distance +and angular measure do not enter otherwise than in cross ratios +can now be enunciated and proved. Accordingly the greater part of +the analytical theory of conics and quadrics belongs to geometry +<span class="pagenum"><a name="page733" id="page733"></a>733</span> +at this stage The theory of distance will be considered after the +principles of descriptive geometry have been developed.</p> +</div> + +<p class="pt2 center"><i>Descriptive Geometry.</i></p> + +<p>Descriptive geometry is essentially the science of multiple +order for open series. The first satisfactory system of axioms +was given by M. Pasch.<a name="fa36d" id="fa36d" href="#ft36d"><span class="sp">36</span></a> An improved version is due to G. +Peano.<a name="fa37d" id="fa37d" href="#ft37d"><span class="sp">37</span></a> Both these authors treat the idea of the class of points +constituting the segment lying <i>between</i> two points as an undefined +fundamental idea. Thus in fact there are in this system two +fundamental ideas, namely, of points and of segments. It is +then easy enough to define the prolongations of the segments, +so as to form the complete straight lines. D. Hilbert’s<a name="fa38d" id="fa38d" href="#ft38d"><span class="sp">38</span></a> formulation +of the axioms is in this respect practically based on the same +fundamental ideas. His work is justly famous for some of the +mathematical investigations contained in it, but his exposition of +the axioms is distinctly inferior to that of Peano. Descriptive +geometry can also be considered<a name="fa39d" id="fa39d" href="#ft39d"><span class="sp">39</span></a> as the science of a class of +relations, each relation being a two-termed serial relation, as +considered in the logic of relations, ranging the points between +which it holds into a linear open order. Thus the relations are +the straight lines, and the terms between which they hold are +the points. But a combination of these two points of view +yields<a name="fa40d" id="fa40d" href="#ft40d"><span class="sp">40</span></a> the simplest statement of all. Descriptive geometry is +then conceived as the investigation of an undefined fundamental +relation between three terms (points); and when the relation +holds between three points A, B, C, the points are said to be “in +the [linear] order ABC.”</p> + +<p>O. Veblen’s axioms and definitions, slightly modified, are as +follows:—</p> + +<p>1. If the points A, B, C are in the order ABC, they are in the +order CBA.</p> + +<p>2. If the points A, B, C are in the order ABC, they are not +in the order BCA.</p> + +<p>3. If the points A, B, C are in the order ABC, A is distinct +from C.</p> + +<p>4. If A and B are any two distinct points, there exists a point +C such that A, B, C are in the order ABC.</p> + +<div class="condensed"> +<p><i>Definition.</i>—The <i>line</i> 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 <i>segment</i> AB.</p> +</div> + +<p>5. If points C and D (C ≠ D) lie on the line AB, then A lies on +the line CD.</p> + +<p>6. There exist three distinct points A, B, C not in any of the +orders ABC, BCA, CAB.</p> + +<table class="flt" style="float: right; width: 260px;" summary="Illustration"> +<tr><td class="figright1"><img style="width:211px; height:136px" src="images/img733a.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 73.</span></td></tr></table> + +<p>7. If three distinct points A, B, C (fig. 73) do not lie on the +same line, and D and E are two distinct points in the orders +BCD and CEA, then a point F exists +in the order AFB, and such that +D, E, F are collinear.</p> + +<div class="condensed"> +<p><i>Definition.</i>—If A, B, C are three +non-collinear points, the <i>plane</i> ABC +is the class of points which lie on any +one of the lines joining any two of the +points belonging to the <i>boundary</i> of +the triangle ABC, the boundary being +formed by the segments BC, CA and +AB. The <i>interior</i> of the triangle ABC is formed by the points in +segments such as PQ, where P and Q are points respectively on +two of the segments BC, CA, AB.</p> +</div> + +<p>8. There exists a plane ABC, which does not contain all the +points.</p> + +<div class="condensed"> +<p><i>Definition.</i>—If A, B, C, D are four non-coplanar points, the space +ABCD is the class of points which lie on any of the lines containing +two points on the surface of the tetrahedron ABCD, the <i>surface</i> +being formed by the interiors of the triangles ABC, BCD, DCA, +DAB.</p> +</div> + +<p>9. There exists a space ABCD which contains all the points.</p> + +<p>10. The Dedekind property holds for the order of the points +on any straight line.</p> + +<p>It follows from axioms 1-9 that the points on any straight line +are arranged in an open serial order. Also all the ordinary +theorems respecting a point dividing a straight line into two +parts, a straight line dividing a plane into two parts, and a plane +dividing space into two parts, follow.</p> + +<div class="condensed"> +<p>Again, in any plane α consider a line l and a point A (fig. 74).</p> + +<table class="flt" style="float: left; width: 250px;" summary="Illustration"> +<tr><td class="figleft1"><img style="width:198px; height:155px" src="images/img733b.jpg" alt="" /></td></tr> +<tr><td class="caption"><span class="sc">Fig. 74.</span></td></tr></table> + +<p>Let any point B divide l into two half-lines l<span class="su">1</span> and l<span class="su">2</span>. Then it can +be proved that the set of half-lines, emanating from A and intersecting +l<span class="su">1</span> (such as m), are bounded by two half-lines, of which ABC +is one. Let r be the other. Then it can be proved that r does not +intersect l<span class="su">1</span>. Similarly for the half-line, +such as n, intersecting l<span class="su">2</span>. Let s be its +bounding half-line. Then two cases are +possible. (1) The half-lines r and s are +collinear, and together form one complete +line. In this case, there is one and +only one line (viz. r + s) through A and +lying in α which does not intersect l. +This is the Euclidean case, and the +assumption that this case holds is the +<i>Euclidean parallel axiom</i>. But (2) the +half-lines r and s may not be collinear. +In this case there will be an infinite +number of lines, such as k for instance, containing A and lying in α, +which do not intersect l. Then the lines through A in α are divided +into two classes by reference to l, namely, the <i>secant</i> lines which +intersect l, and the <i>non-secant</i> lines which do not intersect l. The +two boundary non-secant lines, of which r and s are respectively +halves, may be called the two parallels to l through A.</p> + +<p>The perception of the possibility of case 2 constituted the starting-point +from which Lobatchewsky constructed the first explicit +coherent theory of non-Euclidean geometry, and thus created a +revolution in the philosophy of the subject. For many centuries +the speculations of mathematicians on the foundations of geometry +were almost confined to hopeless attempts to prove the “parallel +axiom” without the introduction of some equivalent axiom.<a name="fa41d" id="fa41d" href="#ft41d"><span class="sp">41</span></a></p> +</div> + +<p><i>Associated Projective and Descriptive Spaces.</i>—A region of a +projective space, such that one, and only one, of the two supplementary +segments between any pair of points within it lies +entirely within it, satisfies the above axioms (1-10) of descriptive +geometry, where the points of the region are the descriptive +points, and the portions of straight lines within the region are +the descriptive lines. If the excluded part of the original projective +space is a single plane, the Euclidean parallel axiom also +holds, otherwise it does not hold for the descriptive space of the +limited region. Again, conversely, starting from an original +descriptive space an associated projective space can be constructed +by means of the concept of <i>ideal points</i>.<a name="fa42d" id="fa42d" href="#ft42d"><span class="sp">42</span></a> These are also +called <i>projective points</i>, where it is understood that the simple +points are the points of the original descriptive space. An +<i>ideal point</i> is the class of straight lines which is composed of two +coplanar lines a and b, together with the lines of intersection of +all pairs of intersecting planes which respectively contain a and b, +together with the lines of intersection with the plane ab of all +planes containing any one of the lines (other than a or b) already +specified as belonging to the ideal point. It is evident that, if +the two original lines a and b intersect, the corresponding ideal +point is nothing else than the whole class of lines which are +concurrent at the point ab. But the essence of the definition is +that an ideal point has an existence when the lines a and b do +not intersect, so long as they are coplanar. An ideal point is +termed <i>proper</i>, if the lines composing it intersect; otherwise it +is <i>improper</i>.</p> + +<p>A theorem essential to the whole theory is the following: if +any two of the three lines a, b, c are coplanar, but the three lines +are not all coplanar, and similarly for the lines a, b, d, then c +and d are coplanar. It follows that any two lines belonging to an +ideal point can be used as the pair of guiding lines in the definition. +An ideal point is said to be <i>coherent</i> with a plane, if any of the +lines composing it lie in the plane. An <i>ideal line</i> is the class of +ideal points each of which is coherent with two given planes. +<span class="pagenum"><a name="page734" id="page734"></a>734</span> +If the planes intersect, the ideal line is termed <i>proper</i>, otherwise +it is <i>improper</i>. It can be proved that any two planes, with which +any two of the ideal points are both coherent, will serve as the +guiding planes used in the definition. The ideal planes are +defined as in projective geometry, and all the other definitions +(for segments, order, &c.) of projective geometry are applied +to the ideal elements. If an ideal plane contains some proper +ideal points, it is called <i>proper</i>, otherwise it is <i>improper</i>. Every +ideal plane contains some improper ideal points.</p> + +<p>It can now be proved that all the axioms of projective geometry +hold of the ideal elements as thus obtained; and also that the +order of the ideal points as obtained by the projective method +agrees with the order of the proper ideal points as obtained from +that of the associated points of the descriptive geometry. Thus +a projective space has been constructed out of the ideal elements, +and the proper ideal elements correspond element by element with +the associated descriptive elements. Thus the proper ideal +elements form a region in the projective space within which the +descriptive axioms hold. Accordingly, by substituting ideal +elements, a descriptive space can always be considered as a +region within a projective space. This is the justification for the +ordinary use of the “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.<a name="fa43d" id="fa43d" href="#ft43d"><span class="sp">43</span></a></p> + +<p><i>Congruence and Measurement.</i>—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,<a name="fa44d" id="fa44d" href="#ft44d"><span class="sp">44</span></a> “Space +is represented as an infinite given <i>quantity</i>.” This quantitative +aspect of space arises from the measurability of distances, of +angles, of surfaces and of volumes. These four types of quantity +depend upon the two first among them as fundamental. The +measurability of space is essentially connected with the idea of +<i>congruence</i>, of which the simplest examples are to be found in +the proofs of equality by the method of superposition, as used +in elementary plane geometry. The mere concepts of “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.</p> + +<p>It would therefore appear as if the idea of the congruence, or +metrical equality, of two portions of space (as empirically suggested +by the motion of rigid bodies) must be considered as a +fundamental idea incapable of definition in terms of those +geometrical concepts which have already been enumerated. +This was in effect the point of view of Pasch.<a name="fa45d" id="fa45d" href="#ft45d"><span class="sp">45</span></a> It has, however, +been proved by Sophus Lie<a name="fa46d" id="fa46d" href="#ft46d"><span class="sp">46</span></a> that congruence is capable of +definition without recourse to a new fundamental idea. This +he does by means of his theory of finite continuous groups (see +<span class="sc"><a href="#artlinks">Groups, Theory of</a></span>), of which the definition is possible in terms +of our established geometrical ideas, remembering that coordinates +have already been introduced. The displacement +of a rigid body is simply a mode of defining to the senses a one-one +transformation of all space into itself. For at any point of +space a particle may be conceived to be placed, and to be rigidly +connected with the rigid body; and thus there is a definite +correspondence of any point of space with the new point occupied +by the associated particle after displacement. Again two successive +displacements of a rigid body from position A to position +B, and from position B to position C, are the same in effect as one +displacement from A to C. But this is the characteristic “group” +property. Thus the transformations of space into itself defined +by displacements of rigid bodies form a group.</p> + +<p>Call this group of transformations a congruence-group. Now +according to Lie a congruence-group is defined by the following +characteristics:—</p> + +<p>1. A congruence-group is a finite continuous group of one-one +transformations, containing the identical transformation.</p> + +<p>2. It is a sub-group of the general projective group, <i>i.e.</i> of +the group of which any transformation converts planes into +planes, and straight lines into straight lines.</p> + +<p>3. An infinitesimal transformation can always be found satisfying +the condition that, at least throughout a certain enclosed +region, any definite line and any definite point on the line are +latent, <i>i.e.</i> correspond to themselves.</p> + +<p>4. No infinitesimal transformation of the group exists, such +that, at least in the region for which (3) holds, a straight line, +a point on it, and a plane through it, shall all be latent.</p> + +<p>The property enunciated by conditions (3) and (4), taken +together, is named by Lie “Free mobility in the infinitesimal.” +Lie proves the following theorems for a projective space:—</p> + +<div class="condensed"> +<p>1. If the above four conditions are only satisfied by a group +throughout part of projective space, this part either (α) must be the +region enclosed by a real closed quadric, or (β) must be the whole of +the projective space with the exception of a single plane. In case +(α) the corresponding congruence group is the continuous group for +which the enclosing quadric is latent; and in case (β) an imaginary +conic (with a real equation) lying in the latent plane is also latent, +and the congruence group is the continuous group for which the +plane and conic are latent.</p> + +<p>2. If the above four conditions are satisfied by a group throughout +the whole of projective space, the congruence group is the continuous +group for which some imaginary quadric (with a real equation) is +latent.</p> + +<p>By a proper choice of non-homogeneous co-ordinates the equation +of any quadrics of the types considered, either in theorem 1 (α), or in +theorem 2, can be written in the form 1 + c(x² + y² + z²) = 0, where c is +negative for a real closed quadric, and positive for an imaginary +quadric. Then the general infinitesimal transformation is defined +by the three equations:</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">dx/dt = u − ω<span class="su">3</span>y + ω<span class="su">2</span>z + cx (ux + vy + wz),</td> <td class="tccm" rowspan="3">(A)</td></tr> + +<tr><td class="tcl">dy/dt = v − ω<span class="su">1</span>z + ω<span class="su">3</span>x + cy (ux + vy + wz),</td></tr> + +<tr><td class="tcl">dz/dt = w − ω<span class="su">2</span>x + ω<span class="su">1</span>y + cz (ux + vy + wz).</td></tr> +</table> + +<p class="noind">In the ease considered in theorem 1 (β), with the proper choice of +co-ordinates the three equations defining the general infinitesimal +transformation are:</p> + +<table class="ws" summary="Contents"> +<tr><td class="tcl">dx/dt = u − ω<span class="su">3</span>y + ω<span class="su">2</span>z,</td> <td class="tccm" rowspan="3">(B)</td></tr> + +<tr><td class="tcl">dy/dt = v − ω<span class="su">1</span>z + ω<span class="su">3</span>x,</td></tr> + +<tr><td class="tcl">dz/dt = w − ω<span class="su">2</span>x + ω<span class="su">1</span>y.</td></tr> +</table> + +<p class="noind">In this case the latent plane is the plane for which at least one of +x, y, z are infinite, that is, the plane 0.x + 0.y + 0.z + a = 0; and the +latent conic is the conic in which the cone x² + y² + z² = 0 intersects +the latent plane.</p> +</div> + +<p>It follows from theorems 1 and 2 that there is not one unique +congruence-group, but an indefinite number of them. There is +one congruence-group corresponding to each closed real quadric, +one to each imaginary quadric with a real equation, and one to +each imaginary conic in a real plane and with a real equation. +The quadric thus associated with each congruence-group is +called the <i>absolute</i> for that group, and in the degenerate case +of 1 (β) the absolute is the latent plane together with the latent +imaginary conic. If the absolute is real, the congruence-group +is <i>hyperbolic</i>; if imaginary, it is <i>elliptic</i>; if the absolute is a +plane and imaginary conic, the group is parabolic. Metrical +geometry is simply the theory of the properties of some particular +congruence-group selected for study.</p> + +<div class="condensed"> +<p>The definition of distance is connected with the corresponding +congruence-group by two considerations in respect to a range of five +points (A<span class="su">1</span>, A<span class="su">2</span>, P<span class="su">1</span>, P<span class="su">2</span>, P<span class="su">3</span>), of which A<span class="su">1</span> and A<span class="su">2</span> are on the absolute.</p> + +<p>Let {A<span class="su">1</span>P<span class="su">1</span>A<span class="su">2</span>P<span class="su">2</span>} stand for the cross ratio (as defined above) of the +range (A<span class="su">1</span>P<span class="su">1</span>A<span class="su">2</span>P<span class="su">2</span>), with a similar notation for the other ranges. +Then</p> + +<p class="noind">(1)</p> + +<p class="center">log {A<span class="su">1</span>P<span class="su">1</span>A<span class="su">2</span>P<span class="su">2</span>} + log {A<span class="su">1</span>P<span class="su">2</span>A<span class="su">2</span>P<span class="su">3</span>} = log {A<span class="su">1</span>P<span class="su">1</span>A<span class="su">2</span>P<span class="su">3</span>},</p> + +<p class="noind">and</p> + +<p class="noind">(2), if the points A<span class="su">1</span>, A<span class="su">2</span>, P<span class="su">1</span>, P<span class="su">2</span> are transformed into A′<span class="su">1</span>, A′<span class="su">2</span>, P′<span class="su">1</span>, P′<span class="su">2</span> +by any transformation of the congruence-group, (α) {A<span class="su">1</span>P<span class="su">1</span>A<span class="su">2</span>P<span class="su">2</span> = +{A′<span class="su">1</span>P′<span class="su">1</span>A′<span class="su">2</span>P′<span class="su">2</span>}, since the transformation is projective, and (β) A′<span class="su">1</span>, A′<span class="su">2</span> +are on the absolute since A<span class="su">1</span> and A<span class="su">2</span> are on it. Thus if we define +<span class="pagenum"><a name="page735" id="page735"></a>735</span> +the distance P<span class="su">1</span>P<span class="su">2</span> to be ½k log {A<span class="su">1</span>P<span class="su">1</span>A<span class="su">2</span>P<span class="su">2</span>}, where A<span class="su">1</span> and A<span class="su">2</span> are the +points in which the line P<span class="su">1</span>P<span class="su">2</span> cuts the absolute, and k is some constant, +the two characteristic properties of distance, namely, (1) the +addition of consecutive lengths on a straight line, and (2) the invariability +of distances during a transformation of the congruence-group, +are satisfied. This is the well-known Cayley-Klein projective +definition<a name="fa47d" id="fa47d" href="#ft47d"><span class="sp">47</span></a> of distance, which was elaborated in view of the addition +property alone, previously to Lie’s discovery of the theory of congruence-groups. +For a hyperbolic group when P<span class="su">1</span> and P<span class="su">2</span> are in the +region enclosed by the absolute, log {A<span class="su">1</span>P<span class="su">1</span>A<span class="su">2</span>P<span class="su">2</span>} is real, and therefore +k must be real. For an elliptic group A<span class="su">1</span> and A<span class="su">2</span> are conjugate +imaginaries, and log {A<span class="su">1</span>P<span class="su">1</span>A<span class="su">2</span>P<span class="su">2</span>} is a pure imaginary, and k is chosen +to be κ/ι, where κ is real and ι = √ −.</p> + +<p>Similarly the angle between two planes, p<span class="su">1</span> and p<span class="su">2</span>, is defined to be +(1/2ι) log (t<span class="su">1</span>p<span class="su">1</span>t<span class="su">2</span>p<span class="su">2</span>), where t<span class="su">1</span> and t<span class="su">2</span> are tangent planes to the absolute +through the line p<span class="su">1</span>p<span class="su">2</span>. The planes t<span class="su">1</span> and t<span class="su">2</span> are imaginary for an +elliptic group, and also for an hyperbolic group when the planes p<span class="su">1</span> +and p<span class="su">2</span> intersect at points within the region enclosed by the absolute. +The development of the consequences of these metrical definitions +is the <span class="correction" title="amended from subjct">subject</span> of non-Euclidean geometry.</p> + +<p>The definitions for the parabolic case can be arrived at as limits +of those obtained in either of the other two cases by making k +ultimately to vanish. It is also obvious that, if P<span class="su">1</span> and P<span class="su">2</span> be the +points (x<span class="su">1</span>, y<span class="su">1</span>, z<span class="su">1</span>) and (x<span class="su">2</span>, y<span class="su">2</span>, z<span class="su">2</span>), it follows from equations (B) above +that {(x<span class="su">1</span> − x<span class="su">2</span>)² + (y<span class="su">1</span> − y<span class="su">2</span>)² + (z<span class="su">1</span> − z<span class="su">2</span>)²}<span class="sp">1/2</span> is unaltered by a congruence +transformation and also satisfies the addition property for collinear +distances. Also the previous definition of an angle can be adapted +to this case, by making t<span class="su">1</span> and t<span class="su">2</span> to be the tangent planes through +the line p<span class="su">1</span>p<span class="su">2</span> to the imaginary conic. Similarly if p<span class="su">1</span> and p<span class="su">2</span> are intersecting +lines, the same definition of an angle holds, where t<span class="su">1</span> and t<span class="su">2</span> +are now the lines from the point p<span class="su">1</span>p<span class="su">2</span> to the two points where the +plane p<span class="su">1</span>p<span class="su">2</span> cuts the imaginary conic. These points are in fact the +“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.</p> +</div> + +<p>Thus the only metrical geometry for the whole of projective +space is of the elliptic type. But the actual measure-relations +(though not their general properties) differ according to the +elliptic congruence-group selected for study. In a descriptive +space a congruence-group should possess the four characteristics +of such a group throughout the whole of the space. Then form +the associated ideal projective space. The associated congruence-group +for this ideal space must satisfy the four conditions +throughout the region of the proper ideal points. Thus the +boundary of this region is the absolute. Accordingly there can +be no metrical geometry for the whole of a descriptive space +unless its boundary (in the associated ideal space) is a closed +quadric or a plane. If the boundary is a closed quadric, there +is one possible congruence-group of the hyperbolic type. If +the boundary is a plane (the plane at infinity), the possible +congruence-groups are parabolic; and there is a congruence-group +corresponding to each imaginary conic in this plane, +together with a Euclidean metrical geometry corresponding to +each such group. Owing to these alternative possibilities, it +would appear to be more accurate to say that systems of quantities +can be found in a space, rather than that space is a quantity.</p> + +<p>Lie has also deduced<a name="fa48d" id="fa48d" href="#ft48d"><span class="sp">48</span></a> the same results with respect to congruence-groups +from another set of defining properties, which +explicitly assume the existence of a quantitative relation (the +distance) between any two points, which is invariant for any +transformation of the congruence-group.<a name="fa49d" id="fa49d" href="#ft49d"><span class="sp">49</span></a></p> + +<p>The above results, in respect to congruence and metrical +geometry, considered in relation to existent space, have led to the +doctrine<a name="fa50d" id="fa50d" href="#ft50d"><span class="sp">50</span></a> that it is intrinsically unmeaning to ask which system +of metrical geometry is true of the physical world. Any one of +these systems can be applied, and in an indefinite number of ways. +The only question before us is one of convenience in respect to +simplicity of statement of the physical laws. This point of view +seems to neglect the consideration that science is to be relevant +to the definite perceiving minds of men; and that (neglecting +the ambiguity introduced by the invariable slight inexactness +of observation which is not relevant to this special doctrine) +we have, in fact, presented to our senses a definite set of transformations +forming a congruence-group, resulting in a set of +measure relations which are in no respect arbitrary. Accordingly +our scientific laws are to be stated relevantly to that particular +congruence-group. Thus the investigation of the type (elliptic, +hyperbolic or parabolic) of this special congruence-group is a +perfectly definite problem, to be decided by experiment. The +consideration of experiments adapted to this object requires some +development of non-Euclidean geometry (see section VI., +<i>Non-Euclidean Geometry</i>). But if the doctrine means that, +assuming some sort of objective reality for the material universe, +beings can be imagined, to whom <i>either</i> all congruence-groups +are equally important, <i>or</i> some other congruence-group is specially +important, the doctrine appears to be an immediate deduction +from the mathematical facts. Assuming a definite congruence-group, +the investigation of surfaces (or three-dimensional loci +in space of four dimensions) with geodesic geometries of the form +of metrical geometries of other types of congruence-groups forms +an important chapter of non-Euclidean geometry. Arising +from this investigation there is a widely-spread fallacy, which +has found its way into many philosophic writings, namely, that +the possibility of the geometry of existent three-dimensional +space being other than Euclidean depends on the physical +existence of Euclidean space of four or more dimensions. The +foregoing exposition shows the baselessness of this idea.</p> + +<div class="condensed"> +<p><span class="sc">Bibliography</span>.—For an account of the investigations on the +axioms of geometry during the Greek period, see M. Cantor, <i>Vorlesungen +über die Geschichte der Mathematik</i>, Bd. i. and iii.; T.L. +Heath, <i>The Thirteen Books of Euclid’s Elements, a New Translation +from the Greek, with Introductory Essays and Commentary, Historical, +Critical, and Explanatory</i> (Cambridge, 1908)—this work is the standard +source of information; W.B. Frankland, <i>Euclid, Book I., with a +Commentary</i> (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., <i>Euclides ab omni naevo vindicatus</i> +(Milan, 1733). Saccheri was an Italian Jesuit who unconsciously +discovered non-Euclidean geometry in the course of his efforts to +prove its impossibility. J.H. Lambert, <i>Theorie der Parallellinien</i> +(1766); A.M. Legendre, <i>Éléments de géométrie</i> (1794). An adequate +account of the above authors is given by P. Stäckel and F. Engel, +<i>Die Theorie der Parallellinien von Euklid bis auf Gauss</i> (Leipzig, +1895). The next period of time (roughly from 1800 to 1870) contains +two streams of thought, both of which are essential to the modern +analysis of the subject. The first stream is that which produced the +discovery and investigation of non-Euclidean geometries, the second +stream is that which has produced the geometry of position, comprising +both projective and descriptive geometry not very accurately +discriminated. The leading authors on non-Euclidean geometry +are K.F. Gauss, in private letters to Schumacher, cf. Stäckel and +Engel, <i>loc. cit.</i>; N. Lobatchewsky, rector of the university of Kazan, +to whom the honour of the effective discovery of non-Euclidean +geometry must be assigned. His first publication was at Kazan +in 1826. His various memoirs have been re-edited by Engel; +cf. <i>Urkunden zur Geschichte der nichteuklidischen Geometrie</i> by +Stäckel and Engel, vol. i. “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, <i>Tentamen juventutem....</i> This +memoir has been separately edited by J. Frischauf, <i>Absolute Geometrie +nach J. Bolyai</i> (Leipzig, 1872); B. Riemann, <i>Über die Hypothesen, +welche der Geometrie zu Grunde liegen</i> (1854); cf. <i>Gesamte Werke</i>, a +translation in The Collected Papers of W.K. Clifford. This is a +fundamental memoir on the subject and must rank with the work of +Lobatchewsky. Riemann discovered elliptic metrical geometry, +and Lobatchewsky hyperbolic geometry. A full account of Riemann’s +ideas, with the subsequent developments due to Clifford, +F. Klein and W. Killing, will be found in <i>The Boston Colloquium for +1903</i> (New York, 1905), article “Forms of Non-Euclidean Space,” +by F.S. Woods. A. Cayley, <i>loc. cit.</i> (1859), and F. Klein, “Über die +sogenannte nichteuklidische Geometrie,” <i>Math. Annal.</i> vols. iv. +and vi. (1871 and 1872), between them elaborated the projective +theory of distance; H. Helmholtz, “Über die thatsächlichen +Grundlagen der Geometrie” (1866), and “Über die Thatsachen, die +der Geometrie zu Grunde liegen” (1868), both in his <i>Wissenschaftliche +Abhandlungen</i>, vol. ii., and S. Lie, <i>loc. cit.</i> (1890 and 1893), between +them elaborated the group theory of congruence.</p> + +<p>The numberless works which have been written to suggest equivalent +alternatives to Euclid’s parallel axioms may be neglected as +being of trivial importance, though many of them are marvels of +geometric ingenuity.</p> + +<p>The second stream of thought confined itself within the circle of +ideas of Euclidean geometry. Its origin was mainly due to a +<span class="pagenum"><a name="page736" id="page736"></a>736</span> +succession of great French mathematicians, for example, G. Monge, +<i>Géométrie descriptive</i> (1800); J.V. Poncelet, <i>Traité des proprietés +projectives des figures</i> (1822); M. Chasles, <i>Aperçu historique sur +l’origine et le développement des méthodes en géométrie</i> (Bruxelles, 1837), +and <i>Traité de géométrie supérieure</i> (Paris, 1852); and many others. +But the works which have been, and are still, of decisive influence on +thought as a store-house of ideas relevant to the foundations of +geometry are K.G.C. von Staudt’s two works, <i>Geometrie der Lage</i> +(Nürnberg, 1847); and <i>Beiträge zur Geometrie der Lage</i> (Nürnberg, +1856, 3rd ed. 1860).</p> + +<p>The final period is characterized by the successful production of +exact systems of axioms, and by the final solution of problems +which have occupied mathematicians for two thousand years. The +successful analysis of the ideas involved in serial continuity is due to +R. Dedekind, <i>Stetigkeit und irrationale Zahlen</i> (1872), and to G. +Cantor, <i>Grundlagen einer allgemeinen Mannigfaltigkeitslehre</i> (Leipzig, +1883), and <i>Acta math.</i> vol. 2.</p> + +<p>Complete systems of axioms have been stated by M. Pasch, <i>loc. +cit.</i>; G. Peano, <i>loc. cit.</i>; M. Pieri, loc. cit.; B. Russell, <i>Principles of +Mathematics</i>; O. Veblen, <i>loc. cit.</i>; and by G. Veronese in his treatise, +<i>Fondamenti di geometria</i> (Padua, 1891; German transl. by A. Schepp, +<i>Grundzüge der Geometrie</i>, Leipzig, 1894). Most of the leading memoirs +on special questions involved have been cited in the text; in addition +there may be mentioned M. Pieri, “Nuovi principii di geometria +projettiva complessa,” <i>Trans. Accad. R. d. Sci.</i> (Turin, 1905); +E.H. Moore, “On the Projective Axioms of Geometry,” <i>Trans. +Amer. Math. Soc.</i>, 1902; O. Veblen and W.H. Bussey, “Finite +Projective Geometries,” <i>Trans. Amer. Math. Soc.</i>, 1905; A.B. +Kempe, “On the Relation between the Logical Theory of Classes +and the Geometrical Theory of Points,” <i>Proc. Lond. Math. Soc.</i>, +1890; J. Royce, “The Relation of the Principles of Logic to the +Foundations of Geometry,” <i>Trans. of Amer. Math. Soc.</i>, 1905; +A. Schoenflies, “Über die Möglichkeit einer projectiven Geometrie +bei transfiniter (nichtarchimedischer) Massbestimmung,” Deutsch. +<i>M.-V. Jahresb.</i>, 1906.</p> + +<p>For general expositions of the bearings of the above investigations, +cf. Hon. Bertrand Russell, <i>loc. cit.</i>; L. Couturat, <i>Les Principes +des mathématiques</i> (Paris, 1905); H. Poincaré, <i>loc. cit.</i>; Russell +and Whitehead, <i>Principia mathematica</i> (Cambridge, Univ. Press). +The philosophers whose views on space and geometric truth deserve +especial study are Descartes, Leibnitz, Hume, Kant and J.S. +Mill.</p> +</div> +<div class="author">(A. N. W.)</div> + +<hr class="foot" /> <div class="note"> + +<p><a name="ft1d" id="ft1d" href="#fa1d"><span class="fn">1</span></a> For Egyptian geometry see <span class="sc"><a href="#artlinks">Egypt</a></span>, § <i>Science and Mathematics</i>.</p> + +<p><a name="ft2d" id="ft2d" href="#fa2d"><span class="fn">2</span></a> Cf. A.N. Whitehead, <i>Universal Algebra</i>, Bk. vi. (Cambridge, +1898).</p> + +<p><a name="ft3d" id="ft3d" href="#fa3d"><span class="fn">3</span></a> Cf. A.N. Whitehead, <i>loc. cit.</i></p> + +<p><a name="ft4d" id="ft4d" href="#fa4d"><span class="fn">4</span></a> Cf. A.N. Whitehead, “The Geodesic Geometry of Surfaces in +non-Euclidean Space,” <i>Proc. Lond. Math. Soc.</i> vol. xxix.</p> + +<p><a name="ft5d" id="ft5d" href="#fa5d"><span class="fn">5</span></a> Cf. Klein, “Zur nicht-Euklidischen Geometrie,” <i>Math. Annal.</i> +vol. xxxvii.</p> + +<p><a name="ft6d" id="ft6d" href="#fa6d"><span class="fn">6</span></a> On the theory of parallels before Lobatchewsky, see Stäckel und +Engel, <i>Theorie der Parallellinien von Euklid bis auf Gauss</i> (Leipzig, +1895). The foregoing remarks are based upon the materials collected +in this work.</p> + +<p><a name="ft7d" id="ft7d" href="#fa7d"><span class="fn">7</span></a> See Stäckel und Engel, <i>op. cit.</i>, and “Gauss, die beiden Bolyai, +und die nicht-Euklidische Geometrie,” <i>Math. Annalen</i>, Bd. xlix.; +also Engel’s translation of Lobatchewsky (Leipzig, 1898), pp. 378 ff.</p> + +<p><a name="ft8d" id="ft8d" href="#fa8d"><span class="fn">8</span></a> Lobatchewsky’s works on the subject are the following:—“On +the Foundations of Geometry,” <i>Kazañ Messenger</i>, 1829-1830; +“New Foundations of Geometry, with a complete Theory of +Parallels,” <i>Proceedings of the University of Kazañ</i>, 1835 (both in +Russian, but translated into German by Engel, Leipzig, 1898); +“Géométrie imaginaire,” Crelle’s Journal, 1837; <i>Theorie der +Parallellinien</i> (Berlin, 1840; 2nd ed., 1887; translated by Halsted, +Austin, Texas, 1891). His results appear to have been set forth in a +paper (now lost) which he read at Kazañ in 1826.</p> + +<p><a name="ft9d" id="ft9d" href="#fa9d"><span class="fn">9</span></a> Translated by Halsted (Austin, Texas, 4th ed., 1896.)</p> + +<p><a name="ft10d" id="ft10d" href="#fa10d"><span class="fn">10</span></a> <i>Abhandlungen d. Königl. Ges. d. Wiss. zu Göttingen</i>, Bd. xiii.; +<i>Ges. math. Werke</i>, pp. 254-269; translated by Clifford, <i>Collected +Mathematical Papers</i>.</p> + +<p><a name="ft11d" id="ft11d" href="#fa11d"><span class="fn">11</span></a> Cf. <i>Gesamm. math. und phys. Werke</i>, vol. i. (Leipzig, 1894).</p> + +<p><a name="ft12d" id="ft12d" href="#fa12d"><span class="fn">12</span></a> <i>Wiss. Abh.</i> vol. ii. pp. 610, 618 (1866, 1868).</p> + +<p><a name="ft13d" id="ft13d" href="#fa13d"><span class="fn">13</span></a> <i>Mind</i>, O.S., vols. i. and iii.; <i>Vorträge und Reden</i>, vol. ii. pp. 1, +256.</p> + +<p><a name="ft14d" id="ft14d" href="#fa14d"><span class="fn">14</span></a> His papers are “Saggio di interpretazione della geometria non-Euclidea,” +<i>Giornale di matematiche</i>, vol. vi. (1868); “Teoria fondamentale +degli spazii di curvatura costante,” <i>Annali di matematica</i>, +vol. ii. (1868-1869). Both were translated into French by J. Hoüel, +<i>Annales scientifiques de l’École Normale supérieure</i>, vol. vi. (1869).</p> + +<p><a name="ft15d" id="ft15d" href="#fa15d"><span class="fn">15</span></a> Beltrami shows also that this definition agrees with that of Gauss.</p> + +<p><a name="ft16d" id="ft16d" href="#fa16d"><span class="fn">16</span></a> “Sur la théorie des foyers,” <i>Nouv. Ann.</i> vol. xii.</p> + +<p><a name="ft17d" id="ft17d" href="#fa17d"><span class="fn">17</span></a> <i>Math. Annalen</i>, iv. vi., 1871-1872.</p> + +<p><a name="ft18d" id="ft18d" href="#fa18d"><span class="fn">18</span></a> For an investigation of these and similar properties, see Whitehead, +<i>Universal Algebra</i> (Cambridge, 1898), bk. vi. ch. ii. The polar +form was independently discovered by Simon Newcomb in 1877.</p> + +<p><a name="ft19d" id="ft19d" href="#fa19d"><span class="fn">19</span></a> For an analysis of Leibnitz’s ideas on space, cf. B. Russell, <i>The +Philosophy of Leibnitz</i>, chs. viii.-x.</p> + +<p><a name="ft20d" id="ft20d" href="#fa20d"><span class="fn">20</span></a> Cf. Hon. Bertrand Russell, “Is Position in Time and Space +Absolute or Relative?” <i>Mind</i>, n.s. vol. 10 (1901), and A.N. Whitehead, +“Mathematical Concepts of the Material World,” <i>Phil. Trans.</i> +(1906), p. 205.</p> + +<p><a name="ft21d" id="ft21d" href="#fa21d"><span class="fn">21</span></a> Cf. <i>Critique of Pure Reason</i>, 1st section: “Of Space,” conclusion +A, Max Müller’s translation.</p> + +<p><a name="ft22d" id="ft22d" href="#fa22d"><span class="fn">22</span></a> Cf. Ernst Mach, <i>Erkenntniss und Irrtum</i> (Leipzig); the relevant +chapters are translated by T.J. McCormack, <i>Space and Geometry</i> +(London, 1906); also A. Meinong, <i>Über die Stellung der Gegenstandstheorie +im System der Wissenschaften</i> (Leipzig, 1907).</p> + +<p><a name="ft23d" id="ft23d" href="#fa23d"><span class="fn">23</span></a> Cf. Russell, <i>Principles of Mathematics</i>, § 352 (Cambridge, 1903).</p> + +<p><a name="ft24d" id="ft24d" href="#fa24d"><span class="fn">24</span></a> Cf. A.N. Whitehead, <i>The Axioms of Projective Geometry</i>, § 3 +(Cambridge, 1906).</p> + +<p><a name="ft25d" id="ft25d" href="#fa25d"><span class="fn">25</span></a> Cf. Russell, <i>Princ. of Math.</i>, ch. i.</p> + +<p><a name="ft26d" id="ft26d" href="#fa26d"><span class="fn">26</span></a> Cf. Russell, <i>loc. cit.</i>, and G. Frege, “Über die Grundlagen der +Géométrie,” <i>Jahresber. der Deutsch. Math. Ver.</i> (1906).</p> + +<p><a name="ft27d" id="ft27d" href="#fa27d"><span class="fn">27</span></a> This formulation—though not in respect to number—is in all +essentials that of M. Pieri, cf. “I principii della Geometria di Posizione,” +<i>Accad. R. di Torino</i> (1898); also cf. Whitehead, <i>loc. cit.</i></p> + +<p><a name="ft28d" id="ft28d" href="#fa28d"><span class="fn">28</span></a> Cf. G. Peano, “Sui fondamenti della Geometria,” p. 73, <i>Rivista +di matematica</i>, vol. iv. (1894), and D. Hilbert, <i>Grundlagen der Geometrie</i> +(Leipzig, 1899); and R.F. Moulton, “A Simple non-Desarguesian +Plane Geometry,” <i>Trans. Amer. Math. Soc.</i>, vol. iii. (1902).</p> + +<p><a name="ft29d" id="ft29d" href="#fa29d"><span class="fn">29</span></a> Cf. “Sui postulati fondamentali della geometria projettiva,” +<i>Giorn. di matematica</i>, vol. xxx. (1891); also of Pieri, loc. cit., and +Whitehead, <i>loc. cit.</i></p> + +<p><a name="ft30d" id="ft30d" href="#fa30d"><span class="fn">30</span></a> Cf. Hilbert, <i>loc. cit.</i>; for a fuller exposition of Hilbert’s proof +cf. K.T. Vahlen, <i>Abstrakte Geometrie</i> (Leipzig, 1905), also Whitehead, +<i>loc. cit.</i></p> + +<p><a name="ft31d" id="ft31d" href="#fa31d"><span class="fn">31</span></a> Cf. H. Wiener, <i>Jahresber. der Deutsch. Math. Ver.</i> vol. i. (1890); +and F. Schur, “Über den Fundamentalsatz der projectiven Geometrie,” +<i>Math. Ann.</i> vol. li. (1899).</p> + +<p><a name="ft32d" id="ft32d" href="#fa32d"><span class="fn">32</span></a> Cf. Hilbert, <i>loc. cit.</i>, and Whitehead, <i>loc. cit.</i></p> + +<p><a name="ft33d" id="ft33d" href="#fa33d"><span class="fn">33</span></a> Cf. Dedekind, <i>Stetigkeit und irrationale Zahlen</i> (1872).</p> + +<p><a name="ft34d" id="ft34d" href="#fa34d"><span class="fn">34</span></a> Cf. v. Staudt, <i>Geometrie der Lage</i> (1847).</p> + +<p><a name="ft35d" id="ft35d" href="#fa35d"><span class="fn">35</span></a> Cf. Pasch, <i>Vorlesungen über neuere Geometrie</i> (Leipzig, 1882), a +classic work; also Fiedler, <i>Die darstellende Geometrie</i> (1st ed., 1871, +3rd ed., 1888); Clebsch, <i>Vorlesungen über Geometrie</i>, vol. iii.; +Hilbert, <i>loc. cit.</i>; F. Schur, <i>Math. Ann. Bd.</i> lv. (1902); Vahlen, +<i>loc. cit.</i>; Whitehead, <i>loc. cit.</i></p> + +<p><a name="ft36d" id="ft36d" href="#fa36d"><span class="fn">36</span></a> Cf. <i>loc. cit.</i></p> + +<p><a name="ft37d" id="ft37d" href="#fa37d"><span class="fn">37</span></a> Cf. <i>I Principii di geometria</i> (Turin, 1889) and “Sui fondamenti +della geometria,” <i>Rivista di mat.</i> vol. iv. (1894).</p> + +<p><a name="ft38d" id="ft38d" href="#fa38d"><span class="fn">38</span></a> Cf. <i>loc. cit.</i></p> + +<p><a name="ft39d" id="ft39d" href="#fa39d"><span class="fn">39</span></a> Cf. Vailati, <i>Rivista di mat.</i> vol. iv. and Russell, <i>loc. cit.</i> § 376.</p> + +<p><a name="ft40d" id="ft40d" href="#fa40d"><span class="fn">40</span></a> Cf. O. Veblen, “On the Projective Axioms of Geometry,” +<i>Trans. Amer. Math. Soc.</i> vol. iii. (1902).</p> + +<p><a name="ft41d" id="ft41d" href="#fa41d"><span class="fn">41</span></a> Cf. P. Stäckel and F. Engel, <i>Die Theorie der Parallellinien von +Euklid bis auf Gauss</i> (Leipzig, 1895).</p> + +<p><a name="ft42d" id="ft42d" href="#fa42d"><span class="fn">42</span></a> Cf. Pasch, loc. cit., and R. Bonola, “Sulla introduzione degli +enti improprii in geometria projettive,” <i>Giorn. di mat.</i> vol. xxxviii. +(1900); and Whitehead, <i>Axioms of Descriptive Geometry</i> (Cambridge, +1907).</p> + +<p><a name="ft43d" id="ft43d" href="#fa43d"><span class="fn">43</span></a> The original idea (confined to this particular case) of ideal +points is due to von Staudt (<i>loc. cit.</i>).</p> + +<p><a name="ft44d" id="ft44d" href="#fa44d"><span class="fn">44</span></a> Cf. <i>Critique</i>, “Trans. Aesth.” Sect. I.</p> + +<p><a name="ft45d" id="ft45d" href="#fa45d"><span class="fn">45</span></a> Cf. <i>loc. cit.</i></p> + +<p><a name="ft46d" id="ft46d" href="#fa46d"><span class="fn">46</span></a> Cf. <i>Über die Grundlagen der Geometrie</i> (Leipzig, Ber., 1890); +and <i>Theorie der Transformationsgruppen</i> (Leipzig, 1893), vol. iii.</p> + +<p><a name="ft47d" id="ft47d" href="#fa47d"><span class="fn">47</span></a> Cf. A. Cayley, “A Sixth Memoir on Quantics,” <i>Trans. Roy. Soc.</i>, +1859, and <i>Coll. Papers</i>, vol. ii.; and F. Klein, <i>Math. Ann.</i> vol. iv., +1871.</p> + +<p><a name="ft48d" id="ft48d" href="#fa48d"><span class="fn">48</span></a> Cf. <i>loc. cit.</i></p> + +<p><a name="ft49d" id="ft49d" href="#fa49d"><span class="fn">49</span></a> For similar deductions from a third set of axioms, suggested in +essence by Peano, Riv. mat. vol. iv. loc. cit. cf. Whitehead, <i>Desc. +Geom.</i> <i>loc. cit.</i></p> + +<p><a name="ft50d" id="ft50d" href="#fa50d"><span class="fn">50</span></a> Cf. H. Poincaré, <i>La Science et l’hypothèse</i>, ch. iii.</p> +</div> + +<hr class="art" /> + + + + + + + + + +<pre> + + + + + +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-h.htm or 37461-h.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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