,transcriber's notes3 7#a7 ,numbers following letters 7without space7 like ,c#b were originally printed in subscript4 ,letter subscripts are preceded by an underscore1 like ,cn4 7#b7 ,characters following a carat 77 were printed in superscript4 7#c7 ,side-notes were relocated to function as titles of their respective paragraphs4 7#d7 ,macrons and breves above letters and dots below letters were not inserted4 7#e7 root stands for the root symbol2 alpha1 beta1 etc4 for greek letters4 7#f7 ,the following typographical errors have been corrected3 ,,article ,,geoffrey3 444 his history in chiefly one of quarrels1 with the see of ,canterbury1 with the chancellor ,william ,longchamp1 with his half-brothers ,richard and ,john1 and especially with his canons at ,york4 ',william' amended from ',willian'4 ,,article ,,geology3 444 and at the same time greater appreciation has been shown of the signification and strength of the geological proofs of the high antiquity of our planet4 'strength' amended from 'stength'4 ,,article ,,geology3 444 it can be demonstrated that sometimes an inch or two of sediment might1 on certain horizons1 represent the deposit of an enormously longer period than a hundred or a thousand times the same amount of sediment on other horizons4 'might' amended from 'much'4 ,,article ,,geology3 ,but no such extensive disturbance of the normal conditions of the distribution of life can take place without carrying with it many secondary effects1 and setting in motion a wide cycle of change and of reaction in the animal and vegetable kingdoms4 'kingdoms' amended from 'kindgoms'4 ,,article ,,geometry3 ,the bases and altitudes of equal solid parallelepipeds are reciprocally proportional2 and if the bases and altitudes be reciprocally proportional1 the solid parallelepipeds are equal4 'are' amended from 'and'4 ,,article ,,geometry3 ,an alternative method of testing a relation is illustrated in the following example3--,if ,a1 ,b1 ,c1 ,d1 ,e1 ,f be six collinear points1 then 'following example3--' amended from 'example3 following'4 ,,article ,,geometry3 #c4 ,in an hyperbolic involution any two conjugate points are harmonic conjugates with regard to the two foci4 'an' amended from 'a'4 ,,article ,,geometry3 ,if two lines1 given by their projections1 intersect1 the intersection of their planes and the intersection of their elevations must lie in a line perpendicular to the axis1 because they must be the projections of the point common to the two lines4 'planes' amended from 'plans'4 ,,article ,,geometry3 ,where this is the case1 if alpha be the measure of curvature1 the linear element can be put into the form 'if' amended from 'it'4 ,,article ,,geometry3 ,the development of the consequences of these metrical definitions is the subject of non-,euclidean geometry4 'subject' amended from 'subjct'4 ,,encyclopaedia ,,britannica ,a ,,dictionary ,,of ,,arts1 ,,sciences1 ,,literature ,,and ,,general ,,information ,,eleventh ,,edition ,,volume ,,xi1 ,,slice ,,vi ,,geodesy to ,,geometry ,,articles ,,in ,,this ,,slice3 ,,geodesy ,,geoffroy1 ,,etienne ,,francois ,,geoffrey 7,martel7 ,,geoffroy1 ,,julien ,,louis ,,geoffrey 7,plantagenet7 ,,geoffroy ,,saint-,,hilaire1 ,,etienne ,,geoffrey 7duke of ,brittany7 ,,geoffroy ,,saint-,,hilaire1 ,,isidore ,,geoffrey 7archbishop of ,york7 ,,geography ,,geoffrey ,,de ,,montbray ,,geoid ,,geoffrey ,,of ,,monmouth ,,geok-,,tepe ,,geoffrey ,,of ,,paris ,,geology ,,geoffrey ,,the ,,baker ,,geometrical ,,continuity ,,geoffrin1 ,,marie ,,therese ,,rodet ,,geometry ,,geodesy 7from the ,gr4 ,greek3 ge1 the earth1 and ,greek3 daiein1 to divide71 the science of surveying 7q4v47 extended to large tracts of country1 having in view not only the production of a system of maps of very great accuracy1 but the determination of the curvature of the surface of the earth1 and eventually of the figure and dimensions of the earth4 ,this last1 indeed1 may be the sole object in view1 as was the case in the operations conducted in ,peru and in ,lapland by the celebrated ,french astronomers ,p4 ,bouguer1 ,c4,m4 de la ,condamine1 ,p4,l4,m4 de ,maupertuis1 ,a4,c4 ,clairault and others2 and the measurement of the meridian arc of ,france by ,p4,f4,a4 ,mechain and ,j4,b4,j4 ,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 7see ,,earth1 ,,figure ,,of ,,the74 ,the basis of every extensive survey is an accurate triangulation1 and the operations of geodesy consist in the measurement1 by theodolites1 of the angles of the triangles2 the measurement of one or more sides of these triangles on the ground2 the determination by astronomical observations of the azimuth of the whole network of triangles2 the determination of the actual position of the same on the surface of the earth by observations1 first for latitude at some of the stations1 and secondly for longitude2 the determination of altitude for all stations4 ,for the computation1 the points of the actual surface of the earth are imagined as projected along their plumb lines on the mathematical figure1 which is given by the stationary sea-level1 and the extension of the sea through the continents by a system of imaginary canals4 ,for many purposes the mathematical surface is assumed to be a plane2 in other cases a sphere of radius #fcga kilometres 7#bj1ijj1jjj ft474 ,in the case of extensive operations the surface must be considered as a compressed ellipsoid of rotation1 whose minor axis coincides with the earth's axis1 and whose compression1 flattening1 or ellipticity is about #a_/bih4 ,measurement of ,base ,lines4 ,to determine by actual measurement on the ground the length of a side of one of the triangles 7base line71 wherefrom to infer the lengths of all the other sides in the triangulation1 is not the least difficult operation of a trigonometrical survey4 ,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 asunder1 so that the error of the result may be pronounced to lie between certain very narrow limits1--then the question demands very serious consideration4 ,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 error1 owing to the difficulty of knowing at any moment the precise temperature of the bar2 and the transference of this unit1 or a multiple of it1 to a measuring bar will be affected not only with errors of observation1 but with errors arising from uncertainty of temperature of both bars4 ,if the measuring bar be not self-compensating for temperature1 its expansion must be determined by very careful experiments4 ,the thermometers required for this purpose must be very carefully studied1 and their errors of division and index error determined4 ,in order to avoid the difficulty in exactly determining the temperature of a bar by the mercury thermometer1 ,f4,w4 ,bessel introduced in #ahcd near ,konigsberg a compound bar which constituted a metallic thermometer4#a ,a zinc bar is laid on an iron bar two toises long1 both bars being perfectly planed and in free contact1 the zinc bar being slightly shorter and the two bars rigidly united at one end4 ,as the temperature varies1 the difference of the lengths of the bars1 as perceived by the other end1 also varies1 and affords a quantitative correction for temperature variations1 which is applied to reduce the length to standard temperature4 ,during the measurement of the base line the bars were not allowed to come into contact1 the interval being measured by the insertion of glass wedges4 ,the results of the comparisons of four measuring rods with one another and with the standards were elaborately computed by the method of least-squares4 ,the probable error of the measured length of #ice toises 7about #fjjj ft47 has been estimated as #a_/hfcejj or #a4b mu 7mu denoting a millionth74 ,with this apparatus fourteen base lines were measured in ,prussia and some neighbouring states2 in these cases a somewhat higher degree of accuracy was obtained4 ,the principal triangulation of ,great ,britain and ,ireland has seven base lines3 five have been measured by steel chains1 and two1 more exactly1 by the compensation bars of ,general ,t4,f4 ,colby1 an apparatus introduced in #ahbg-ahbh at ,lough ,foyle in ,ireland4 ,ten base lines were measured in ,india in #ahca-ahfi by the same apparatus4 ,this is a system of six compound-bars self-correcting for temperature4 ,the bars may be thus described3 ,two bars1 one of brass and the other of iron1 are laid in parallelism side by side1 firmly united at their centres1 from which they may freely expand or contract2 at the standard temperature they are of the same length4 ,let ,,ab be one bar1 ,a',b' the other2 draw lines through the corresponding extremities ,,aa' 7to ,p7 and ,,bb' 7to ,q71 and make ,a',p ,b',q1 ,,aa' being equal to ,,bb'4 ,if the ratio ,a',p_/,,ap equals the ratio of the coefficients of expansion of the bars ,a',b' and ,,ab1 then1 obviously1 the distance ,,pq is constant 7or nearly so74 ,in the actual instrument ,p and ,q are finely engraved dots #aj ft4 apart4 ,in practice the bars1 when aligned1 are not in contact1 an interval of #f in4 being allowed between each bar and its neighbour4 ,this distance is accurately measured by an ingenious micrometrical arrangement constructed on exactly the same principle as the bars themselves4 ,the last base line measured in ,india had a length of #hiac ft4 ,in consequence of some suspicion as to the accuracy of the compensation apparatus1 the measurement was repeated four times1 the operations being conducted so as to determine the actual values of the probable errors of the apparatus4 ,the direction of the line 7which is at ,cape ,comorin7 is north and south4 ,in two of the measurements the brass component was to the west1 in the others to the east2 the differences between the individual measurements and the mean of the four were #j4jjag1 -#j4jjdi1 -#j4jjae1 #j4jjde ft4 ,these differences are very small2 an elaborate investigation of all sources of error shows that the probable error of a base line in ,india is on the average -#b4h mu4 ,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 ,cape4 ,the account of this operation will be found in a volume entitled ,verification and ,extension of ,lacaille's ,arc of ,meridian at the ,cape of ,good ,hope1 by ,sir ,thomas ,maclear1 published in #ahff4 ,a rediscussion has been given by ,sir ,david ,gill in his ,report on the ,geodetic ,survey of ,south ,africa1 c41 #ahif4 ,a very simple base apparatus was employed by ,w4 ,struve in his triangulations in ,russia from #ahag to #ahee4 ,this consisted of four wrought-iron bars1 each two toises 7rather more than #ac ft47 long2 one end of each bar is terminated in a small steel cylinder presenting a slightly convex surface for contact1 the other end carries a contact lever rigidly connected with the bar4 ,the shorter arm of the lever terminates below in a polished hemisphere1 the upper and longer arm traversing a vertical divided arc4 ,in measuring1 the plane end of one bar is brought into contact with the short arm of the contact lever 7pushed forward by a weak spring7 of the next bar4 ,each bar has two thermometers1 and a level for determining the inclination of the bar in measuring4 ,the manner of transferring the end of a bar to the ground is simply this3 under the end of the bar a stake is driven very firmly into the ground1 carrying on its upper surface a disk1 capable of movement in the direction of the measured line by means of slow-motion screws4 ,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 #be ft4 from the stake in a direction perpendicular to the base4 ,struve investigated for each base the probable errors of the measurement arising from each of these seven causes3 ,alignment1 inclination1 comparisons with standards1 readings of index1 personal errors1 uncertainties of temperature1 and the probable errors of adopted rates of expansion4 ,he found that -#j4h mu was the mean of the probable errors of the seven bases measured by him4 ,the ,austro-,hungarian apparatus is similar2 the distance of the rods is measured by a slider1 which rests on one of the ends of each rod4 ,twenty-two base lines were measured in #ahdj-ahii4 ,general ,carlos ,ibanez employed in #aheh-ahgi1 for the measurement of nine base lines in ,spain1 two apparatus similar to the apparatus previously employed by ,porro in ,italy2 one is complicated1 the other simplified4 ,the first1 an apparatus of the brothers ,brunner of ,paris1 was a thermometric combination of two bars1 one of platinum and one of brass1 in length #d metres1 furnished with three levels and four thermometers4 ,suppose ,a1 ,b1 ,c three micrometer microscopes very firmly supported at intervals of #d metres with their axes vertical1 and aligned in the plane of the base line by means of a transit instrument1 their micrometer screws being in the line of measurement4 ,the measuring bar is brought under say ,a and ,b1 and those micrometers read2 the bar is then shifted and brought under ,b and ,c4 ,by repetition of this process1 the reading of a micrometer indicating the end of each position of the bar1 the measurement is made4 ,quite similar apparatus 7among others7 has been employed by the ,french and ,germans4 ,since1 however1 it only permitted a distance of about #cjj m4 to be measured daily1 ,ibanez introduced a simplification2 the measuring rod being made simply of steel1 and provided with inlaid mercury thermometers4 ,this apparatus was used in ,switzerland for the measurement of three base lines4 ,the accuracy is shown by the estimated probable errors3 -#j4b mu to -#j4h mu4 ,the distance measured daily amounts at least to #hjj m4 ,a greater daily distance can be measured with the same accuracy by means of ,bessel's apparatus2 this permits the ready measurement of #bjjj m4 daily4 ,for this1 however1 it is important to notice that a large staff and favourable ground are necessary4 ,an important improvement was introduced by ,edward ,jaderin of ,stockholm1 who measures with stretched wires of about #bd metres long2 these wires are about #a4fe mm4 in diameter1 and when in use are stretched by an accurate spring balance with a tension of #aj kg4#b ,the nature of the ground has a very trifling effect on this method4 ,the difficulty of temperature determinations is removed by employing wires made of invar1 an alloy of steel 7#fd7 and nickel 7#cf7 which has practically no linear expansion for small thermal changes at ordinary temperatures2 this alloy was discovered in #ahif by ,benoit and ,guillaume of the ,international ,bureau of ,weights and ,measures at ,breteuil4 ,apparently the future of base-line measurements rests with the invar wires of the ,jaderin apparatus2 next comes ,porro's apparatus with invar bars #d to #e metres long4 ,results have been obtained in the ,united ,states1 of great importance in view of their accuracy1 rapidity of determination and economy4 ,for the measurement of the arc of meridian in longitude #ih deg4 ,e41 in #aijj1 nine base lines of a total length of #fi4b km4 were measured in six months4 ,the total cost of one base was #abca4 ,at the beginning and at the end of the field-season a distance of exactly #ajj m4 was measured with ,r4,s4 ,woodward's #e-m4 ice-bar 7invented in #ahia72 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 measurements4 ,for the measurements there were employed two steel tapes of #ajj m4 long1 provided with supports at distances of #be m41 two of #ej m41 and the duplex apparatus of ,eimbeck1 consisting of four #e-m4 rods4 ,each base was divided into sections of about #ajjj m42 one of these1 the test kilometre1 was measured with all the five apparatus1 the others only with two apparatus1 mostly tapes4 ,the probable error was about -#j4h mu1 and the day's work a distance of about #bjjj m4 ,each of the four rods of the duplex apparatus consists of two bars of brass and steel4 ,mercury thermometers are inserted in both bars2 these serve for the measurement of the length of the base lines by each of the bars1 as they are brought into their consecutive positions1 the contact being made by an elastic-sliding contact4 ,the length of the base lines may be calculated for each bar only1 and also by the supposition that both bars have the same temperature4 ,the apparatus thus affords three sets of results1 which mutually control themselves1 and the contact adjustments permit rapid work4 ,the same device has been applied to the older bimetallic-compensating apparatus of ,bache-,wurdemann 7six bases1 #ahdg-aheg7 and of ,schott4 ,there was also employed a single rod bimetallic apparatus on ,f4 ,porro's principle1 constructed by the brothers ,repsold for some base lines4 ,excellent results have been more recently obtained with invar tapes4 ,the following results show the lengths of the same ,german base lines as measured by different apparatus3 metres4 ,base at ,berlin #ahfd ,apparatus of ,bessel #bccf4cibj #ahhj ,brunner 4#cibd ,base at ,strehlen #ahed ,bessel #bgfb4ehbd #ahgi ,brunner 4#eheb ,old base at ,bonn #ahdg ,bessel #bacc4ijie #ahib 4#ijig ,new base at ,bonn #ahib #beab4ifab #ahib ,brunner 4#ifif ,it is necessary that the altitude above the level of the sea of every part of a base line be ascertained by spirit levelling1 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 sea1 produced in imagination4 ,thus if l be the length of a measuring bar1 h its height at any given position in the measurement1 r the radius of the earth1 then the length radially projected on to the level of the sea is l7#a - h_/r74 ,in the ,salisbury ,plain base line the reduction to the level of the sea is -#j4fbid ft4 ,the total number of base lines measured in ,europe up to the present time is about one hundred and ten1 nineteen of which do not exceed in length #bejj metres1 or about #a-a_/b miles1 and three--one in ,france1 the others in ,bavaria--exceed #ai1jjj metres4 ,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 requires1 or whether as much precision is not obtainable in the end by careful triangulation from a short base4 ,but the answer cannot be given generally2 it must depend on the circumstances of each particular case4 ,with ,jaderin's apparatus1 provided with invar wires1 bases of #bj to #cj km4 long are obtained without difficulty4 ,illustration3 ,,fig4 #a4 ,in working away from a base line ab1 stations c1 d1 e1 f are carefully selected so as to obtain from well-shaped triangles gradually increasing sides4 ,before1 however1 finally leaving the base line1 it is usual to verify it by triangulation thus3 during the measurement two or more points1 as p1 q 7fig4 #a71 are marked in the base in positions such that the lengths of the different segments of the line are known2 then1 taking suitable external stations1 as h1 k1 the angles of the triangles bhp1 phq1 hqk1 kqa are measured4 ,from these angles can be computed the ratios of the segments1 which must agree1 if all operations are correctly performed1 with the ratios resulting from the measures4 ,leaving the base line1 the sides increase up to #aj1 #cj or #ej miles occasionally1 but seldom reaching #ajj miles4 ,the triangulation points may either be natural objects presenting themselves in suitable positions1 such as church towers2 or they may be objects specially constructed in stone or wood on mountain tops or other prominent ground4 ,in every case it is necessary that the precise centre of the station be marked by some permanent mark4 ,in ,india no expense is spared in making permanent the principal trigonometrical stations--costly towers in masonry being erected4 ,it is essential that every trigonometrical station shall present a fine object for observation from surrounding stations4 ,horizontal ,angles4 ,in placing the theodolite over a station to be observed from1 the first point to be attended to is that it shall rest upon a perfectly solid foundation4 ,the method of obtaining this desideratum must depend entirely on the nature of the ground2 the instrument must if possible be supported on rock1 or if that be impossible a solid foundation must be obtained by digging4 ,when the theodolite is required to be raised above the surface of the ground in order to command particular points1 it is necessary to build two scaffolds1--the outer one to carry the observatory1 the inner one to carry the instrument1--and these two edifices must have no point of contact4 ,many cases of high scaffolding have occurred on the ,english ,ordnance ,survey1 as for instance at ,thaxted church1 where the tower1 #hj ft4 high1 is surmounted by a spire of #ij ft4 ,the scaffold for the observatory was carried from the base to the top of the spire2 that for the instrument was raised from a point of the spire #adj ft4 above the ground1 having its bearing upon timbers passing through the spire at that height4 ,thus the instrument1 at a height of #agh ft4 above the ground1 was insulated1 and not affected by the action of the wind on the observatory4 ,at every station it is necessary to examine and correct the adjustments of the theodolite1 which are these3 the line of collimation of the telescope must be perpendicular to its axis of rotation2 this axis perpendicular to the vertical axis of the instrument2 and the latter perpendicular to the plane of the horizon4 ,the micrometer microscopes must also measure correct quantities on the divided circle or circles4 ,the method of observing is this4 ,let ,a1 ,b1 ,c 444 be the stations to be observed taken in order of azimuth2 the telescope is first directed to ,a and the cross-hairs of the telescope made to bisect the object presented by ,a1 then the microscopes or verniers of the horizontal circle 7also of the vertical circle if necessary7 are read and recorded4 ,the telescope is then turned to ,b1 which is observed in the same manner2 then ,c and the other stations4 ,coming round by continuous motion to ,a1 it is again observed1 and the agreement of this second reading with the first is some test of the stability of the instrument4 ,in taking this round of angles--or arc1 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 care4 ,before taking the next arc the horizontal circle is moved through #bj deg4 or #cj deg42 thus a different set of divisions of the circle is used in each arc1 which tends to eliminate the errors of division4 ,it is very desirable that all arcs at a station should contain one point in common1 to which all angular measurements are thus referred1--the observations on each arc commencing and ending with this point1 which is on the ,ordnance ,survey called the referring object4 ,it is usual for this purpose to select1 from among the points which have to be observed1 that one which affords the best object for precise observation4 ,for mountain tops a referring object is constructed of two rectangular plates of metal in the same vertical plane1 their edges parallel and placed at such a distance apart that the light of the sky seen through appears as a vertical line about #aj in width4 ,the best distance for this object is from #a to #b miles4 ,this method seems at first sight very advantageous2 but if1 however1 it be desired to attain the highest accuracy1 it is better1 as shown by ,general ,schreiber of ,berlin in #ahgh1 to measure only single angles1 and as many of these as possible between the directions to be determined4 ,division-errors are thus more perfectly eliminated1 and errors due to the variation in the stability1 c41 of the instruments are diminished4 ,this method is rapidly gaining precedence4 ,the theodolites used in geodesy vary in pattern and in size--the horizontal circles ranging from #aj in4 to #cf in4 in diameter4 ,in ,ramsden's #cf-;in4 theodolite the telescope has a focal length of #cf in4 and an aperture of #b4e in41 the ordinarily used magnifying power being #ed2 this last1 however1 can of course be changed at the requirements of the observer or of the weather4 ,the probable error of a single observation of a fine object with this theodolite is about #j4b4 ,fig4 #b represents an altazimuth theodolite of an improved pattern used on the ,ordnance ,survey4 ,the horizontal circle of #ad-;in4 diameter is read by three micrometer microscopes2 the vertical circle has a diameter of #ab in41 and is read by two microscopes4 ,in the great trigonometrical survey of ,india the theodolites used in the more important parts of the work have been of #b and #c ft4 diameter--the circle read by five equidistant microscopes4 ,every angle is measured twice in each position of the zero of the horizontal circle1 of which there are generally ten2 the entire number of measures of an angle is never less than #bj4 ,an examination of #adjg angles showed that the probable error of an observed angle is on the average - #j4bh4 ,for the observations of very distant stations it is usual to employ a heliotrope 7from the ,gr4 ,greek3 helios1 sun2 ,greek3 tropos1 a turn71 invented by ,gauss at ,gottingen in #ahba4 ,in its simplest form this is a plane mirror1 #d1 #f1 or #h in4 in diameter1 capable of rotation round a horizontal and a vertical axis4 ,this mirror is placed at the station to be observed1 and in fine weather it is kept so directed that the rays of the sun reflected by it strike the distant observing telescope4 ,to the observer the heliotrope presents the appearance of a star of the first or second magnitude1 and is generally a pleasant object for observing4 ,observations at night1 with the aid of light-signals1 have been repeatedly made1 and with good results1 particularly in ,france by ,general ,francois ,perrier1 and more recently in the ,united ,states by the ,coast and ,geodetic ,survey2 the signal employed being an acetylene bicycle-lamp1 with a lens #e in4 in diameter4 ,particularly noteworthy are the trigonometrical connexions of ,spain and ,algeria1 which were carried out in #ahgi by ,generals ,ibanez and ,perrier 7over a distance of #bgj km471 of ,sicily and ,malta in #aijj1 and of the islands of ,elba and ,sardinia in #aijb by ,dr ,guarducci 7over distances up to #bcj km472 in these cases artificial light was employed3 in the first case electric light and in the two others acetylene lamps4 ,illustration3 ,,fig4 #b4--;,altazimuth ,theodolite4 ,astronomical ,observations4 ,the direction of the meridian is determined either by a theodolite or a portable transit instrument4 ,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 azimuth1 or1 at any rate1 when very near that position4 ,if the observation be made t minutes of time before or after the time of greatest azimuth1 the azimuth then will differ from its maximum value by 7#dejt7#b sin #a sin #b;delta_/ sin z1 in seconds of angle1 omitting smaller terms1 delta being the star's declination and z its zenith distance4 ,the collimation and level errors are very carefully determined before and after these observations1 and it is usual to arrange the observations by the reversal of the telescope so that collimation error shall disappear4 ,if b1 c be the level and collimation errors1 the correction to the circle reading is b cot z - c cosec z1 b being positive when the west end of the axis is high4 ,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 azimuth1--an uncertainty which increases with the latitude and is very large in high latitudes4 ,this may be partly remedied by observing in connexion with the star its reflection in mercury4 ,in determining the value of one division of a level tube1 it is necessary to bear in mind that in some the value varies considerably with the temperature4 ,by experiments on the level of ,ramsden's #c-;foot theodolite1 it was found that though at the ordinary temperature of #ff deg4 the value of a division was about one second1 yet at #cb deg4 it was about five seconds4 ,in a very excellent portable transit used on the ,ordnance ,survey1 the uprights carrying the telescope are constructed of mahogany1 each upright being built of several pieces glued and screwed together2 the base1 which is a solid and heavy plate of iron1 carries a reversing apparatus for lifting the telescope out of its bearings1 reversing it and letting it down again4 ,thus is avoided the change of temperature which the telescope would incur by being lifted by the hands of the observer4 ,another form of transit is the ,german diagonal form1 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 telescope1 at the extremity of which arm is the eye-piece4 ,the unused half of the ordinary telescope being cut away is replaced by a counterpoise4 ,in this instrument there is the advantage that the observer without moving the position of his eye commands the whole meridian1 and that the level may remain on the pivots whatever be the elevation of the telescope4 ,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 directed2 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 telescope4 ,to determine the direction of the meridian1 it is well to erect two marks at nearly equal angular distances on either side of the north meridian line1 so that the pole star crosses the vertical of each mark a short time before and after attaining its greatest eastern and western azimuths4 ,if now the instrument1 perfectly levelled1 is adjusted to have its centre wire on one of the marks1 then when elevated to the star1 the star will traverse the wire1 and its exact position in the field at any moment can be measured by the micrometer wire4 ,alternate observations of the star and the terrestrial mark1 combined with careful level readings and reversals of the instrument1 will enable one1 even with only one mark1 to determine the direction of the meridian in the course of an hour with a probable error of less than a second4 ,the second mark enables one to complete the station more rapidly and gives a check upon the work4 ,as an instance1 at ,findlay ,seat1 in latitude #eg deg4 #ce'1 the resulting azimuths of the two marks were #agg deg4 #de' #cg4bi - #j4bj and #ahb deg4 #ag' #ae4fa - #j4ac1 while the angle between the two marks directly measured by a theodolite was found to be #d deg4 #ca' #cg4dc - #j4bc4 ,illustration3 ,,fig4 #c4 ,we now come to the consideration of the determination of time with the transit instrument4 ,let fig4 #c represent the sphere stereographically projected on the plane of the horizon1--ns being the meridian1 we the prime vertical1 ,z1 ,p the zenith and the pole4 ,let p be the point in which the production of the axis of the instrument meets the celestial sphere1 ,s the position of a star when observed on a wire whose distance from the collimation centre is c4 ,let a be the azimuthal deviation1 namely1 the angle w,zp1 b the level error so that ,zp #ij deg4 - b4 ,let also the hour angle corresponding to p be #ij deg4 - n1 and the declination of the same m1 the star's declination being delta1 and the latitude phi4 ,then to find the hour angle ,,zps tau of the star when observed1 in the triangles p,p,s1 p,p,z we have1 since p,p,s #ij tau - n1 -,sin c sin m sin delta cos m cos delta sin 7n - tau71 ,sin m sin b sin phi - cos b cos phi sin a1 ,cos m sin n sin b cos phi cos b sin phi sin a4 ,and these equations solve the problem1 however large be the errors of the instrument4 ,supposing1 as usual1 a1 b1 m1 n to be small1 we have at once tau n c sec delta m tan delta1 which is the correction to the observed time of transit4 ,or1 eliminating m and n by means of the second and third equations1 and putting z for the zenith distance of the star1 t for the observed time of transit1 the corrected time is t 7a sin z b cos z c7 _/ cos delta4 ,another very convenient form for stars near the zenith is tau b sec phi c sec delta m 7tan delta - tan phi74 ,suppose that in commencing to observe at a station the error of the chronometer is not known2 then having secured for the instrument a very solid foundation1 removed as far as possible level and collimation errors1 and placed it by estimation nearly in the meridian1 let two stars differing considerably in declination be observed--the instrument not being reversed between them4 ,from these two stars1 neither of which should be a close circumpolar star1 a good approximation to the chronometer error can be obtained2 thus let epsilon#a1 epsilon#b1 be the apparent clock errors given by these stars if delta#a1 delta#b be their declinations the real error is epsilon epsilon#a 7epsilon#a - epsilon#b7 7tan phi - tan delta#a7 _/ 7tan delta#a - tan delta#b74 ,of course this is still only approximate1 but it will enable the observer 7who by the help of a table of natural tangents can compute epsilon in a few minutes7 to find the meridian by placing at the proper time1 which he now knows approximately1 the centre wire of his instrument on the first star that passes--not near the zenith4 ,the transit instrument is always reversed at least once in the course of an evening's observing1 the level being frequently read and recorded4 ,it is necessary in most instruments to add a correction for the difference in size of the pivots4 ,the transit instrument is also used in the prime vertical for the determination of latitudes4 ,in the preceding figure let q be the point in which the northern extremity of the axis of the instrument produced meets the celestial sphere4 ,let n,zq be the azimuthal deviation a1 and b being the level error1 ,zq #ij deg4 - b2 let also n,pq tau and ,pq psi4 ,let ,s' be the position of a star when observed on a wire whose distance from the collimation centre is c1 positive when to the south1 and let h be the observed hour angle of the star1 viz4 ,,zps'4 ,then the triangles q,p,s'1 g,p,z give -,sin c sin delta cos psi - cos delta sin psi cos 7h tau71 ,cos psi sin b sin phi cos b cos phi cos a1 ,sin psi sin tau cos b sin a4 ,now when a and b are very small1 we see from the last two equations that psi phi - b1 a tau sin psi1 and if we calculate phi' by the formula cot phi' cot delta cos h1 the first equation leads us to this result-- phi phi' 7a sin z b cos z c7_/cos z1 the correction for instrumental error being very similar to that applied to the observed time of transit in the case of meridian observations4 ,when a is not very small and z is small1 the formulae required are more complicated4 ,illustration3 ,,fig4 #d4--,zenith ,telescope constructed for the ,international ,stations at ,mizusawa1 ,carloforte1 ,gaithersburg and ,ukiah1 by ,hermann ,wanschaff1 ,berlin4 ,the method of determining latitude by transits in the prime vertical has the disadvantage of being a somewhat slow process1 and of requiring a very precise knowledge of the time1 a disadvantage from which the zenith telescope is free4 ,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 equal1 the latitude is the mean of their declinations2 or1 if the zenith distance of a star culminating to the south of the zenith be ,z1 its declination being delta1 and that of another culminating to the north with zenith distance ,z' and declination delta'1 then clearly the latitude is #a_/b7;delta delta'7 #a_/b7,z - ,z'74 ,now the zenith telescope does away with the divided circle1 and substitutes the measurement micrometrically of the quantity ,z' - ,z4 ,in fig4 #d is shown a zenith telescope by ,h4 ,wanschaff of ,berlin1 which is the type used 7according to the ,central ,bureau at ,potsdam7 since about #ahij for the determination of the variations of latitude due to different1 but as yet imperfectly understood1 influences4 ,the instrument is supported on a strong tripod1 fitted with levelling screws2 to this tripod is fixed the azimuth circle and a long vertical steel axis4 ,fitting on this axis is a hollow axis which carries on its upper end a short transverse horizontal axis with a level4 ,this latter carries the telescope1 which1 supported at the centre of its length1 is free to rotate in a vertical plane4 ,the telescope is thus mounted eccentrically with respect to the vertical axis around which it revolves4 ,two extremely sensitive levels are attached to the telescope1 which latter carries a micrometer in its eye-piece1 with a screw of long range for measuring differences of zenith distance4 ,two levels are employed for controlling and increasing the accuracy4 ,for this instrument stars are selected in pairs1 passing north and south of the zenith1 culminating within a few minutes of time and within about twenty minutes 7angular7 of zenith distance of each other4 ,when a pair of stars is to be observed1 the telescope is set to the mean of the zenith distances and in the plane of the meridian4 ,the first star on passing the central meridional wire is bisected by the micrometer2 then the telescope is rotated very carefully through #ahj deg4 round the vertical axis1 and the second star on passing through the field is bisected by the micrometer on the centre wire4 ,the micrometer has thus measured the difference of the zenith distances1 and the calculation to get the latitude is most simple4 ,of course it is necessary to read the level1 and the observations are not necessarily confined to the centre wire4 ,in fact if n1 s be the north and south readings of the level for the south star1 n'1 s' the same for the north star1 l the value of one division of the level1 m the value of one division of the micrometer1 r1 r' the refraction corrections1 mu1 mu' the micrometer readings of the south and north star1 the micrometer being supposed to read from the zenith1 then1 supposing the observation made on the centre wire1-- phi #a_/b7;delta delta'7 #a_/b7mu - mu'7m #a_/d7n n' - s - s'7l #a_/b7r - r'74 ,it is of course of the highest importance that the value m of the screw be well determined4 ,this is done most effectually by observing the vertical movement of a close circumpolar star when at its greatest azimuth4 ,in a single night with this instrument a very accurate result1 say with a probable error of about #j4b1 could be obtained for latitude from1 say1 twenty pair of stars2 but when the latitude is required to be obtained with the highest possible precision1 two nights at least are necessary4 ,the weak point of the zenith telescope lies in the circumstance that its requirements prevent the selection of stars whose positions are well fixed2 very frequently it is necessary to have the declinations of the stars selected for this instrument specially observed at fixed observatories4 ,the zenith telescope is made in various sizes from #cj to #ed in4 in focal length2 a #cj-;in4 telescope is sufficient for the highest purposes and is very portable4 ,the net observation probable-error for one pair of stars is only -#j4a4 ,the zenith telescope is a particularly pleasant instrument to work with1 and an observer has been known 7a sergeant of ,royal ,engineers1 on one occasion7 to take every star in his list during eleven hours on a stretch1 namely1 from #f o'clock ,p4,m4 until #e ,a4,m41 and this on a very cold ,november night on one of the highest points of the ,grampians4 ,observers accustomed to geodetic operations attain considerable powers of endurance4 ,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 roofless4 ,three observatory roofs were subsequently demolished1 and for some time the observatory was used without a roof1 being filled with snow every night and emptied every morning4 ,quite different1 however1 was the experience of the same party when on the top of ,ben ,nevis1 #ddjf ft4 high4 ,for about a fortnight the state of the atmosphere was unusually calm1 so much so1 that a lighted candle could often be carried between the tents of the men and the observatory1 whilst at the foot of the hill the weather was wild and stormy4 ,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 stations1 and the comparison by signals of the clocks at ,a and ,b4 ,whenever telegraphic lines are available these comparisons are made by telegraphy4 ,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 second4 ,in order to record the minutes as well as seconds1 one second in each minute1 namely that numbered #j or #fj1 is omitted4 ,the seconds are recorded on a chronograph1 which consists of a cylinder revolving uniformly at the rate of one revolution per minute covered with white paper1 on which a pen having a slow movement in the direction of the axis of the cylinder describes a continuous spiral4 ,this pen is deflected through the agency of an electromagnet every second1 and thus the seconds of the clock are recorded on the chronograph by offsets from the spiral curve4 ,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 stars4 ,the method of determination of difference of longitude is1 therefore1 virtually as follows4 ,after the necessary observations for instrumental corrections1 which are recorded only at the station of observation1 the clock at ,a is put in connexion with the circuit so as to write on both chronographs1 namely1 that at ,a and that at ,b4 ,then the clock at ,b is made to write on both chronographs4 ,it is clear that by this double operation one can eliminate the effect of the small interval of time consumed in the transmission of signals1 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 defect4 ,the determination of the personal errors of the observers in this delicate operation is a matter of the greatest importance1 as therein lies probably the chief source of residual error4 ,these errors can nevertheless be almost entirely avoided by using the impersonal micrometer of ,dr ,repsold 7,hamburg1 #ahhi74 ,in this device there is a movable micrometer wire which is brought by hand into coincidence with the star and moved along with it2 at fixed points there are electrical contacts1 which replace the fixed wires4 ,experiments at the ,geodetic ,institute and ,central ,bureau at ,potsdam in #ahia gave the following personal equations in the case of four observers3-- ,older ,procedure4 ,new ,procedure4 ,a-,b -#js4#ajh -#js4#jjd ,a-,g -#js4#cad -#js4#jce ,a-,s -#js4#ahd -#js4#jbg ,b-,g -#js4#bbe #js4#jac ,b-,s -#js4#jhf -#js4#jbc ,g-,s #js4#aji -#js4#jjf ,these results show that in the later method the personal equation is small and not so variable2 and consequently the repetition of longitude determinations with exchanged observers and apparatus entirely eliminates the constant errors1 the probable error of such determinations on ten nights being scarcely -#js4#ja4 ,calculation of ,triangulation4 ,the surface of ,great ,britain and ,ireland is uniformly covered by triangulation1 of which the sides are of various lengths from #aj to #aaa miles4 ,the largest triangle has one angle at ,snowdon in ,wales1 another on ,slieve ,donard in ,ireland1 and a third at ,scaw ,fell in ,cumberland2 each side is over a hundred miles and the spherical excess is #fd4 ,the more ordinary method of triangulation is1 however1 that of chains of triangles1 in the direction of the meridian and perpendicular thereto4 ,the principal triangulations of ,france1 ,spain1 ,austria and ,india are so arranged4 ,oblique chains of triangles are formed in ,italy1 ,sweden and ,norway1 also in ,germany and ,russia1 and in the ,united ,states4 ,chains are composed sometimes merely of consecutive plain triangles2 sometimes1 and more frequently in ,india1 of combinations of triangles forming consecutive polygonal figures4 ,in this method of triangulating1 the sides of the triangles are generally from #bj to #cj miles in length--seldom exceeding #dj4 ,the inevitable errors of observation1 which are inseparable from all angular as well as other measurements1 introduce a great difficulty into the calculation of the sides of a triangulation4 ,starting from a given base in order to get a required distance1 it may generally be obtained in several different ways--that is1 by using different sets of triangles4 ,the results will certainly differ one from another1 and probably no two will agree4 ,the experience of the computer will then come to his aid1 and enable him to say which is the most trustworthy result2 but no experience or ability will carry him through a large network of triangles with anything like assurance4 ,the only way to obtain trustworthy results is to employ the method of least squares4 ,we cannot here give any illustration of this method as applied to general triangulation1 for it is most laborious1 even for the simplest cases4 ,three stations1 projected on the surface of the sea1 give a spherical or spheroidal triangle according to the adoption of the sphere or the ellipsoid as the form of the surface4 ,a spheroidal triangle differs from a spherical triangle1 not only in that the curvatures of the sides are different one from another1 but more especially in this that1 while in the spherical triangle the normals to the surface at the angular points meet at the centre of the sphere1 in the spheroidal triangle the normals at the angles ,a1 ,b1 ,c meet the axis of revolution of the spheroid in three different points1 which we may designate alpha1 beta1 gamma respectively4 ,now the angle ,a of the triangle as measured by a theodolite is the inclination of the planes ,,baalpha and ,,caalpha1 and the angle at ,b is that contained by the planes ,,abbeta and ,,cbbeta4 ,but the planes ,,abalpha and ,,abbeta containing the line ,,ab in common cut the surface in two distinct plane curves4 ,in order1 therefore1 that a spheroidal triangle may be exactly defined1 it is necessary that the nature of the lines joining the three vertices be stated4 ,in a mathematical point of view the most natural definition is that the sides be geodetic or shortest lines4 ,c4,c4,g4 ,andrae1 of ,copenhagen1 has also shown that other lines give a less convenient computation4 ,k4,f4 ,gauss1 in his treatise1 ,disquisitiones generales circa superficies curvas1 entered fully into the subject of geodetic 7or geodesic7 triangles1 and investigated expressions for the angles of a geodetic triangle whose sides are given1 not certainly finite expressions1 but approximations inclusive of small quantities of the fourth order1 the side of the triangle or its ratio to the radius of the nearly spherical surface being a small quantity of the first order4 ,the terms of the fourth order1 as given by ,gauss for any surface in general1 are very complicated even when the surface is a spheroid4 ,if we retain small quantities of the second order only1 and put ,a1 ,b1 ,c for the angles of the geodetic triangle1 while ,a1 ,b1 ,c are those of a plane triangle having sides equal respectively to those of the geodetic triangle1 then1 sigma being the area of the plane triangle and a1 b1 c the measures of curvature at the angular points1 ,a ,a sigma7#b;a b c7_/#ab1 ,b ,b sigma7a #b;b c7_/#ab1 ,c ,c sigma7a b #b;c7_/#ab4 ,for the sphere a b r1 and making this simplification1 we obtain the theorem previously given by ,a4,m4 ,legendre4 ,with the terms of the fourth order1 we have 7after ,andrae73 epsilon sigma _/m#b - a#b a - k ,a - ,a --------- -------k 7 ---------k ------- 71 #c #c #bj #dk _/ epsilon sigma _/m#b - b#b b - k ,b - ,b --------- -------k 7 ---------k -------- 71 #c #c #bj #dk _/ epsilon sigma _/m#b - c#b c - k ,c - ,c --------- -------k 7 ---------k -------- 71 #c #c #bj #dk _/ in which epsilon sigma k #a 7m#bk _/ #h71 #cm#b a#b b#b c#b1 #ck a b c4 ,for the ellipsoid of rotation the measure of curvature is equal to #a _/ rhon1 rho and n being the radii of curvature of the meridian and perpendicular4 ,it is rarely that the terms of the fourth order are required4 ,as a rule spheroidal triangles are calculated as spherical 7after ,legendre71 i4e4 like plane triangles with a decrease of each angle of about epsilon _/ #c2 epsilon must1 however1 be calculated for each triangle separately with its mean measure of curvature k4 ,the geodetic line being the shortest that can be drawn on any surface between two given points1 we may be conducted to its most important characteristics by the following considerations3 let p1 q be adjacent points on a curved surface2 through s the middle point of the chord pq imagine a plane drawn perpendicular to pq1 and let ,s be any point in the intersection of this plane with the surface2 then p,s ,sq is evidently least when s,s is a minimum1 which is when s,s is a normal to the surface2 hence it follows that of all plane curves on the surface joining p1 q1 when those points are indefinitely near to one another1 that is the shortest which is made by the normal plane4 ,that is to say1 the osculating plane at any point of a geodetic line contains the normal to the surface at that point4 ,imagine now three points in space1 ,a1 ,b1 ,c1 such that ,,ab ,,bc c2 let the direction cosines of ,,ab be l1 m1 n1 those of ,,bc l'1 m'1 n'1 then x1 y1 z being the co-ordinates of ,b1 those of ,a and ,c will be respectively-- x - cl 3 y - cm 3 z - cn x cl'3 y cm'3 z cn'4 ,hence the co-ordinates of the middle point ,m of ,,ac are x #a_/b;c7l' - l71 y #a_/b;c7m' - m71 z #a_/b;c7n' - n71 and the direction cosines of ,,bm are therefore proportional to l' - l 3 m' - m 3 n' - n4 ,if the angle made by ,,bc with ,,ab be indefinitely small1 the direction cosines of ,,bm are as deltal 3 deltam 3 deltan4 ,now if ,,ab1 ,,bc be two contiguous elements of a geodetic1 then ,,bm must be a normal to the surface1 and since deltal1 deltam1 deltan are in this case represented by delta7dx_/ds71 delta7dy_/ds71 delta7dz_/ds71 and if the equation of the surface be u #j1 we have d#bx _/ du d#by _/ du d#bz _/ du ---- _/ -- ---- _/ -- ---- _/ --1 ds#b _/ dx ds#b _/ dy ds#b _/ dz which1 however1 are equivalent to only one equation4 ,in the case of the spheroid this equation becomes d#bx d#by y ---- - x ---- #j1 ds#b ds#b which integrated gives ydx - xdy ,cds4 ,this again may be put in the form r sin a ,c1 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 revolution4 ,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 ,aalpha,b4 ,generally speaking1 the geodetic lies between the two plane section curves joining ,a and ,b which are formed by the two vertical planes1 supposing these points not far apart4 ,if1 however1 ,a and ,b are nearly in the same latitude1 the geodetic may cross 7between ,a and ,b7 that plane curve which lies nearest the adjacent pole of the spheroid4 ,the condition of crossing is this4 ,suppose that for a moment we drop the consideration of the earth's non-sphericity1 and draw a perpendicular from the pole ,c on ,,ab1 meeting it in ,s between ,a and ,b4 ,then ,a being that point which is nearest the pole1 the geodetic will cross the plane curve if ,,as be between #a_/d;,,ab and #c_/h ,,ab4 ,if ,,as lie between this last value and #a_/b;,,ab1 the geodetic will lie wholly to the north of both plane curves1 that is1 supposing both points to be in the northern hemisphere4 ,the difference of the azimuths of the vertical section ,,ab and of the geodetic ,,ab1 i4e4 the astronomical and geodetic azimuths1 is very small for all observable distances1 being approximately3-- ,geod4 azimuth ,astr4 azimuth -7#a_/ab7 e#b_/7a - e#b7 7s#b_/rhon7 7cos#bphi sin #b;alpha 7s_/#d;a7sin #bphi sin alpha71 in which3 e and a are the numerical eccentricity and semi-major axis respectively of the meridian ellipse1 phi and alpha are the latitude and azimuth at ,a1 s ,,ab1 and rho and n are the radii of curvature of the meridian and perpendicular at ,a4 ,for s #ajj kilometres1 only the first term is of moment2 its value is #j4jbh cos#b phi sin #b;alpha1 and it lies well within the errors of observation4 ,if we imagine the geodetic ,,ab1 it will generally trisect the angles between the vertical sections at ,a and ,b1 so that the geodetic at ,a is near the vertical section ,,ab1 and at ,b near the section ,,ba4#c ,the greatest distance of the vertical sections one from another is e#bs#c cos#b phi#j sin #b;alpha#j_/af;a#b1 in which phi#j and alpha#j are the mean latitude and azimuth respectively of the middle point of ,,ab4 ,for the value s #fd kilometres1 the maximum distance is #c mm4 ,an idea of the course of a longer geodetic line may be gathered from the following example4 ,let the line be that joining ,cadiz and ,st ,petersburg1 whose approximate positions are-- ,cadiz4 ,st ,petersburg4 ,lat4 #cf deg4 #bb' ,n4 #ei deg4 #ef' ,n4 ,long4 #f deg4 #ah' ,w4 #cj deg4 #ag' ,e4 ,if ,g be the point on the geodetic corresponding to ,f on that one of the plane curves which contains the normal at ,cadiz 7by corresponding we mean that ,f and ,g are on a meridian7 then ,g is to the north of ,f2 at a quarter of the whole distance from ,cadiz ,,gf is #deh ft41 at half the distance it is #fcg ft41 and at three-quarters it is #dgc ft4 ,the azimuth of the geodetic at ,cadiz differs #bj from that of the vertical plane1 which is the astronomical azimuth4 ,the azimuth of a geodetic line cannot be observed1 so that the line does not enter of necessity into practical geodesy1 although many formulae connected with its use are of great simplicity and elegance4 ,the geodetic line has always held a more important place in the science of geodesy among the mathematicians of ,france1 ,germany and ,russia than has been assigned to it in the operations of the ,english and ,indian triangulations4 ,although the observed angles of a triangulation are not geodetic angles1 yet in the calculation of the distance and reciprocal bearings of two points which are far apart1 and are connected by a long chain of triangles1 we may fall upon the geodetic line in this manner3-- ,if ,a1 ,z be the points1 then to start the calculation from ,a1 we obtain by some preliminary calculation the approximate azimuth of ,z1 or the angle made by the direction of ,z with the side ,,ab or ,,ac of the first triangle4 ,let ,p#a be the point where this line intersects ,,bc2 then1 to find ,p#b1 where the line cuts the next triangle side ,,cd1 we make the angle ,,bp#a,p#b such that ,,bp#a,p#b ,,bp#a;,a #ahj deg4 ,this fixes ,p#b1 and ,p#c is fixed by a repetition of the same process2 so for ,p#d1 ,p#e 4444 ,now it is clear that the points ,p#a1 ,p#b1 ,p#c so computed are those which would be actually fixed by an observer with a theodolite1 proceeding in the following manner4 ,having set the instrument up at ,a1 and turned the telescope in the direction of the computed bearing1 an assistant places a mark ,p#a on the line ,,bc1 adjusting it till bisected by the cross-hairs of the telescope at ,a4 ,the theodolite is then placed over ,p#a1 and the telescope turned to ,a2 the horizontal circle is then moved through #ahj deg4 ,the assistant then places a mark ,p#b on the line ,,cd1 so as to be bisected by the telescope1 which is then moved to ,p#b1 and in the same manner ,p#c is fixed4 ,now it is clear that the series of points ,p#a1 ,p#b1 ,p#c approaches to the geodetic line1 for the plane of any two consecutive elements ,p7n-#a7 ,pn1 ,pn ,p7n#a7 contains the normal at ,pn4 ,if the objection be raised that not the geodetic azimuths but the astronomical azimuths are observed1 it is necessary to consider that the observed vertical sections do not correspond to points on the sea-level but to elevated points4 ,since the normals of the ellipsoid of rotation do not in general intersect1 there consequently arises an influence of the height on the azimuth4 ,in the case of the measurement of the azimuth from ,a to ,b1 the instrument is set to a point ,a' over the surface of the ellipsoid 7the sea-level71 and it is then adjusted to a point ,b'1 also over the surface1 say at a height h'4 ,the vertical plane containing ,a' and ,b' also contains ,a but not ,b3 it must therefore be rotated through a small azimuth in order to contain ,b4 ,the correction amounts approximately to -e#b;h' cos#bphi sin #b;alpha_/#b;a2 in the case of h' #ajjj m41 its value is #j4ajh cos#bphi sin #b;alpha4 ,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 azimuths4 ,the observed azimuths and horizontal angles must therefore also be corrected in the case1 where it is required to dispense with geodetic lines4 ,when the angles of a triangulation have been adjusted by the method of least squares1 and the sides are calculated1 the next process is to calculate the latitudes and longitudes of all the stations starting from one given point4 ,the calculated latitudes1 longitudes and azimuths1 which are designated geodetic latitudes1 longitudes and azimuths1 are not to be confounded with the observed latitudes1 longitudes and azimuths1 for these last are subject to somewhat large errors4 ,supposing the latitudes of a number of stations in the triangulation to be observed1 practically the mean of these determines the position in latitude of the network1 taken as a whole4 ,so the orientation or general azimuth of the whole is inferred from all the azimuth observations4 ,the triangulation is then supposed to be projected on a spheroid of given elements1 representing as nearly as one knows the real figure of the earth4 ,then1 taking the latitude of one point and the direction of the meridian there as given--obtained1 namely1 from the astronomical observations there--one can compute the latitudes of all the other points with any degree of precision that may be considered desirable4 ,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 seconds1 otherwise there will arise an accumulation of errors from imperfect calculation which should always be avoided4 ,for very long distances1 eight places of decimals should be employed in logarithmic calculations2 if seven places only are available very great care will be required to keep the last place true4 ,now let phi1 phi' be the latitudes of two stations ,a and ,b2 alpha1 alpha99 their mutual azimuths counted from north by east continuously from #j deg4 to #cfj deg42 omega their difference of longitude measured from west to east2 and s the distance ,,ab4 ,first compute a latitude phi#a by means of the formula phi#a phi 7s cos alpha7 _/ rho1 where rho is the radius of curvature of the meridian at the latitude phi2 this will require but four places of logarithms4 ,then1 in the first two of the following1 five places are sufficient-- s#b s#b epsilon ------- sin alpha cos a1 eta ------- sin#b;alpha tanphi#a1 #brhon #brhon s phi' - phi ---- cos 7alpha - #b_/c;epsilon7 - eta1 rho#j s sin 7alpha - #a_/c;epsilon7 omega ----------------------------1 n cos 7phi' #a_/c;eta7 alpha99 - alpha omega sin 7phi' #b_/c;eta7 - epsilon #ahj deg4 ,here n is the normal or radius of curvature perpendicular to the meridian2 both n and rho correspond to latitude phi#a1 and rho#j to latitude #a_/b7phi phi'74 ,for calculations of latitude and longitude1 tables of the logarithmic values of rho sin #a1 n sin #a1 and #b n rho sin #a are necessary4 ,the following table contains these logarithms for every ten minutes of latitude from #eb deg4 to #ec deg4 computed with the elements a #bjibfjfj and a 3 b #bie 3 #bid 3-- ---------------------------------------- ------------------ #a #a #a ,lat4 ,log4------------4 ,log4--------4 ,log4--------------4 rho sin #a n sin #a #brhon sin #a ---------------------------------------- ------------------ deg4 ' #eb #j #g4iicidcd #g4iibhbca #j4cgaca #aj #icji #haij #bi #bj #iahe #hadh #bh #cj #ijfj #hajg #bf #dj #hicf #hjfe #bd #ej #hhab #hjbd #bc #ec #j #hfhh #gihb #bb ---------------------------------------- ------------------ ,the logarithm in the last column is that required also for the calculation of spherical excesses1 the spherical excess of a triangle being expressed by a b sin 7,c_/#brhon7 sin #a4 ,it is frequently necessary to obtain the co-ordinates of one point with reference to another point2 that is1 let a perpendicular arc be drawn from ,b to the meridian of ,a meeting it in ,p1 then1 alpha being the azimuth of ,b at ,a1 the co-ordinates of ,b with reference to ,a are ,,ap s cos 7alpha - #b_/c;epsilon71 ,,bp s sin 7alpha - #a_/c;epsilon71 where epsilon is the spherical excess of ,,apb1 viz4 s#b sin alpha cos alpha multiplied by the quantity whose logarithm is in the fourth column of the above table4 ,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 formulae1 we make use of the following formula3 given the latitude phi of ,a1 and the azimuth alpha and the distance s of ,b1 to determine the latitude phi' and longitude omega of ,b1 and the back azimuth alpha'4 ,here it is understood that alpha' is symmetrical to alpha1 so that alpha99 alpha' #cfj deg4 ,let theta s ,delta _/ a1 where ,delta 7#a - e#b sin#b phi7#a_/b and e#b theta#b xi ------------- cos#b phi sin #b;alpha1 7#d 7#a - e#b7 e#b theta#c xi' ------------- cos#b phi cos#b alpha2 7#f 7#a - e#b7 xi1 xi' are always very minute quantities even for the longest distances2 then1 putting kappa #ij deg4 - phi1 alpha' xi - omega sin #a_/b7kappa - theta - xi'7 alpha tan------------------------- ---------------------------------- cot ------- #b sin #a_/b7kappa theta xi'7 #b alpha' xi omega cos #a_/b7kappa - theta - xi'7 alpha tan------------------------- ---------------------------------- cot ------- #b cos #a_/b7kappa theta xi'7 #b s sin #a_/b7;alpha' xi - alpha7 _/ theta#b alpha' - alpha phi' - phi ---------------------------------------- - 7 #a ---------cos#b ------------------ 72 rho#j sin #a_/b7;alpha' xi alpha7 #ab #b _/ here rho#j is the radius of curvature of the meridian for the mean latitude #a_/b7phi phi'74 ,these formulae are approximate only1 but they are sufficiently precise even for very long distances4 ,for lines of any length the formulae of ,f4,w4 ,bessel 7,astr4 ,nach41 #ahbc1 iv4 #bda7 are suitable4 ,if the two points ,a and ,b be defined by their geographical co-ordinates1 we can accurately calculate the corresponding astronomical azimuths1 i4e4 those of the vertical section1 and then proceed1 in the case of not too great distances1 to determine the length and the azimuth of the shortest lines4 ,for any distances recourse must again be made to ,bessel's formula4#d ,let alpha1 alpha' be the mutual azimuths of two points ,a1 ,b on a spheroid1 k the chord line joining them1 mu1 mu' the angles made by the chord with the normals at ,a and ,b1 phi1 phi'1 omega their latitudes and difference of longitude1 and 7x#b y#b7_/;a#b z#b b#b #a the equation of the surface2 then if the plane xz passes through ,a the co-ordinates of ,a and ,b will be x 7a_/,delta7 cos phi1 x' 7a_/,delta'7 cos phi' cos omega1 y #j y' 7a_/,delta'7 cos phi' sin omega1 z 7a_/,delta7 7#a - e#b7 sin phi1 z' 7a_/,delta'7 7#a - e#b7 sin phi'1 where ,delta 7#a - e#b sin#b phi7#a_/b1 ,delta' 7#a - e#b sin#b phi'7#a_/b1 and e is the eccentricity4 ,let f1 g1 h be the direction cosines of the normal to that plane which contains the normal at ,a and the point ,b1 and whose inclinations to the meridian plane of ,a is alpha2 let also l1 m1 n and l'1 m'1 n' be the direction cosines of the normal at ,a1 and of the tangent to the surface at ,a which lies in the plane passing through ,b1 then since the first line is perpendicular to each of the other two and to the chord k1 whose direction cosines are proportional to x' - x1 y' - y1 z' - z1 we have these three equations f7x' - x7 gy' h7z' - z7 #j fl gm hn #j fl' gm' hn' #j4 ,eliminate f1 g1 h from these equations1 and substitute l cos phi l' - sin phi cos alpha m #j m' sin alpha n sin phi n' cos phi cos alpha1 and we get 7x' - x7 sin phi y' cot alpha - 7z' - z7 cos phi #j4 ,the substitution of the values of x1 z1 x'1 y'1 z' in this equation will give immediately the value of cot alpha2 and if we put zeta1 zeta' for the corresponding azimuths on a sphere1 or on the supposition e #j1 the following relations exist cos phi ,q cot alpha - cot zeta e#b ------------------ cos phi' ,delta cos phi' ,q cot alpha' - cot zeta' e#b ------------------ cos phi ,delta' ,delta' sin phi - ,delta sin phi' ,q sin omega4 ,if from ,b we let fall a perpendicular on the meridian plane of ,a1 and from ,a let fall a perpendicular on the meridian plane of ,b1 then the following equations become geometrically evident3 k sin mu sin alpha 7a_/,delta'7 cos phi' sin omega k sin mu' sin alpha' 7a_/,delta7 cos phi sin omega4 ,now in any surface u #j we have k#b 7x' - x7#b 7y' - y7#b 7z' - z7#b du du du _/ _/ du#b du#b du#b #a_/b -cos mu 7x' - x7 -- 7y' - y7 -- 7z' - z7 -- _/ k 7 ---- ---- ---- 7 dx dy dz_/ dx#b dy#b dz#b _/ du du du _/ _/ du#b du#b du#b #a_/b -cos mu' 7x' - x7 --- 7y' - y7 --- 7z' - z7 --- _/ k 7 ----- ----- ----- 74 dx' dy' dz'_/ dx'#b dy'#b dz'#b _/ ,in the present case1 if we put xx' zz' #a - --- - --- ,u1 a#b b#b then k#b _/z' - z #b --- #b,u - e#b 7 ------ 7 a#b b _/ cos mu 7a_/k7 ,delta,u2 cos mu' 7a_/k7 ,delta',u4 ,let u be such an angle that 7#a - e#b7a_/b sin phi ,delta sin u cos phi ,delta cos u1 then on expressing x1 x'1 z1 z' in terms of u and u'1 ,u #a - cos u cos u' cos omega - sin u sin u'2 also1 if v be the third side of a spherical triangle1 of which two sides are #a_/bpi - u and #a_/bpi - u' and the included angle omega1 using a subsidiary angle psi such that sin psi sin #a_/bv e sin #a_/b7u' - u7 cos #a_/b7u' u71 we obtain finally the following equations3-- k #b;a cos psi sin #a_/bv cos mu ,delta sec psi sin #a_/bv cos mu' ,delta' sec psi sin #a_/bv sin mu sin alpha 7a_/k7 cos u' sin omega sin mu' sin alpha' 7a_/k7 cos u sin omega4 ,these determine rigorously the distance1 and the mutual zenith distances and azimuths1 of any two points on a spheroid whose latitudes and difference of longitude are given4 ,by a series of reductions from the equations containing zeta1 zeta' it may be shown that alpha alpha' zeta zeta' #a_/d;e#domega7phi' - phi7#b cos#d phi#j sin phi#j 4441 where phi#j is the mean of phi and phi'1 and the higher powers of e are neglected4 ,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 #j4a;a1 which is1 practically speaking1 zero2 consequently the sum of the azimuths alpha alpha' on the spheroid is equal to the sum of the spherical azimuths1 whence follows this very important theorem 7known as ,dalby's theorem74 ,if phi1 phi' be the latitudes of two points on the surface of a spheroid1 omega their difference of longitude1 alpha1 alpha' their reciprocal azimuths1 tan #a_/bomega cot #a_/b7;alpha alpha'7 cos #a_/b7phi' - phi7_/ sin #a_/b7phi' phi74 ,the computation of the geodetic from the astronomical azimuths has been given above4 ,from k we can now compute the length s of the vertical section1 and from this the shortest length4 ,the difference of length of the geodetic line and either of the plane curves is e#d s#e cos#d phi#j sin#b #b;alpha#j_/cfj a#d4 ,at least this is an approximate expression4 ,supposing s #j4a;a1 this quantity would be less than one-hundredth of a millimetre4 ,the line s is now to be calculated as a circular arc with a mean radius r along ,,ab4 ,if phi#j #a_/b7phi phi'71 alpha#j #a_/b7ahj deg4 alpha - alpha'71 ,delta#j 7#a - e#b sin#b phi#j7a_/b1 then #a_/r ,delta#j_/;a #a e#b_/7a - e#b7 7cos#b phi#j cos#b alpha#j71 and approximately sin 7s_/#br7 k_/#br4 ,these formulae give1 in the case of k #j4a;a1 values certain to eight logarithmic decimal places4 ,an excellent series of formulae for the solution of the problem1 to determine the azimuths1 chord and distance along the surface from the geographical co-ordinates1 was given in #ahhb by ,ch4 ,m4 ,schols 7,archives ,neerlandaises1 vol4 xvii474 ,irregularities of the ,earth's ,surface4 ,in considering the effect of unequal distribution of matter in the earth's crust on the form of the surface1 we may simplify the matter by disregarding the considerations of rotation and eccentricity4 ,in the first place1 supposing the earth a sphere covered with a film of water1 let the density rho be a function of the distance from the centre so that surfaces of equal density are concentric spheres4 ,let now a disturbance of the arrangement of matter take place1 so that the density is no longer to be expressed by rho1 a function of r only1 but is expressed by rho rho'1 where rho' is a function of three co-ordinates theta1 phi1 r4 ,then rho' is the density of what may be designated disturbing matter2 it is positive in some places and negative in others1 and the whole quantity of matter whose density is rho' is zero4 ,the previously spherical surface of the sea of radius a now takes a new form4 ,let ,p be a point on the disturbed surface1 ,p' the corresponding point vertically below it on the undisturbed surface1 ,,pp' ,n4 ,the knowledge of ,n over the whole surface gives us the form of the disturbed or actual surface of the sea2 it is an equipotential surface1 and if ,v be the potential at ,p of the disturbing matter rho'1 ,m the mass of the earth 7the attraction-constant is assumed equal to unity7 ,m ,m ,m ----- ,v ,c -- - --- ,n ,v4 a ,n a a#b ,as far as we know1 ,n is always a very small quantity1 and we have with sufficient approximation ,n #c,v_/#dpideltaa1 where delta is the mean density of the earth4 ,thus we have the disturbance in elevation of the sea-level expressed in terms of the potential of the disturbing matter4 ,if at any point ,p the value of ,n remain constant when we pass to any adjacent point1 then the actual surface is there parallel to the ideal spherical surface2 as a rule1 however1 the normal at ,p is inclined to that at ,p'1 and astronomical observations have shown that this inclination1 the deflection or deviation1 amounting ordinarily to one or two seconds1 may in some cases exceed #aj1 or1 as at the foot of the ,himalayas1 even #fj4 ,by the expression mathematical figure of the earth we mean the surface of the sea produced in imagination so as to percolate the continents4 ,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 spheroidal4 ,no geodesist can proceed far in his work without encountering the irregularities of the mathematical surface1 and it is necessary that he should know how they affect his astronomical observations4 ,the whole of this subject is dealt with in his usual elegant manner by ,bessel in the ,astronomische ,nachrichten1 ,nos4 #cbi1 #ccj1 #cca1 in a paper entitled ,ueber den ,einfluss der ,unregelmassigkeiten der ,figur der ,erde auf geodatische ,arbeiten1 c4 ,but without entering into further details it is not difficult to see how local attraction at any station affects the determinations of latitude1 longitude and azimuth there4 ,let there be at the station an attraction to the north-east throwing the zenith to the south-west1 so that it takes in the celestial sphere a position ,z'1 its undisturbed position being ,z4 ,let the rectangular components of the displacement ,,zz' be xi measured southwards and eta measured westwards4 ,now the great circle joining ,z' with the pole of the heavens ,p makes there an angle with the meridian ,,pz eta cosec ,,pz' eta sec phi1 where phi is the latitude of the station4 ,also this great circle meets the horizon in a point whose distance from the great circle ,,pz is eta sec phi sin phi eta tan phi4 ,that is1 a meridian mark1 fixed by observations of the pole star1 will be placed that amount to the east of north4 ,hence the observed latitude requires the correction xi2 the observed longitude a correction eta sec phi2 and any observed azimuth a correction eta tan phi4 ,here it is supposed that azimuths are measured from north by east1 and longitudes eastwards4 ,the horizontal angles are also influenced by the deflections of the plumb-line1 in fact1 just as if the direction of the vertical axis of the theodolite varied by the same amount4 ,this influence1 however1 is slight1 so long as the sights point almost horizontally at the objects1 which is always the case in the observation of distant points4 ,the expression given for ,n enables one to form an approximate estimate of the effect of a compact mountain in raising the sea-level4 ,take1 for instance1 ,ben ,nevis1 which contains about a couple of cubic miles2 a simple calculation shows that the elevation produced would only amount to about #c in4 ,in the case of a mountain mass like the ,himalayas1 stretching over some #aejj miles of country with a breadth of #cjj and an average height of #c miles1 although it is difficult or impossible to find an expression for ,v1 yet we may ascertain that an elevation amounting to several hundred feet may exist near their base4 ,the geodetical operations1 however1 rather negative this idea1 for it was shown by ,colonel ,clarke 7,phil4 ,mag41 #ahgh7 that the form of the sea-level along the ,indian arc departs but slightly from that of the mean figure of the earth4 ,if this be so1 the action of the ,himalayas must be counteracted by subterranean tenuity4 ,suppose now that ,a1 ,b1 ,c1 444 are the stations of a network of triangulation projected on or lying on a spheroid of semiaxis major and eccentricity a1 e1 this spheroid having its axis parallel to the axis of rotation of the earth1 and its surface coinciding with the mathematical surface of the earth at ,a4 ,then basing the calculations on the observed elements at ,a1 the calculated latitudes1 longitudes and directions of the meridian at the other points will be the true latitudes1 c41 of the points as projected on the spheroid4 ,on comparing these geodetic elements with the corresponding astronomical determinations1 there will appear a system of differences which represent the inclinations1 at the various points1 of the actual irregular surface to the surface of the spheroid of reference4 ,these differences will suggest two things1--first1 that we may improve the agreement of the two surfaces1 by not restricting the spheroid of reference by the condition of making its surface coincide with the mathematical surface of the earth at ,a2 and secondly1 by altering the form and dimensions of the spheroid4 ,with respect to the first circumstance1 we may allow the spheroid two degrees of freedom1 that is1 the normals of the surfaces at ,a may be allowed to separate a small quantity1 compounded of a meridional difference and a difference perpendicular to the same4 ,let the spheroid be so placed that its normal at ,a lies to the north of the normal to the earth's surface by the small quantity xi and to the east by the quantity eta4 ,then in starting the calculation of geodetic latitudes1 longitudes and azimuths from ,a1 we must take1 not the observed elements phi1 alpha1 but for phi1 phi xi1 and for alpha1 alpha eta tan phi1 and zero longitude must be replaced by eta sec phi4 ,at the same time suppose the elements of the spheroid to be altered from a1 e to a da1 e de4 ,confining our attention at first to the two points ,a1 ,b1 let 7phi'71 7alpha'71 7omega7 be the numerical elements at ,b as obtained in the first calculation1 viz4 before the shifting and alteration of the spheroid2 they will now take the form 7phi'7 fxi geta hda kde1 7alpha'7 f'xi g'eta h'da k'de1 omega fxi geta hda kde1 where the coefficients f1 g1 444 c4 can be numerically calculated4 ,now these elements1 corresponding to the projection of ,b on the spheroid of reference1 must be equal severally to the astronomically determined elements at ,b1 corrected for the inclination of the surfaces there4 ,if xi'1 eta' be the components of the inclination at that point1 then we have xi' 7phi'7 - phi' fxi geta hda kde1 eta' tan phi' 7alpha'7 - alpha' f'xi g'eta h'da k'de1 eta' sec phi' 7omega7 - omega fxi geta hda kde1 where phi'1 alpha'1 omega are the observed elements at ,b4 ,here it appears that the observation of longitude gives no additional information1 but is available as a check upon the azimuthal observations4 ,if now there be a number of astronomical stations in the triangulation1 and we form equations such as the above for each point1 then we can from them determine those values of xi1 eta1 da1 de1 which make the quantity xi#b eta#b xi'#b eta'#b 444 a minimum4 ,thus we obtain that spheroid which best represents the surface covered by the triangulation4 ,in the ,account of the ,principal ,triangulation of ,great ,britain and ,ireland will be found the determination1 from #ge equations1 of the spheroid best representing the surface of the ,british ,isles4 ,its elements are a #bjibgjje - #bie ft41 b 3 a - b #bhj - #h2 and it is so placed that at ,greenwich ,observatory xi #a4hfd1 eta -#j4edf4 ,taking ,durham ,observatory as the origin1 and the tangent plane to the surface 7determined by xi -#j4ffd1 eta -#d4aag7 as the plane of x and y1 the former measured northwards1 and z measured vertically downwards1 the equation to the surface is 4#iiebdiec x#b 4#iibhhjje y#b 4#iigfcjeb z#b - #j4jjfgajjcxz - #dafeejgjz #j4 ,altitudes4 ,the precise determination of the altitude of his station is a matter of secondary importance to the geodesist2 nevertheless it is usual to observe the zenith distances of all trigonometrical points4 ,of great importance is a knowledge of the height of the base for its reduction to the sea-level4 ,again the height of a station does influence a little the observation of terrestrial angles1 for a vertical line at ,b does not lie generally in the vertical plane of ,a 7see above74 ,the height above the sea-level also influences the geographical latitude1 inasmuch as the centrifugal force is increased and the magnitude and direction of the attraction of the earth are altered1 and the effect upon the latitude is a very small term expressed by the formula h 7g'- g7 sin #b phi _/ ag1 where g1 g' are the values of gravity at the equator and at the pole4 ,this is h sin #b phi _/ #ehbj seconds1 h being in metres1 a quantity which may be neglected1 since for ordinary mountain heights it amounts to only a few hundredths of a second4 ,we can assume this amount as joined with the northern component of the plumb-line perturbations4 ,the uncertainties of terrestrial refraction render it impossible to determine accurately by vertical angles the heights of distant points4 ,generally speaking1 refraction is greatest at about daybreak2 from that time it diminishes1 being at a minimum for a couple of hours before and after mid-day2 later in the afternoon it again increases4 ,this at least is the general march of the phenomenon1 but it is by no means regular4 ,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 amount1 lasting from #a ,p4,m4 to #e ,p4,m4 ,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 4#jgib - 4#jjdg2 and if we separate those rays which for a considerable portion of their length cross the sea from those which do not1 the former give k 4#jhac and the latter k 4#jgec4 ,these values are determined from high stations and long distances2 when the distance is short1 and the rays graze the ground1 the amount of refraction is extremely uncertain and variable4 ,a case is noted in the ,indian survey where the zenith distance of a station #aj4e miles off varied from a depression of #d' #eb4f at #d4cj ,p4,m4 to an elevation of #b' #bd4j at #aj4ej ,p4,m4 ,if h1 h' be the heights above the level of the sea of two stations1 #ij deg4 delta1 #ij deg4 delta' their mutual zenith distances 7delta being that observed at h71 s their distance apart1 the earth being regarded as a sphere of radius a1 then1 with sufficient precision1 _/ #a - #bk _/ #a - #bk h' - h s tan 7 s -------- - delta 71 h - h' s tan 7 -------- - delta' 74 #b;a _/ #b;a _/ ,if from a station whose height is h the horizon of the sea be observed to have a zenith distance #ij deg4 delta1 then the above formula gives for h the value a tan#b delta h -- -------------4 #b #a - #bk ,suppose the depression delta to be n minutes1 then h #a4jedn#b if the ray be for the greater part of its length crossing the sea2 if otherwise1 h #a4jdjn#b4 ,to take an example3 the mean of eight observations of the zenith distance of the sea horizon at the top of ,ben ,nevis is #ia deg4 #d' #dh1 or delta #fd4h2 the ray is pretty equally disposed over land and water1 and hence h #a4jdgn#b #dcif ft4 ,the actual height of the hill by spirit-levelling is #ddjf ft41 so that the error of the height thus obtained is only #aj ft4 ,the determination of altitudes by means of spirit-levelling is undoubtedly the most exact method1 particularly in its present development as precise-levelling1 by which there have been determined in all civilized countries close-meshed nets of elevated points covering the entire land4 7,a4 ,r4 ,c2 ,f4 ,r4 ,h47 ,,footnotes3 #a ,an arrangement acting similarly had been previously introduced by ,borda4 #b ,geodetic ,survey of ,south ,africa1 vol4 iii4 7#aije71 p4 viii2 ,les ,nouveaux ,appareils pour la mesure rapide des bases geod41 par ,j4 ,rene ,benoit et ,ch4 ,ed4 ,guillaume 7#aijf74 #c ,see a paper ,on the ,course of ,geodetic ,lines on the ,earth's ,surface in the ,phil4 ,mag4 #ahgj2 ,helmert1 ,theorien der hoheren ,geodasie1 #a4 #cba4 #d ,helmert1 ,theorien der hoheren ,geodasie1 #a4 #bcb1 #bdg4 ,,geoffrey1 surnamed ,,martel 7#ajjf-ajfj71 count of ,anjou1 son of the count ,fulk ,nerra 7q4v47 and of the countess ,hildegarde or ,audegarde1 was born on the #adth of ,october #ajjf4 ,during his father's lifetime he was recognized as suzerain by ,fulk l',oison 7the ,gosling71 count of ,vendome1 the son of his half-sister ,adela4 ,fulk having revolted1 he confiscated the countship1 which he did not restore till #ajej4 ,on the #ast of ,january #ajcb he married ,agnes1 widow of ,william the ,great1 duke of ,aquitaine1 and taking arms against ,william the ,fat1 eldest son and successor of ,william the ,great1 defeated him and took him prisoner at ,mont-,couer near ,saint-,jouin-de-,marnes on the #bjth of ,september #ajcc4 ,he then tried to win recognition as dukes of ,aquitaine for the sons of his wife ,agnes by ,william the ,great1 who were still minors1 but ,fulk ,nerra promptly took up arms to defend his suzerain ,william the ,fat1 from whom he held the ,loudunois and ,saintonge in fief against his son4 ,in #ajcf ,geoffrey ,martel had to liberate ,william the ,fat1 on payment of a heavy ransom1 but the latter having died in #ajch1 and the second son of ,william the ,great1 ,odo1 duke of ,gascony1 having fallen in his turn at the siege of ,mauze 7#ajth of ,march #ajci7 ,geoffrey made peace with his father in the autumn of #ajci1 and had his wife's two sons recognized as dukes4 ,about this time1 also1 he had interfered in the affairs of ,maine1 though without much result1 for having sided against ,gervais1 bishop of ,le ,mans1 who was trying to make himself guardian of the young count of ,maine1 ,hugh1 he had been beaten and forced to make terms with ,gervais in #ajch4 ,in #ajdj he succeeded his father in ,anjou and was able to conquer ,touraine 7#ajdd7 and assert his authority over ,maine 7see ,,anjou74 ,about #ajej he repudiated ,agnes1 his first wife1 and married ,grecie1 the widow of ,bellay1 lord of ,montreuil-,bellay 7before ,august #ajeb71 whom he subsequently left in order to marry ,adela1 daughter of a certain ,count ,odo4 ,later he returned to ,grecie1 but again left her to marry ,adelaide the ,german4 ,when1 however1 he died on the #adth of ,november #ajfj1 at the monastery of ,st ,nicholas at ,angers1 he left no children1 and transmitted the countship to ,geoffrey the ,bearded1 the eldest of his nephews 7see ,,anjou74 ,see ,louis ,halphen1 ,le ,comte d',anjou au ,,xie siecle 7,paris1 #aijf74 ,a summary biography is given by ,celestin ,port1 ,dictionnaire historique1 geographique et biographique de ,maine-et-,loire 7#c vols41 ,paris-,angers1 #ahgd-ahgh71 vol4 ii4 pp4 #beb-bec1 and a sketch of the wars by ,kate ,norgate1 ,england under the ,angevin ,kings 7#b vols41 ,london1 #ahhg71 vol4 i4 chs4 iii4 iv4 7,l4 ,h4997 ,,geoffrey1 surnamed ,,plantagenet or ,,plantegenet 7#aaac-aaea71 count of ,anjou1 was the son of ,count ,fulk the ,young and of ,eremburge 7or ,arembourg of ,la ,fleche72 he was born on the #bdth of ,august #aaac4 ,he is also called le bel or the handsome1 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 7genet74 ,in #aabg he was made a knight1 and on the #bnd of ,june #aabi married ,matilda1 daughter of ,henry ,i4 of ,england1 and widow of the emperor ,henry ,v4 ,some months afterwards he succeeded to his father1 who gave up the countship when he definitively went to the kingdom of ,jerusalem4 ,the years of his government were spent in subduing the ,angevin barons and in conquering ,normandy 7see ,,anjou74 ,in #aaea1 while returning from the siege of ,montreuil-,bellay1 he took cold1 in consequence of bathing in the ,loir at ,chateau-du-,loir1 and died on the #gth of ,september4 ,he was buried in the cathedral of ,le ,mans4 ,by his wife ,matilda he had three sons3 ,henry ,plantagenet1 born at ,le ,mans on ,sunday1 the #eth of ,march #aacc2 ,geoffrey1 born at ,argentan on the #ast of ,june #aacd2 and ,william ,long-,sword1 born on the #bbnd of ,july #aacf4 ,see ,kate ,norgate1 ,england under the ,angevin ,kings 7#b vols41 ,london1 #ahhg71 vol4 i4 chs4 v4-viii42 ,celestin ,port1 ,dictionnaire historique1 geographique et biographique de ,maine-et-,loire 7#c vols41 ,paris-,angers1 #ahgd-ahgh71 vol4 ii4 pp4 #bed-bef4 ,a history of ,geoffrey le ,bel has yet to be written2 there is a biography of him written in the #abth century by ,jean1 a monk of ,marmoutier1 ,historia ,gaufredi1 ducis ,normannorum et comitis ,andegavorum1 published by ,marchegay et ,salmon2 ,chroniques des comtes d',anjou 7,societe de l'histoire de ,france1 ,paris1 #ahef71 pp4 #bbi-caj4 7,l4 ,h4997 ,,geoffrey 7#aaeh-aahf71 duke of ,brittany1 fourth son of the ,english king ,henry ,,ii4 and his wife ,eleanor of ,aquitaine1 was born on the #bcrd of ,september #aaeh4 ,in #aafg ,henry suggested a marriage between ,geoffrey and ,constance 7d4 #abja71 daughter and heiress of ,conan ,,iv41 duke of ,brittany 7d4 #aaga72 and ,conan not only assented1 perhaps under compulsion1 to this proposal1 but surrendered the greater part of his unruly duchy to the ,english king4 ,having received the homage of the ,breton nobles1 ,geoffrey joined his brothers1 ,henry and ,richard1 who1 in alliance with ,louis ,,vii4 of ,france1 were in revolt against their father2 but he made his peace in #aagd1 afterwards helping to restore order in ,brittany and ,normandy1 and aiding the new ,french king1 ,philip ,augustus1 to crush some rebellious vassals4 ,in ,july #aaha his marriage with ,constance was celebrated1 and practically the whole of his subsequent life was spent in warfare with his brother ,richard4 ,in #aahc he made peace with his father1 who had come to ,richard's assistance2 but a fresh struggle soon broke out for the possession of ,anjou1 and ,geoffrey was in ,paris treating for aid with ,philip ,augustus1 when he died on the #aith of ,august #aahf4 ,he left a daughter1 ,eleanor1 and his wife bore a posthumous son1 the unfortunate ,arthur4 ,,geoffrey 7c4 #aaeb-abab71 archbishop of ,york1 was a bastard son of ,henry ,,ii41 king of ,england4 ,he was distinguished from his legitimate half-brothers by his consistent attachment and fidelity to his father4 ,he was made bishop of ,lincoln at the age of twenty-one 7#aagc72 but though he enjoyed the temporalities he was never consecrated and resigned the see in #aahc4 ,he then became his father's chancellor1 holding a large number of lucrative benefices in plurality4 ,richard nominated him archbishop of ,york in #aahi1 but he was not consecrated till #aaia1 or enthroned till #aaid4 ,geoffrey1 though of high character1 was a man of uneven temper2 his history in chiefly one of quarrels1 with the see of ,canterbury1 with the chancellor ,william ,longchamp1 with his half-brothers ,richard and ,john1 and especially with his canons at ,york4 ,this last dispute kept him in litigation before ,richard and the pope for many years4 ,he led the clergy in their refusal to be taxed by ,john and was forced to fly the kingdom in #abjg4 ,he died in ,normandy on the #abth of ,december #abab4 ,see ,giraldus ,cambrensis1 ,vita ,galfridi2 ,stubbs's prefaces to ,roger de ,hoveden1 vols4 iii4 and iv4 7,rolls ,series74 7,h4 ,w4 ,c4 ,d47 ,,geoffrey ,,de ,,montbray 7d4 #ajic71 bishop of ,coutances 7,constantiensis71 a right-hand man of ,william the ,conqueror1 was a type of the great feudal prelate1 warrior and administrator at need4 ,he knew1 says ,orderic1 more about marshalling mailed knights than edifying psalm-singing clerks4 ,obtaining1 as a young man1 in #ajdh1 the see of ,coutances1 by his brother's influence 7see ,,mowbray71 he raised from his fellow nobles and from their ,sicilian spoils funds for completing his cathedral1 which was consecrated in #ajef4 ,with bishop ,odo1 a warrior like himself1 he was on the battle-field of ,hastings1 exhorting the ,normans to victory2 and at ,william's coronation it was he who called on them to acclaim their duke as king4 ,his reward in ,england was a mighty fief scattered over twelve counties4 ,he accompanied ,william on his visit to ,normandy 7#ajfg71 but1 returning1 led a royal force to the relief of ,montacute in ,september #ajfi4 ,in #ajge he again took the field1 leading with ,bishop ,odo a vast host against the rebel earl of ,norfolk1 whose stronghold at ,norwich they besieged and captured4 ,meanwhile the ,conqueror had invested him with important judicial functions4 ,in #ajgb he had presided over the great ,kentish suit between the primate and ,bishop ,odo1 and about the same time over those between the abbot of ,ely and his despoilers1 and between the bishop of ,worcester and the abbot of ,ely1 and there is some reason to think that he acted as a ,domesday commissioner 7#ajhf71 and was placed about the same time in charge of ,northumberland4 ,the bishop1 who attended the ,conqueror's funeral1 joined in the great rising against ,william ,rufus next year 7#ajhh71 making ,bristol1 with which 7as ,domesday shows7 he was closely connected and where he had built a strong castle1 his base of operations4 ,he burned ,bath and ravaged ,somerset1 but had submitted to the king before the end of the year4 ,he appears to have been at ,dover with ,william in ,january #ajij1 but1 withdrawing to ,normandy1 died at ,coutances three years later4 ,in his fidelity to ,duke ,robert he seems to have there held out for him against his brother ,henry1 when the latter obtained the ,cotentin4 ,see ,e4,a4 ,freeman1 ,norman ,conquest and ,william ,rufus2 ,j4,h4 ,round1 ,feudal ,england2 and1 for original authorities1 the works of ,orderic ,vitalis and ,william of ,poitiers1 and of ,florence of ,worcester2 the ,anglo-,saxon ,chronicle2 ,william of ,malmesbury's ,gesta pontificum1 and ,lanfranc's works1 ed4 ,giles2 ,domesday ,book4 7,j4 ,h4 ,r47 ,,geoffrey ,,of ,,monmouth 7d4 #aaed71 bishop of ,st ,asaph and writer on early ,british history1 was born about the year #aajj4 ,of his early life little is known1 except that he received a liberal education under the eye of his paternal uncle1 ,uchtryd1 who was at that time archdeacon1 and subsequently bishop1 of ,llandaff4 ,in #aabi ,geoffrey appears at ,oxford among the witnesses of an ,oseney charter4 ,he subscribes himself ,geoffrey ,arturus2 from this we may perhaps infer that he had already begun his experiments in the manufacture of ,celtic mythology4 ,a first edition of his ,historia ,britonum was in circulation by the year #aaci1 although the text which we possess appears to date from #aadg4 ,this famous work1 which the author has the audacity to place on the same level with the histories of ,william of ,malmesbury and ,henry of ,huntingdon1 professes to be a translation from a ,celtic source2 a very old book in the ,british tongue which ,walter1 archdeacon of ,oxford1 had brought from ,brittany4 ,walter the archdeacon is a historical personage2 whether his book has any real existence may be fairly questioned4 ,there is nothing in the matter or the style of the ,historia to preclude us from supposing that ,geoffrey drew partly upon confused traditions1 partly on his own powers of invention1 and to a very slight degree upon the accepted authorities for early ,british history4 ,his chronology is fantastic and incredible2 ,william of ,newburgh justly remarks that1 if we accepted the events which ,geoffrey relates1 we should have to suppose that they had happened in another world4 ,william of ,newburgh wrote1 however1 in the reign of ,richard ,i4 when the reputation of ,geoffrey's work was too well established to be shaken by such criticisms4 ,the fearless romancer had achieved an immediate success4 ,he was patronized by ,robert1 earl of ,gloucester1 and by two bishops of ,lincoln2 he obtained1 about #aadj1 the archdeaconry of ,llandaff on account of his learning2 and in #aaea was promoted to the see of ,st ,asaph4 ,before his death the ,historia ,britonum had already become a model and a quarry for poets and chroniclers4 ,the list of imitators begins with ,geoffrey ,gaimar1 the author of the ,estorie des ,engles 7c4 #aadg71 and ,wace1 whose ,roman de ,brut 7#aaee7 is partly a translation and partly a free paraphrase of the ,historia4 ,in the next century the influence of ,geoffrey is unmistakably attested by the ,brut of ,layamon1 and the rhyming ,english chronicle of ,robert of ,gloucester4 ,among later historians who were deceived by the ,historia ,britonum it is only needful to mention ,higdon1 ,hardyng1 ,fabyan 7#aeab71 ,holinshed 7#aehj7 and ,john ,milton4 ,still greater was the influence of ,geoffrey upon those writers who1 like ,warner in ,albion's ,england 7#aehf71 and ,drayton in ,polyolbion 7#afac71 deliberately made their accounts of ,english history as poetical as possible4 ,the stories which ,geoffrey preserved or invented were not infrequently a source of inspiration to literary artists4 ,the earliest ,english tragedy1 ,gorboduc 7#aefe71 the ,mirror for ,magistrates 7#aehg71 and ,shakespeare's ,lear1 are instances in point4 ,it was1 however1 the ,arthurian legend which of all his fabrications attained the greatest vogue4 ,in the work of expanding and elaborating this theme the successors of ,geoffrey went as far beyond him as he had gone beyond ,nennius2 but he retains the credit due to the founder of a great school4 ,marie de ,france1 who wrote at the court of ,henry ,,ii41 and ,chretien de ,troyes1 her ,french contemporary1 were the earliest of the avowed romancers to take up the theme4 ,the succeeding age saw the ,arthurian story popularized1 through translations of the ,french romances1 as far afield as ,germany and ,scandinavia4 ,it produced in ,england the ,roman du ,saint ,graal and the ,roman de ,merlin1 both from the pen of ,robert de ,borron2 the ,roman de ,lancelot2 the ,roman de ,tristan1 which is attributed to a fictitious ,lucas de ,gast4 ,in the reign of ,edward ,,iv4 ,sir ,thomas ,malory paraphrased and arranged the best episodes of these romances in ,english prose4 ,his ,morte d',arthur1 printed by ,caxton in #adhe1 epitomizes the rich mythology which ,geoffrey's work had first called into life1 and gave the ,arthurian story a lasting place in the ,english imagination4 ,the influence of the ,historia ,britonum may be illustrated in another way1 by enumerating the more familiar of the legends to which it first gave popularity4 ,of the twelve books into which it is divided only three 7,bks4 ,,ix41 ,x41 ,,xi47 are concerned with ,arthur4 ,earlier in the work1 however1 we have the adventures of ,brutus2 of his follower ,corineus1 the vanquisher of the ,cornish giant ,goemagol 7,gogmagog72 of ,locrinus and his daughter ,sabre 7immortalized in ,milton's ,comus72 of ,bladud the builder of ,bath2 of ,lear and his daughters2 of the three pairs of brothers1 ,ferrex and ,porrex1 ,brennius and ,belinus1 ,elidure and ,peridure4 ,the story of ,vortigern and ,rowena takes its final form in the ,historia ,britonum2 and ,merlin makes his first appearance in the prelude to the ,arthur legend4 ,besides the ,historia ,britonum ,geoffrey is also credited with a ,life of ,merlin composed in ,latin verse4 ,the authorship of this work has1 however1 been disputed1 on the ground that the style is distinctly superior to that of the ,historia4 ,a minor composition1 the ,prophecies of ,merlin1 was written before #aacf1 and afterwards incorporated with the ,historia1 of which it forms the seventh book4 ,for a discussion of the manuscripts of ,geoffrey's work1 see ,sir ,t4,d4 ,hardy's ,descriptive ,catalogue 7,rolls ,series71 i4 pp4 #cda ff4 ,the ,historia ,britonum has been critically edited by ,san ,marte 7,halle1 #ahed74 ,there is an ,english translation by ,j4,a4 ,giles 7,london1 #ahdb74 ,the ,vita ,merlini has been edited by ,f4 ,michel and ,t4 ,wright 7,paris1 #ahcg74 ,see also the ,dublin ,univ4 ,magazine for ,april #ahgf1 for an article by ,t4 ,gilray on the literary influence of ,geoffrey2 ,g4 ,heeger's ,trojanersage der ,britten 7#ahhi72 and ,la ,borderie's ,etudes historiques bretonnes 7#ahhc74 7,h4 ,w4 ,c4 ,d47 ,,geoffrey ,,of ,,paris 7d4 c4 #acbj71 ,french chronicler1 was probably the author of the ,chronique metrique de ,philippe le ,bel1 or ,chronique rimee de ,geoffroi de ,paris4 ,this work1 which deals with the history of ,france from #acjj to #acaf1 contains #giah verses1 and is valuable as that of a writer who had a personal knowledge of many of the events which he relates4 ,various short historical poems have also been attributed to ,geoffrey1 but there is no certain information about either his life or his writings4 ,the ,chronique was published by ,j4,a4 ,buchon in his ,collection des chroniques1 tome ix4 7,paris1 #ahbg71 and it has also been printed in tome xxii4 of the ,recueil des historiens des ,gaules et de la ,france 7,paris1 #ahfe74 ,see ,g4 ,paris1 ,histoire de la litterature francaise au moyen age 7,paris1 #ahij72 and ,a4 ,molinier1 ,les ,sources de l'histoire de ,france1 tome iii4 7,paris1 #aijc74 ,,geoffrey ,,the ,,baker 7d4 c4 #acfj71 ,english chronicler1 is also called ,walter of ,swinbroke1 and was probably a secular clerk at ,swinbrook in ,oxfordshire4 ,he wrote a ,chronicon ,angliae temporibus ,edwardi ,,ii4 et ,edwardi ,,iii41 which deals with the history of ,england from #acjc to #acef4 ,from the beginning until about #acbd this work is based upon ,adam ,murimuth's ,continuatio chronicarum1 but after this date it is valuable and interesting1 containing information not found elsewhere1 and closing with a good account of the battle of ,poitiers4 ,the author obtained his knowledge about the last days of ,edward ,,ii4 from ,william ,bisschop1 a companion of the king's murderers1 ,thomas ,gurney and ,john ,maltravers4 ,geoffrey also wrote a ,chroniculum from the creation of the world until #accf1 the value of which is very slight4 ,his writings have been edited with notes by ,sir ,e4,m4 ,thompson as the ,chronicon ,galfridi le ,baker de ,swynebroke 7,oxford1 #ahhi74 ,some doubt exists concerning ,geoffrey's share in the compilation of the ,vita et mors ,edwardi ,,ii41 usually attributed to ,sir ,thomas de la ,more1 or ,moor1 and printed by ,camden in his ,anglica scripta4 ,it has been maintained by ,camden and others that ,more wrote an account of ,edward's reign in ,french1 and that this was translated into ,latin by ,geoffrey and used by him in compiling his ,chronicon4 ,recent scholarship1 however1 asserts that ,more was no writer1 and that the ,vita et mors is an extract from ,geoffrey's ,chronicon1 and was attributed to ,more1 who was the author's patron4 ,in the main this conclusion substantiates the verdict of ,stubbs1 who has published the ,vita et mors in his ,chronicles of the reigns of ,edward ,i4 and ,edward ,,ii4 7,london1 #ahhc74 ,the manuscripts of ,geoffrey's works are in the ,bodleian library at ,oxford4 ,,geoffrin1 ,,marie ,,therese ,,rodet 7#afii-aggg71 a ,frenchwoman who played an interesting part in ,french literary and artistic life1 was born in ,paris in #afii4 ,she married1 on the #aith of ,july #agac1 ,pierre ,francois ,geoffrin1 a rich manufacturer and lieutenant-colonel of the ,national ,guard1 who died in #agej4 ,it was not till ,mme ,geoffrin was nearly fifty years of age that we begin to hear of her as a power in ,parisian society4 ,she had learned much from ,mme de ,tencin1 and about #agdh began to gather round her a literary and artistic circle4 ,she had every week two dinners1 on ,monday for artists1 and on ,wednesday for her friends the ,encyclopaedists and other men of letters4 ,she received many foreigners of distinction1 ,hume and ,horace ,walpole among others4 ,walpole spent much time in her society before he was finally attached to ,mme du ,deffand1 and speaks of her in his letters as a model of common sense4 ,she was indeed somewhat of a small tyrant in her circle4 ,she had adopted the pose of an old woman earlier than necessary1 and her coquetry1 if such it can be called1 took the form of being mother and mentor to her guests1 many of whom were indebted to her generosity for substantial help4 ,although her aim appears to have been to have the ,encyclopedie in conversation and action around her1 she was extremely displeased with any of her friends who were so rash as to incur open disgrace4 ,marmontel lost her favour after the official censure of ,belisaire1 and her advanced views did not prevent her from observing the forms of religion4 ,a devoted ,parisian1 ,mme ,geoffrin rarely left the city1 so that her journey to ,poland in #agff to visit the king1 ,stanislas ,poniatowski1 whom she had known in his early days in ,paris1 was a great event in her life4 ,her experiences induced a sensible gratitude that she had been born ,francaise and particuliere4 ,in her last illness her daughter1 ,therese1 marquise de la ,ferte ,imbault1 excluded her mother's old friends so that she might die as a good ,christian1 a proceeding wittily described by the old lady3 ,my daughter is like ,godfrey de ,bouillon1 she wished to defend my tomb from the infidels4 ,mme ,geoffrin died in ,paris on the #fth of ,october #aggg4 ,see ,correspondance inedite du roi ,stanislas ,auguste ,poniatowski et de ,madame ,geoffrin1 edited by the comte de ,mouy 7#ahge72 ,p4 de ,segur1 ,le ,royaume de la rue ,saint-,honore1 ,madame ,geoffrin et sa fille 7#ahig72 ,a4 ,tornezy1 ,un ,bureau d'esprit au ,,xviiie siecle3 le salon de ,madame ,geoffrin 7#ahie72 and ,janet ,aldis1 ,madame ,geoffrin1 her ,salon and her ,times1 #agej-aggg 7#aije74 ,,geoffroy1 ,,etienne ,,francois 7#afgb-agca71 ,french chemist1 born in ,paris on the #acth of ,february #afgb1 was first an apothecary and then practised medicine4 ,after studying at ,montpellier he accompanied ,marshal ,tallard on his embassy to ,london in #afih and thence travelled to ,holland and ,italy4 ,returning to ,paris he became professor of chemistry at the ,jardin du ,roi and of pharmacy and medicine at the ,college de ,france1 and dean of the faculty of medicine4 ,he died in ,paris on the #fth of ,january #agca4 ,his name is best known in connexion with his tables of affinities 7tables des rapports71 which he presented to the ,french ,academy in #agah and #agbj4 ,these were lists1 prepared by collating observations on the actions of substances one upon another1 showing the varying degrees of affinity exhibited by analogous bodies for different reagents1 and they retained their vogue for the rest of the century1 until displaced by the profounder conceptions introduced by ,c4,l4 ,berthollet4 ,another of his papers dealt with the delusions of the philosopher's stone1 but nevertheless he believed that iron could be artificially formed in the combustion of vegetable matter4 ,his ,tractatus de materia medica1 published posthumously in #agda1 was long celebrated4 ,his brother ,,claude ,,joseph1 known as ,geoffroy the younger 7#afhe-ageb71 was also an apothecary and chemist who1 having a considerable knowledge of botany1 devoted himself especially to the study of the essential oils in plants4 ,,geoffroy1 ,,julien ,,louis 7#agdc-ahad71 ,french critic1 was born at ,rennes in #agdc4 ,he studied in the school of his native town and at the ,college ,louis le ,grand in ,paris4 ,he took orders and fulfilled for some time the humble functions of an usher1 eventually becoming professor of rhetoric at the ,college ,mazarin4 ,a bad tragedy1 ,caton1 was accepted at the ,theatre ,francais1 but was never acted4 ,on the death of ,elie ,freron in #aggf the other collaborators in the ,annee litteraire asked ,geoffroy to succeed him1 and he conducted the journal until in #agib it ceased to appear4 ,geoffroy was a bitter critic of ,voltaire and his followers1 and made for himself many enemies4 ,an enthusiastic royalist1 he published with ,freron's brother-in-law1 the abbe ,thomas ,royou 7#agda-agib71 a journal1 ,l',ami du roi 7#agij-agib71 which possibly did more harm than good to the king's cause by its ill-advised partisanship4 ,during the ,terror ,geoffroy hid in the neighbourhood of ,paris1 only returning in #agii4 ,an attempt to revive the ,annee litteraire failed1 and ,geoffroy undertook the dramatic feuilleton of the ,journal des debats4 ,his scathing criticisms had a success of notoriety1 but their popularity was ephemeral1 and the publication of them 7#e vols41 #ahai-ahbj7 as ,cours de litterature dramatique proved a failure4 ,he was also the author of a perfunctory ,commentaire on the works of ,racine prefixed to ,lenormant's edition 7#ahjh74 ,he died in ,paris on the #bgth of ,february #ahad4 ,,geoffroy ,,saint-,,hilaire1 ,,etienne 7#aggb-ahdd71 ,french naturalist1 was the son of ,jean ,gerard ,geoffroy1 procurator and magistrate of ,etampes1 ,seine-et-,oise1 where he was born on the #aeth of ,april #aggb4 ,destined for the church he entered the college of ,navarre1 in ,paris1 where he studied natural philosophy under ,m4,j4 ,brisson2 and in #aghh he obtained one of the canonicates of the chapter of ,sainte ,croix at ,etampes1 and also a benefice4 ,science1 however1 offered him a more congenial career1 and he gained from his father permission to remain in ,paris1 and to attend the lectures at the ,college de ,france and the ,jardin des ,plantes1 on the condition that he should also read law4 ,he accordingly took up his residence at ,cardinal ,lemoine's college1 and there became the pupil and soon the esteemed associate of ,brisson's friend1 the abbe ,hauy1 the mineralogist4 ,having1 before the close of the year #agij1 taken the degree of bachelor in law1 he became a student of medicine1 and attended the lectures of ,a4,f4 de ,fourcroy at the ,jardin des ,plantes1 and of ,l4,j4,m4 ,daubenton at the ,college de ,france4 ,his studies at ,paris were at length suddenly interrupted1 for1 in ,august #agib1 ,hauy and the other professors of ,lemoine's college1 as also those of the college of ,navarre1 were arrested by the revolutionists as priests1 and confined in the prison of ,st ,firmin4 ,through the influence of ,daubenton and others ,geoffroy on the #adth of ,august obtained an order for the release of ,hauy in the name of the ,academy2 still the other professors of the two colleges1 save ,c4,f4 ,lhomond1 who had been rescued by his pupil ,j4,l4 ,tallien1 remained in confinement4 ,geoffroy1 foreseeing their certain destruction if they remained in the hands of the revolutionists1 determined if possible to secure their liberty by stratagem4 ,by bribing one of the officials at ,st ,firmin1 and disguising himself as a commissioner of prisons1 he gained admission to his friends1 and entreated them to effect their escape by following him4 ,all1 however1 dreading lest their deliverance should render the doom of their fellow-captives the more certain1 refused the offer1 and one priest only1 who was unknown to ,geoffroy1 left the prison4 ,already on the night of the #bnd of ,september the massacre of the proscribed had begun1 when ,geoffroy1 yet intent on saving the life of his friends and teachers1 repaired to ,st ,firmin4 ,at #d o'clock on the morning of the #crd of ,september1 after eight hours' waiting1 he by means of a ladder assisted the escape of twelve ecclesiastics1 not of the number of his acquaintance1 and then the approach of dawn and the discharge of a gun directed at him warned him1 his chief purpose unaccomplished1 to return to his lodgings4 ,leaving ,paris he retired to ,etampes1 where1 in consequence of the anxieties of which he had lately been the prey1 and the horrors which he had witnessed1 he was for some time seriously ill4 ,at the beginning of the winter of #agib he returned to his studies in ,paris1 and in ,march of the following year ,daubenton1 through the interest of ,bernardin de ,saint ,pierre1 procured him the office of sub-keeper and assistant demonstrator of the cabinet of natural history1 vacant by the resignation of ,b4,g4,e4 ,lacepede4 ,by a law passed in ,june #agic1 ,geoffroy was appointed one of the twelve professors of the newly constituted museum of natural history1 being assigned the chair of zoology4 ,in the same year he busied himself with the formation of a menagerie at that institution4 ,in #agid through the introduction of ,a4,h4 ,tessier he entered into correspondence with ,georges ,cuvier1 to whom1 after the perusal of some of his manuscripts1 he wrote3 ,venez jouer parmi nous le role de ,linne1 d'un autre legislateur de l'histoire naturelle4 ,shortly after the appointment of ,cuvier as assistant at the ,museum d',histoire ,naturelle1 ,geoffroy received him into his house4 ,the two friends wrote together five memoirs on natural history1 one of which1 on the classification of mammals1 puts forward the idea of the subordination of characters upon which ,cuvier based his zoological system4 ,it was in a paper entitled ,histoire des ,makis1 ou singes de ,madagascar1 written in #agie1 that ,geoffroy first gave expression to his views on the unity of organic composition1 the influence of which is perceptible in all his subsequent writings2 nature1 he observes1 presents us with only one plan of construction1 the same in principle1 but varied in its accessory parts4 ,in #agih ,geoffroy was chosen a member of the great scientific expedition to ,egypt1 and on the capitulation of ,alexandria in ,august #ahja1 he took part in resisting the claim made by the ,british general to the collections of the expedition1 declaring that1 were that demand persisted in1 history would have to record that he also had burnt a library in ,alexandria4 ,early in ,january #ahjb ,geoffroy returned to his accustomed labours in ,paris4 ,he was elected a member of the academy of sciences of that city in ,september #ahjg4 ,in ,march of the following year the emperor1 who had already recognized his national services by the award of the cross of the legion of honour1 selected him to visit the museums of ,portugal1 for the purpose of procuring collections from them1 and in the face of considerable opposition from the ,british he eventually was successful in retaining them as a permanent possession for his country4 ,in #ahji1 the year after his return to ,france1 he was made professor of zoology at the faculty of sciences at ,paris1 and from that period he devoted himself more exclusively than before to anatomical study4 ,in #ahah he gave to the world the first part of his celebrated ,philosophie anatomique1 the second volume of which1 published in #ahbb1 and subsequent memoirs account for the formation of monstrosities on the principle of arrest of development1 and of the attraction of similar parts4 ,when1 in #ahcj1 ,geoffroy proceeded to apply to the invertebrata his views as to the unity of animal composition1 he found a vigorous opponent in ,georges ,cuvier1 and the discussion between them1 continued up to the time of the death of the latter1 soon attracted the attention of the scientific throughout ,europe4 ,geoffroy1 a synthesist1 contended1 in accordance with his theory of unity of plan in organic composition1 that all animals are formed of the same elements1 in the same number1 and with the same connexions3 homologous parts1 however they differ in form and size1 must remain associated in the same invariable order4 ,with ,goethe he held that there is in nature a law of compensation or balancing of growth1 so that if one organ take on an excess of development1 it is at the expense of some other part2 and he maintained that1 since nature takes no sudden leaps1 even organs which are superfluous in any given species1 if they have played an important part in other species of the same family1 are retained as rudiments1 which testify to the permanence of the general plan of creation4 ,it was his conviction that1 owing to the conditions of life1 the same forms had not been perpetuated since the origin of all things1 although it was not his belief that existing species are becoming modified4 ,cuvier1 who was an analytical observer of facts1 admitted only the prevalence of laws of co-existence or harmony in animal organs1 and maintained the absolute invariability of species1 which he declared had been created with a regard to the circumstances in which they were placed1 each organ contrived with a view to the function it had to fulfil1 thus putting1 in ,geoffroy's considerations1 the effect for the cause4 ,in ,july #ahdj ,geoffroy became blind1 and some months later he had a paralytic attack4 ,from that time his strength gradually failed him4 ,he resigned his chair at the museum in #ahda1 and died at ,paris on the #aith of ,june #ahdd4 ,geoffroy wrote3 ,catalogue des mammiferes du ,museum ,national d',histoire ,naturelle 7#ahac71 not quite completed2 ,philosophie anatomique--t4 i41 ,des organes respiratoires 7#ahah71 and t4 ii41 ,des monstruosites humaines 7#ahbb72 ,systeme dentaire des mammiferes et des oiseaux 7#ast pt41 #ahbd72 ,sur le principe de l'unite de composition organique 7#ahbh72 ,cours de l'histoire naturelle des mammiferes 7#ahbi72 ,principes de philosophie zoologique 7#ahcj72 ,etudes progressives d'un naturaliste 7#ahce72 ,fragments biographiques 7#ahcb72 ,notions synthetiques1 historiques et physiologiques de philosophie naturelle 7#ahch71 and other works2 also part of the ,description de l',egypte par la commission des sciences 7#ahba-ahcj72 and1 with ,frederic ,cuvier 7#aggc-ahch71 a younger brother of ,g4 ,cuvier1 ,histoire naturelle des mammiferes 7#d vols41 #ahbj-ahdb72 besides numerous papers on such subjects as the anatomy of marsupials1 ruminants and electrical fishes1 the vertebrate theory of the skull1 the opercula of fishes1 teratology1 palaeontology and the influence of surrounding conditions in modifying animal forms4 ,see ,vie1 travaux1 et doctrine scientifique d',etienne ,geoffroy ,saint-,hilaire1 par son fils ,m4 ,isidore ,geoffroy ,saint-,hilaire 7,paris and ,strasburg1 #ahdg71 to which is appended a list of ,geoffroy's works2 and ,joly1 in ,biog4 universelle1 t4 xvi4 7#ahef74 ,,geoffroy ,,saint-,,hilaire1 ,,isidore 7#ahje-ahfa71 ,french zoologist1 son of the preceding1 was born at ,paris on the #afth of ,december #ahje4 ,in his earlier years he showed an aptitude for mathematics1 but eventually he devoted himself to the study of natural history and of medicine1 and in #ahbd he was appointed assistant naturalist to his father4 ,on the occasion of his taking the degree of doctor of medicine in ,september #ahbi1 he read a thesis entitled ,propositions sur la monstruosite1 consideree chez l'homme et les animaux2 and in #ahcb-ahcg was published his great teratological work1 ,histoire generale et particuliere des anomalies de l'organisation chez l'homme et les animaux1 #c vols4 #hvo4 with #bj plates4 ,in #ahbi he delivered for his father the second part of a course of lectures on ornithology1 and during the three following years he taught zoology at the ,athenee1 and teratology at the ,ecole pratique4 ,he was elected a member of the academy of sciences at ,paris in #ahcc1 was in #ahcg appointed to act as deputy for his father at the faculty of sciences in ,paris1 and in the following year was sent to ,bordeaux to organize a similar faculty there4 ,he became successively inspector of the academy of ,paris 7#ahdj71 professor of the museum on the retirement of his father 7#ahda71 inspector-general of the university 7#ahdd71 a member of the royal council for public instruction 7#ahde71 and on the death of ,h4,m4,d4 de ,blainville1 professor of zoology at the faculty of sciences 7#ahej74 ,in #ahed he founded the ,acclimatization ,society of ,paris1 of which he was president4 ,he died at ,paris on the #ajth of ,november #ahfa4 ,besides the above-mentioned works1 he wrote3 ,essais de zoologie generale 7#ahda72 ,vie 444 d',etienne ,geoffroy ,saint-,hilaire 7#ahdg72 ,acclimatation et domestication des animaux utiles 7#ahdi2 #dth ed41 #ahfa72 ,lettres sur les substances alimentaires et particulierement sur la viande de cheval 7#ahef72 and ,histoire naturelle generale des regnes organiques 7#c vols41 #ahed-ahfb71 which was not quite completed4 ,he was the author also of various papers on zoology1 comparative anatomy and palaeontology4 ,,geography 7,gr4 ,greek3 ge1 earth1 and ,greek3 graphein1 to write71 the exact and organized knowledge of the distribution of phenomena on the surface of the earth4 ,the fundamental basis of geography is the vertical relief of the earth's crust1 which controls all mobile distributions4 ,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 hollows1 filling them up to a height corresponding to the volume1 and thus producing the important practical division of the surface into land and water4 ,the distribution of the mass of the atmosphere over the surface of the earth is also controlled by the relief of the crust1 its greater or lesser density at the surface corresponding to the lesser or greater elevation of the surface4 ,the simplicity of the zonal distribution of solar energy on the earth's surface1 which would characterize a uniform globe1 is entirely destroyed by the dissimilar action of land and water with regard to radiant heat1 and by the influence of crust-forms on the direction of the resulting circulation4 ,the influence of physical environment becomes clearer and stronger when the distribution of plant and animal life is considered1 and if it is less distinct in the case of man1 the reason is found in the modifications of environment consciously produced by human effort4 ,geography is a synthetic science1 dependent for the data with which it deals on the results of specialized sciences such as astronomy1 geology1 oceanography1 meteorology1 biology and anthropology1 as well as on topographical description4 ,the physical and natural sciences are concerned in geography only so far as they deal with the forms of the earth's surface1 or as regards the distribution of phenomena4 ,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 distributions4 ,this gives to it unity and definiteness1 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 other4 ,it is essential to classify the subject-matter of geography in such a manner as to give prominence not only to facts1 but to their mutual relations and their natural and inevitable order4 ,the fundamental conception of geography is form1 including the figure of the earth and the varieties of crustal relief4 ,hence mathematical geography 7see ,,map71 including cartography as a practical application1 comes first4 ,it merges into physical geography1 which takes account of the forms of the lithosphere 7geomorphology71 and also of the distribution of the hydrosphere and the rearrangements resulting from the workings of solar energy throughout the hydrosphere and atmosphere 7oceanography and climatology74 ,next follows the distribution of plants and animals 7biogeography71 and finally the distribution of mankind and the various artificial boundaries and redistributions 7anthropogeography74 ,the applications of anthropogeography to human uses give rise to political and commercial geography1 in the elucidation of which all the earlier departments or stages have to be considered1 together with historical and other purely human conditions4 ,the evolutionary idea has revolutionized and unified geography as it did biology1 breaking down the old hard-and-fast partitions between the various departments1 and substituting the study of the nature and influence of actual terrestrial environments for the earlier motive1 the discovery and exploration of new lands4 ,,history ,,of ,,geographical ,,theory ,the earliest conceptions of the earth1 like those held by the primitive peoples of the present day1 are difficult to discover and almost impossible fully to grasp4 ,early generalizations1 as far as they were made from known facts1 were usually expressed in symbolic language1 and for our present purpose it is not profitable to speculate on the underlying truths which may sometimes be suspected in the old mythological cosmogonies4 ,early ,greek ideas4 ,flat earth of ,homer4 ,hecataeus4 ,herodotus4 ,the idea of symmetry4 ,the first definite geographical theories to affect the western world were those evolved1 or at least first expressed1 by the ,greeks4#a ,the earliest theoretical problem of geography was the form of the earth4 ,the natural supposition that the earth is a flat disk1 circular or elliptical in outline1 had in the time of ,homer acquired a special definiteness by the introduction of the idea of the ocean river bounding the whole1 an application of imperfectly understood observations4 ,thales of ,miletus is claimed as the first exponent of the idea of a spherical earth2 but1 although this does not appear to be warranted1 his disciple ,anaximander 7c4 #ehj ,b4,c47 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 heavens4 ,the ,pythagorean school of philosophers adopted the theory of a spherical earth1 but from metaphysical rather than scientific reasons2 their convincing argument was that a sphere being the most perfect solid figure was the only one worthy to circumscribe the dwelling-place of man4 ,the division of the sphere into parallel zones and some of the consequences of this generalization seem to have presented themselves to ,parmenides 7c4 #dej ,b4,c472 but these ideas did not influence the ,ionian school of philosophers1 who in their treatment of geography preferred to deal with facts demonstrable by travel rather than with speculations4 ,thus ,hecataeus1 claimed by ,h4,f4 ,tozer#b as the father of geography on account of his ,periodos1 or general treatise on the earth1 did not advance beyond the primitive conception of a circular disk4 ,he systematized the form of the land within the ring of ocean--the ,greek3 oikoumene1 or habitable world--by recognizing two continents3 ,europe to the north1 and ,asia to the south of the midland sea4 ,herodotus1 equally oblivious of the sphere1 criticized and ridiculed the circular outline of the oekumene1 which he knew to be longer from east to west than it was broad from north to south4 ,he also pointed out reasons for accepting a division of the land into three continents--,europe1 ,asia and ,africa4 ,beyond the limits of his personal travels ,herodotus applied the characteristically ,greek theory of symmetry to complete1 in the unknown1 outlines of lands and rivers analogous to those which had been explored4 ,symmetry was in fact the first geographical theory1 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 ,park4#c ,aristotle and the sphere4 ,to ,aristotle 7#chd-cbb ,b4,c47 must be given the distinction of founding scientific geography4 ,he demonstrated the sphericity of the earth by three arguments1 two of which could be tested by observation4 ,these were3 7#a7 that the earth must be spherical1 because of the tendency of matter to fall together towards a common centre2 7#b7 that only a sphere could always throw a circular shadow on the moon during an eclipse2 and 7#c7 that the shifting of the horizon and the appearance of new constellations1 or the disappearance of familiar stars1 as one travelled from north to south1 could only be explained on the hypothesis that the earth was a sphere4 ,aristotle1 too1 gave greater definiteness to the idea of zones conceived by ,parmenides1 who had pictured a torrid zone uninhabitable by reason of heat1 two frigid zones uninhabitable by reason of cold1 and two intermediate temperate zones fit for human occupation4 ,aristotle defined the temperate zone as extending from the tropic to the arctic circle1 but there is some uncertainty as to the precise meaning he gave to the term arctic circle4 ,soon after his time1 however1 this conception was clearly established1 and with so large a generalization the mental horizon was widened to conceive of a geography which was a science4 ,aristotle had himself shown that in the southern temperate zone winds similar to those of the northern temperate zone should blow1 but from the opposite direction4 ,fitting the oekumene to the sphere4 ,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 oekumene1 or habitable world1 the only portion of the terrestrial surface known to the ancients and to the medieval peoples1 and still retaining a shadow of its old monopoly of geographical attention in its modern name of the ,old ,world4 ,the fitting of the oekumene to the sphere was the second theoretical problem4 ,the circular outline had given way in geographical opinion to the elliptical with the long axis lying east and west1 and ,aristotle was inclined to view it as a very long and relatively narrow band almost encircling the globe in the temperate zone4 ,his argument as to the narrowness of the sea between ,west ,africa and ,east ,asia1 from the occurrence of elephants at both extremities1 is difficult to understand1 although it shows that he looked on the distribution of animals as a problem of geography4 ,problem of the ,antipodes4 ,pythagoras had speculated as to the existence of antipodes1 but it was not until the first approximately accurate measurements of the globe and estimates of the length and breadth of the oekumene were made by ,eratosthenes 7c4 #bej ,b4,c47 that the fact that1 as then known1 it occupied less than a quarter of the surface of the sphere was clearly recognized4 ,it was natural1 if not strictly logical1 that the ocean river should be extended from a narrow stream to a world-embracing sea1 and here again ,greek theory1 or rather fancy1 gave its modern name to the greatest feature of the globe4 ,the old instinctive idea of symmetry must often have suggested other oekumene balancing the known world in the other quarters of the globe4 ,the ,stoic philosophers1 especially ,crates of ,mallus1 arguing from the love of nature for life1 placed an oekumene in each quarter of the sphere1 the three unknown world-islands being those of the ,antoeci1 ,perioeci and ,antipodes4 ,this was a theory not only attractive to the philosophical mind1 but eminently adapted to promote exploration4 ,it had its opponents1 however1 for ,herodotus showed that sea-basins existed cut off from the ocean1 and it is still a matter of controversy how far the pre-,ptolemaic geographers believed in a water-connexion between the ,atlantic and ,indian oceans4 ,it is quite clear that ,pomponius ,mela 7c4 ,a4,d4 #dj71 following ,strabo1 held that the southern temperate zone contained a habitable land1 which he designated by the name ,antichthones4 ,aristotle's geographical views4 ,aristotle left no work on geography1 so that it is impossible to know what facts he associated with the science of the earth's surface4 ,the word geography did not appear before ,aristotle1 the first use of it being in the ,greek3 ,peri kosmon1 which is one of the writings doubtfully ascribed to him1 and ,h4 ,berger considers that the expression was introduced by ,eratosthenes4#d ,aristotle was certainly conversant with many facts1 such as the formation of deltas1 coast-erosion1 and to a certain extent the dependence of plants and animals on their physical surroundings4 ,he formed a comprehensive theory of the variations of climate with latitude and season1 and was convinced of the necessity of a circulation of water between the sea and rivers1 though1 like ,plato1 he held that this took place by water rising from the sea through crevices in the rocks1 losing its dissolved salts in the process4 ,he speculated on the differences in the character of races of mankind living in different climates1 and correlated the political forms of communities with their situation on a seashore1 or in the neighbourhood of natural strongholds4 ,strabo4 ,strabo 7c4 #ej ,b4,c4-,a4,d4 #bd7 followed ,eratosthenes rather than ,aristotle1 but with sympathies which went out more to the human interests than the mathematical basis of geography4 ,he compiled a very remarkable work dealing1 in large measure from personal travel1 with the countries surrounding the ,mediterranean4 ,he may be said to have set the pattern which was followed in succeeding ages by the compilers of political geographies dealing less with theories than with facts1 and illustrating rather than formulating the principles of the science4 ,ptolemy4 ,claudius ,ptolemaeus 7c4 ,a4,d4 #aej7 concentrated in his writings the final outcome of all ,greek geographical learning1 and passed it across the gulf of the middle ages by the hands of the ,arabs1 to form the starting-point of the science in modern times4 ,his geography was based more immediately on the work of his predecessor1 ,marinus of ,tyre1 and on that of ,hipparchus1 the follower and critic of ,eratosthenes4 ,it was the ambition of ,ptolemy to describe and represent accurately the surface of the oekumene1 for which purpose he took immense trouble to collect all existing determinations of the latitude of places1 all estimates of longitude1 and to make every possible rectification in the estimates of distances by land or sea4 ,his work was mainly cartographical in its aim1 and theory was as far as possible excluded4 ,the symmetrically placed hypothetical islands in the great continuous ocean disappeared1 and the oekumene acquired a new form by the representation of the ,indian ,ocean as a larger ,mediterranean completely cut off by land from the ,atlantic4 ,the terra incognita uniting ,africa and ,farther ,asia was an unfortunate hypothesis which helped to retard exploration4 ,ptolemy used the word geography to signify the description of the whole oekumene on mathematical principles1 while chorography signified the fuller description of a particular region1 and topography the very detailed description of a smaller locality4 ,he introduced the simile that geography represented an artist's sketch of a whole portrait1 while chorography corresponded to the careful and detailed drawing of an eye or an ear4#e ,the ,caliph al-,mam un 7c4 ,a4,d4 #hae71 the son and successor of ,h ar un al-,rash id1 caused an ,arabic version of ,ptolemy's great astronomical work 7,greek3 ,suntaxis megiste7 to be made1 which is known as the ,almagest1 the word being nothing more than the ,gr4 ,greek3 megiste with the ,arabic article al prefixed4 ,the geography of ,ptolemy was also known and is constantly referred to by ,arab writers4 ,the ,arab astronomers measured a degree on the plains of ,mesopotamia1 thereby deducing a fair approximation to the size of the earth4 ,the caliph's librarian1 ,abu ,jafar ,muhammad ,ben ,musa1 wrote a geographical work1 now unfortunately lost1 entitled ,rasm el ,arsi 7,a ,description of the ,world71 which is often referred to by subsequent writers as having been composed on the model of that of ,ptolemy4 ,geography in the middle ages4 ,the middle ages saw geographical knowledge die out in ,christendom1 although it retained1 through the ,arabic translations of ,ptolemy1 a certain vitality in ,islam4 ,the verbal interpretation of ,scripture led ,lactantius 7c4 ,a4,d4 #cbj7 and other ecclesiastics to denounce the spherical theory of the earth as heretical4 ,the wretched subterfuge of ,cosmas 7c4 ,a4,d4 #eej7 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 ,hecataeus4#f ,the journey of ,marco ,polo1 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 #adaj4 ,revival of geography4 ,the old arguments of ,aristotle and the old measurements of ,ptolemy were used by ,toscanelli and ,columbus in urging a westward voyage to ,india2 and mainly on this account did the crossing of the ,atlantic rank higher in the history of scientific geography than the laborious feeling out of the coast-line of ,africa4 ,but not until the voyage of ,magellan shook the scales from the eyes of ,europe did modern geography begin to advance4 ,discovery had outrun theory2 the rush of new facts made ,ptolemy practically obsolete in a generation1 after having been the fount and origin of all geography for a millennium4 ,apianus4 ,the earliest evidence of the reincarnation of a sound theoretical geography is to be found in the text-books by ,peter ,apian and ,sebastian ,munster4 ,apian in his ,cosmographicus liber1 published in #aebd1 and subsequently edited and added to by ,gemma ,frisius under the title of ,cosmographia1 based the whole science on mathematics and measurement4 ,he followed ,ptolemy closely1 enlarging on his distinction between geography and chorography1 and expressing the artistic analogy in a rough diagram4 ,this slender distinction was made much of by most subsequent writers until ,nathanael ,carpenter in #afbe pointed out that the difference between geography and chorography was simply one of degree1 not of kind4 ,munster4 ,sebastian ,munster1 on the other hand1 in his ,cosmographia universalis of #aedd1 paid no regard to the mathematical basis of geography1 but1 following the model of ,strabo1 described the world according to its different political divisions1 and entered with great zest into the question of the productions of countries1 and into the manners and costumes of the various peoples4 ,thus early commenced the separation between what were long called mathematical and political geography1 the one subject appealing mainly to mathematicians1 the other to historians4 ,throughout the #afth and #agth centuries the rapidly accumulating store of facts as to the extent1 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 chief2 but the writings of ,apian and ,munster held the field for a hundred years without a serious rival1 unless the many annotated editions of ,ptolemy might be so considered4 ,meanwhile the new facts were the subject of original study by philosophers and by practical men without reference to classical traditions4 ,bacon argued keenly on geographical matters and was a lover of maps1 in which he observed and reasoned upon such resemblances as that between the outlines of ,south ,america and ,africa4 ,cluverius4 ,philip ,cluver's ,introductio in geographiam universam tam veterem quam novam was published in #afbd4 ,geography he defined as the description of the whole earth1 so far as it is known to us4 ,it is distinguished from cosmography by dealing with the earth alone1 not with the universe1 and from chorography and topography by dealing with the whole earth1 not with a country or a place4 ,the first book1 of fourteen short chapters1 is concerned with the general properties of the globe2 the remaining six books treat in considerable detail of the countries of ,europe and of the other continents4 ,each country is described with particular regard to its people as well as to its surface1 and the prominence given to the human element is of special interest4 ,carpenter4 ,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 #afbe by ,nathanael ,carpenter1 fellow of ,exeter ,college1 with the title ,geographie delineated forth in ,two ,bookes1 containing the ,sphericall and ,topicall parts thereof4 ,it is discursive in its style and verbose2 but1 considering the period at which it appeared1 it is remarkable for the strong common sense displayed by the author1 his comparative freedom from prejudice1 and his firm application of the methods of scientific reasoning to the interpretation of phenomena4 ,basing his work on the principles of ,ptolemy1 he brings together illustrations from the most recent travellers1 and does not hesitate to take as illustrative examples the familiar city of ,oxford and his native county of ,devon4 ,he divides geography into ,the ,spherical ,part1 or that for the study of which mathematics alone is required1 and ,the ,topical ,part1 or the description of the physical relations of parts of the earth's surface1 preferring this division to that favoured by the ancient geographers--into general and special4 ,it is distinguished from other ,english geographical books of the period by confining attention to the principles of geography1 and not describing the countries of the world4 ,varenius4 ,a much more important work in the history of geographical method is the ,geographia generalis of ,bernhard ,varenius1 a ,german medical doctor of ,leiden1 who died at the age of twenty-eight in #afej1 the year of the publication of his book4 ,although for a time it was lost sight of on the continent1 ,sir ,isaac ,newton thought so highly of this book that he prepared an annotated edition which was published in ,cambridge in #afgb1 with the addition of the plates which had been planned by ,varenius1 but not produced by the original publishers4 ,the reason why this great man took so much care in correcting and publishing our author was1 because he thought him necessary to be read by his audience1 the young gentlemen of ,cambridge1 while he was delivering lectures on the same subject from the ,lucasian ,chair4#g ,the treatise of ,varenius is a model of logical arrangement and terse expression2 it is a work of science and of genius2 one of the few of that age which can still be studied with profit4 ,the ,english translation renders the definition thus3 ,geography is that part of mixed mathematics which explains the state of the earth and of its parts1 depending on quantity1 viz4 its figure1 place1 magnitude and motion1 with the celestial appearances1 c4 ,by some it is taken in too limited a sense1 for a bare description of the several countries2 and by others too extensively1 who along with such a description would have their political constitution4 ,varenius was reluctant to include the human side of geography in his system1 and only allowed it as a concession to custom1 and in order to attract readers by imparting interest to the sterner details of the science4 ,his division of geography was into two parts--7i47 ,general or universal1 dealing with the earth in general1 and explaining its properties without regard to particular countries2 and 7ii47 ,special or particular1 dealing with each country in turn from the chorographical or topographical point of view4 ,general geography was divided into--7#a7 the ,absolute part1 dealing with the form1 dimensions1 position and substance of the earth1 the distribution of land and water1 mountains1 woods and deserts1 hydrography 7including all the waters of the earth7 and the atmosphere2 7#b7 the ,relative part1 including the celestial properties1 i4e4 latitude1 climate zones1 longitude1 c42 and 7#c7 the ,comparative part1 which considers the particulars arising from comparing one part with another2 but under this head the questions discussed were longitude1 the situation and distances of places1 and navigation4 ,varenius does not treat of special geography1 but gives a scheme for it under three heads--7#a7 ,terrestrial1 including position1 outline1 boundaries1 mountains1 mines1 woods and deserts1 waters1 fertility and fruits1 and living creatures2 7#b7 ,celestial1 including appearance of the heavens and the climate2 7#c7 ,human1 but this was added out of deference to popular usage4 ,this system of geography founded a new epoch1 and the book--translated into ,english1 ,dutch and ,french--was the unchallenged standard for more than a century4 ,the framework was capable of accommodating itself to new facts1 and was indeed far in advance of the knowledge of the period4 ,the method included a recognition of the causes and effects of phenomena as well as the mere fact of their occurrence1 and for the first time the importance of the vertical relief of the land was fairly recognized4 ,the physical side of geography continued to be elaborated after ,varenius's methods1 while the historical side was developed separately4 ,both branches1 although enriched by new facts1 remained stationary so far as method is concerned until nearly the end of the #ahth century4 ,the compilation of geography books by uninstructed writers led to the pernicious habit1 which is not yet wholly overcome1 of reducing the general or physical part to a few pages of concentrated information1 and expanding the particular or political part by including unrevised travellers' stories and uncritical descriptions of the various countries of the world4 ,such books were in fact not geography1 but merely compressed travel4 ,bergman4 ,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 chemist1 ,torbern ,bergman1 acting under the impulse of ,linnaeus1 and by the ,german philosopher1 ,immanuel ,kant4 ,bergman's ,physical ,description of the ,earth was published in ,swedish in #agff1 and translated into ,english in #aggb and into ,german in #aggd4 ,it is a plain1 straightforward description of the globe1 and of the various phenomena of the surface1 dealing only with definitely ascertained facts in the natural order of their relationships1 but avoiding any systematic classification or even definitions of terms4 ,kant4 ,the problems of geography had been lightened by the destructive criticism of the ,french cartographer ,d',anville 7who had purged the map of the world of the last remnants of traditional fact unverified by modern observations7 and rendered richer by the dawn of the new era of scientific travel1 when ,kant brought his logical powers to bear upon them4 ,kant's lectures on physical geography were delivered in the university of ,konigsberg from #agfe onwards4#h ,geography appealed to him as a valuable educational discipline1 the joint foundation with anthropology of that knowledge of the world which was the result of reason and experience4 ,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 geographical2 both history and geography being viewed as descriptions1 the former a description in order of time1 the latter a description in order of space4 ,physical geography he viewed as a summary of nature1 the basis not only of history but also of all the other possible geographies1 of which he enumerates five1 viz4 7#a7 ,mathematical geography1 which deals with the form1 size and movements of the earth and its place in the solar system2 7#b7 ,moral geography1 or an account of the different customs and characters of mankind according to the region they inhabit2 7#c7 ,political geography1 the divisions according to their organized governments2 7#d7 ,mercantile geography1 dealing with the trade in the surplus products of countries2 7#e7 ,theological geography1 or the distribution of religions4 ,here there is a clear and formal statement of the interaction and causal relation of all the phenomena of distribution on the earth's surface1 including the influence of physical geography upon the various activities of mankind from the lowest to the highest4 ,notwithstanding the form of this classification1 ,kant himself treats mathematical geography as preliminary to1 and therefore not dependent on1 physical geography4 ,physical geography itself is divided into two parts3 a general1 which has to do with the earth and all that belongs to it--water1 air and land2 and a particular1 which deals with special products of the earth--mankind1 animals1 plants and minerals4 ,particular importance is given to the vertical relief of the land1 on which the various branches of human geography are shown to depend4 ,humboldt4 ,alexander von ,humboldt 7#agfi-ahei7 was the first modern geographer to become a great traveller1 and thus to acquire an extensive stock of first-hand information on which an improved system of geography might be founded4 ,the impulse given to the study of natural history by the example of ,linnaeus2 the results brought back by ,sir ,joseph ,banks1 ,dr ,solander and the two ,forsters1 who accompanied ,cook in his voyages of discovery2 the studies of ,de ,saussure in the ,alps1 and the lists of desiderata in physical geography drawn up by that investigator1 combined to prepare the way for ,humboldt4 ,the theory of geography was advanced by ,humboldt mainly by his insistence on the great principle of the unity of nature4 ,he brought all the observable things1 which the eager collectors of the previous century had been heaping together regardless of order or system1 into relation with the vertical relief and the horizontal forms of the earth's surface4 ,thus he demonstrated that the forms of the land exercise a directive and determining influence on climate1 plant life1 animal life and on man himself4 ,this was no new idea2 it had been familiar for centuries in a less definite form1 deduced from a priori considerations1 and so far as regards the influence of surrounding circumstances upon man1 ,kant had already given it full expression4 ,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 effect4 ,the treatises on physical geography by ,mrs ,mary ,somerville and ,sir ,john ,herschel 7the latter written for the eighth edition of the ,encyclopaedia ,britannica7 showed the effect produced in ,great ,britain by the stimulus of ,humboldt's work4 ,ritter4 ,humboldt's contemporary1 ,carl ,ritter 7#aggi-ahei71 extended and disseminated the same views1 and in his interpretation of ,comparative ,geography he laid stress on the importance of forming conclusions1 not from the study of one region by itself1 but from the comparison of the phenomena of many places4 ,impressed by the influence of terrestrial relief and climate on human movements1 ,ritter was led deeper and deeper into the study of history and archaeology4 ,his monumental ,vergleichende ,geographie1 which was to have made the whole world its theme1 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 ,africa4 ,some of his followers showed a tendency to look on geography rather as an auxiliary to history than as a study of intrinsic worth4 ,geography as a natural science4 ,during the rapid development of physical geography many branches of the study of nature1 which had been included in the cosmography of the early writers1 the physiography of ,linnaeus and even the ,erdkunde of ,ritter1 had been so much advanced by the labours of specialists that their connexion was apt to be forgotten4 ,thus geology1 meteorology1 oceanography and anthropology developed into distinct sciences4 ,the absurd attempt was1 and sometimes is still1 made by geographers to include all natural science in geography2 but it is more common for specialists in the various detailed sciences to think1 and sometimes to assert1 that the ground of physical geography is now fully occupied by these sciences4 ,political geography has been too often looked on from both sides as a mere summary of guide-book knowledge1 useful in the schoolroom1 a poor relation of physical geography that it was rarely necessary to recognize4 ,the science of geography1 passed on from antiquity by ,ptolemy1 re-established by ,varenius and ,newton1 and systematized by ,kant1 included within itself definite aspects of all those terrestrial phenomena which are now treated exhaustively under the heads of geology1 meteorology1 oceanography and anthropology2 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 itself4 ,the study of geography was advanced by improvements in cartography 7see ,,map71 not only in the methods of survey and projection1 but in the representation of the third dimension by means of contour lines introduced by ,philippe ,buache in #agcg1 and the more remarkable because less obvious invention of isotherms introduced by ,humboldt in #ahag4 ,the teleological argument in geography4 ,the argument from design had been a favourite form of reasoning amongst ,christian theologians1 and1 as worked out by ,paley in his ,natural ,theology1 it served the useful purpose of emphasizing the fitness which exists between all the inhabitants of the earth and their physical environment4 ,it was held that the earth had been created so as to fit the wants of man in every particular4 ,this argument was tacitly accepted or explicitly avowed by almost every writer on the theory of geography1 and ,carl ,ritter distinctly recognized and adopted it as the unifying principle of his system4 ,as a student of nature1 however1 he did not fail to see1 and as professor of geography he always taught1 that man was in very large measure conditioned by his physical environment4 ,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 man4 ,this was the central theme of ,ritter's philosophy2 his religion and his geography were one1 and the consequent fervour with which he pursued his mission goes far to account for the immense influence he acquired in ,germany4 ,the theory of evolution in geography4 ,the evolutionary theory1 more than hinted at in ,kant's ,physical ,geography1 has1 since the writings of ,charles ,darwin1 become the unifying principle in geography4 ,the conception of the development of the plan of the earth from the first cooling of the surface of the planet throughout the long geological periods1 the guiding power of environment on the circulation of water and of air1 on the distribution of plants and animals1 and finally on the movements of man1 give to geography a philosophical dignity and a scientific completeness which it never previously possessed4 ,the influence of environment on the organism may not be quite so potent as it was once believed to be1 in the writings of ,buckle1 for instance1#i and certainly man1 the ultimate term in the series1 reacts upon and greatly modifies his environment2 yet the fact that environment does influence all distributions is established beyond the possibility of doubt4 ,in this way also the position of geography1 at the point where physical science meets and mingles with mental science1 is explained and justified4 ,the change which took place during the #aith century in the substance and style of geography may be well seen by comparing the eight volumes of ,malte-,brun's ,geographie universelle 7,paris1 #ahab-ahbi7 with the twenty-one volumes of ,reclus's ,geographie universelle 7,paris1 #ahgf-ahie74 ,in estimating the influence of recent writers on geography it is usual to assign to ,oscar ,peschel 7#ahbf-ahge7 the credit of having corrected the preponderance which ,ritter gave to the historical element1 and of restoring physical geography to its old pre-eminence4#aj ,as a matter of fact1 each of the leading modern exponents of theoretical geography--such as ,ferdinand von ,richthofen1 ,hermann ,wagner1 ,friedrich ,ratzel1 ,william ,m4 ,davis1 ,a4 ,penck1 ,a4 de ,lapparent and ,elisee ,reclus--has his individual point of view1 one devoting more attention to the results of geological processes1 another to anthropological conditions1 and the rest viewing the subject in various blendings of the extreme lights4 ,the two conceptions which may now be said to animate the theory of geography are the genetic1 which depends upon processes of origin1 and the morphological1 which depends on facts of form and distribution4 ,,progress ,,of ,,geographical ,,discovery ,exploration and geographical discovery must have started from more than one centre1 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 intercommunication4 ,at the least there should be some consideration of four separate systems of discovery--the ,eastern1 in which ,chinese and ,japanese explorers acquired knowledge of the geography of ,asia1 and felt their way towards ,europe and ,america2 the ,western1 in which the dominant races of the ,mexican and ,south ,american plateaus extended their knowledge of the ,american continent before ,columbus2 the ,polynesian1 in which the conquering races of the ,pacific ,islands found their way from group to group2 and the ,mediterranean4 ,for some of these we have no certain information1 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 history4 ,so it has come about that the only practicable history of geographical exploration starts from the ,mediterranean centre1 the first home of that civilization which has come to be known as ,european1 though its field of activity has long since overspread the habitable land of both temperate zones1 eastern ,asia alone in part excepted4 ,from all centres the leading motives of exploration were probably the same--commercial intercourse1 warlike operations1 whether resulting in conquest or in flight1 religious zeal expressed in pilgrimages or missionary journeys1 or1 from the other side1 the avoidance of persecution1 and1 more particularly in later years1 the advancement of knowledge for its own sake4 ,at different times one or the other motive predominated4 ,before the #adth century ,b4,c4 the warrior kings of ,egypt had carried the power of their arms southward from the delta of the ,nile well-nigh to its source1 and eastward to the confines of ,assyria4 ,the hieroglyphic inscriptions of ,egypt and the cuneiform inscriptions of ,assyria are rich in records of the movements and achievements of armies1 the conquest of towns and the subjugation of peoples2 but though many of the recorded sites have been identified1 their discovery by wandering armies was isolated from their subsequent history and need not concern us here4 ,the ,phoenicians4 ,the ,phoenicians are the earliest ,mediterranean people in the consecutive chain of geographical discovery which joins pre-historic time with the present4 ,from ,sidon1 and later from its more famous rival ,tyre1 the merchant adventurers of ,phoenicia explored and colonized the coasts of the ,mediterranean and fared forth into the ocean beyond4 ,they traded also on the ,red sea1 and opened up regular traffic with ,india as well as with the ports of the south and west1 so that it was natural for ,solomon to employ the merchant navies of ,tyre in his oversea trade4 ,the western emporium known in the scriptures as ,tarshish was probably situated in the south of ,spain1 possibly at ,cadiz1 although some writers contend that it was ,carthage in ,north ,africa4 ,still more diversity of opinion prevails as to the southern gold-exporting port of ,ophir1 which some scholars place in ,arabia1 others at one or another point on the east coast of ,africa4 ,whether associated with the exploitation of ,ophir 7q4v47 or not the first great voyage of ,african discovery appears to have been accomplished by the ,phoenicians sailing the ,red ,sea4 ,herodotus 7himself a notable traveller in the #eth century ,b4,c47 relates that the ,egyptian king ,necho of the ,x,x,v,ith ,dynasty 7c4 #fjj ,b4,c47 built a fleet on the ,red ,sea1 and confided it to ,phoenician sailors with the orders to sail southward and return to ,egypt by the ,pillars of ,hercules and the ,mediterranean sea4 ,according to the tradition1 which ,herodotus quotes sceptically1 this was accomplished2 but the story is too vague to be accepted as more than a possibility4 ,the great ,phoenician colony of ,carthage1 founded before #hjj ,b4,c41 perpetuated the commercial enterprise of the parent state1 and extended the sphere of practical trade to the ocean shores of ,africa and ,europe4 ,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 ,hanno1 with the intention of founding new colonies along the west coast of ,africa4 ,according to ,pliny1 the only authority on this point1 the period of the voyage was that of the greatest prosperity of ,carthage1 which may be taken as somewhere between #egj and #dhj ,b4,c4 ,the extent of this voyage is doubtful1 but it seems probable that the farthest point reached was on the east-running coast which bounds the ,gulf of ,guinea on the north4 ,himilco1 a contemporary of ,hanno1 was charged with an expedition along the west coast of ,iberia northward1 and as far as the uncertain references to this voyage can be understood1 he seems to have passed the ,bay of ,biscay and possibly sighted the coast of ,england4 ,the ,greeks4 ,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 ,himilco1 and peculiar interest attaches to the first recorded ,greek voyage beyond the ,pillars of ,hercules4 ,pytheas1 a navigator of the ,phocean colony of ,massilia 7,marseilles71 determined the latitude of that port with considerable precision by the somewhat clumsy method of ascertaining the length of the longest day1 and when1 about #ccj ,b4,c41 he set out on exploration to the northward in search of the lands whence came gold1 tin and amber1 he followed this system of ascertaining his position from time to time4 ,if on each occasion he himself made the observations his voyage must have extended over six years2 but it is not impossible that he ascertained the approximate length of the longest day in some cases by questioning the natives4 ,pytheas1 whose own narrative is not preserved1 coasted the ,bay of ,biscay1 sailed up the ,english ,channel and followed the coast of ,britain to its most northerly point4 ,beyond this he spoke of a land called ,thule1 which1 if his estimate of the length of the longest day is correct1 may have been ,shetland1 but was possibly ,iceland2 and from some confused statements as to a sea which could not be sailed through1 it has been assumed that ,pytheas was the first of the ,greeks to obtain direct knowledge of the ,arctic regions4 ,during this or a second voyage ,pytheas entered the ,baltic1 discovered the coasts where amber is obtained and returned to the ,mediterranean4 ,it does not seem that any maritime trade followed these discoveries1 and indeed it is doubtful whether his contemporaries accepted the truth of ,pytheas's narrative2 ,strabo four hundred years later certainly did not1 but the critical studies of modern scholars have rehabilitated the ,massilian explorer4 ,alexander the ,great4 ,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 ,great1 but the campaigns of that conqueror from #cbi to #cbe ,b4,c4 opened up the greater ,asia to the knowledge of ,europe4 ,his armies crossed the plains beyond the ,caspian1 penetrated the wild mountain passes north-west of ,india1 and did not turn back until they had entered on the ,indo-,gangetic plain4 ,this was one of the few great epochs of geographical discovery4 ,the world was henceforth viewed as a very large place stretching far on every side beyond the ,midland or ,mediterranean ,sea1 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 ,tigris1 thus opening direct intercourse between ,grecian and ,hindu civilization4 ,the ,greeks who accompanied ,alexander described with care the towns and villages1 the products and the aspect of the country4 ,the conqueror also intended to open up trade by sea between ,europe and ,india1 and the narrative of his general ,nearchus records this famous voyage of discovery1 the detailed accounts of the chief pilot ,onesicritus being lost4 ,at the beginning of ,october #cbf ,b4,c4 ,nearchus left the ,indus with his fleet1 and the anchorages sought for each night are carefully recorded4 ,he entered the ,persian ,gulf1 and rejoined ,alexander at ,susa1 when he was ordered to prepare another expedition for the circumnavigation of ,arabia4 ,alexander died at ,babylon in #cbc ,b4,c41 and the fleet was dispersed without making the voyage4 ,the dynasties founded by ,alexander's generals1 ,seleucus1 ,antiochus and ,ptolemy1 encouraged the same spirit of enterprise which their master had fostered1 and extended geographical knowledge in several directions4 ,seleucus ,nicator established the ,greco-,bactrian empire and continued the intercourse with ,india4 ,authentic information respecting the great valley of the ,ganges was supplied by ,megasthenes1 an ambassador sent by ,seleucus1 who reached the remote city of ,patali-putra1 the modern ,patna4 ,the ,ptolemies4 ,the ,ptolemies in ,egypt showed equal anxiety to extend the bounds of geographical knowledge4 ,ptolemy ,euergetes 7#bdg-bbb ,b4,c47 rendered the greatest service to geography by the protection and encouragement of ,eratosthenes1 whose labours gave the first approximate knowledge of the true size of the spherical earth4 ,the second ,euergetes and his successor ,ptolemy ,lathyrus 7#aah-aae ,b4,c47 furnished ,eudoxus with a fleet to explore the ,arabian sea4 ,after two successful voyages1 ,eudoxus1 impressed with the idea that ,africa was surrounded by ocean on the south1 left the ,egyptian service1 and proceeded to ,cadiz and other ,mediterranean centres of trade seeking a patron who would finance an expedition for the purpose of ,african discovery2 and we learn from ,strabo that the veteran explorer made at least two voyages southward along the coast of ,africa4 ,the ,ptolemies continued to send fleets annually from their ,red ,sea ports of ,berenice and ,myos ,hormus to ,arabia1 as well as to ports on the coasts of ,africa and ,india4 ,the ,romans4 ,the ,romans did not encourage navigation and commerce with the same ardour as their predecessors2 still the luxury of ,rome1 which gave rise to demands for the varied products of all the countries of the known world1 led to an active trade both by ships and caravans4 ,but it was the military genius of ,rome1 and the ambition for universal empire1 which led1 not only to the discovery1 but also to the survey of nearly all ,europe1 and of large tracts in ,asia and ,africa4 ,every new war produced a new survey and itinerary of the countries which were conquered1 and added one more to the imperishable roads that led from every quarter of the known world to ,rome4 ,in the height of their power the ,romans had surveyed and explored all the coasts of the ,mediterranean1 ,italy1 ,greece1 the ,balkan ,peninsula1 ,spain1 ,gaul1 western ,germany and southern ,britain4 ,in ,africa their empire included ,egypt1 ,carthage1 ,numidia and ,mauritania4 ,in ,asia they held ,asia ,minor and ,syria1 had sent expeditions into ,arabia1 and were acquainted with the more distant countries formerly invaded by ,alexander1 including ,persia1 ,scythia1 ,bactria and ,india4 ,roman intercourse with ,india especially led to the extension of geographical knowledge4 ,before the ,roman legions were sent into a new region to extend the limits of the empire1 it was usual to send out exploring expeditions to report as to the nature of the country4 ,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 7probably about ,a4,d4 #fj71 and that the travellers pushed southwards until they reached vast marshes through which they could not make their way either on foot or in boats4 ,this seems to indicate that they had penetrated to about #i deg4 ,n4 ,shortly before ,a4,d4 #gi ,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 land4 ,even though this sea-route was known1 the author of the ,periplus of the ,erythraean ,sea1 published after the time of ,pliny1 recites the old itinerary around the coast of the ,arabian ,gulf4 ,it was1 however1 in the reigns of ,severus and his immediate successors that ,roman intercourse with ,india was at its height1 and from the writings of ,pausanias 7c4 #agd7 it appears that direct communication between ,rome and ,china had already taken place4 ,after the division of the ,roman empire1 ,constantinople became the last refuge of learning1 arts and taste2 while ,alexandria continued to be the emporium whence were imported the commodities of the ,east4 ,the emperor ,justinian 7#dhc-efe71 in whose reign the greatness of the ,eastern empire culminated1 sent two ,nestorian monks to ,china1 who returned with eggs of the silkworm concealed in a hollow cane1 and thus silk manufactures were established in the ,peloponnesus and the ,greek islands4 ,it was also in the reign of ,justinian that ,cosmas ,indicopleustes1 an ,egyptian merchant1 made several voyages1 and afterwards composed his ,greek3 ,christianike topographia 7,christian ,topography71 containing1 in addition to his absurd cosmogony1 a tolerable description of ,india4 ,the ,arabs4 ,the great outburst of ,mahommedan conquest in the #gth century was followed by the ,arab civilization1 having its centres at ,bagdad and ,cordova1 in connexion with which geography again received a share of attention4 ,the works of the ancient ,greek geographers were translated into ,arabic1 and starting with a sound basis of theoretical knowledge1 exploration once more made progress4 ,from the #ith to the #acth century intelligent ,arab travellers wrote accounts of what they had seen and heard in distant lands4 ,the earliest ,arabian traveller whose observations have come down to us is the merchant ,sulaiman1 who embarked in the ,persian ,gulf and made several voyages to ,india and ,china1 in the middle of the #ith century4 ,abu ,zaid also wrote on ,india1 and his work is the most important that we possess before the epoch-making discoveries of ,marco ,polo4 ,masudi1 a great traveller who knew from personal experience all the countries between ,spain and ,china1 described the plains1 mountains and seas1 the dynasties and peoples1 in his ,meadows of ,gold1 an abstract made by himself of his larger work ,news of the ,time4 ,he died in #ief1 and was known1 from the comprehensiveness of his survey1 as the ,pliny of the ,east4 ,amongst his contemporaries were ,istakhri1 who travelled through all the ,mahommedan countries and wrote his ,book of ,climates in #iej1 and ,ibn ,haukal1 whose ,book of ,roads and ,kingdoms1 based on the work of ,istakhri1 was written in #igf4 ,idrisi1 the best known of the ,arabian geographical authors1 after travelling far and wide in the first half of the #abth century1 settled in ,sicily1 where he wrote a treatise descriptive of an armillary sphere which he had constructed for ,roger ,,ii41 the ,norman king1 and in this work he incorporated all accessible results of contemporary travel4 ,the ,northmen4 ,the ,northmen of ,denmark and ,norway1 whose piratical adventures were the terror of all the coasts of ,europe1 and who established themselves in ,great ,britain and ,ireland1 in ,france and ,sicily1 were also geographical explorers in their rough but practical way during the darkest period of the middle ages4 ,all ,northmen were not bent on rapine and plunder2 many were peaceful merchants4 ,alfred the ,great1 king of the ,saxons in ,england1 not only educated his people in the learning of the past ages2 he inserted in the geographical works he translated many narratives of the travel of his own time4 ,thus he placed on record the voyages of the merchant ,ulfsten in the ,baltic1 including particulars of the geography of ,germany4 ,and in particular he told of the remarkable voyage of ,other1 a ,norwegian of ,helgeland1 who was the first authentic ,arctic explorer1 the first to tell of the rounding of the ,north ,cape and the sight of the midnight sun4 ,this voyage of the middle of the #ith century deserves to be held in happy memory1 for it unites the first ,norwegian polar explorer with the first ,english collector of travels4 ,scandinavian merchants brought the products of ,india to ,england and ,ireland4 ,from the #hth to the #aath century a commercial route from ,india passed through ,novgorod to the ,baltic1 and ,arabian coins found in ,sweden1 and particularly in the island of ,gotland1 prove how closely the enterprise of the ,northmen and of the ,arabs intertwined4 ,five-sixths of these coins preserved at ,stockholm were from the mints of the ,samanian dynasty1 which reigned in ,khorasan and ,transoxiana from about ,a4,d4 #ijj to #ajjj4 ,it was the trade with the ,east that originally gave importance to the city of ,visby in ,gotland4 ,in the end of the #ith century ,iceland was colonized from ,norway2 and about #ihe the intrepid viking1 ,eric the ,red1 discovered ,greenland1 and induced some of his ,icelandic countrymen to settle on its inhospitable shores4 ,his son1 ,leif ,ericsson1 and others of his followers were concerned in the discovery of the ,north ,american coast 7see ,,vinland71 which1 but for the isolation of ,iceland from the centres of ,european awakening1 would have had momentous consequences4 ,as things were1 the importance of this discovery passed unrecognized4 ,the story of two ,venetians1 ,nicolo and ,antonio ,zeno1 who gave a vague account of voyages in the northern seas in the end of the #acth century1 is no longer to be accepted as history4 ,close of the dark ages4 ,at length the long period of barbarism which accompanied and followed the fall of the ,roman empire drew to a close in ,europe4 ,the ,crusades had a favourable influence on the intellectual state of the ,western nations4 ,interesting regions1 known only by the scant reports of pilgrims1 were made the objects of attention and study2 while religious zeal1 and the hope of gain1 combined with motives of mere curiosity1 induced several persons to travel by land into remote regions of the ,east1 far beyond the countries to which the operations of the crusaders extended4 ,among these was ,benjamin of ,tudela1 who set out from ,spain in #aafj1 travelled by land to ,constantinople1 and having visited ,india and some of the eastern islands1 returned to ,europe by way of ,egypt after an absence of thirteen years4 ,asiatic journeys4 ,joannes de ,plano ,carpini1 a ,franciscan monk1 was the head of one of the missions despatched by ,pope ,innocent to call the chief and people of the ,tatars to a better mind4 ,he reached the headquarters of ,batu1 on the ,volga1 in ,february #abdf2 and1 after some stay1 went on to the camp of the great khan near ,karakorum in central ,asia1 and returned safely in the autumn of #abdg4 ,a few years afterwards1 a ,fleming named ,rubruquis was sent on a similar mission1 and had the merit of being the first traveller of this era who gave a correct account of the ,caspian ,sea4 ,he ascertained that it had no outlet4 ,at nearly the same time ,hayton1 king of ,armenia1 made a journey to ,karakorum in #abed1 by a route far to the north of that followed by ,carpini and ,rubruquis4 ,he was treated with honour and hospitality1 and returned by way of ,samarkand and ,tabriz1 to his own territory4 ,the curious narrative of ,king ,hayton was translated by ,klaproth4 ,while the republics of ,italy1 and above all the state of ,venice1 were engaged in distributing the rich products of ,india and the ,far ,east over the ,western world1 it was impossible that motives of curiosity1 as well as a desire of commercial advantage1 should not be awakened to such a degree as to impel some of the merchants to visit those remote lands4 ,among these were the brothers ,polo1 who traded with the ,east and themselves visited ,tatary4 ,the recital of their travels fired the youthful imagination of young ,marco ,polo1 son of ,nicolo1 and he set out for the court of ,kublai ,khan1 with his father and uncle1 in #abfe4 ,marco remained for seventeen years in the service of the ,great ,khan1 and was employed on many important missions4 ,besides what he learnt from his own observation1 he collected much information from others concerning countries which he did not visit4 ,he returned to ,europe possessed of a vast store of knowledge respecting the eastern parts of the world1 and1 being afterwards made a prisoner by the ,genoese1 he dictated the narrative of his travels during his captivity4 ,the work of ,marco ,polo is the most valuable narrative of travels that appeared during the middle ages1 and despite a cold reception and many denials of the accuracy of the record1 its substantial truthfulness has been abundantly proved4 ,missionaries continued to do useful geographical work4 ,among them were ,john of ,monte ,corvino1 a ,franciscan monk1 ,andrew of ,perugia1 ,john ,marignioli and ,friar ,jordanus1 who visited the west coast of ,india1 and above all ,friar ,odoric of ,pordenone4 ,odoric set out on his travels about #acah1 and his journeys embraced parts of ,india1 the ,malay ,archipelago1 ,china and even ,tibet1 where he was the first ,european to enter ,lhasa1 not yet a forbidden city4 ,ibn ,batuta1 the great ,arab traveller1 is separated by a wide space of time from his countrymen already mentioned1 and he finds his proper place in a chronological notice after the days of ,marco ,polo1 for he did not begin his wanderings until #acbe1 his career thus coinciding in time with the fabled journeyings of ,sir ,john ,mandeville4 ,while ,arab learning flourished during the darkest ages of ,european ignorance1 the last of the ,arab geographers lived to see the dawn of the great period of the ,european awakening4 ,ibn ,batuta went by land from ,tangier to ,cairo1 then visited ,syria1 and performed the pilgrimages to ,medina and ,mecca4 ,after exploring ,persia1 and again residing for some time at ,mecca1 he made a voyage down the ,red sea to ,yemen1 and travelled through that country to ,aden4 ,thence he visited the ,african coast1 touching at ,mombasa and ,quiloa1 and then sailed across to ,ormuz and the ,persian ,gulf4 ,he crossed ,arabia from ,bahrein to ,jidda1 traversed the ,red sea and the desert to ,syene1 and descended the ,nile to ,cairo4 ,after this he revisited ,syria and ,asia ,minor1 and crossed the ,black sea1 the desert from ,astrakhan to ,bokhara1 and the ,hindu ,kush4 ,he was in the service of ,muhammad ,tughluk1 ruler of ,delhi1 about eight years1 and was sent on an embassy to ,china1 in the course of which the ambassadors sailed down the west coast of ,india to ,calicut1 and then visited the ,maldive ,islands and ,ceylon4 ,ibn ,batuta made the voyage through the ,malay ,archipelago to ,china1 and on his return he proceeded from ,malabar to ,bagdad and ,damascus1 ultimately reaching ,fez1 the capital of his native country1 in ,november #acdi4 ,after a journey into ,spain he set out once more for ,central ,africa in #aceb1 and reached ,timbuktu and the ,niger1 returning to ,fez in #acec4 ,his narrative was committed to writing from his dictation4 ,spanish exploration4 ,the ,european country which had come the most completely under the influence of ,arab culture now began to send forth explorers to distant lands1 though the impulse came not from the ,moors but from ,italian merchant navigators in ,spanish service4 ,the peaceful reign of ,henry ,,iii4 of ,castile is famous for the attempts of that prince to extend the diplomatic relations of ,spain to the remotest parts of the earth4 ,he sent embassies to all the princes of ,christendom and to the ,moors4 ,in #adjc the ,spanish king sent a knight of ,madrid1 ,ruy ,gonzalez de ,clavijo1 to the distant court of ,timur1 at ,samarkand4 ,he returned in #adjf1 and wrote a valuable narrative of his travels4 ,italians continued to make important journeys in the ,east during the #aeth century4 ,among them was ,nicolo ,conti1 who passed through ,persia1 sailed along the coast of ,malabar1 visited ,sumatra1 ,java and the south of ,china1 returned by the ,red sea1 and got home to ,venice in #addd after an absence of twenty-five years4 ,he related his adventures to ,poggio ,bracciolini1 secretary to ,pope ,eugenius ,,iv42 and the narrative contains much interesting information4 ,one of the most remarkable of the ,italian travellers was ,ludovico di ,varthema1 who left his native land in #aejb4 ,he went to ,egypt and ,syria1 and for the sake of visiting the holy cities became a ,mahommedan4 ,he was the first ,european who gave an account of the interior of ,yemen4 ,he afterwards visited and described many places in ,persia1 ,india and the ,malay ,archipelago1 returning to ,europe in a ,portuguese ship after an absence of five years4 ,portuguese exploration--,prince ,henry the ,navigator4 ,in the #aeth century the time was approaching when the discovery of the ,cape of ,good ,hope was to widen the scope of geographical enterprise4 ,this great event was preceded by the general utilization in ,europe of the polarity of the magnetic needle in the construction of the mariner's compass4 ,portugal took the lead along this new path1 and foremost among her pioneers stands ,prince ,henry the ,navigator 7#acid-adfj71 who was a patron both of exploration and of the study of geographical theory4 ,the great westward projection of the coast of ,africa1 and the islands to the north-west of that continent1 were the principal scene of the work of the mariners sent out at his expense2 but his object was to push onward and reach ,india from the ,atlantic4 ,the progress of discovery received a check on his death1 but only for a time4 ,in #adfb ,pedro de ,cintra extended ,portuguese exploration along the ,african coast and discovered ,sierra ,leone4 ,fernan ,gomez followed in #adfi1 and opened trade with the ,gold ,coast2 and in #adhd ,diogo ,cao discovered the mouth of the ,congo4 ,the king of ,portugal next despatched ,bartolomeu ,diaz in #adhf to continue discoveries southwards2 while1 in the following year1 he sent ,pedro de ,covilhao and ,affonso de ,payva to discover the country of ,prester ,john4 ,diaz succeeded in rounding the southern point of ,africa1 which he named ,cabo ,tormentoso--the ,cape of ,storms--but ,king ,joao ,,ii41 foreseeing the realization of the long-sought passage to ,india1 gave it the stimulating and enduring name of the ,cape of ,good ,hope4 ,payva died at ,cairo2 but ,covilhao1 having heard that a ,christian ruler reigned in the mountains of ,ethiopia1 penetrated into ,abyssinia in #adij4 ,he delivered the letter which ,joao ,,ii4 had addressed to ,prester ,john to the ,negus ,alexander of ,abyssinia1 but he was detained by that prince and never allowed to leave the country4 ,columbus4 ,the ,portuguese1 following the lead of ,prince ,henry1 continued to look for the road to ,india by the ,cape of ,good ,hope4 ,the same end was sought by ,christopher ,columbus1 following the suggestion of ,toscanelli1 and under-estimating the diameter of the globe1 by sailing due west4 ,the voyages of ,columbus 7#adib-adih7 resulted in the discovery of the ,west ,indies and ,north ,america which barred the way to the ,far ,east4 ,in #adic the pope1 ,alexander ,,vi41 issued a bull instituting the famous line of demarcation running from ,n4 to ,s4 #ajj leagues ,w4 of the ,azores1 to the west of which the ,spaniards were authorized to explore and to the east of which the ,portuguese received the monopoly of discovery4 ,the direct line of ,portuguese exploration resulted in the discovery of the ,cape route to ,india by ,vasco da ,gama 7#adih71 and in #aejj to the independent discovery of ,south ,america by ,pedro ,alvarez ,cabral4 ,the voyages of ,columbus and of ,vasco da ,gama were so important that it is unnecessary to detail their results in this place4 ,see ,,columbus1 ,,christopher2 ,,gama1 ,,vasco ,,da4 ,vasco da ,gama4 ,the three voyages of ,vasco da ,gama 7who died on the scene of his labours1 at ,cochin1 in #aebd7 revolutionized the commerce of the ,east4 ,until then the ,venetians held the carrying trade of ,india1 which was brought by the ,persian ,gulf and ,red sea into ,syria and ,egypt1 the ,venetians receiving the products of the ,east at ,alexandria and ,beirut and distributing them over ,europe4 ,this commerce was a great source of wealth to ,venice2 but after the discovery of the new passage round the ,cape1 and the conquests of the ,portuguese1 the trade of the ,east passed into other hands4 ,spaniards in ,america4 ,the discoveries of ,columbus awakened a spirit of enterprise in ,spain which continued in full force for a century2 adventurers flocked eagerly across the ,atlantic1 and discovery followed discovery in rapid succession4 ,many of the companions of ,columbus continued his work4 ,vicente ,yanez ,pinzon in #aejj reached the mouth of the ,amazon4 ,in the same year ,alonso de ,ojeda1 accompanied by ,juan de la ,cosa1 from whose maps we learn much of the discoveries of the #afth century navigators1 and by a ,florentine named ,amerigo ,vespucci1 touched the coast of ,south ,america somewhere near ,surinam1 following the shore as far as the ,gulf of ,maracaibo4 ,vespucci afterwards made three voyages to the ,brazilian coast2 and in #aejd he wrote an account of his four voyages1 which was widely circulated1 and became the means of procuring for its author at the hands of the cartographer ,waldseemuller in #aejg the disproportionate distinction of giving his name to the whole continent4 ,in #aejh ,alonso de ,ojeda obtained the government of the coast of ,south ,america from ,cabo de la ,vela to the ,gulf of ,darien2 ,ojeda landed at ,cartagena in #aeaj1 and sustained a defeat from the natives1 in which his lieutenant1 ,juan de la ,cosa1 was killed4 ,after another reverse on the east side of the ,gulf of ,darien ,ojeda returned to ,hispaniola and died there4 ,the ,spaniards in the ,gulf of ,darien were left by ,ojeda under the command of ,francisco ,pizarro1 the future conqueror of ,peru4 ,after suffering much from famine and disease1 ,pizarro resolved to leave1 and embarked the survivors in small vessels1 but outside the harbour they met a ship which proved to be that of ,martin ,fernandez ,enciso1 ,ojeda's partner1 coming with provisions and reinforcements4 ,one of the crew of ,enciso's ship1 ,vasco ,nunez de ,balboa1 the future discoverer of the ,pacific ,ocean1 induced his commander to form a settlement on the other side of the ,gulf of ,darien4 ,the soldiers became discontented and deposed ,enciso1 who was a man of learning and an accomplished cosmographer4 ,his work ,suma de ,geografia1 which was printed in #aeai1 is the first ,spanish book which gives an account of ,america4 ,vasco ,nunez1 the new commander1 entered upon a career of conquest in the neighbourhood of ,darien1 which ended in the discovery of the ,pacific ,ocean on the #beth of ,september #aeac4 ,vasco ,nunez was beheaded in #aeag by ,pedrarias de ,avila1 who was sent out to supersede him4 ,this was one of the greatest calamities that could have happened to ,south ,america2 for the discoverer of the ,south sea was on the point of sailing with a little fleet into his unknown ocean1 and a humane and judicious man would probably have been the conqueror of ,peru1 instead of the cruel and ignorant ,pizarro4 ,in the year #aeai ,panama was founded by ,pedrarias2 and the conquest of ,peru by ,pizarro followed a few years afterwards4 ,hernan ,cortes overran and conquered ,mexico from #aeah to #aeba1 and the discovery and conquest of ,guatemala by ,alvarado1 the invasion of ,florida by ,de ,soto1 and of ,nueva ,granada by ,quesada1 followed in rapid succession4 ,the first detailed account of the west coast of ,south ,america was written by a keenly observant old soldier1 ,pedro de ,cieza de ,leon1 who was travelling in ,south ,america from #aecc to #aeej1 and published his story at ,seville in #aeec4 ,pacific ,ocean4 ,the great desire of the ,spanish government at that time was to find a westward route to the ,moluccas4 ,for this purpose ,juan ,diaz de ,solis was despatched in ,october #aeae1 and in ,january #aeaf he discovered the mouth of the ,rio de la ,plata4 ,he was1 however1 killed by the natives1 and his ships returned4 ,in the following year the ,portuguese ,ferdinando ,magalhaes1 familiarly known as ,magellan1 laid before ,charles ,v41 at ,valladolid1 a scheme for reaching the ,spice ,islands by sailing westward4 ,he started on the #bast of ,september #aeai1 entered the strait which now bears his name in ,october #aebj1 worked his way through between ,patagonia and ,tierra del ,fuego1 and entered on the vast ,pacific which he crossed without sighting any of its innumerable island groups4 ,this was unquestionably the greatest of the voyages which followed from the impulse of ,prince ,henry1 and it was rendered possible only by the magnificent courage of the commander in spite of rebellion1 mutiny and starvation4 ,it was the #fth of ,march #aeba when he reached the ,ladrone ,islands4 ,thence ,magellan proceeded to the ,philippines1 and there his career ended in an unimportant encounter with hostile natives4 ,eventually a ,biscayan named ,sebastian del ,cano1 sailing home by way of the ,cape of ,good ,hope1 reached ,san ,lucar in command of the ,victoria on the #fth of ,september #aebb1 with eighteen survivors2 this one ship of the squadron which sailed on the quest succeeded in accomplishing the first circumnavigation of the globe4 ,del ,cano was received with great distinction by the emperor1 who granted him a globe for his crest1 and the motto ,primus circumdedisti me4 ,portuguese in ,africa and the ,east4 ,while the ,spaniards were circumnavigating the world and completing their knowledge of the coasts of ,central and ,south ,america1 the ,portuguese were actively engaged on similar work as regards ,africa and the ,east ,indies4 ,with ,abyssinia the mission of ,covilhao led to further intercourse4 ,in ,april #aebj ,vasco da ,gama1 as viceroy of the ,indies1 took a fleet into the ,red sea1 and landed an embassy consisting of ,dom ,rodriguez de ,lima and ,father ,francisco ,alvarez1 a priest whose detailed narrative is the earliest and not the least interesting account we possess of ,abyssinia4 ,it was not until #aebf that the embassy was dismissed2 and not many years afterwards the negus entreated the help of the ,portuguese against ,mahommedan invaders1 and the viceroy sent an expeditionary force1 commanded by his brother ,cristoforo da ,gama1 with #dej musketeers4 ,da ,gama was taken prisoner and killed1 but his followers enabled the ,christians of ,abyssinia to regain their power1 and a ,jesuit mission remained in the country4 ,the ,portuguese also established a close connexion with the kingdom of ,congo on the west side of ,africa1 and obtained much information respecting the interior of the continent4 ,duarte ,lopez1 a ,portuguese settled in the country1 was sent on a mission to ,rome by the king of ,congo1 and ,pope ,sixtus ,v4 caused him to recount to his chamberlain1 ,felipe ,pigafetta1 all he had learned during the nine years he had been in ,africa1 from #aegh to #aehg4 ,this narrative1 under the title of ,description of the ,kingdom of ,congo1 was published at ,rome by ,pigafetta in #aeia4 ,a map was attached on which several great equatorial lakes are shown1 and the empire of ,monomwezi or ,unyamwezi is laid down4 ,the most valuable work on ,africa about this time is1 however1 that written by the ,moor ,leo ,africanus in the early part of the #afth century4 ,leo travelled extensively in the north and west of ,africa1 and was eventually taken by pirates and sold to a master who presented him to ,pope ,leo ,x4 ,at the pope's desire he translated his work on ,africa into ,italian4 ,in ,further ,india and the ,malay ,archipelago the ,portuguese acquired predominating influence at sea1 establishing factories on the ,malabar coast1 in the ,persian ,gulf1 at ,malacca1 and in the ,spice ,islands1 and extending their commercial enterprises from the ,red sea to ,china4 ,their missionaries were received at the court of ,akbar1 and ,benedict ,goes1 a native of the ,azores1 was despatched on a journey overland from ,agra to ,china4 ,he started in #afjc1 and1 after traversing the least-known parts of ,central ,asia1 he reached the confines of ,china4 ,he appears to have ascended from ,kabul to the plateau of the ,pamir1 and thence onwards by ,yarkand1 ,khotan and ,aksu4 ,he died on the journey in ,march #afjg2 and thus1 as one of the brethren pronounced his epitaph1 seeking ,cathay he found heaven4 ,english1 ,dutch and ,french4 ,the activity and love of adventure1 which became a passion for two or three generations in ,spain and ,portugal1 spread to other countries4 ,it was the spirit of the age2 and ,england1 ,holland and ,france were fired by it4 ,english enterprise was first aroused by ,john and ,sebastian ,cabot1 father and son1 who came from ,venice and settled at ,bristol in the time of ,henry ,,vii4 ,the ,cabots received a patent in #adif1 empowering them to seek unknown lands2 and ,john ,cabot discovered ,newfoundland and part of the coast of ,america4 ,sebastian afterwards made a voyage to ,rio de la ,plata in the service of ,spain1 but he returned to ,england in #aedh and received a pension from ,edward ,,vi4 ,at his suggestion a voyage was undertaken for the discovery of a north-east passage to ,cathay1 with ,sir ,hugh ,willoughby as captain-general of the fleet and ,richard ,chancellor as pilot-major4 ,they sailed in ,may #aeec1 but ,willoughby and all his crew perished on the ,lapland coast4 ,chancellor1 however1 was more fortunate4 ,he reached the ,white ,sea1 performed the journey overland to ,moscow1 where he was well received1 and may be said to have been the founder of the trade between ,russia and ,england4 ,he returned to ,archangel and brought his ship back in safety to ,england4 ,on a second voyage1 in #aeef1 ,chancellor was drowned2 and three subsequent voyages1 led by ,stephen ,burrough1 ,arthur ,pet and ,charles ,jackman1 in small craft of #ej tons and under1 carried on an examination of the straits which lead into the ,kara sea4 ,the ,french followed closely on the track of ,john ,cabot1 and ,norman and ,breton fishermen frequented the banks of ,newfoundland at the beginning of the #afth century4 ,in #aebd ,francis ,i4 sent ,giovanni da ,verazzano of ,florence on an expedition of discovery to the coast of ,north ,america2 and the details of his voyage were embodied in a letter addressed by him to the king of ,france from ,dieppe1 in ,july #aebd4 ,in #aecd ,jacques ,cartier set out to continue the discoveries of ,verazzano1 and visited ,newfoundland and the ,gulf of ,st ,lawrence4 ,in the following year he made another voyage1 discovered the island of ,anticosti1 and ascended the ,st ,lawrence to ,hochelaga1 now ,montreal4 ,he returned1 after passing two winters in ,canada2 and on another occasion he also failed to establish a colony4 ,admiral de ,coligny made several unsuccessful endeavours to form a colony in ,florida under ,jean ,ribault of ,dieppe1 ,rene de ,laudonniere and others1 but the settlers were furiously assailed by the ,spaniards and the attempt was abandoned4 ,the ,elizabethan era4 ,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 earth4 ,the great promoter of geographical discovery in the ,elizabethan period was ,richard ,hakluyt 7#aeec-afaf71 who was active in the formation of the two companies for colonizing ,virginia in #afjf2 and devoted his life to encouraging and recording similar undertakings4 ,he published much1 and left many valuable papers at his death1 most of which1 together with many other narratives1 were published in #afbb in the great work of the ,rev4 ,samuel ,purchas1 entitled ,hakluytus ,posthumus1 or ,purchas his ,pilgrimes4 ,it is from these works that our knowledge of the gallant deeds of the ,english and other explorers of the ,elizabethan age is mainly derived4 ,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 ,europe4 ,one important object of ,english maritime adventurers of those days was to discover a route to ,cathay by the north-west1 a second was to settle ,virginia1 and a third was to raid the ,spanish settlements in the ,west ,indies4 ,nor was the trade to ,muscovy and ,turkey neglected2 while latterly a resolute and successful attempt was made to establish direct commercial relations with ,india4 ,the conception of the north-western route to ,cathay now leads the story of exploration1 for the first time as far as important and sustained efforts are concerned1 towards the ,arctic seas4 ,this part of the story is fully told under the heading of ,,polar ,,regions1 and only the names of ,martin ,frobisher 7#aegf71 ,john ,davis 7#aehe71 ,henry ,hudson 7#afjg7 and ,william ,baffin 7#afaf7 need be mentioned here in order to preserve the complete conspectus of the history of discovery4 ,the ,dutch emulated the ,british in the ,arctic seas during this period1 directing their efforts mainly towards the discovery of a north-east passage round the northern end of ,novaya ,zemlya2 and ,william ,barents or ,barendsz 7#aeid-aeig7 is the most famous name in this connexion1 his boat voyage along the coast of ,novaya ,zemlya after losing his ship and wintering in a high latitude1 being one of the most remarkable achievements in polar annals4 ,many ,english voyages were also made to ,guinea and the ,west ,indies1 and twice ,english vessels followed in the track of ,magellan1 and circumnavigated the globe4 ,in #aegg ,francis ,drake1 who had previously served with ,hawkins in the ,west ,indies1 undertook his celebrated voyage round the world4 ,reaching the ,pacific through the ,strait of ,magellan1 ,drake proceeded northward along the west coast of ,america1 resolved to attempt the discovery of a northern passage from the ,pacific to the ,atlantic4 ,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 #aeci4 ,drake's discoveries extended from ,cape ,mendocino to #dh deg4 ,n41 in which latitude he gave up his quest1 sailed across the ,pacific and reached the ,philippine ,islands1 returning home round the ,cape of ,good ,hope in #aehj4 ,thomas ,cavendish1 emulous of ,drake's example1 fitted out three vessels for an expedition to the ,south sea in #aehf4 ,he took the same route as ,drake along the west coast of ,america4 ,from ,cape ,san ,lucas ,cavendish steered across the ,pacific1 seeing no land until he reached the ,ladrone ,islands4 ,he returned to ,england in #aehh4 ,the third ,english voyage into the ,pacific was not so fortunate4 ,sir ,richard ,hawkins 7#aeic7 on reaching the bay of ,atacames1 in #a deg4,n4 in #aeid1 was attacked by a ,spanish fleet1 and1 after a desperate naval engagement1 was forced to surrender4 ,hawkins declared his object to be discovery and the survey of unknown lands1 and his voyage1 though terminating in disaster1 bore good fruit4 ,the ,observations of ,sir ,richard ,hawkins in his ,voyage into the ,south ,sea1 published in #afbb1 are very valuable4 ,it was long before another ,british ship entered the ,pacific ,ocean4 ,sir ,john ,narborough took two ships through the ,strait of ,magellan in #afgj and touched on the coast of ,chile1 but it was not until #afhe that ,dampier sailed over the part of the ,pacific where ,hawkins met his defeat4 ,the exploring enterprise of the ,spanish nation did not wane after the conquest of ,peru and ,mexico1 and the acquisition of the vast empire of the ,indies4 ,it was spurred into renewed activity by the audacity of ,sir ,john ,hawkins in the ,west ,indies1 and by the appearance of ,drake1 ,cavendish and ,richard ,hawkins in the ,pacific4 ,in the interior of ,south ,america the ,spanish conquerors had explored the region of the ,andes from the isthmus of ,panama to ,chile4 ,pedro de ,valdivia in #aedj made an expedition into the country of the ,araucanian ,indians of ,chile1 and was the first to explore the eastern base of the ,andes in what is now ,argentine ,patagonia4 ,in #aeda ,francisco de ,orellana discovered the whole course of the ,amazon from its source in the ,andes to the ,atlantic4 ,a second voyage on the ,amazon was made in #aefa by the mad pirate ,lope de ,aguirre2 but it was not until #afci that a full account was written of the great river by ,father ,cristoval de ,acuna1 who ascended it from its mouth and reached the city of ,quito4 ,spaniards in the ,pacific4 ,the voyage of ,drake across the ,pacific was preceded by that of ,alvaro de ,mendana1 who was despatched from ,peru in #aefg to discover the great ,antarctic continent which was believed to extend far northward into the ,south sea1 the search for which now became one of the leading motives of exploration4 ,after a voyage of eighty days across the ,pacific1 ,mendana discovered the ,solomon ,islands2 and the expedition returned in safety to ,callao4 ,the appearance of ,drake on the ,peruvian coast led to an expedition being fitted out at ,callao1 to go in chase of him1 under the command of ,pedro ,sarmiento4 ,he sailed from ,callao in ,october #aegi1 and made a careful survey of the ,strait of ,magellan1 with the object of fortifying that entrance to the ,south sea4 ,the colony which he afterwards took out from ,spain was a complete failure1 and is only remembered now from the name of ,port ,famine1 which ,cavendish gave to the site at which he found the starving remnant of ,sarmiento's settlers4 ,in ,june #aeie ,mendana sailed from the coast of ,peru in command of a second expedition to colonize the ,solomon ,islands4 ,after discovering the ,marquesas1 he reached the island of ,santa ,cruz of evil memory1 where he and many of the settlers died4 ,his young widow took command of the survivors and brought them safely to ,manila4 ,the viceroys of ,peru still persevered in their attempts to plant a colony in the hypothetical southern continent4 ,pedro ,fernandez de ,quiros1 who was pilot under ,mendana and ,luis ,vaez de ,torres1 were sent in command of two ships to continue the work of exploration4 ,they sailed from ,callao in ,december #afje1 and discovered several islands of the ,new ,hebrides group4 ,they anchored in a bay of a large island which ,quiros named ,australia del ,espiritu ,santo4 ,from this place ,quiros returned to ,america1 but ,torres continued the voyage1 passed through the strait between ,australia and ,new ,guinea which bears his name1 and explored and mapped the southern and eastern coasts of ,new ,guinea4 ,the ,portuguese1 in the early part of the #agth century 7#aegh-afdj71 were under the dominion of ,spain1 and their enterprise was to some extent damped2 but their missionaries extended geographical knowledge in ,africa4 ,father ,francisco ,paez acquired great influence in ,abyssinia1 and explored its highlands from #afjj to #afbb4 ,fathers ,mendez and ,lobo traversed the deserts between the coast of the ,red sea and the mountains1 became acquainted with ,lake ,tsana1 and discovered the sources of the ,blue ,nile in #afbd-afcc4 ,rivalry in the ,east4 ,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 ,dutch4 ,the ,english enterprises were persevering1 continuous and successful4 ,james ,lancaster made a voyage to the ,indian ,ocean from #aeia to #aeid2 and in #aeii the merchants and adventurers of ,london resolved to form a company1 with the object of establishing a trade with the ,east ,indies4 ,on the #cast of ,december #aeii ,queen ,elizabeth granted the charter of incorporation to the ,east ,india ,company1 and ,sir ,james ,lancaster1 one of the directors1 was appointed general of their first fleet4 ,he was accompanied by ,john ,davis1 the great ,arctic navigator1 as pilot-major4 ,this voyage was eminently successful4 ,the ships touched at ,achin in ,sumatra and at ,java1 returning with full ladings of pepper in #afjc4 ,the second voyage was commanded by ,sir ,henry ,middleton2 but it was in the third voyage1 under ,keelinge and ,hawkins1 that the mainland of ,india was first reached in #afjg4 ,captain ,hawkins landed at ,surat and travelled overland to ,agra1 passing some time at the court of the ,great ,mogul4 ,in the voyage of ,sir ,edward ,michelborne in #afje1 ,john ,davis lost his life in a fight with a ,japanese junk4 ,the eighth voyage1 led by ,captain ,saris1 extended the operations of the company to ,japan2 and in #afac the ,japanese government granted privileges to the company2 but the ,british retired in #afbc1 giving up their factory4 ,the chief result of this early intercourse between ,great ,britain and ,japan was the interesting series of letters written by ,william ,adams from #afaa to #afag4 ,from the tenth voyage of the ,east ,india ,company1 commanded by ,captain ,best1 who left ,england in #afab1 dates the establishment of permanent ,british factories on the coast of ,india4 ,it was ,captain ,best who secured a regular firman for trade from the ,great ,mogul4 ,from that time a fleet was despatched every year1 and the company's operations greatly increased geographical knowledge of ,india and the ,eastern ,archipelago4 ,british visits to ,eastern countries1 at this time1 were not confined to the voyages of the company4 ,journeys were also made by land1 and1 among others1 the entertaining author of the ,crudities1 ,thomas ,coryate1 of ,odcombe in ,somersetshire1 wandered on foot from ,france to ,india1 and died 7#afag7 in the company's factory at ,surat4 ,in #aefa ,anthony ,jenkinson arrived in ,persia with a letter from ,queen ,elizabeth to the shah4 ,he travelled through ,russia to ,bokhara1 and returned by the ,caspian and ,volga4 ,in #aegi ,christopher ,burroughs built a ship at ,nizhniy ,novgorod and traded across the ,caspian to ,baku2 and in #aeih ,sir ,anthony and ,robert ,shirley arrived in ,persia1 and ,robert was afterwards sent by the shah to ,europe as his ambassador4 ,he was followed by a ,spanish mission under ,garcia de ,silva1 who wrote an interesting account of his travels2 and to ,sir ,dormer ,cotton's mission1 in #afbh1 we are indebted for ,sir ,thomas ,herbert's charming narrative4 ,in like manner ,sir ,thomas ,roe's mission to ,india resulted not only in a large collection of valuable reports and letters of his own1 but also in the detailed account of his chaplain ,terry4 ,but the most learned and intelligent traveller in the ,east1 during the #agth century1 was the ,german1 ,engelbrecht ,kaempfer1 who accompanied an embassy to ,persia1 in #afhd1 and was afterwards a surgeon in the service of the ,dutch ,east ,india ,company4 ,he was in the ,persian ,gulf1 ,india and ,java1 and resided for more than two years in ,japan1 of which he wrote a history4 ,dutch exploration1 #afth-#agth centuries4 ,the ,dutch nation1 as soon as it was emancipated from ,spanish tyranny1 displayed an amount of enterprise1 which1 for a long time1 was fully equal to that of the ,british4 ,the ,arctic voyages of ,barents were quickly followed by the establishment of a ,dutch ,east ,india ,company2 and the ,dutch1 ousting the ,portuguese1 not only established factories on the mainland of ,india and in ,japan1 but acquired a preponderating influence throughout the ,malay ,archipelago4 ,in #aehc ,jan ,hugen van ,linschoten made a voyage to ,india with a ,portuguese fleet1 and his full and graphic descriptions of ,india1 ,africa1 ,china and the ,malay ,archipelago must have been of no small use to his countrymen in their distant voyages4 ,the first of the ,dutch ,indian voyages was performed by ships which sailed in ,april #aeie1 and rounded the ,cape of ,good ,hope4 ,a second large ,dutch fleet sailed in #aeih2 and1 so eager was the republic to extend her commerce over the world that another fleet1 consisting of five ships of ,rotterdam1 was sent in the same year by way of ,magellan's ,strait1 under ,jacob ,mahu as admiral1 with ,william ,adams as pilot4 ,mahu died on the passage out1 and was succeeded by ,simon de ,cordes1 who was killed on the coast of ,chile4 ,in ,september #aeii the fleet had entered the ,pacific4 ,the ships were then steered direct for ,japan1 and anchored off ,bungo in ,april #afjj4 ,in the same year1 #aeih1 a third expedition was despatched under ,oliver van ,noort1 a native of ,utrecht1 but the voyage contributed nothing to geography4 ,the ,dutch ,company in #afad again resolved to send a fleet to the ,moluccas by the westward route1 and ,joris ,spilbergen was appointed to the command as admiral1 with a commission from the ,states-,general4 ,he was furnished with four ships of ,amsterdam1 two of ,rotterdam and one from ,zeeland4 ,on the #fth of ,may #afae ,spilbergen entered the ,pacific ,ocean1 and touched at several places on the coast of ,chile and ,peru1 defeating the ,spanish fleet in a naval engagement off ,chilca4 ,after plundering ,payta and making requisitions at ,acapulco1 the ,dutch fleet crossed the ,pacific and reached the ,moluccas in ,march #afaf4 ,the ,dutch now resolved to discover a passage into the ,pacific to the south of ,tierra del ,fuego1 the insular nature of which had been ascertained by ,sir ,francis ,drake4 ,the vessels fitted out for this purpose were the ,eendracht1 of #cfj tons1 commanded by ,jacob ,lemaire1 and the ,hoorn1 of #aaj tons1 under ,willem ,schouten4 ,they sailed from the ,texel on the #adth of ,june #afae1 and by the #bjth of ,january #afaf they were south of the entrance of ,magellan's ,strait4 ,passing through the strait of ,lemaire they came to the southern extremity of ,tierra del ,fuego1 which was named ,cape ,horn1 in honour of the town of ,hoorn in ,west ,friesland1 of which ,schouten was a native4 ,they passed the cape on the #cast of ,january1 encountering the usual westerly winds4 ,the great merit of this discovery of a second passage into the ,south sea lies in the fact that it was not accidental or unforeseen1 but was due to the sagacity of those who designed the voyage4 ,on the #ast of ,march the ,dutch fleet sighted the island of ,juan ,fernandez2 and1 having crossed the ,pacific1 the explorers sailed along the north coast of ,new ,guinea and arrived at the ,moluccas on the #agth of ,september #afaf4 ,there were several early indications of the existence of the great ,australian continent1 and the ,dutch endeavoured to obtain further knowledge concerning the country and its extent2 but only its northern and western coasts had been visited before the time of ,governor van ,diemen4 ,dirk ,hartog had been on the west coast in latitude #bf deg4 #cj' ,s4 in #afaf4 ,pelsert struck on a reef called ,houtman's ,abrolhos on the #dth of ,june #afbi4 ,in #afig the ,dutch captain ,vlamingh landed on the west coast of ,australia1 then called ,new ,holland1 in #ca deg4 #dc' ,s41 and named the ,swan river from the black swans he discovered there4 ,in #afdb the governor and council of ,batavia fitted out two ships to prosecute the discovery of the south land1 then believed to be part of a vast ,antarctic continent1 and entrusted the command to ,captain ,abel ,jansen ,tasman4 ,this voyage proved to be the most important to geography that had been undertaken since the first circumnavigation of the globe4 ,tasman sailed from ,batavia in #afdb1 and on the #bdth of ,november sighted high land in #db deg4 #cj' ,s41 which was named van ,diemen's ,land1 and after landing there proceeded to the discovery of the western coast of ,new ,zealand2 at first called ,staten ,land1 and supposed to be connected with the ,antarctic continent from which this voyage proved ,new ,holland to be separated4 ,he then reached ,tongatabu1 one of the ,friendly ,islands of ,cook2 and returned by the north coast of ,new ,guinea to ,batavia4 ,in #afdd ,tasman made a second voyage to effect a fuller discovery of ,new ,guinea4 ,french in ,north ,america4 ,the ,french directed their enterprise more in the direction of ,north ,america than of the ,indies4 ,one of their most distinguished explorers was ,samuel ,champlain1 a captain in the navy1 who1 after a remarkable journey through ,mexico and the ,west ,indies from #aeii to #afjb1 established his historic connexion with ,canada1 to the geographical knowledge of which he made a very large addition4 ,missionaries in the ,east4 ,the principles and methods of surveying and position finding had by this time become well advanced1 and the most remarkable example of the early application of these improvements is to be found in the survey of ,china by ,jesuit missionaries4 ,they first prepared a map of the country round ,peking1 which was submitted to the emperor ,kang-hi1 and1 being satisfied with the accuracy of the ,european method of surveying1 he resolved to have a survey made of the whole empire on the same principles4 ,this great work was begun in ,july #agjh1 and the completed maps were presented to the emperor in #agah4 ,the records preserved in each city were examined1 topographical information was diligently collected1 and the ,jesuit fathers checked their triangulation by meridian altitudes of the sun and pole star and by a system of remeasurements4 ,the result was a more accurate map of ,china than existed1 at that time1 of any country in ,europe4 ,kang-hi next ordered a similar map to be made of ,tibet1 the survey being executed by two lamas who were carefully trained as surveyors by the ,jesuits at ,peking4 ,from these surveys were constructed the well-known maps which were forwarded to ,duhalde1 and which ,d',anville utilized for his atlas4 ,the #ahth century4 ,asia4 ,several ,european missionaries had previously found their way from ,india to ,tibet4 ,antonio ,andrada1 in #afbd1 was the first ,european to enter ,tibet since the visit of ,friar ,odoric in #acbe4 ,the next journey was that of ,fathers ,grueber and ,dorville about #affj1 who succeeded in passing from ,china1 through ,tibet1 into ,india4 ,in #agae ,fathers ,desideri and ,freyre made their way from ,agra1 across the ,himalayas1 to ,lhasa1 and the ,capuchin ,friar ,orazio della ,penna resided in that city from #agce until #agdg4 ,but the most remarkable journey in this direction was performed by a ,dutch traveller named ,samuel van de ,putte4 ,he left ,holland in #agah1 went by land through ,persia to ,india1 and eventually made his way to ,lhasa1 where he resided for a long time4 ,he went thence to ,china1 returned to ,lhasa1 and was in ,india in time to be an eye-witness of the sack of ,delhi by ,nadir ,shah in #agcg4 ,in #agdc he left ,india and died at ,batavia on the #bgth of ,september #agde4 ,the premature death of this illustrious traveller is the more to be lamented because his vast knowledge died with him4 ,two ,english missions sent by ,warren ,hastings to ,tibet1 one led by ,george ,bogle in #aggd1 and the other by ,captain ,turner in #aghc1 complete ,tibetan exploration in the #ahth century4 ,from ,persia much new information was supplied by ,jean ,chardin1 ,jean ,tavernier1 ,charles ,hamilton1 ,jean de ,thevenot and ,father ,jude ,krusinski1 and by ,english traders on the ,caspian4 ,in #agch ,john ,elton traded between ,astrakhan and the ,persian port of ,enzeli on the ,caspian1 and undertook to build a fleet for ,nadir ,shah4 ,another ,english merchant1 named ,jonas ,hanway1 arrived at ,astrabad from ,russia1 and travelled to the camp of ,nadir at ,kazvin4 ,one lasting and valuable result of ,hanway's wanderings was a charming book of travels4 ,in #agjj ,guillaume ,delisle published his map of the continents of the ,old ,world2 and his successor ,d',anville produced his map of ,india in #ageb4 ,d',anville's map contained all that was then known1 but ten years afterwards ,major ,rennell began his surveying labours1 which extended over the period from #agfc to #aghb4 ,his survey covered an area #ijj m4 long by #cjj wide1 from the eastern confines of ,bengal to ,agra1 and from the ,himalayas to ,calpi4 ,rennell was indefatigable in collecting geographical information2 his ,bengal atlas appeared in #agha1 his famous map of ,india in #aghh and the memoir in #agib4 ,surveys were also made along the ,indian coasts4 ,arabia received very careful attention1 in the #ahth century1 from the ,danish scientific mission1 which included ,carsten ,niebuhr among its members4 ,niebuhr landed at ,loheia1 on the coast of ,yemen1 in ,december #agfb1 and went by land to ,sana4 ,all the other members of the mission died1 but he proceeded from ,mokha to ,bombay4 ,he then made a journey through ,persia and ,syria to ,constantinople1 returning to ,copenhagen in #agfg4 ,his valuable work1 the ,description of ,arabia1 was published in #aggb1 and was followed in #aggd-aggh by two volumes of travels in ,asia4 ,the great traveller survived until #ahae1 when he died at the age of eighty-two4 ,africa4 ,james ,bruce of ,kinnaird1 the contemporary of ,niebuhr1 was equally devoted to ,eastern travel2 and his principal geographical work was the tracing of the ,blue ,nile from its source to its junction with the ,white ,nile4 ,before the death of ,bruce an ,african ,association was formed1 in #aghh1 for collecting information respecting the interior of that continent1 with ,major ,rennell and ,sir ,joseph ,banks as leading members4 ,the association first employed ,john ,ledyard 7who had previously made an extraordinary journey into ,siberia7 to cross ,africa from east to west on the parallel of the ,niger1 and ,william ,lucas to cross the ,sahara to ,fezzan4 ,lucas went from ,tripoli to ,mesurata1 obtained some information respecting ,fezzan and returned in #aghi4 ,one of the chief problems the association wished to solve was that of the existence and course of the river ,niger1 which was believed by some authorities to be identical with the ,congo4 ,mungo ,park1 then an assistant surgeon of an ,indiaman1 volunteered his services1 which were accepted by the association1 and in #agie he succeeded in reaching the town of ,segu on the ,niger1 but was prevented from continuing his journey to ,timbuktu4 ,five years later he accepted an offer from the government to command an expedition into the interior of ,africa1 the plan being to cross from the ,gambia to the ,niger and descend the latter river to the sea4 ,after losing most of his companions he himself and the rest perished in a rapid on the ,niger at ,busa1 having been attacked from the shore by order of a chief who thought he had not received suitable presents4 ,his work1 however1 had established the fact that the ,niger was not identical with the ,congo4 ,while the ,british were at work in the direction of the ,niger1 the ,portuguese were not unmindful of their old exploring fame4 ,in #agih ,dr ,f4,j4,m4 de ,lacerda1 an accomplished astronomer1 was appointed to command a scientific expedition of discovery to the north of the ,zambesi4 ,he started in ,july1 crossed the ,muchenja ,mountains1 and reached the capital of the ,cazembe1 where he died of fever4 ,lacerda left a valuable record of his adventurous journey2 but with ,mungo ,park and ,lacerda the history of ,african exploration in the #ahth century closes4 ,south ,america4 ,in ,south ,america scientific exploration was active during this period4 ,the great geographical event of the century1 as regards that continent1 was the measurement of an arc of the meridian4 ,the undertaking was proposed by the ,french ,academy as part of an investigation with the object of ascertaining the length of the degree near the equator and near the pole respectively so as to determine the figure of the earth4 ,a commission left ,paris in #agce1 consisting of ,charles ,marie de la ,condamine1 ,pierre ,bouguer1 ,louis ,godin and ,joseph de ,jussieu the naturalist4 ,spain appointed two accomplished naval officers1 the brothers ,ulloa1 as coadjutors4 ,the operations were carried on during eight years on a plain to the south of ,quito2 and1 in addition to his memoir on this memorable measurement1 ,la ,condamine collected much valuable geographical information during a voyage down the ,amazon4 ,the arc measured was #c deg4 #g' #c in length2 and the work consisted of two measured bases connected by a series of triangles1 one north and the other south of the equator1 on the meridian of ,quito4 ,contemporaneously1 in #agch1 ,pierre ,louis ,moreau de ,maupertuis1 ,alexis ,claude ,clairaut1 ,charles ,etienne ,louis ,camus1 ,pierre ,charles ,lemonnier and the ,swedish physicist ,celsius measured an arc of the meridian in ,lapland4 ,the ,pacific ,ocean4 ,the ,british and ,french governments despatched several expeditions of discovery into the ,pacific and round the world during the #ahth century4 ,they were preceded by the wonderful and romantic voyages of the buccaneers4 ,the narratives of such men as ,woodes ,rogers1 ,edward ,davis1 ,george ,shelvocke1 ,clipperton and ,william ,dampier1 can never fail to interest1 while they are not without geographical value4 ,the works of ,dampier are especially valuable1 and the narratives of ,william ,funnell and ,lionel ,wafer furnished the best accounts then extant of the ,isthmus of ,darien4 ,dampier's literary ability eventually secured for him a commission in the king's service2 and he was sent on a voyage of discovery1 during which he explored part of the coasts of ,australia and ,new ,guinea1 and discovered the strait which bears his name between ,new ,guinea and ,new ,britain1 returning in #agja4 ,in #agba ,jacob ,roggewein was despatched on a voyage of some importance across the ,pacific by the ,dutch ,west ,india ,company1 during which he discovered ,easter ,island on the #fth of ,april #agbb4 ,the voyage of ,lord ,anson to the ,pacific in #agdj-agdd was of a predatory character1 and he lost more than half his men from scurvy2 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 ,quito1 the ,british under ,anson were pillaging along the coast of the ,pacific and burning the town of ,payta4 ,but a romantic interest attaches to the wreck of the ,wager1 one of ,anson's fleet1 on a desert island near ,chiloe1 for it bore fruit in the charming narrative of ,captain ,john ,byron1 which will endure for all time4 ,in #agfd ,byron himself was sent on a voyage of discovery round the world1 which led immediately after his return to the despatch of another to complete his work1 under the command of ,captain ,samuel ,wallis4 ,the expedition1 consisting of the ,dolphin commanded by ,wallis1 and the ,swallow under ,captain ,philip ,carteret1 sailed in ,september #agff1 but the ships were separated on entering the ,pacific from the ,strait of ,magellan4 ,wallis discovered ,tahiti on the #aith of ,june #agfg1 and he gave a detailed account of that island4 ,he returned to ,england in ,may #agfh4 ,carteret discovered the ,charlotte and ,gloucester ,islands1 and ,pitcairn ,island on the #bnd of ,july #agfg2 revisited the ,santa ,cruz group1 which was discovered by ,mendana and ,quiros2 and discovered the strait separating ,new ,britain from ,new ,ireland4 ,he reached ,spithead again in ,february #agfi4 ,wallis and ,carteret were followed very closely by the ,french expedition of ,bougainville1 which sailed from ,nantes in ,november #agff4 ,bougainville had first to perform the unpleasant task of delivering up the ,falkland ,islands1 where he had encouraged the formation of a ,french settlement1 to the ,spaniards4 ,he then entered the ,pacific1 and reached ,tahiti in ,april #agfh4 ,passing through the ,new ,hebrides group he touched at ,batavia1 and arrived at ,st ,malo after an absence of two years and four months4 ,captain ,cook4 ,the three voyages of ,captain ,james ,cook form an era in the history of geographical discovery4 ,in #agfg he sailed for ,tahiti1 with the object of observing the transit of ,venus1 accompanied by two naturalists1 ,sir ,joseph ,banks and ,dr ,solander1 a pupil of ,linnaeus1 as well as by two astronomers4 ,the transit was observed on the #crd of ,june #agfi4 ,after exploring ,tahiti and the ,society group1 ,cook spent six months surveying ,new ,zealand1 which he discovered to be an island1 and the coast of ,new ,south ,wales from latitude #ch deg4 ,s4 to the northern extremity4 ,the belief in a vast ,antarctic continent stretching far into the temperate zone had never been abandoned1 and was vehemently asserted by ,charles ,dalrymple1 a disappointed candidate nominated by the ,royal ,society for the command of the ,transit expedition of #agfi4 ,in #aggb 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 island1 and naturally believed it to be part of the southern continent4 ,cook's second voyage was mainly intended to settle the question of the existence of such a continent once for all1 and to define the limits of any land that might exist in navigable seas towards the ,antarctic circle4 ,james ,cook at his first attempt reached a south latitude of #eg deg4 #ae'4 ,on a second cruise from the ,society ,islands1 in #aggc1 he1 first of all men1 crossed the ,antarctic circle1 and was stopped by ice in #ga deg4 #aj' ,s4 ,during the second voyage ,cook visited ,easter ,island1 discovered several islands of the ,new ,hebrides and ,new ,caledonia2 and on his way home by ,cape ,horn1 in ,march #aggd1 he discovered the ,sandwich ,island group and described ,south ,georgia4 ,he proved conclusively that any southern continent that might exist lay under the polar ice4 ,the third voyage was intended to attempt the passage from the ,pacific to the ,atlantic by the north-east4 ,the ,resolution and ,discovery sailed in #aggf1 and ,cook again took the route by the ,cape of ,good ,hope4 ,on reaching the ,north ,american coast1 he proceeded northward1 fixed the position of the western extremity of ,america and surveyed ,bering ,strait4 ,he was stopped by the ice in #gj deg4 #da' ,n41 and named the farthest visible point on the ,american shore ,icy ,cape4 ,he then visited the ,asiatic shore and discovered ,cape ,north4 ,returning to ,hawaii1 ,cook was murdered by the natives4 ,on the #adth of ,february #aggi1 his second1 ,captain ,edward ,clerke1 took command1 and proceeding to ,petropavlovsk in the following summer1 he again examined the edge of the ice1 but only got as far as #gj deg4 #cc' ,n4 ,the ships returned to ,england in ,october #aghj4 ,in #aghe the ,french government carefully fitted out an expedition of discovery at ,brest1 which was placed under the command of ,francois ,la ,perouse1 an accomplished and experienced officer4 ,after touching at ,concepcion in ,chile and at ,easter ,island1 ,la ,perouse proceeded to ,hawaii and thence to the coast of ,california1 of which he has given a very interesting account4 ,he then crossed the ,pacific to ,macao1 and in ,july #aghg he proceeded to explore the ,gulf of ,tartary and the shores of ,sakhalin1 remaining some time at ,castries ,bay1 so named after the ,french minister of marine4 ,thence he went to the ,kurile ,islands and ,kamchatka1 and sailed from the far north down the meridian to the ,navigator and ,friendly ,islands4 ,he was in ,botany ,bay in ,january #aghh2 and sailing thence1 the explorer1 his ship and crew were never seen again4 ,their fate was long uncertain4 ,in ,september #agia ,captain ,antoine d',entrecasteaux sailed from ,brest with two vessels to seek for tidings4 ,he visited the ,new ,hebrides1 ,santa ,cruz1 ,new ,caledonia and ,solomon ,islands1 and made careful though rough surveys of the ,louisiade ,archipelago1 islands north of ,new ,britain and part of ,new ,guinea4 ,d',entrecasteaux died on board his ship on the #bjth of ,july #agic1 without ascertaining the fate of ,la ,perouse4 ,captain ,peter ,dillon at length ascertained1 in #ahbh1 that the ships of ,la ,perouse had been wrecked on the island of ,vanikoro during a hurricane4 ,the work of ,captain ,cook bore fruit in many ways4 ,his master1 ,captain ,william ,bligh1 was sent in the ,bounty to convey breadfruit plants from ,tahiti to the ,west ,indies4 ,he reached ,tahiti in ,october #aghh1 and in ,april #aghi a mutiny broke out1 and he1 with several officers and men1 was thrust into an open boat in mid-ocean4 ,during the remarkable voyage he then made to ,timor1 ,bligh passed amongst the northern islands of the ,new ,hebrides1 which he named the ,banks ,group1 and made several running surveys4 ,he reached ,england in ,march #agij4 ,the ,pandora1 under ,captain ,edwards1 was sent out in search of the ,bounty1 and discovered the islands of ,cherry and ,mitre1 east of the ,santa ,cruz group1 but she was eventually lost on a reef in ,torres ,strait4 ,in #agif-agig ,captain ,wilson1 in the missionary ship ,duff1 discovered the ,gambier and other islands1 and rediscovered the islands known to and seen by ,quiros1 but since called the ,duff ,group4 ,another result of ,captain ,cook's work was the colonization of ,australia4 ,on the #ahth of ,january #aghh ,admiral ,phillip and ,captain ,hunter arrived in ,botany ,bay in the ,supply and ,sirius1 followed by six transports1 and established a colony at ,port ,jackson4 ,surveys were then undertaken in several directions4 ,in #agie and #agif ,matthew ,flinders and ,george ,bass were engaged on exploring work in a small boat called the ,tom ,thumb4 ,in #agig ,bass1 who had been a surgeon1 made an expedition southwards1 continued the work of ,cook from ,ram ,head1 and explored the strait which bears his name1 and in #agih he and ,flinders were surveying on the east coast of ,van ,diemen's land4 ,yet another outcome of ,captain ,cook's work was the voyage of ,george ,vancouver1 who had served as a midshipman in ,cook's second and third voyages4 ,the ,spaniards under ,quadra had begun a survey of north-western ,america and occupied ,nootka ,sound1 which their government eventually agreed to surrender4 ,captain ,vancouver was sent out to receive the cession1 and to survey the coast from ,cape ,mendocino northwards4 ,he commanded the old ,discovery1 and was at work during the seasons of #agib1 #agic and #agid1 wintering at ,hawaii4 ,returning home in #agie1 he completed his narrative and a valuable series of charts4 ,arctic regions4 ,the #ahth century saw the ,arctic coast of ,north ,america reached at two points1 as well as the first scientific attempt to reach the ,north ,pole4 ,the ,hudson ,bay ,company had been incorporated in #afgj1 and its servants soon extended their operations over a wide area to the north and west of ,canada4 ,in #agda ,captain ,christopher ,middleton was ordered to solve the question of a passage from ,hudson ,bay to the westward4 ,leaving ,fort ,churchill in ,july #agdb1 he discovered the ,wager river and ,repulse ,bay4 ,he was followed by ,captain ,w4 ,moor in #agdf1 and ,captain ,coats in #agea1 who examined the ,wager ,inlet up to the end4 ,in ,november #agfi ,samuel ,hearne was sent by the ,hudson ,bay ,company to discover the sea on the north side of ,america1 but was obliged to return4 ,in ,february #aggj he set out again from ,fort ,prince of ,wales2 but1 after great hardships1 he was again forced to return to the fort4 ,he started once more in ,december #agga1 and at length reached the ,coppermine river1 which he surveyed to its mouth1 but his observations are unreliable4 ,with the same object ,alexander ,mackenzie1 with a party of ,canadians1 set out from ,fort ,chippewyan on the #crd of ,june #aghi1 and descending the great river which now bears the explorer's name reached the ,arctic sea4 ,in ,february #aggc the ,royal ,society submitted a proposal to the king for an expedition towards the ,north ,pole4 ,the expedition was fitted out under ,captains ,constantine ,phipps and ,skeffington ,lutwidge1 and the highest latitude reached was #hj deg4 #dh' ,n41 but no opening was discovered in the heavy ,polar pack4 ,the most important ,arctic work in the #ahth century was performed by the ,russians1 for they succeeded in delineating the whole of the northern coast of ,siberia4 ,some of this work was possibly done at a still earlier date4 ,the ,cossack ,simon ,dezhneff is thought to have made a voyage1 in the summer of #afdh1 from the river ,kolyma1 through ,bering ,strait 7which was rediscovered by ,vitus ,bering in #agbh7 to ,anadyr4 ,between #agch and #agej ,manin and ,sterlegoff made their way in small sloops from the mouth of the ,yenesei as far north as #ge deg4 #ae' ,n4 ,the land from ,taimyr to ,cape ,chelyuskin1 the most northern extremity of ,siberia1 was mapped in many years of patient exploration by ,chelyuskin1 who reached the extreme point 7#gg deg4 #cd' ,n47 in ,may #agdb4 ,to the east of ,cape ,chelyuskin the ,russians encountered greater difficulties4 ,they built small vessels at ,yakutsk on the ,lena1 #ijj m4 from its mouth1 whence the first expedition was despatched under ,lieut4 ,prontschichev in #agce4 ,he sailed from the mouth of the ,lena to the mouth of the ,olonek1 where he wintered1 and on the #ast of ,september #agcf he got as far as #gg deg4 #bi' ,n41 within #e m4 of ,cape ,chelyuskin4 ,both he and his young wife died of scurvy1 and the vessel returned4 ,a second expedition1 under ,lieut4 ,laptyev1 started from the ,lena in #agci1 but encountered masses of drift ice in ,chatanga bay1 and with this ended the voyages to the westward of the ,lena4 ,several attempts were also made to navigate the sea from the ,lena to the ,kolyma4 ,in #agcf ,lieut4 ,laptyev sailed1 but was stopped by the drift ice in ,august1 and in #agci1 during another trial1 he reached the mouth of the ,indigirka1 where he wintered4 ,in the season of #agdj he continued his voyage to beyond the ,kolyma1 wintering at ,nizhni ,kolymsk4 ,in ,september #agdj ,vitus ,bering sailed from ,okhotsk on a second ,arctic voyage with ,george ,william ,steller on board as naturalist4 ,in ,june #agda he named the magnificent peak on the coast of ,north ,america ,mount ,st ,elias and explored the ,aleutian ,islands4 ,in ,november the ship was wrecked on ,bering ,island2 and the gallant ,dane1 worn out with scurvy1 died there on the #hth of ,december #agda4 ,in ,march #aggj a merchant named ,liakhov saw a large herd of reindeer coming from the north to the ,siberian coast1 which induced him to start in a sledge in the direction whence they came4 ,thus he reached the ,new ,siberian or ,liakhov ,islands1 and for years afterwards the seekers for fossil ivory resorted to them4 ,the ,russian ,captain ,vassili ,chitschakov in #agfe and #agff made two persevering attempts to penetrate the ice north of ,spitsbergen1 and reached #hj deg4 #cj' ,n41 while ,russian parties twice wintered at ,bell ,sound4 ,geographical societies4 ,in reviewing the progress of geographical discovery thus far1 it has been possible to keep fairly closely to a chronological order4 ,but in the #aith century and after exploring work was so generally and steadily maintained in all directions1 and was in so many cases narrowed down from long journeys to detailed surveys within relatively small areas1 that it becomes desirable to cover the whole period at one view for certain great divisions of the world4 7,see ,,africa2 ,,asia2 ,,australia2 ,,polar ,,regions2 c47 ,here1 however1 may be noticed the development of geographical societies devoted to the encouragement of exploration and research4 ,the first of the existing geographical societies was that of ,paris1 founded in #ahbe under the title of ,la ,societe de ,geographie4 ,the ,berlin ,geographical ,society 7,gesellschaft fur ,erdkunde7 is second in order of seniority1 having been founded in #ahbg4 ,the ,royal ,geographical ,society1 which was founded in ,london in #ahcj1 comes third on the list2 but it may be viewed as a direct result of the earlier ,african ,association founded in #aghh4 ,sir ,john ,barrow1 ,sir ,john ,cam ,hobhouse 7,lord ,broughton71 ,sir ,roderick ,murchison1 ,mr ,robert ,brown and ,mr ,bartle ,frere formed the foundation committee of the ,royal ,geographical ,society1 and the first president was ,lord ,goderich4 ,the action of the society in supplying practical instruction to intending travellers1 in astronomy1 surveying and the various branches of science useful to collectors1 has had much to do with advancement of discovery4 ,since the war of #ahgj many geographical societies have been established on the continent of ,europe4 ,at the close of the #aith century there were upwards of #ajj such societies in the world1 with more than #ej1jjj members1 and over #aej journals were devoted entirely to geographical subjects4#aa ,the great development of photography has been a notable aid to explorers1 not only by placing at their disposal a faithful and ready means of recording the features of a country and the types of inhabitants1 but by supplying a method of quick and accurate topographical surveying4 ,,the ,,principles ,,of ,,geography ,as regards the scope of geography1 the order of the various departments and their inter-relation1 there is little difference of opinion1 and the principles of geography#ab are now generally accepted by modern geographers4 ,the order in which the various subjects are treated in the following sketch is the natural succession from fundamental to dependent facts1 which corresponds also to the evolution of the diversities of the earth's crust and of its inhabitants4 ,mathematical geography4 ,the fundamental geographical conceptions are mathematical1 the relations of space and form4 ,the figure and dimensions of the earth are the first of these4 ,they are ascertained by a combination of actual measurement of the highest precision on the surface and angular observations of the positions of the heavenly bodies4 ,the science of geodesy is part of mathematical geography1 of which the arts of surveying and cartography are applications4 ,the motions of the earth as a planet must be taken into account1 as they render possible the determination of position and direction by observations of the heavenly bodies4 ,the diurnal rotation of the earth furnishes two fixed points or poles1 the axis joining which is fixed or nearly so in its direction in space4 ,the rotation of the earth thus fixes the directions of north and south and defines those of east and west4 ,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 climate4 ,another important consequence of rotation is the deviation produced in moving bodies relatively to the surface4 ,in the form known as ,ferrell's ,law this runs3 ,if a body moves in any direction on the earth's surface1 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 hemisphere4 ,the deviation is of importance in the movement of air1 of ocean currents1 and to some extent of rivers4#ac ,physical geography4 ,in popular usage the words physical geography have come to mean geography viewed from a particular standpoint rather than any special department of the subject4 ,the popular meaning is better conveyed by the word physiography1 a term which appears to have been introduced by ,linnaeus1 and was reinvented as a substitute for the cosmography of the middle ages by ,professor ,huxley4 ,although the term has since been limited by some writers to one particular part of the subject1 it seems best to maintain the original and literal meaning4 ,in the stricter sense1 physical geography is that part of geography which involves the processes of contemporary change in the crust and the circulation of the fluid envelopes4 ,it thus draws upon physics for the explanation of the phenomena with the space-relations of which it is specially concerned4 ,physical geography naturally falls into three divisions1 dealing respectively with the surface of the lithosphere--geomorphology2 the hydrosphere--oceanography2 and the atmosphere--climatology4 ,all these rest upon the facts of mathematical geography1 and the three are so closely inter-related that they cannot be rigidly separated in any discussion4 ,geomorphology4 ,geomorphology is the part of geography which deals with terrestrial relief1 including the submarine as well as the subaerial portions of the crust4 ,the history of the origin of the various forms belongs to geology1 and can be completely studied only by geological methods4 ,but the relief of the crust is not a finished piece of sculpture2 the forms are for the most part transitional1 owing their characteristic outlines to the process by which they are produced2 therefore the geographer must1 for strictly geographical purposes1 take some account of the processes which are now in action modifying the forms of the crust4 ,opinion still differs as to the extent to which the geographer's work should overlap that of the geologist4 ,the primary distinction of the forms of the crust is that between elevations and depressions4 ,granting that the geoid or mean surface of the ocean is a uniform spheroid1 the distribution of land and water approximately indicates a division of the surface of the globe into two areas1 one of elevation and one of depression4 ,the increasing number of measurements of the height of land in all continents and islands1 and the very detailed levellings in those countries which have been thoroughly surveyed1 enable the average elevation of the land above sea-level to be fairly estimated1 although many vast gaps in accurate knowledge remain1 and the estimate is not an exact one4 ,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 #ajj fathoms or #bjj metres1 i4e4 those parts which sailors speak of as in soundings4 ,actual or projected routes for telegraph cables across the deep sea have also been sounded with extreme accuracy in many cases2 but beyond these lines of sounding the vast spaces of the ocean remain unplumbed save for the rare researches of scientific expeditions1 such as those of the ,challenger1 the ,valdivia1 the ,albatross and the ,scotia4 ,thus the best approximation to the average depth of the ocean is little more than an expert guess2 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 them4 ,crustal relief4 ,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 ,antarctic1 of which we know nothing4 ,we know that the earth's surface if unveiled of water would exhibit a great region of elevation arranged with a certain rough radiate symmetry round the north pole1 and extending southwards in three unequal arms which taper to points in the south4 ,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 area4 ,so far only is it possible to speak with certainty1 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-heap4 ,the boundary between these should obviously be the mean surface of the sphere4 ,sir ,john ,murray deduced the mean height of the land of the globe as about #bbej ft4 above sea-level1 and the mean depth of the oceans as #bjhj fathoms or #ab1dhj ft4 below sea-level4#ad ,calculating the area of the land at #ee1jjj1jjj sq4 m4 7or #bh4f of the surface71 and that of the oceans as #acg1bjj1jjj sq4 m4 7or #ga4d of the surface71 he found that the volume of the land above sea-level was #bc1dej1jjj cub4 m41 the volume of water below sea-level #cbc1hjj1jjj1 and the total volume of the water equal to about #a_/fffth of the volume of the whole globe4 ,from these data1 as revised by ,a4 ,supan1#ae ,h4,r4 ,mill calculated the position of mean sphere-level at about #aj1jjj ft4 or #agjj fathoms below sea-level4 ,he showed that an imaginary spheroidal shell1 concentric with the earth and cutting the slope between the elevated and depressed areas at the contour-line of #agjj fathoms1 would not only leave above it a volume of the crust equal to the volume of the hollow left below it1 but would also divide the surface of the earth so that the area of the elevated region was equal to that of the depressed region4#af ,areas of the crust according to ,murray4 ,a similar observation was made almost simultaneously by ,romieux1#ag who further speculated on the equilibrium between the weight of the elevated land mass and that of the total waters of the ocean1 and deduced some interesting relations between them4 ,murray1 as the result of his study1 divided the earth's surface into three zones--the continental area containing all dry land1 the transitional area including the submarine slopes down to #ajjj fathoms1 and the abysmal area consisting of the floor of the ocean beyond that depth2 and ,mill proposed to take the line of mean-sphere level1 instead of the empirical depth of #ajjj fathoms1 as the boundary between the transitional and abysmal areas4 ,an elaborate criticism of all the existing data regarding the volume relations of the vertical relief of the globe was made in #ahid by ,professor ,hermann ,wagner1 whose recalculations of volumes and mean heights--the best results which have yet been obtained--led to the following conclusions4#ah ,areas of the crust according to ,wagner4 ,the area of the dry land was taken as #bh4c of the surface of the globe1 and that of the oceans as #ga4g4 ,the mean height deduced for the land was #bcjj ft4 above sea-level1 the mean depth of the sea #aa1ejj ft4 below1 while the position of mean-sphere level comes out as #gejj ft4 7#abej fathoms7 below sea-level4 ,from this it would appear that #dc of the earth's surface was above and #eg below the mean level4 ,it must be noted1 however1 that since #ahie the soundings of ,nansen in the north polar area1 of the ,valdivia1 ,belgica1 ,gauss and ,scotia in the ,southern ,ocean1 and of various surveying ships in the ,north and ,south ,pacific1 have proved that the mean depth of the ocean is considerably greater than had been supposed1 and mean-sphere level must therefore lie deeper than the calculations of #ahie show2 possibly not far from the position deduced from the freer estimate of #ahhh4 ,the whole of the available data were utilized by the prince of ,monaco in #aije in the preparation of a complete bathymetrical map of the oceans on a uniform scale1 which must long remain the standard work for reference on ocean depths4 ,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 elevation1 according to the system introduced by ,supan1#ai ,wagner showed his results graphically4 ,this curve with the values reduced from metres to feet is reproduced below4 ,wagner subdivides the earth's surface1 according to elevation1 into the following five regions3 ,wagner's ,divisions of the ,earth's ,crust3 ---------------------------------------- ------------------ ,name4 ,per cent of ,from ,to ,surface4 ---------------------------------------- ------------------ ,depressed area #c ,deepest4 -#af1djj feet4 ,oceanic plateau #ed -#af1djj feet4- #g1djj ,continental slope #i - #g1djj - #ffj ,continental plateau #bh - #ffj #c1jjj ,culminating area #f #c1cjj ,highest4 ---------------------------------------- ------------------ ,illustration ,the continental plateau might for purposes of detailed study be divided into the continental shelf from -#ffj ft4 to sea-level1 and lowlands from sea-level to #ffj ft4 7corresponding to the mean level of the whole globe74#bj ,uplands reaching from #ffj ft4 to #bcjj 7the approximate mean level of the land71 and highlands1 from #bcjj upwards1 might also be distinguished4 ,arrangement of world-ridges and hollows4 ,a striking fact in the configuration of the crust is that each continent1 or elevated mass of the crust1 is diametrically opposite to an ocean basin or great depression2 the only partial exception being in the case of southern ,south ,america1 which is antipodal to eastern ,asia4 ,professor ,c4 ,lapworth has generalized the grand features of crustal relief in a scheme of attractive simplicity4 ,he sees throughout all the chaos of irregular crust-forms the recurrence of a certain harmony1 a succession of folds or waves which build up all the minor features4#ba ,one great series of crust waves from east to west is crossed by a second great series of crust waves from north to south1 giving rise by their interference to six great elevated masses 7the continents71 arranged in three groups1 each consisting of a northern and a southern member separated by a minor depression4 ,these elevated masses are divided from one another by similar great depressions4 ,lapworth's fold-theory4 ,he says3 ,the surface of each of our great continental masses of land resembles that of a long and broad arch-like form1 of which we see the simplest type in the ,new ,world4 ,the surface of the ,north ,american arch is sagged downwards in the middle into a central depression which lies between two long marginal plateaus1 and these plateaus are finally crowned by the wrinkled crests which form its two modern mountain systems4 ,the surface of each of our ocean floors exactly resembles that of a continent turned upside down4 ,taking the ,atlantic as our simplest type1 we may say that the surface of an ocean basin resembles that of a mighty trough or syncline1 buckled up more or less centrally in a medial ridge1 which is bounded by two long and deep marginal hollows1 in the cores of which still deeper grooves sink to the profoundest depths4 ,this complementary relationship descends even to the minor features of the two4 ,where the great continental sag sinks below the ocean level1 we have our gulfs and our ,mediterraneans1 seen in our type continent1 as the ,mexican ,gulf and ,hudson ,bay4 ,where the central oceanic buckle attains the water-line we have our oceanic islands1 seen in our type ocean1 as ,st ,helena and the ,azores4 ,although the apparent crust-waves are neither equal in size nor symmetrical in form1 this complementary relationship between them is always discernible4 ,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 ,america4 ,suess's theory4 ,the most thorough discussion of the great features of terrestrial relief in the light of their origin is that by ,professor ,e4 ,suess1#bb who points out that the plan of the earth is the result of two movements of the crust--one1 subsidence over wide areas1 giving rise to oceanic depressions and leaving the continents protuberant2 the other1 folding along comparatively narrow belts1 giving rise to mountain ranges4 ,this theory of crust blocks dropped by subsidence is opposed to ,lapworth's theory of vast crust-folds1 but geology is the science which has to decide between them4 ,geomorphology is concerned1 however1 in the suggestions which have been made as to the cause of the distribution of heap and hollow in the larger features of the crust4 ,elie de ,beaumont1 in his speculations on the relation between the direction of mountain ranges and their geological age and character1 was feeling towards a comprehensive theory of the forms of crustal relief2 but his ideas were too geometrical1 and his theory that the earth is a spheroid built up on a rhombic dodecahedron1 the pentagonal faces of which determined the direction of mountain ranges1 could not be proved4#bc ,the tetrahedral theory brought forward by ,lowthian ,green1#bd that the form of the earth is a spheroid based on a regular tetrahedron1 is more serviceable1 because it accounts for three very interesting facts of the terrestrial plan--7#a7 the antipodal position of continents and ocean basins2 7#b7 the triangular outline of the continents2 and 7#c7 the excess of sea in the southern hemisphere4 ,recent investigations have recalled attention to the work of ,lowthian ,green1 but the question is still in the controversial stage4#be ,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 moon4 ,more recently ,professor ,a4,e4,h4 ,love has shown that the great features of the relief of the lithosphere may be expressed by spherical harmonics of the first1 second and third degrees1 and their formation related to gravitational action in a sphere of unequal density4#bf ,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 detection4 ,the continents4 ,if the level of the sea were to become coincident with the mean level of the lithosphere1 there would result one tri-radiate land-mass of nearly uniform outline and one continuous sheet of water broken by few islands4 ,the actual position of sea-level lies so near the summit of the crust-heap that the varied relief of the upper portion leads to the formation of a complicated coast-line and a great number of detached portions of land4 ,the hydrosphere is1 in fact1 continuous1 and the land is all in insular masses3 the largest is the ,old ,world of ,europe1 ,asia and ,africa2 the next in size1 ,america2 the third1 possibly1 ,antarctica2 the fourth1 ,australia2 the fifth1 ,greenland4 ,after this there is a considerable gap before ,new ,guinea1 ,borneo1 ,madagascar1 ,sumatra and the vast multitude of smaller islands descending in size by regular gradations to mere rocks4 ,the contrast between island and mainland was natural enough in the days before the discovery of ,australia1 and the mainland of the ,old ,world was traditionally divided into three continents4 ,these continents1 parts of the earth1 or quarters of the globe1 proved to be convenient divisions2 ,america was added as a fourth1 and subsequently divided into two1 while ,australia on its discovery was classed sometimes as a new continent1 sometimes merely as an island1 sometimes compromisingly as an island-continent1 according to individual opinion4 ,the discovery of the insularity of ,greenland might again give rise to the argument as to the distinction between island and continent4 ,although the name of continent was not applied to large portions of land for any physical reasons1 it so happens that there is a certain physical similarity or homology between them which is not shared by the smaller islands or peninsulas4 ,homology of continents4 ,the typical continental form is triangular as regards its sea-level outline4 ,the relief of the surface typically includes a central plain1 sometimes dipping below sea-level1 bounded by lateral highlands or mountain ranges1 loftier on one side than on the other1 the higher enclosing a plateau shut in by mountains4 ,south ,america and ,north ,america follow this type most closely2 ,eurasia 7the land mass of ,europe and ,asia7 comes next1 while ,africa and ,australia are farther removed from the type1 and the structure of ,antarctica and ,greenland is unknown4 ,if the continuous1 unbroken1 horizontal extent of land in a continent is termed its trunk1#bg and the portions cut up by inlets or channels of the sea into islands and peninsulas the limbs1 it is possible to compare the continents in an instructive manner4 ,the following table is from the statistics of ,professor ,h4 ,wagner1#bh his metric measurements being transposed into ,british units3 ,comparison of the ,continents4 ---------------------------------------- ---------------------- ,area ,area ,mean ,area penin- ,area ,area ,area total height1 trunk1sulas1islands1limbs1limbs1 mil4 feet4 mil4 mil4 mil4 mil4 per sq4 m4 sq4 m4sq4 m4 sq4 m4 sq4 m4 cent4 ---------------------------------------- ---------------------- ,old ,world #ce4h #bcfj ,new ,world #af4b #bbcj ,eurasia #bj4he #bfbj #ae4db #d4ji #a4cd #e4dc #bf ,africa #aa4df #bacj #aa4bb 44 #j4bd #j4bd #b4a ,north ,america #i4bf #bcjj #f4ib #j4gh #a4ef #b4cd #be ,south ,america #f4hd #aigj #f4gf #j4jb #j4jf #j4jh #a4a ,australia #c4dc #acaj #b4gg #j4af #j4ej #j4ff #ai ,asia #ag4jb #cabj #ab4ic #c4je #a4jd #d4ji #bd ,europe #c4hc #ihj #b4di #a4jd #j4cj #a4cd #ce ---------------------------------------- ---------------------- ,islands4 ,the usual classification of islands is into continental and oceanic4 ,the former class includes all those which rise from the continental shelf1 or show evidence in the character of their rocks of having at one time been continuous with a neighbouring continent4 ,the latter rise abruptly from the oceanic abysses4 ,oceanic islands are divided according to their geological character into volcanic islands and those of organic origin1 including coral islands4 ,more elaborate subdivisions according to structure1 origin and position have been proposed4#bi ,in some cases a piece of land is only an island at high water1 and by imperceptible gradation the form passes into a peninsula4 ,the typical peninsula is connected with the mainland by a relatively narrow isthmus2 the name is1 however1 extended to any limb projecting from the trunk of the mainland1 even when1 as in the ,indian peninsula1 it is connected by its widest part4 ,coasts4 ,small peninsulas are known as promontories or headlands1 and the extremity as a cape4 ,the opposite form1 an inlet of the sea1 is known when wide as a gulf1 bay or bight1 according to size and degree of inflection1 or as a fjord or ria when long and narrow4 ,it is convenient to employ a specific name for a projection of a coast-line less pronounced than a peninsula1 and for an inlet less pronounced than a bay or bight2 outcurve and incurve may serve the turn4 ,the varieties of coast-lines were reduced to an exact classification by ,richthofen1 who grouped them according to the height and slope of the land into cliff-coasts 7,steilkusten7--narrow beach coasts with cliffs1 wide beach coasts with cliffs1 and low coasts1 subdividing each group according as the coast-line runs parallel to or crosses the line of strike of the mountains1 or is not related to mountain structure4 ,a further subdivision depends on the character of the inter-relation of land and sea along the shore producing such types as a fjord-coast1 ria-coast or lagoon-coast4 ,this extremely elaborate subdivision may be reduced1 as ,wagner points out1 to three types--the continental coast where the sea comes up to the solid rock-material of the land2 the marine coast1 which is formed entirely of soft material sorted out by the sea2 and the composite coast1 in which both forms are combined4 ,coast-lines4 ,on large-scale maps it is necessary to show two coast-lines1 one for the highest1 the other for the lowest tide2 but in small-scale maps a single line is usually wider than is required to represent the whole breadth of the inter-tidal zone4 ,the measurement of a coast-line is difficult1 because the length will necessarily be greater when measured on a large-scale map where minute irregularities can be taken into account4 ,it is usual to distinguish between the general coast-line measured from point to point of the headlands disregarding the smaller bays1 and the detailed coast-line which takes account of every inflection shown by the map employed1 and follows up river entrances to the point where tidal action ceases4 ,the ratio between these two coast-lines represents the coastal development of any region4 ,submarine forms4 ,while the forms of the sea-bed are not yet sufficiently well known to admit of exact classification1 they are recognized to be as a rule distinct from the forms of the land1 and the importance of using a distinctive terminology is felt4 ,efforts have been made to arrive at a definite international agreement on this subject1 and certain terms suggested by a committee were adopted by the ,eighth ,international ,geographical ,congress at ,new ,york in #aijd4cj ,the forms of the ocean floor include the shelf1 or shallow sea margin1 the depression1 a general term applied to all submarine hollows1 and the elevation4 ,a depression when of great extent is termed a basin1 when it is of a more or less round form with approximately equal diameters1 a trough when it is wide and elongated with gently sloping borders1 and a trench when narrow and elongated with steeply sloping borders1 one of which rises higher than the other4 ,the extension of a trough or basin penetrating the land or an elevation is termed an embayment when wide1 and a gully when long and narrow2 and the deepest part of a depression is termed a deep4 ,a depression of small extent when steep-sided is termed a caldron1 and a long narrow depression crossing a part of the continental border is termed a furrow4 ,an elevation of great extent which rises at a very gentle angle from a surrounding depression is termed a rise1 one which is relatively narrow and steep-sided a ridge1 and one which is approximately equal in length and breadth but steep-sided a plateau1 whether it springs direct from a depression or from a rise4 ,an elevation of small extent is distinguished as a dome when it is more than #ajj fathoms from the surface1 a bank when it is nearer the surface than #ajj fathoms but deeper than #f fathoms1 and a shoal when it comes within #f fathoms of the surface and so becomes a serious danger to shipping4 ,the highest point of an elevation is termed a height1 if it does not form an island or one of the minor forms4 ,land forms4 ,the forms of the dry land are of infinite variety1 and have been studied in great detail4#ca ,from the descriptive or topographical point of view1 geometrical form alone should be considered2 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 distributions4 ,the geographers who have hitherto given most attention to the forms of the land have been trained as geologists1 and consequently there is a general tendency to make origin or structure the basis of classification rather than form alone4 ,the six elementary land forms4 ,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 origin4#cb ,these may be looked upon as being all derived by various modifications or arrangements of the single form-unit1 the slope or inclined plane surface4 ,no one form occurs alone1 but always grouped together with others in various ways to make up districts1 regions and lands of distinctive characters4 ,the form-elements are3 #a4 ,the plain or gently inclined uniform surface4 #b4 ,the scarp or steeply inclined slope2 this is necessarily of small extent except in the direction of its length4 #c4 ,the valley1 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 length4 ,many varieties of this fundamental form may be distinguished4 #d4 ,the mount1 composed of a surface falling away on every side from a particular place4 ,this place may either be a point1 as in a volcanic cone1 or a line1 as in a mountain range or ridge of hills4 #e4 ,the hollow or form produced by a land surface sloping inwards from all sides to a particular lowest place1 the converse of a mount4 #f4 ,the cavern or space entirely surrounded by a land surface4 ,geology and land forms4 ,these forms never occur scattered haphazard over a region1 but always in an orderly subordination depending on their mode of origin4 ,the dominant forms result from crustal movements1 the subsidiary from secondary reactions during the action of the primitive forms on mobile distributions4 ,the geological structure and the mineral composition of the rocks are often the chief causes determining the character of the land forms of a region4 ,thus the scenery of a limestone country depends on the solubility and permeability of the rocks1 leading to the typical ,karst-formations of caverns1 swallow-holes and underground stream courses1 with the contingent phenomena of dry valleys and natural bridges4 ,a sandy beach or desert owes its character to the mobility of its constituent sand-grains1 which are readily drifted and piled up in the form of dunes4 ,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 eminences4#cc ,classification of mountains4 ,it would be impracticable to go fully into the varieties of each specific form2 but1 partly as an example of modern geographical classification1 partly because of the exceptional importance of mountains amongst the features of the land1 one exception may be made4 ,the classification of mountains into types has usually had regard rather to geological structure than to external form1 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 range4 ,a mountain may be described 7it cannot be defined7 as an elevated region of irregular surface rising comparatively abruptly from lower ground4 ,the actual elevation of a summit above sea-level does not necessarily affect its mountainous character2 a gentle eminence1 for instance1 rising a few hundred feet above a tableland1 even if at an elevation of say #ae1jjj ft41 could only be called a hill4#cd ,but it may be said that any abrupt slope of #bjjj ft4 or more in vertical height may justly be called a mountain1 while abrupt slopes of lesser height may be called hills4 ,existing classifications1 however1 do not take account of any difference in kind between mountain and hills1 although it is common in the ,german language to speak of ,hugelland1 ,mittelgebirge and ,hochgebirge with a definite significance4 ,the simple classification employed by ,professor ,james ,geikie#ce into mountains of accumulation1 mountains of elevation and mountains of circumdenudation1 is not considered sufficiently thorough by ,german geographers1 who1 following ,richthofen1 generally adopt a classification dependent on six primary divisions1 each of which is subdivided4 ,the terms employed1 especially for the subdivisions1 cannot be easily translated into other languages1 and the ,english equivalents in the following table are only put forward tentatively3-- ,,richthofen',s ,,classification ,,of ,,mountains#cf ,i4 ,tektonische ,gebirge--,tectonic mountains4 7a7 ,bruchgebirge oder ,schollengebirge--,block mountains4 #a4 ,einseitige ,schollengebirge oder ,schollenrandgebirge-- ,scarp or tilted block mountains4 7i47 ,tafelscholle--,table blocks4 7ii47 ,abrasionsscholle--,abraded blocks4 7iii47 ,transgressionsscholle--,blocks of unconformable strata4 #b4 ,flexurgebirge--,flexure mountains4 #c4 ,horstgebirge--,symmetrical block mountains4 7b7 ,faltungsgebirge--,fold mountains4 #a4 ,homoomorphe ,faltungsgebirge--,homomorphic fold mountains4 #b4 ,heteromorphe ,faltungsgebirge--,heteromorphic fold mountains4 ,,ii4 ,rumpfgebirge oder ,abrasionsgebirge--,trunk or abraded mountains4 ,,iii4 ,ausbruchsgebirge--,eruptive mountains4 ,,iv4 ,aufschuttungsgebirge--,mountains of accumulation4 ,v4 ,flachboden--,plateaux4 7a7 ,abrasionsplatten--,abraded plateaux4 7b7 ,marines ,flachland--,plain of marine erosion4 7c7 ,schichtungstafelland--,horizontally stratified tableland4 7d7 ,ubergusstafelland--,lava plain4 7e7 ,stromflachland--,river plain4 7f7 ,flachboden der atmospharischen ,aufschuttung--,plains of aeolian formation4 ,,vi4 ,erosionsgebirge--,mountains of erosion4 ,mountain forms4 ,from the morphological point of view it is more important to distinguish the associations of forms1 such as the mountain mass or group of mountains radiating from a centre1 with the valleys furrowing their flanks spreading towards every direction2 the mountain chain or line of heights1 forming a long narrow ridge or series of ridges separated by parallel valleys2 the dissected plateau or highland1 divided into mountains of circumdenudation by a system of deeply-cut valleys2 and the isolated peak1 usually a volcanic cone or a hard rock mass left projecting after the softer strata which embedded it have been worn away 7,monadnock of ,professor ,davis74 ,distribution of mountains4 ,the geographical distribution of mountains is intimately associated with the great structural lines of the continents of which they form the culminating region4 ,lofty lines of fold mountains form the backbones of ,north ,america in the ,rocky ,mountains and the west coast systems1 of ,south ,america in the ,cordillera of the ,andes1 of ,europe in the ,pyrenees1 ,alps1 ,carpathians and ,caucasus1 and of ,asia in the mountains of ,asia ,minor1 converging on the ,pamirs and diverging thence in the ,himalaya and the vast mountain systems of central and eastern ,asia4 ,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 globe4 ,functions of land forms4 ,land waste4 ,glaciers4 ,if land forms may be compared to organs1 the part they serve in the economy of the earth may1 without straining the term1 be characterized as functions4 ,the first and simplest function of the land surface is that of guiding loose material to a lower level4 ,the downward pull of gravity suffices to bring about the fall of such material1 but the path it will follow and the distance it will travel before coming to rest depend upon the land form4 ,the loose material may1 and in an arid region does1 consist only of portions of the higher parts of the surface detached by the expansion and contraction produced by heating and cooling due to radiation4 ,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 slope4 ,but where the side is not a uniform scarp1 but made up of a series of ridges and valleys1 the tendency will be to distribute the detritus in an irregular manner1 directing it away from one place and collecting it in great masses in another1 so that in time the land form assumes a new appearance4 ,snow accumulating on the higher portions of the land1 when compacted into ice and caused to flow downwards by gravity1 gives rise1 on account of its more coherent character1 to continuous glaciers1 which mould themselves to the slopes down which they are guided1 different ice-streams converging to send forward a greater volume4 ,gradually coming to occupy definite beds1 which are deepened and polished by the friction1 they impress a characteristic appearance on the land1 which guides them as they traverse it1 and1 although the ice melts at lower levels1 vast quantities of clay and broken stones are brought down and deposited in terminal moraines where the glacier ends4 ,rain4 ,river systems4 ,adjustment of rivers to land4 ,rain is by far the most important of the inorganic mobile distributions upon which land forms exercise their function of guidance and control4 ,the precipitation of rain from the aqueous vapour of the atmosphere is caused in part by vertical movements of the atmosphere involving heat changes and apparently independent of the surface upon which precipitation occurs2 but in greater part it is dictated by the form and altitude of the land surface and the direction of the prevailing winds1 which itself is largely influenced by the land4 ,it is on the windward faces of the highest ground1 or just beyond the summit of less dominant heights upon the leeward side1 that most rain falls1 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 land4 ,more mobile and more searching than ice or rock rubbish1 the trickling drops are guided by the deepest lines of the hillside in their incipient flow1 and as these lines converge1 the stream1 gaining strength1 proceeds in its torrential course to carve its channel deeper and entrench itself in permanent occupation4 ,thus the stream-bed1 from which at first the water might be blown away into a new channel by a gale of wind1 ultimately grows to be the strongest line of the landscape4 ,as the main valley deepens1 the tributary stream-beds are deepened also1 and gradually cut their way headwards1 enlarging the area whence they draw their supplies4 ,thus new land forms are created--valleys of curious complexity1 for example--by the capture and diversion of the water of one river by another1 leading to a change of watershed4#cg ,the minor tributaries become more numerous and more constant1 until the system of torrents has impressed its own individuality on the mountain side4 ,as the river leaves the mountain1 ever growing by the accession of tributaries1 it ceases1 save in flood time1 to be a formidable instrument of destruction2 the gentler slope of the land surface gives to it only power sufficient to transport small stones1 gravel1 sand and ultimately mud4 ,its valley banks are cut back by the erosion of minor tributaries1 or by rain-wash if the climate be moist1 or left steep and sharp while the river deepens its bed if the climate be arid4 ,the outline of the curve of a valley's sides ultimately depends on the angle of repose of the detritus which covers them1 if there has been no subsequent change1 such as the passage of a glacier along the valley1 which tends to destroy the regularity of the cross-section4 ,the slope of the river bed diminishes until the plain compels the river to move slowly1 swinging in meanders proportioned to its size1 and gradually1 controlled by the flattening land1 ceasing to transport material1 but raising its banks and silting up its bed by the dropped sediment1 until1 split up and shoaled1 its distributaries struggle across its delta to the sea4 ,this is the typical river of which there are infinite varieties1 yet every variety would1 if time were given1 and the land remained unchanged in level relatively to the sea1 ultimately approach to the type4 ,movements of the land either of subsidence or elevation1 changes in the land by the action of erosion in cutting back an escarpment or cutting through a col1 changes in climate by affecting the rainfall and the volume of water1 all tend to throw the river valley out of harmony with the actual condition of its stream4 ,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 period4 ,before full adjustment has been attained the river bed may be broken in places by waterfalls or interrupted by lakes2 after adjustment the bed assumes a permanent outline1 the slope diminishing more and more gradually1 without a break in its symmetrical descent4 ,excellent examples of the indecisive drainage of a new land surface1 on which the river system has not had time to impress itself1 are to be seen in northern ,canada and in ,finland1 where rivers are separated by scarcely perceptible divides1 and the numerous lakes frequently belong to more than one river system4 ,the geographical cycle4 ,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 ,w4,m4 ,davis1 who classifies land surfaces in terms of the three factors--structure1 process and time4#ch ,of these time1 during which the process is acting on the structure1 is the most important4 ,a land may thus be characterized by its position in the geographical cycle1 or cycle of erosion1 as young1 mature or old1 the last term being reached when the base-level of erosion is attained1 and the land1 however varied its relief may have been in youth or maturity1 is reduced to a nearly uniform surface or peneplain4 ,by a re-elevation of a peneplain the rivers of an old land surface may be restored to youthful activity1 and resume their shaping action1 deepening the old valleys and initiating new ones1 starting afresh the whole course of the geographical cycle4 ,it is1 however1 not the action of the running water on the land1 but the function exercised by the land on the running water1 that is considered here to be the special province of geography4 ,at every stage of the geographical cycle the land forms1 as they exist at that stage1 are concerned in guiding the condensation and flow of water in certain definite ways4 ,thus1 for example1 in a mountain range at right angles to a prevailing sea-wind1 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 valleys1 while the other side is dry1 indefinite in its valley systems1 and sends none of its scanty drainage to the sea4 ,the action of rain1 ice and rivers conspires with the movement of land waste to strip the layer of soil from steep slopes as rapidly as it forms1 and to cause it to accumulate on the flat valley bottoms1 on the graceful flattened cones of alluvial fans at the outlet of the gorges of tributaries1 or in the smoothly-spread surface of alluvial plains4 ,the whole question of the regime of rivers and lakes is sometimes treated under the name hydrography1 a name used by some writers in the sense of marine surveying1 and by others as synonymous with oceanography4 ,for the study of rivers alone the name potamology#ci has been suggested by ,penck1 and the subject being of much practical importance has received a good deal of attention4#dj ,lakes and internal drainage4 ,the study of lakes has also been specialized under the name of limnology 7see ,,lake74#da ,the existence of lakes in hollows of the land depends upon the balance between precipitation and evaporation4 ,a stream flowing into a hollow will tend to fill it up1 and the water will begin to escape as soon as its level rises high enough to reach the lowest part of the rim4 ,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 lip1 so that there is no outflow to the sea1 and a basin of internal drainage is the result4 ,this is the case1 for instance1 in the ,caspian sea1 the ,aral and ,balkhash lakes1 the ,tarim basin1 the ,sahara1 inner ,australia1 the great basin of the ,united ,states and the ,titicaca basin4 ,these basins of internal drainage are calculated to amount to #bb of the land surface4 ,the percentages of the land surface draining to the different oceans are approximately--,atlantic1 #cd4c2 ,arctic sea1 #af4e2 ,pacific1 #ad4d2 ,indian ,ocean1 #ab4h4db ,terminology of river systems4 ,the parts of a river system have not been so clearly defined as is desirable1 hence the exaggerated importance popularly attached to the source of a river4 ,a well-developed river system has in fact many equally important and widely-separated sources1 the most distant from the mouth1 the highest1 or even that of largest initial volume not being necessarily of greater geographical interest than the rest4 ,the whole of the land which directs drainage towards one river is known as its basin1 catchment area or drainage area--sometimes1 by an incorrect expression1 as its valley or even its watershed4 ,the boundary line between one drainage area and others is rightly termed the watershed1 but on account of the ambiguity which has been tolerated it is better to call it water-parting or1 as in ,america1 divide4 ,the only other important term which requires to be noted here is talweg1 a word introduced from the ,german into ,french and ,english1 and meaning the deepest line along the valley1 which is necessarily occupied by a stream unless the valley is dry4 ,the functions of land forms extend beyond the control of the circulation of the atmosphere1 the hydrosphere and the water which is continually being interchanged between them2 they are exercised with increased effect in the higher departments of biogeography and anthropogeography4 ,biogeography4 ,the sum of the organic life on the globe is termed by some geographers the biosphere1 and it has been estimated that the whole mass of living substance in existence at one time would cover the surface of the earth to a depth of one-fifth of an inch4#dc ,the distribution of living organisms is a complex problem1 a function of many factors1 several of which are yet but little known4 ,they include the biological nature of the organism and its physical environment1 the latter involving conditions in which geographical elements1 direct or indirect1 preponderate4 ,the direct geographical elements are the arrangement of land and sea 7continents and islands standing in sharp contrast7 and the vertical relief of the globe1 which interposes barriers of a less absolute kind between portions of the same land area or oceanic depression4 ,the indirect geographical elements1 which1 as a rule1 act with and intensify the direct1 are mainly climatic2 the prevailing winds1 rainfall1 mean and extreme temperatures of every locality depending on the arrangement of land and sea and of land forms4 ,climate thus guided affects the weathering of rocks1 and so determines the kind and arrangement of soil4 ,different species of organisms come to perfection in different climates2 and it may be stated as a general rule that a species1 whether of plant or animal1 once established at one point1 would spread over the whole zone of the climate congenial to it unless some barrier were interposed to its progress4 ,in the case of land and fresh-water organisms the sea is the chief barrier2 in the case of marine organisms1 the land4 ,differences in land forms do not exert great influence on the distribution of living creatures directly1 but indirectly such land forms as mountain ranges and internal drainage basins are very potent through their action on soil and climate4 ,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 sea4 ,in this way the surface of the land is divided into numerous natural regions1 the flora and fauna of each of which include some distinctive species not shared by the others4 ,the distribution of life is discussed in the various articles in this ,encyclopaedia dealing with biological1 botanical and zoological subjects4#dd ,floral zones4 ,the classification of the land surface into areas inhabited by distinctive groups of plants has been attempted by many phyto-geographers1 but without resulting in any scheme of general acceptance4 ,the simplest classification is perhaps that of ,drude according to climatic zones1 subdivided according to continents4 ,this takes account of--7#a7 the ,arctic-,alpine zone1 including all the vegetation of the region bordering on perpetual snow2 7#b7 the ,boreal zone1 including the temperate lands of ,north ,america1 ,europe and ,asia1 all of which are substantially alike in botanical character2 7#c7 the ,tropical zone1 divided sharply into 7a7 the tropical zone of the ,new ,world1 and 7b7 the tropical zone of the ,old ,world1 the forms of which differ in a significant degree2 7#d7 the ,austral zone1 comprising all continental land south of the equator1 and sharply divided into three regions the floras of which are strikingly distinct--7a7 ,south ,american1 7b7 ,south ,african and 7c7 ,australian2 7#e7 the ,oceanic1 comprising all oceanic islands1 the flora of which consists exclusively of forms whose seeds could be drifted undestroyed by ocean currents or carried by birds4 ,to these might be added the antarctic1 which is still very imperfectly known4 ,many subdivisions and transitional zones have been suggested by different authors4 ,vegetation areas4 ,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 environment4 ,the following are the chief areas of vegetational activity usually recognized3 7#a7 ,the ice-deserts of the arctic and antarctic and the highest mountain regions1 where there is no vegetation except the lowest forms1 like that which causes red snow4 7#b7 ,the tundra or region of intensely cold winters1 forbidding tree-growth1 where mosses and lichens cover most of the ground when unfrozen1 and shrubs occur of species which in other conditions are trees1 here stunted to the height of a few inches4 ,a similar zone surrounds the permanent snow on lofty mountains in all latitudes4 ,the tundra passes by imperceptible gradations into the moor1 bog and heath of warmer climates4 7#c7 ,the temperate forests of evergreen or deciduous trees1 according to circumstances1 which occupy those parts of both temperate zones where rainfall and sunlight are both abundant4 7#d7 ,the grassy steppes or prairies where the rainfall is diminished and temperatures are extreme1 and grass is the prevailing form of vegetation4 ,these pass imperceptibly into--7#e7 the arid desert1 where rainfall is at a minimum1 and the only plants are those modified to subsist with the smallest supply of water4 7#f7 ,the tropical forest1 which represents the maximum of plant luxuriance1 stimulated by the heaviest rainfall1 greatest heat and strongest light4 ,these divisions merge one into the other1 and admit of almost indefinite subdivision1 while they are subject to great modifications by human interference in clearing and cultivating4 ,plants exhibit the controlling power of environment to a high degree1 and thus vegetation is usually in close adjustment to the bolder geographical features of a region4 ,faunal realms4 ,the divisions of the earth into faunal regions by ,dr ,p4,l4 ,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 mammals1 and they may thus be accepted as based on nature4 ,they are six in number3 7#a7 ,palaearctic1 including ,europe1 ,asia north of the ,himalaya1 and ,africa north of the ,sahara2 7#b7 ,ethiopian1 consisting of ,africa south of the ,atlas range1 and ,madagascar2 7#c7 ,oriental1 including ,india1 ,indo-,china and the ,malay ,archipelago north of ,wallace's line1 which runs between ,bali and ,lombok2 7#d7 ,australian1 including ,australia1 ,new ,zealand1 ,new ,guinea and ,polynesia2 7#e7 ,nearctic or ,north ,america1 north of ,mexico2 and 7#f7 ,neotropical or ,south ,america4 ,each of these divisions is the home of a special fauna1 many species of which are confined to it alone2 in the ,australian region1 indeed1 practically the whole fauna is peculiar and distinctive1 suggesting a prolonged period of complete biological isolation4 ,in some cases1 such as the ,ethiopian and ,neotropical and the ,palaearctic and ,nearctic regions1 the faunas1 although distinct1 are related1 several forms on opposite sides of the ,atlantic being analogous1 e4g4 the lion and puma1 ostrich and rhea4 ,where two of the faunal realms meet there is usually1 though not always1 a mixing of faunas4 ,these facts have led some naturalists to include the ,palaearctic and ,nearctic regions in one1 termed ,holarctic1 and to suggest transitional regions1 such as the ,sonoran1 between ,north and ,south ,america1 and the ,mediterranean1 between ,europe and ,africa1 or to create sub-regions1 such as ,madagascar and ,new ,zealand4 ,oceanic islands have1 as a rule1 distinctive faunas and floras which resemble1 but are not identical with1 those of other islands in similar positions4 ,biological distribution as a means of geographical research4 ,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 research1 which enables earlier geographical relations of land and sea to be traced out1 and the approximate period1 or at least the chronological order of the larger changes1 to be estimated4 ,in this way1 for example1 it has been suggested that a land1 ,lemuria1 once connected ,madagascar with the ,malay ,archipelago1 and that a northern extension of the antarctic land once united the three southern continents4 ,the distribution of fossils frequently makes it possible to map out approximately the general features of land and sea in long-past geological periods1 and so to enable the history of crustal relief to be traced4#de ,reaction of organisms on environment4 ,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 change1 it is to be observed that the action of organisms themselves often tends to change their environment4 ,corals and other quick-growing calcareous marine organisms are the most powerful in this respect by creating new land in the ocean4 ,vegetation of all sorts acts in a similar way1 either in forming soil and assisting in breaking up rocks1 in filling up shallow lakes1 and even1 like the mangrove1 in reclaiming wide stretches of land from the sea4 ,plant life1 utilizing solar light to combine the inorganic elements of water1 soil and air into living substance1 is the basis of all animal life4 ,this is not by the supply of food alone1 but also by the withdrawal of carbonic acid from the atmosphere1 by which vegetation maintains the composition of the air in a state fit for the support of animal life4 ,man in the primitive stages of culture is scarcely to be distinguished from other animals as regards his subjection to environment1 but in the higher grades of culture the conditions of control and reaction become much more complicated1 and the department of anthropogeography is devoted to their consideration4 ,anthropogeography4 ,the first requisites of all human beings are food and protection1 in their search for which men are brought into intimate relations with the forms and productions of the earth's surface4 ,the degree of dependence of any people upon environment varies inversely as the degree of culture or civilization1 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 community4 ,the development of culture is to a certain extent a question of race1 and although forming one species1 the varieties of man differ in almost imperceptible gradations with a complexity defying classification 7see ,,anthropology74 ,professor ,keane groups man round four leading types1 which may be named the black1 yellow1 red and white1 or the ,ethiopic1 ,mongolic1 ,american and ,caucasic4 ,each may be subdivided1 though not with great exactness1 into smaller groups1 either according to physical characteristics1 of which the form of the head is most important1 or according to language4 ,types of man4 ,the black type is found only in tropical or sub-tropical countries1 and is usually in a primitive condition of culture1 unless educated by contact with people of the white type4 ,they follow the most primitive forms of religion 7mainly fetishism71 live on products of the woods or of the chase1 with the minimum of work1 and have only a loose political organization4 ,the red type is peculiar to ,america1 inhabiting every climate from polar to equatorial1 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 #aeth century4 ,the yellow type is capable of a higher culture1 cherishes higher religious beliefs1 and inhabits as a rule the temperate zone1 although extending to the tropics on one side and to the arctic regions on the other4 ,the white type1 originating in the north temperate zone1 has spread over the whole world4 ,they have attained the highest culture1 profess the purest forms of monotheistic religion1 and have brought all the people of the black type and many of those of the yellow under their domination4 ,the contrast between the yellow and white types has been softened by the remarkable development of the ,japanese following the assimilation of western methods4 ,the actual number of human inhabitants in the world has been calculated as follows3 ,by ,continents4#df ,asia #hge1jjj1jjj ,europe #cib1jjj1jjj ,africa #agj1jjj1jjj ,america #adc1jjj1jjj ,australia and ,polynesia #g1jjj1jjj ------------- ,total #a1ehg1jjj1jjj ,by ,race4#dg ,white 7,caucasic7 #ggj1jjj1jjj ,yellow 7,mong47 #edj1jjj1jjj ,black 7,ethiopic7 #age1jjj1jjj ,red 7,american7 #bb1jjj1jjj ------------- ,total #a1ejg1jjj1jjj ,in round numbers the population of the world is about #a1fjj1jjj1jjj1 and1 according to an estimate by ,ravenstein1#dh the maximum population which it will be possible for the earth to maintain is #fjjj millions1 a number which1 if the average rate of increase in #ahia continued1 would be reached within #bjj years4 ,while highly civilized communities are able to evade many of the restrictions of environment1 to overcome the barriers to intercommunication interposed by land or sea1 to counteract the adverse influence of climate1 and by the development of trade even to inhabit countries which cannot yield a food-supply1 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 race4 ,influence of environment on man4 ,in tropical forests primitive tribes depend on the collection of wild fruits1 and in a minor degree on the chase of wild animals1 for their food4 ,clothing is unnecessary2 hence there is little occasion for exercising the mental faculties beyond the sense of perception to avoid enemies1 or the inventive arts beyond what is required for the simplest weapons and the most primitive fortifications4 ,when the pursuit of game becomes the chief occupation of a people there is of necessity a higher development of courage1 skill1 powers of observation and invention2 and these qualities are still further enhanced in predatory tribes who take by force the food1 clothing and other property prepared or collected by a feebler people4 ,the fruit-eating savage cannot stray beyond his woods which bound his life as the water bounds that of a fish2 the hunter is free to live on the margin of forests or in open country1 while the robber or warrior from some natural stronghold of the mountains sweeps over the adjacent plains and carries his raids into distant lands4 ,wide grassy steppes lead to the organization of the people as nomads whose wealth consists in flocks and herds1 and their dwellings are tents4 ,the nomad not only domesticates and turns to his own use the gentler and more powerful animals1 such as sheep1 cattle1 horses1 camels1 but even turns some predatory creatures1 like the dog1 into a means of defending their natural prey4 ,they hunt the beasts of prey destructive to their flocks1 and form armed bands for protection against marauders or for purposes of aggression on weaker sedentary neighbours4 ,on the fertile low grounds along the margins of rivers or in clearings of forests1 agricultural communities naturally take their rise1 dwelling in villages and cultivating the wild grains1 which by careful nurture and selection have been turned into rich cereals4 ,the agriculturist as a rule is rooted to the soil4 ,the land he tills he holds1 and acquires a closer connexion with a particular patch of ground than either the hunter or the herdsman4 ,in the temperate zone1 where the seasons are sharply contrasted1 but follow each other with regularity1 foresight and self-denial were fostered1 because if men did not exercise these qualities seed-time or harvest might pass into lost opportunities and the tribes would suffer4 ,the more extreme climates of arid regions on the margins of the tropics1 by the unpredictable succession of droughts and floods1 confound the prevision of uninstructed people1 and make prudence and industry qualities too uncertain in their results to be worth cultivating4 ,thus the civilization of agricultural peoples of the temperate zone grew rapidly1 yet in each community a special type arose adapted to the soil1 the crop and the climate4 ,on the seashore fishing naturally became a means of livelihood1 and dwellers by the sea1 in virtue of the dangers to which they are exposed from storm and unseaworthy craft1 are stimulated to a higher degree of foresight1 quicker observation1 prompter decision and more energetic action in emergencies than those who live inland4 ,the building and handling of vessels also1 and the utilization of such uncontrollable powers of nature as wind and tide1 helped forward mechanical invention4 ,to every type of coast there may be related a special type of occupation and even of character2 the deep and gloomy fjord1 backed by almost impassable mountains1 bred bold mariners whose only outlet for enterprise was seawards towards other lands--the viks created the vikings4 ,on the gently sloping margin of the estuary of a great river a view of tranquil inland life was equally presented to the shore-dweller1 and the ocean did not present the only prospect of a career4 ,finally the mountain valley1 with its patches of cultivable soil on the alluvial fans of tributary torrents1 its narrow pastures on the uplands only left clear of snow in summer1 its intensified extremes of climates and its isolation1 almost equal to that of an island1 has in all countries produced a special type of brave and hardy people1 whose utmost effort may bring them comfort1 but not wealth1 by honest toil1 who know little of the outer world1 and to whom the natural outlet for ambition is marauding on the fertile plains4 ,the highlander and viking1 products of the valleys raised high amid the mountains or half-drowned in the sea1 are everywhere of kindred spirit4 ,it is in some such manner as these that the natural conditions of regions1 which must be conformed to by prudence and utilized by labour to yield shelter and food1 have led to the growth of peoples differing in their ways of life1 thought and speech4 ,the initial differences so produced are confirmed and perpetuated by the same barriers which divide the faunal or floral regions1 the sea1 mountains1 deserts and the like1 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 account4#di ,density of population4 ,the specialization which accompanies the division of labour has important geographical consequences1 for it necessitates communication between communities and the interchange of their products4 ,trade makes it possible to work mineral resources in localities where food can only be grown with great difficulty and expense1 or which are even totally barren and waterless1 entirely dependent on supplies from distant sources4 ,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 people4 ,pastoral communities are always scattered very thinly over large areas2 agricultural populations may be almost equally sparse where advanced methods of agriculture and labour-saving machinery are employed2 but where a frugal people are situated on a fertile and inexhaustible soil1 such as the deltas and river plains of ,egypt1 ,india and ,china1 an enormous population may be supported on a small area4 ,in most cases1 however1 a very dense population can only be maintained in regions where mineral resources have fixed the site of great manufacturing industries4 ,the maximum density of population which a given region can support is very difficult to determine2 it depends partly on the race and standard of culture of the people1 partly on the nature and origin of the resources on which they depend1 partly on the artificial burdens imposed and very largely on the climate4 ,density of population is measured by the average number of people residing on a unit of area2 but in order to compare one part of the world with another the average should1 strictly speaking1 be taken for regions of equal size or of equal population2 and the portions of the country which are permanently uninhabitable ought to be excluded from the calculation4#ej ,considering the average density of population within the political limits of countries1 the following list is of some value2 the figures for a few smaller divisions of large countries are added 7in brackets7 for comparison3 ,average ,population on #a sq4 m4 7,for #aijj or #aija47 ---------------------------------------- ----------------- ,country4 ,density ,country4 ,density of pop4 of pop4 ---------------------------------------- ----------------- 7,saxony7 #gdc99 ,ceylon #ada9999 ,belgium #ehi99 ,greece #ig ,java #efh9999 ,european ,turkey #ij 7,england and ,wales7 #eeh ,spain #ig 7,bengal7 #die9999 ,european ,russia #ee9999 ,holland #dcf ,sweden #cj ,united ,kingdom #cdd ,united ,states #be ,japan #cag ,mexico #ah ,italy #bic ,norway #ah ,china proper #bgj9999 ,persia #ae ,german ,empire #bgj ,new ,zealand #g ,austria #bbf ,argentina #e ,switzerland #bjg ,brazil #d4e ,france #ahh ,eastern ,states of ,indian ,empire #afg9999 ,australia #c ,denmark #afj9999 ,dominion of ,canada #a4e ,hungary #aed9999 ,siberia #a ,portugal #adf ,west ,australia #j4b ---------------------------------------- ----------------- 99 ,almost exclusively industrial4 9999 ,almost exclusively agricultural4 ,migration4 ,the movement of people from one place to another without the immediate intention of returning is known as migration1 and according to its origin it may be classed as centrifugal 7directed from a particular area7 and centripetal 7directed towards a particular area74 ,centrifugal migration is usually a matter of compulsion2 it may be necessitated by natural causes1 such as a change of climate leading to the withering of pastures or destruction of agricultural land1 to inundation1 earthquake1 pestilence or to an excess of population over means of support2 or to artificial causes1 such as the wholesale deportation of a conquered people2 or to political or religious persecution4 ,in any case the people are driven out by some adverse change2 and when the urgency is great they may require to drive out in turn weaker people who occupy a desirable territory1 thus propagating the wave of migration1 the direction of which is guided by the forms of the land into inevitable channels4 ,many of the great historic movements of peoples were doubtless due to the gradual change of geographical or climatic conditions2 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 ,world1 the theatres of which were critical points on the great natural lines of communication between east and west4 ,in the case of centripetal migrations people flock to some particular place where exceptionally favourable conditions have been found to exist4 ,the rushes to gold-fields and diamond-fields are typical instances2 the growth of towns on coal-fields and near other sources of power1 and the rapid settlement of such rich agricultural districts as the wheat-lands of the ,american prairies and great plains are other examples4 ,there is1 however1 a tendency for people to remain rooted to the land of their birth1 when not compelled or induced by powerful external causes to seek a new home4 ,political geography4 ,thus arises the spirit of patriotism1 a product of purely geographical conditions1 thereby differing from the sentiment of loyalty1 which is of racial origin4 ,where race and soil conspire to evoke both loyalty and patriotism in a people1 the moral qualities of a great and permanent nation are secured4 ,it is noticeable that the patriotic spirit is strongest in those places where people are brought most intimately into relation with the land2 dwellers in the mountain or by the sea1 and1 above all1 the people of rugged coasts and mountainous archipelagoes1 have always been renowned for love of country1 while the inhabitants of fertile plains and trading communities are frequently less strongly attached to their own land4 ,amongst nomads the tribe is the unit of government1 the political bond is personal1 and there is no definite territorial association of the people1 who may be loyal but cannot be patriotic4 ,the idea of a country arises only when a nation1 either homogeneous or composed of several races1 establishes itself in a region the boundaries of which may be defined and defended against aggression from without4 ,political geography takes account of the partition of the earth amongst organized communities1 dealing with the relation of races to regions1 and of nations to countries1 and considering the conditions of territorial equilibrium and instability4 ,boundaries4 ,the definition of boundaries and their delimitation is one of the most important parts of political geography4 ,natural boundaries are always the most definite and the strongest1 lending themselves most readily to defence against aggression4 ,the sea is the most effective of all1 and an island state is recognized as the most stable4 ,next in importance comes a mountain range1 but here there is often difficulty as to the definition of the actual crest-line1 and mountain ranges being broad regions1 it may happen that a small independent state1 like ,switzerland or ,andorra1 occupies the mountain valleys between two or more great countries4 ,rivers do not form effective international boundaries1 although between dependent self-governing communities they are convenient lines of demarcation4 ,a desert1 or a belt of country left purposely without inhabitants1 like the mark1 marches or debatable lands of the middle ages1 was once a common means of separating nations which nourished hereditary grievances4 ,the buffer-state of modern diplomacy is of the same ineffectual type4 ,a less definite though very practical boundary is that formed by the meeting-line of two languages1 or the districts inhabited by two races4 ,the line of fortresses protecting ,austria from ,italy lies in some places well back from the political boundary1 but just inside the linguistic frontier1 so as to separate the ,german and ,italian races occupying ,austrian territory4 ,arbitrary lines1 either traced from point to point and marked by posts on the ground1 or defined as portions of meridians and parallels1 are now the most common type of boundaries fixed by treaty4 ,in ,europe and ,asia frontiers are usually strongly fortified and strictly watched in times of peace as well as during war4 ,in ,south ,america strictly defined boundaries are still the exception1 and the claims of neighbouring nations have very frequently given rise to war1 though now more commonly to arbitration4#ea ,forms of government4 ,the modes of government amongst civilized peoples have little influence on political geography2 some republics are as arbitrary and exacting in their frontier regulations as some absolute monarchies4 ,it is1 however1 to be noticed that absolute monarchies are confined to the east of ,europe and to ,asia1 ,japan being the only established constitutional monarchy east of the ,carpathians4 ,limited monarchies are 7with the exception of ,japan7 peculiar to ,europe1 and in these the degree of democratic control may be said to diminish as one passes eastwards from the ,united ,kingdom4 ,republics1 although represented in ,europe1 are the peculiar form of government of ,america and are unknown in ,asia4 ,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-government4 ,the latter occurs only in the temperate possessions of the ,british empire1 in which there is no great preponderance of a coloured native population4 ,new colonial forms have been developed during the partition of ,africa amongst ,european powers1 the sphere of influence being especially worthy of notice4 ,this is a vaguer form of control than a protectorate1 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 area1 if it should decide to do so at any future time4 ,the central governments of all civilized countries concerned with external relations are closely similar in their modes of action1 but the internal administration may be very varied4 ,in this respect a country is either centralized1 like the ,united ,kingdom or ,france1 or federated of distinct self-governing units like ,germany 7where the units include kingdoms1 at least three minor types of monarchies1 municipalities and a crown land under a nominated governor71 or the ,united ,states1 where the units are democratic republics4 ,the ultimate cause of the predominant form of federal government may be the geographical diversity of the country1 as in the cantons occupying the once isolated mountain valleys of ,switzerland1 the racial diversity of the people1 as in ,austria-,hungary1 or merely political expediency1 as in republics of the ,american type4 ,the minor subdivisions into provinces1 counties and parishes1 or analogous areas1 may also be related in many cases to natural features or racial differences perpetuated by historical causes4 ,the territorial divisions and subdivisions often survive the conditions which led to their origin2 hence the study of political geography is allied to history as closely as the study of physical geography is allied to geology1 and for the same reason4 ,towns4 ,the aggregation of population in towns was at one time mainly brought about by the necessity for defence1 a fact indicated by the defensive sites of many old towns4 ,in later times1 towns have been more often founded in proximity to valuable mineral resources1 and at critical points or nodes on lines of communication4 ,these are places where the mode of travelling or of transport is changed1 such as seaports1 river ports and railway termini1 or natural resting-places1 such as a ford1 the foot of a steep ascent on a road1 the entrance of a valley leading up from a plain into the mountains1 or a crossing-place of roads or railways4#eb ,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 communication4 ,lines of communication4 ,in countries of uniform surface or faint relief1 roads and railways may be constructed in any direction without regard to the configuration4 ,in places where the low ground is marshy1 roads and railways often follow the ridge-lines of hills1 or1 as in ,finland1 the old glacial eskers1 which run parallel to the shore4 ,wherever the relief of the land is pronounced1 roads and railways are obliged to occupy the lowest ground winding along the valleys of rivers and through passes in the mountains4 ,in exceptional cases obstructions which it would be impossible or too costly to turn are overcome by a bridge or tunnel1 the magnitude of such works increasing with the growth of engineering skill and financial enterprise4 ,similarly the obstructions offered to water communication by interruption through land or shallows are overcome by cutting canals or dredging out channels4 ,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 power4#ec ,commercial geography4 ,commercial geography may be defined as the description of the earth's surface with special reference to the discovery1 production1 transport and exchange of commodities4 ,the transport concerns land routes and sea routes1 the latter being the more important4 ,while steam has been said to make a ship independent of wind and tide1 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 currents1 and this involves the application of oceanographical and meteorological knowledge4 ,the older navigation by utilizing the power of the wind demands a very intimate knowledge of these conditions1 and it is probable that a revival of sailing ships may in the present century vastly increase the importance of the study of maritime meteorology4 ,the discovery and production of commodities require a knowledge of the distribution of geological formations for mineral products1 of the natural distribution1 life-conditions and cultivation or breeding of plants and animals and of the labour market4 ,attention must also be paid to the artificial restrictions of political geography1 to the legislative restrictions bearing on labour and trade as imposed in different countries1 and1 above all1 to the incessant fluctuations of the economic conditions of supply and demand and the combinations of capitalists or workers which affect the market4#ed ,the term applied geography has been employed to designate commercial geography1 the fact being that every aspect of scientific geography may be applied to practical purposes1 including the purposes of trade4 ,but apart from the applied science1 there is an aspect of pure geography which concerns the theory of the relation of economics to the surface of the earth4 ,conclusion4 ,it will be seen that as each successive aspect of geographical science is considered in its natural sequence the conditions become more numerous1 complex1 variable and practically important4 ,from the underlying abstract mathematical considerations all through the superimposed physical1 biological1 anthropological1 political and commercial development of the subject runs the determining control exercised by crust-forms acting directly or indirectly on mobile distributions2 and this is the essential principle of geography4 7,h4 ,r4 ,m47 ,,footnotes3 #a ,a concise sketch of the whole history of geographical method or theory as distinguished from the history of geographical discovery 7see later section of this article7 is only to be found in the introduction to ,h4 ,wagner's ,lehrbuch der ,geographie1 vol4 i4 7,leipzig1 #aijj71 which is in every way the most complete treatise on the principles of geography4 #b ,history of ,ancient ,geography 7,cambridge1 #ahig71 p4 #gj4 #c ,see ,j4,l4 ,myres1 ,an ,attempt to reconstruct the ,maps used by ,herodotus1 ,geographical ,journal1 viii4 7#ahif71 p4 #fje4 #d ,geschichte der wissenschaftlichen ,erdkunde der ,griechen 7,leipzig1 #ahia71 ,abt4 #c1 p4 #fj4 #e ,bunbury's ,history of ,ancient ,geography 7#b vols41 ,london1 #ahgi71 ,muller's ,geographi ,graeci minores 7#b vols41 ,paris1 #ahee1 #ahfa7 and ,berger's ,geschichte der wissenschaftlichen ,erdkunde der ,griechen 7#d vols41 ,leipzig1 #ahhg-ahic7 are standard authorities on the ,greek geographers4 #f ,the period of the early middle ages is dealt with in ,beazley's ,dawn of ,modern ,geography 7,london2 part i41 #ahig2 part ii41 #aija2 part iii41 #aijf72 see also ,winstedt1 ,cosmos ,indicopleustes 7#aiaj74 #g ,from translator's preface to the ,english version by ,mr ,dugdale 7#agcc71 entitled ,a ,complete ,system of ,general ,geography1 revised by ,dr ,peter ,shaw 7,london1 #agef74 #h ,printed in ,schriften zur physischen ,geographie1 vol4 vi4 of ,schubert's edition of the collected works of ,kant 7,leipzig1 #ahci74 ,first published with notes by ,rink in #ahjb4 #i ,history of ,civilization1 vol4 i4 7#aheg74 #aj ,see ,h4,j4 ,mackinder in ,british ,association ,report 7,ipswich71 #ahie1 p4 #gch1 for a summary of ,german opinion1 which has been expressed by many writers in a somewhat voluminous literature4 #aa ,h4 ,wagner's year-book1 ,geographische ,jahrbuch1 published at ,gotha1 is the best systematic record of the progress of geography in all departments2 and ,haack's ,geographen ,kalender1 also published annually at ,gotha1 gives complete lists of the geographical societies and geographers of the world4 #ab ,this phrase is old1 appearing in one of the earliest ,english works on geography1 ,william ,cuningham's ,cosmographical ,glasse conteinyng the pleasant ,principles of ,cosmographie1 ,geographie1 ,hydrographie or ,navigation 7,london1 #aeei74 #ac ,see also ,s4 ,gunther1 ,handbuch der mathematischen ,geographie 7,stuttgart1 #ahij74 #ad ,on the ,height of the ,land and the ,depth of the ,ocean1 ,scot4 ,geog4 ,mag4 iv4 7#ahhh71 p4 #a4 ,estimates had been made previously by ,humboldt1 ,de ,lapparent1 ,h4 ,wagner1 and subsequently by ,penck and ,heiderich1 and for the oceans by ,karstens4 #ae ,petermanns ,mitteilungen1 xxv4 7#ahhi71 p4 #ag4 #af ,proc4 ,roy4 ,soc4 ,edin4 xvii4 7#ahij7 p4 #ahe4 #ag ,comptes rendus ,acad4 ,sci4 7,paris1 #ahij71 vol4 iii4 p4 #iid4 #ah ,areal und mittlere ,erhebung der ,landflachen sowie der ,erdkruste in ,gerland's ,beitrage zur ,geophysik1 ii4 7#ahie7 p4 #ffg4 ,see also ,nature1 #ed 7#ahif71 p4 #aab4 #ai ,petermanns ,mitteilungen1 xxxv4 7#ahhi7 p4 #ai4 #bj ,the areas of the continental shelf and lowlands are approximately equal1 and it is an interesting circumstance that1 taken as a whole1 the actual coast-line comes just midway on the most nearly level belt of the earth's surface1 excepting the ocean floor4 ,the configuration of the continental slope has been treated in detail by ,nansen in ,scientific ,results of ,norwegian ,north ,polar ,expedition1 vol4 iv4 7#aijd71 where full references to the literature of the subject will be found4 #ba ,british ,association ,report 7,edinburgh1 #ahib71 p4 #fii4 #bb ,das ,antlitz der ,erde 7#d vols41 ,leipzig1 #ahhe1 #ahhh1 #aija74 ,translated under the editorship of ,e4 de ,margerie1 with much additional matter1 as ,la ,face de la terre1 vols4 i4 and ii4 7,paris1 #ahig1 #aijj71 and into ,english by ,dr ,hertha ,sollas as ,the ,face of the ,earth1 vols4 i4 and ii4 7,oxford1 #aijd1 #aijf74 #bc ,elie de ,beaumont1 ,notice sur les systemes de montagnes 7#c vols41 ,paris1 #aheb74 #bd ,vestiges of the ,molten ,globe 7,london1 #ahge74 #be ,see ,j4,w4 ,gregory1 ,the ,plan of the ,earth and its ,causes1 ,geog4 ,journal1 xiii4 7#ahii7 p4 #bbe2 ,lord ,avebury1 ibid4 xv4 7#aijj7 p4 #df2 ,marcel ,bertrand1 ,deformation tetraedrique de la terre et deplacement du pole1 ,comptes rendus ,acad4 ,sci4 7,paris1 #aijj71 vol4 cxxx4 p4 #ddi2 and ,a4 de ,lapparent1 ibid4 p4 #fad4 #bf ,see ,a4,e4,h4 ,love1 ,gravitational ,stability of the ,earth1 ,phil4 ,trans4 ser4 ,a4 vol4 ccvii4 7#aijg7 p4 #aga4 #bg ,rumpf1 in ,german1 the language in which this distinction was first made4 #bh ,lehrbuch der ,geographie 7,hanover and ,leipzig1 #aijj71 ,bd4 i4 ,s4 #bde1 #bdi4 #bi ,see1 for example1 ,f4,g4 ,hahn's ,insel-,studien 7,leipzig1 #ahhc74 #cj ,see ,geographical ,journal1 xxii4 7#aijc7 pp4 #aia-aid4 #ca ,the most important works on the classification of land forms are ,f4 von ,richthofen1 ,fuhrer fur ,forschungsreisende 7,berlin1 #ahhf72 ,g4 de la ,noe and ,e4 de ,margerie1 ,les ,formes du terrain 7,paris1 #ahhh72 and above all ,a4 ,penck1 ,morphologie der ,erdoberflache 7#b vols41 ,stuttgart1 #ahid74 ,compare also ,a4 de ,lapparent1 ,lecons de geographie physique 7#bnd ed41 ,paris1 #ahih71 and ,w4,m4 ,davis1 ,physical ,geography 7,boston1 #ahii74 #cb ,geomorphologie als genetische ,wissenschaft1 in ,report of ,sixth ,international ,geog4 ,congress 7,london1 #ahie71 p4 #gce 7,english ,abstract1 p4 #gdh74 #cc ,on this subject see ,j4 ,geikie1 ,earth ,sculpture 7,london1 #ahih72 ,j4,e4 ,marr1 ,the ,scientific ,study of ,scenery 7,london1 #aijj72 ,sir ,a4 ,geikie1 ,the ,scenery and ,geology of ,scotland 7,london1 #bnd ed41 #ahhg72 ,lord ,avebury 7,sir ,j4 ,lubbock71 ,the ,scenery of ,switzerland 7,london1 #ahif7 and ,the ,scenery of ,england 7,london1 #aijb74 #cd ,some geographers distinguish a mountain from a hill by origin2 thus ,professor ,seeley says a mountain implies elevation and a hill implies denudation1 but the external forms of both are often identical4 ,report ,,vi4 ,int4 ,geog4 ,congress 7,london1 #ahie71 p4 #gea4 #ce ,mountains1 in ,scot4 ,geog4 ,mag4 ii4 7#ahif7 p4 #ade4 #cf ,fuhrer fur ,forschungsreisende1 pp4 #feb-fhe4 #cg ,see1 for a summary of river-action1 ,a4 ,phillipson1 ,studien uber ,wasserscheiden 7,leipzig1 #ahhf72 also ,i4,c4 ,russell1 ,river ,development1 7,london1 #ahih7 7published as ,the ,rivers of ,north ,america1 ,new ,york1 #ahih74 #ch ,w4,m4 ,davis1 ,the ,geographical ,cycle1 ,geog4 ,journ4 xiv4 7#ahii7 p4 #dhd4 #ci ,a4 ,penck1 ,potamology as a ,branch of ,physical ,geography1 ,geog4 ,journ4 x4 7#ahig7 p4 #fai4 #dj ,see1 for instance1 ,e4 ,wisotzki1 ,hauptfluss und ,nebenfluss 7,stettin1 #ahhi74 ,for practical studies see official reports on the ,mississippi1 ,rhine1 ,seine1 ,elbe and other great rivers4 #da ,f4,a4 ,forel1 ,handbuch der ,seenkunde3 allgemeine ,limnologie 7,stuttgart1 #aija72 ,f4,a4 ,forel1 ,la ,limnologie1 branche de la geographie1 ,report ,,vi4 ,int4 ,geog4 ,congress 7,london1 #ahie71 p4 #eic2 also ,le ,leman 7#b vols41 ,lausanne1 #ahib1 #ahid72 ,h4 ,lullies1 ,studien uber ,seen1 ,jubilaumsschrift der ,albertus-,universitat 7,konigsberg1 #ahid72 and ,g4,r4 ,credner1 ,die ,reliktenseen1 ,petermanns ,mitteilungen1 ,erganzungshefte #hf and #hi 7,gotha41 #ahhg1 #ahhh74 #db ,j4 ,murray1 ,drainage ,areas of the ,continents1 ,scot4 ,geog4 ,mag4 ii4 7#ahhf7 p4 #edh4 #dc ,wagner1 ,lehrbuch der ,geographie 7#aijj71 i4 #ehf4 #dd ,for details1 see ,a4,r4 ,wallace1 ,geographical ,distribution of ,animals and ,island ,life2 ,a4 ,heilprin1 ,geographical and ,geological ,distribution of ,animals 7#ahhg72 ,o4 ,drude1 ,handbuch der ,pflanzengeographie2 ,a4 ,engler1 ,entwickelungsgeschichte der ,pflanzenwelt2 also ,beddard1 ,zoogeography 7,cambridge1 #ahie72 and ,sclater1 ,the ,geography of ,mammals 7,london1 #ahii74 #de ,see particularly ,a4 de ,lapparent1 ,traite de geologie 7#dth ed41 ,paris1 #aijj74 #df ,estimate for #aijj4 ,h4 ,wagner1 ,lehrbuch der ,geographie1 i4 ,p4 #feh4 #dg ,estimate for year not stated4 ,a4,h4 ,keane in ,international ,geography1 p4 #ajh4 #dh ,in ,proc4 ,r4 ,g4 ,s4 xiii4 7#ahia7 p4 #bg4 #di ,on the influence of land on people see ,shaler1 ,nature and ,man in ,america 7,new ,york and ,london1 #ahib72 and ,ellen ,c4 ,semple's ,american ,history and its ,geographic ,conditions 7,boston1 #aijc74 #ej ,see maps of density of population in ,bartholomew's great large-scale atlases1 ,atlas of ,scotland and ,atlas of ,england4 #ea ,for the history of territorial changes in ,europe1 see ,freeman1 ,historical ,geography of ,europe1 edited by ,bury 7,oxford71 #aijc2 and for the official definition of existing boundaries1 see ,hertslet1 ,the ,map of ,europe by ,treaty 7#d vols41 ,london1 #ahge1 #ahia72 ,the ,map of ,africa by ,treaty 7#c vols41 ,london1 #ahif74 ,also ,lord ,curzon's ,oxford address on ,frontiers 7#aijg74 #eb ,for numerous special instances of the determining causes of town sites1 see ,g4,g4 ,chisholm1 ,on the ,distribution of ,towns and ,villages in ,england1 ,geographical ,journal 7#ahig71 ix4 #gf1 x4 #eaa4 #ec ,the whole subject of anthropogeography is treated in a masterly way by ,f4 ,ratzel in his ,anthropogeographie 7,stuttgart1 vol4 i4 #bnd ed41 #ahii1 vol4 ii4 #ahia71 and in his ,politische ,geographie 7,leipzig1 #ahig74 ,the special question of the reaction of man on his environment is handled by ,g4,p4 ,marsh in ,man and ,nature1 or ,physical ,geography as modified by ,human ,action 7,london1 #ahfd74 #ed ,for commercial geography see ,g4,g4 ,chisholm1 ,manual of ,commercial ,geography 7#ahij74 ,,geoid 7from ,gr4 ,greek3 ge1 the earth71 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 it4 ,compared with the spheroid of reference the surface of the geoid is in general depressed over the oceans and raised over the great land masses4 7,see ,,earth1 ,,figure ,,of ,,the47 ,,geok-,,tepe1 a former fortress of the ,turkomans1 in ,russian ,transcaspia1 in the oasis of ,akhal-tekke1 on the ,transcaspian railway1 #bh m4 ,n4,w4 of ,askabad4 ,it consisted of a walled enclosure #a-c_/d m4 in circuit1 the wall being #ah ft4 high and #bj to #cj ft4 thick4 ,in ,december #ahhj the place was attacked by #fjjj ,russians under ,general ,skobelev1 and after a siege of twenty-three days was carried by storm1 although the defenders numbered #be1jjj4 ,a monument and a small museum commemorate the event4 ,,geology 7from ,gr4 ,greek3 ge1 the earth1 and ,greek3 logos1 science71 the science which investigates the physical history of the earth4 ,its object is to trace the structural progress of our planet from the earliest beginnings of its separate existence1 through its various stages of growth1 down to the present condition of things4 ,it seeks to determine the manner in which the evolution of the earth's great surface features has been effected4 ,it unravels the complicated processes by which each continent has been built up4 ,it follows1 even into detail1 the varied sculpture of mountain and valley1 crag and ravine4 ,nor does it confine itself merely to changes in the inorganic world4 ,geology shows that the present races of plants and animals are the descendants of other and very different races which once peopled the earth4 ,it teaches that there has been a progressive development of the inhabitants1 as well as one of the globe on which they have dwelt2 that each successive period in the earth's history1 since the introduction of living things1 has been marked by characteristic types of the animal and vegetable kingdoms2 and that1 however imperfectly the remains of these organisms have been preserved or may be deciphered1 materials exist for a history of life upon the planet4 ,the geographical distribution of existing faunas and floras is often made clear and intelligible by geological evidence2 and in the same way light is thrown upon some of the remoter phases in the history of man himself4 ,a subject so comprehensive as this must require a wide and varied basis of evidence4 ,it is one of the characteristics of geology to gather evidence from sources which at first sight seem far removed from its scope1 and to seek aid from almost every other leading branch of science4 ,thus1 in dealing with the earliest conditions of the planet1 the geologist must fully avail himself of the labours of the astronomer4 ,whatever is ascertainable by telescope1 spectroscope or chemical analysis1 regarding the constitution of other heavenly bodies1 has a geological bearing4 ,the experiments of the physicist1 undertaken to determine conditions of matter and of energy1 may sometimes be taken as the starting-points of geological investigation4 ,the work of the chemical laboratory forms the foundation of a vast and increasing mass of geological inquiry4 ,to the botanist1 the zoologist1 even to the unscientific1 if observant1 traveller by land or sea1 the geologist turns for information and assistance4 ,but while thus culling freely from the dominions of other sciences1 geology claims as its peculiar territory the rocky framework of the globe4 ,in the materials composing that framework1 their composition and arrangement1 the processes of their formation1 the changes which they have undergone1 and the terrestrial revolutions to which they bear witness1 lie the main data of geological history4 ,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 planet4 ,he finds that they have in large measure arranged themselves in chronological sequence1--the oldest lying at the bottom and the newest at the top4 ,relics of an ancient sea-floor are overlain by traces of a vanished land-surface2 these are in turn covered by the deposits of a former lake1 above which once more appear proofs of the return of the sea4 ,among these rocky records lie the lavas and ashes of long-extinct volcanoes4 ,the ripple left upon the shore1 the cracks formed by the sun's heat upon the muddy bottom of a dried-up pool1 the very imprint of the drops of a passing rainshower1 have all been accurately preserved1 and yield their evidence as to geographical conditions often widely different from those which exist where such markings are now found4 ,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 changes4 ,he has found that a certain order of appearance characterizes these organic remains1 that each great group of rocks is marked by its own special types of life1 and that these types can be recognized1 and the rocks in which they occur can be correlated even in distant countries1 and where no other means of comparison would be possible4 ,at one moment he has to deal with the bones of some large mammal scattered through a deposit of superficial gravel1 at another time with the minute foraminifers and ostracods of an upraised sea-bottom4 ,corals and crinoids crowded and crushed into a massive limestone where they lived and died1 ferns and terrestrial plants matted together into a bed of coal where they originally grew1 the scattered shells of a submarine sand-bank1 the snails and lizards which lived and died within a hollow-tree1 the insects which have been imprisoned within the exuding resin of old forests1 the footprints of birds and quadrupeds1 the trails of worms left upon former shores--these1 and innumerable other pieces of evidence1 enable the geologist to realize in some measure what the faunas and floras of successive periods have been1 and what geographical changes the site of every land has undergone4 ,it is evident that to deal successfully with these varied materials1 a considerable acquaintance with different branches of science is needful4 ,especially necessary is a tolerably wide knowledge of the processes now at work in changing the surface of the earth1 and of at least those forms of plant and animal life whose remains are apt to be preserved in geological deposits1 or which in their structure and habitat enable us to realize what their forerunners were4 ,it has often been insisted that the present is the key to the past2 and in a wide sense this assertion is eminently true4 ,only in proportion as we understand the present1 where everything is open on all sides to the fullest investigation1 can we expect to decipher the past1 where so much is obscure1 imperfectly preserved or not preserved at all4 ,a study of the existing economy of nature ought thus to be the foundation of the geologist's training4 ,while1 however1 the present condition of things is thus employed1 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 time1 that geological changes have always or generally taken place in former ages in the manner and on the scale which we behold to-day1 and that at the present time all the great geological processes1 which have produced changes in the past eras of the earth's history1 are still existent and active4 ,as a working hypothesis we may suppose that the nature of geological processes has remained constant from the beginning2 but we cannot postulate that the action of these processes has never varied in energy4 ,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 time4 ,for aught we can tell the present is an era of quietude and slow change1 compared with some of the eras which have preceded it4 ,nor perhaps can we be quite sure that1 when we have explored every geological process now in progress1 we have exhausted all the causes of change which1 even in comparatively recent times1 have been at work4 ,in dealing with the geological record1 as the accessible solid part of the globe is called1 we cannot too vividly realize that at the best it forms but an imperfect chronicle4 ,geological history cannot be compiled from a full and continuous series of documents4 ,from the very nature of its origin the record is necessarily fragmentary1 and it has been further mutilated and obscured by the revolutions of successive ages4 ,and even where the chronicle of events is continuous1 it is of very unequal value in different places4 ,in one case1 for example1 it may present us with an unbroken succession of deposits many thousands of feet in thickness1 from which1 however1 only a few meagre facts as to geological history can be gleaned4 ,in another instance it brings before us1 within the compass of a few yards1 the evidence of a most varied and complicated series of changes in physical geography1 as well as an abundant and interesting suite of organic remains4 ,these and other characteristics of the geological record become more apparent and intelligible as we proceed in the study of the science4 ,classification4--,for systematic treatment the subject may be conveniently arranged in the following parts3-- #a4 ,the ,historical ,development of ,geological ,science4--,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 times1 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 science4 #b4 ,the ,cosmical ,aspects of ,geology4--,this section embraces the evidence supplied by astronomy and physics regarding the form and motions of the earth1 the composition of the planets and sun1 and the probable history of the solar system4 ,the subjects dealt with under this head are chiefly treated in separate articles4 #c4 ,geognosy4--,an inquiry into the materials of the earth's substance4 ,this division1 which deals with the parts of the earth1 its envelopes of air and water1 its solid crust and the probable condition of its interior1 especially treats of the more important minerals of the crust1 and the chief rocks of which that crust is built up4 ,geognosy thus lays a foundation of knowledge regarding the nature of the materials constituting the mass of the globe1 and prepares the way for an investigation of the processes by which these materials are produced and altered4 #d4 ,dynamical ,geology studies the nature and working of the various geological processes whereby the rocks of the earth's crust are formed and metamorphosed1 and by which changes are effected upon the distribution of sea and land1 and upon the forms of terrestrial surfaces4 ,such an inquiry necessitates a careful examination of the existing geological economy of nature1 and forms a fitting introduction to an inquiry into the geological changes of former periods4 #e4 ,geotectonic or ,structural ,geology has for its object the architecture of the earth's crust4 ,it embraces an inquiry into the manner in which the various materials composing this crust have been arranged4 ,it shows that some have been formed in beds or strata of sediment on the floor of the sea1 that others have been built up by the slow aggregation of organic forms1 that others have been poured out in a molten condition or in showers of loose dust from subterranean sources4 ,it further reveals that1 though originally laid down in almost horizontal beds1 the rocks have subsequently been crumpled1 contorted and dislocated1 that they have been incessantly worn down1 and have often been depressed and buried beneath later accumulations4 #f4 ,palaeontological ,geology4--,this branch of the subject1 starting from the evidence supplied by the organic forms which are found preserved in the crust of the earth1 includes such questions as the relations between extinct and living types1 the laws which appear to have governed the distribution of life in time and in space1 the relative importance of different genera of animals in geological inquiry1 the nature and use of the evidence from organic remains regarding former conditions of physical geography4 ,some of these problems belong also to zoology and botany1 and are more fully discussed in the articles ,,palaeontology and ,,palaeobotany4 #g4 ,stratigraphical ,geology4--,this section might be called geological history4 ,it works out the chronological succession of the great formations of the earth's crust1 and endeavours to trace the sequence of events of which they contain the record4 ,more particularly1 it determines the order of succession of the various plants and animals which in past time have peopled the earth1 and thus ascertains what has been the grand march of life upon this planet4 #h4 ,physiographical ,geology1 proceeding from the basis of fact laid down by stratigraphical geology regarding former geographical changes1 embraces an inquiry into the origin and history of the features of the earth's surface--continental ridges and ocean basins1 plains1 valleys and mountains4 ,it explains the causes on which local differences of scenery depend1 and shows under what very different circumstances1 and at what widely separated intervals1 the hills and mountains1 even of a single country1 have been produced4 ,most of the detail embraced in these several sections is relegated to separate articles1 to which references are here inserted4 ,the following pages thus deal mainly with the general principles and historical development of the science3-- ,,part ,i4--,,historical ,,development ,geological ,ideas among the ,greeks and ,romans4--,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 mankind4 ,such incidents as earthquakes and volcanic eruptions1 destructive storms on land and sea1 disastrous floods and landslips suddenly strewing valleys with ruin1 must have awakened the terror of those who witnessed them4 ,prominent features of landscape1 such as mountain-chains with their snows1 clouds and thunderstorms1 dark river-chasms that seem purposely cleft open in order to give passage to the torrents that rush through them1 crags with their impressive array of pinnacles and recesses must have appealed of old1 as they still do1 to the awe and wonder of those who for the first time behold them4 ,again1 banks of sea-shells in far inland districts would1 in course of time1 arrest the attention of the more intelligent and reflective observers1 and raise in their minds some kind of surmise as to how such shells could ever have come there4 ,these and other conspicuous geological problems found their earliest solution in legends and myths1 wherein the more striking terrestrial features and the elemental forces of nature were represented to be the manifestation of the power of unseen supernatural beings4 ,the basin of the ,mediterranean ,sea was especially well adapted1 from its physical conditions1 to be the birth-place of such fables4 ,it is a region frequently shaken by earthquakes1 and contains two distinct centres of volcanic activity1 one in the ,aegean ,sea and one in ,italy4 ,it is bounded on the north by a long succession of lofty snow-capped mountain-ranges1 whence copious rivers1 often swollen by heavy rains or melted snows1 carry the drainage into the sea4 ,on the south it boasts the ,nile1 once so full of mystery2 likewise wide tracts of arid desert with their dreaded dust storms4 ,the ,mediterranean itself1 though an inland sea1 is subject to gales1 which1 on exposed coasts1 raise breakers quite large enough to give a vivid impression of the power of ocean waves4 ,the countries that surround this great sheet of water display in many places widely-spread deposits full of sea shells1 like those that still live in the neighbouring bays and gulfs4 ,such a region was not only well fitted to supply subjects for mythology1 but also to furnish1 on every side1 materials which1 in their interest and suggestiveness1 would appeal to the reason of observant men4 ,it was natural1 therefore1 that the early philosophers of ,greece should have noted some of these geological features1 and should have sought for other explanations of them than those to be found in the popular myths4 ,the opinions entertained in antiquity on these subjects may be conveniently grouped under two heads3 7#a7 ,geological processes now in operation1 and 7#b7 geological changes in the past4 ,earthquakes and volcanoes4 #a4 ,contemporary ,processes4--,the geological processes of the present time are partly at work underground and partly on the surface of the earth4 ,the former1 from their frequently disastrous character1 received much attention from ,greek and ,roman authors4 ,aristotle1 in his ,meteorics1 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 earth1 under the influence of the warmth of the sun and the internal heat4 ,wind1 being the lightest and most rapidly moving body1 is the cause of motion in other bodies1 and fire1 united with wind1 becomes flame1 which is endowed with great rapidity of motion4 ,aristotle looked upon earthquakes and volcanic eruptions as closely connected with each other1 the discharge of hot materials to the surface being the result of a severe earthquake1 when finally the wind rushes out with violence1 and sometimes buries the surrounding country under sparks and cinders1 as had happened at ,lipari4 ,these crude conceptions of the nature of volcanic action1 and the cause of earthquakes1 continued to prevail for many centuries4 ,they are repeated by ,lucretius1 who1 however1 following ,anaximenes1 includes as one of the causes of earthquakes the fall of mountainous masses of rock undermined by time1 and the consequent propagation of gigantic tremors far and wide through the earth4 ,strabo1 having travelled through the volcanic districts of ,italy1 was able to recognize that ,vesuvius had once been an active volcano1 although no eruption had taken place from it within human memory4 ,he continued to hold the belief that volcanic energy arose from the movement of subterranean wind4 ,he believed that the district around the ,strait of ,messina1 which had formerly suffered from destructive earthquakes1 was seldom visited by them after the volcanic vents of that region had been opened1 so as to provide an escape for the subterranean fire1 wind1 water and burning masses4 ,he cites in his ,geography a number of examples of widespread as well as local sinkings of land1 and alludes also to the uprise of the sea-bottom4 ,he likewise regards some islands as having been thrown up by volcanic agency1 and others as torn from the mainland by such convulsions as earthquakes4 ,the most detailed account of earthquake phenomena which has come down to us from antiquity is that of ,seneca in his ,quaestiones ,naturales4 ,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 ,a4,d4 #fc4 ,he distinguished several distinct movements of the ground3 #ast1 the up and down motion 7succussio72 #bnd1 the oscillatory motion 7inclinatio72 and probably a third1 that of trembling or vibration4 ,while admitting that some earthquakes may arise from the collapse of the walls of subterranean cavities1 he adhered to the old idea1 held by the most numerous and important previous writers1 that these commotions are caused mainly by the movements of wind imprisoned within the earth4 ,as to the origin of volcanic outbursts he supposed that the subterranean wind in struggling for an outlet1 and whirling through the chasms and passages1 meets with great store of sulphur and other combustible substances1 which by mere friction are set on fire4 ,the elder ,pliny reiterates the commonly accepted opinion as to the efficacy of wind underground4 ,in discussing the phenomena of earthquakes he remarks that towns with many culverts and houses with cellars suffer less than others1 and that at ,naples those houses are most shaken which stand on hard ground4 ,it thus appears that with regard to subterranean geological operations1 no advance was made during the time of the ,greeks and ,romans as to the theoretical explanation of these phenomena2 but a considerable body of facts was collected1 especially as to the effects of earthquakes and the occurrence of volcanic eruptions4 ,action of rivers4 ,the superficial processes of geology1 being much less striking than those of subterranean energy1 naturally attracted less attention in antiquity4 ,the operations of rivers1 however1 which so intimately affect a human population1 were watched with more or less care4 ,herodotus1 struck by the amount of alluvial silt brought down annually by the ,nile and spread over the flat inundated land1 inferred that ,egypt is the gift of the river4 ,aristotle1 in discussing some of the features of rivers1 displays considerable acquaintance with the various drainage-systems on the north side of the ,mediterranean basin4 ,he refers to the mountains as condensers of the atmospheric moisture1 and shows that the largest rivers rise among the loftiest high grounds4 ,he shows how sensibly the alluvial deposits carried down to the sea increase the breadth of the land1 and cites some parts of the shores of the ,black ,sea1 where1 in sixty years1 the rivers had brought down such a quantity of material that the vessels then in use required to be of much smaller draught than previously1 the water shallowing so much that the marshy ground would1 in course of time1 become dry land4 ,strabo supplies further interesting information as to the work of rivers in making their alluvial plains and in pushing their deltas seaward4 ,he remarks that these deltas are prevented from advancing farther outward by the ebb and flow of the tides4 ,occurrences of fossils4 #b4 ,past ,processes4--,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 literature4 ,xenophanes of ,colophon 7#fad ,b4,c47 noticed the occurrence of shells and other marine productions inland among the mountains1 and inferred from them that the land had risen out of the sea4 ,a similar conclusion was drawn by ,xanthus the ,lydian 7#dfd ,b4,c47 from shells like scallops and cockles1 which were found far from the sea in ,armenia and ,lower ,phrygia4 ,herodotus1 ,eratosthenes1 ,strato and ,strabo noted the vast quantities of fossil shells in different parts of ,egypt1 together with beds of salt1 as evidence that the sea had once spread over the country4 ,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 cited4 ,reviewing the evidence of these changes1 he recognized that the sea now covers tracts that were once dry land1 and that land will one day reappear where there is now sea4 ,these alternations are to be regarded as following each other in a certain order and periodicity4 ,but they are apt to escape our notice because they require successive periods of time1 which1 compared with our brief existence1 are of enormous duration1 and because they are brought about so imperceptibly that we fail to detect them in progress4 ,in a celebrated passage in his ,metamorphoses1 ,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 age4 ,it affirms the interchange of land and sea1 the erosion of valleys by descending rivers1 the washing down of mountains into the sea1 the disappearance of the rivers and the submergence of land by earthquake movements1 the separation of some islands from1 and the union of others with1 the mainland1 the uprise of hills by volcanic action1 the rise and extinction of burning mountains4 ,there was a time before ,etna began to glow1 and the time is coming when the mountain will cease to burn4 ,from this brief sketch it will be seen that while the ancients had accumulated a good deal of information regarding the occurrence of geological changes1 their interpretations of the phenomena were to a considerable extent mere fanciful speculation4 ,they had acquired only a most imperfect conception of the nature and operation of the geological processes2 and though many writers realized that the surface of the earth has not always been1 and will not always remain1 as it is now1 they had no glimpse of the vast succession of changes of that surface which have been revealed by geology4 ,they built hypotheses on the slenderest basis of fact1 and did not realize the necessity of testing or verifying them4 ,progress of ,geological ,conceptions in the ,middle ,ages4--,during the centuries that succeeded the fall of the ,western empire little progress was made in natural science4 ,the schoolmen in the monasteries and other seminaries were content to take their science from the literature of ,greece and ,rome4 ,the ,arabs1 however1 not only collected and translated that literature1 but in some departments made original observations themselves4 ,to one of the most illustrious of their number1 ,avicenna1 the translator of ,aristotle1 a treatise has been ascribed1 in which singularly modern ideas are expressed regarding mountains1 some of which are there stated to have been produced by an uplifting of the ground1 while others have been left prominent1 owing to the wearing away of the softer rocks around them4 ,in either case1 it is confessed that the process would demand long tracts of time for its completion4 ,after the revival of learning the ancient problem presented by fossil shells imbedded in the rocks of the interior of many countries received renewed attention4 ,but the conditions for its solution were no longer what they had been in the days of the philosophers of antiquity4 ,men were not now free to adopt and teach any doctrine they pleased on the subject4 ,the ,christian church had meanwhile arisen to power all over ,europe1 and adjudged as heretics all who ventured to impugn any of her dogmas4 ,she taught that the land and the sea had been separated on the third day of creation1 before the appearance of any animal life1 which was not created until the fifth day4 ,to assert that the dry land is made up in great part of rocks that were formed in the sea1 and are crowded with the remains of animals1 was plainly to impugn the veracity of the ,bible4 ,again1 it had come to be the orthodox belief that only somewhere about #fjjj years had elapsed since the time of ,adam and ,eve4 ,if any thoughtful observer1 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 accumulation1 ventured to give public expression to his conviction1 he ran considerable risk of being proceeded against as a heretic4 ,it was needful1 therefore1 to find some explanation of the facts of nature1 which would not run counter to the ecclesiastical system of the day4 ,various such interpretations were proposed1 doubtless in an honest endeavour at reconciliation4 ,three of these deserve special notice3 7#a7 ,many able observers and diligent collectors of fossils persuaded themselves that these objects never belonged to organisms of any kind1 but should be regarded as mere freaks of nature1 having no more connexion with any once living creature than the frost patterns on a window4 ,they were styled formed or figured stones1 lapides sui generis1 and were asserted to be due to some inorganic imitative process within the earth or to the influence of the stars4 7#b7 ,observers who could not resist the evidence of their senses that the fossil shells once belonged to living animals1 and who1 at the same time1 felt the necessity of accounting for the presence of marine organisms in the rocks of which the dry land is largely built up1 sought a way out of the difficulty by invoking the ,deluge of ,noah4 ,here was a catastrophe which1 they said1 extended over the whole globe1 and by which the entire dry land was submerged even up to the tops of the high hills4 ,true1 it only lasted one hundred and fifty days1 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 time4 7#c7 ,some more intelligent men in ,italy1 recognizing that these interpretations could not be upheld1 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 centres4 ,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 crater4 ,it was argued that if ,monte ,nuovo near ,naples could have been accumulated to a height of nearly #ejj ft4 in two days1 there seemed to be no reason against believing that1 during the time of the ,flood1 and in the course of the centuries that have elapsed since that event1 the whole of the fossiliferous rocks might have been deposited4 ,unfortunately for this hypothesis it ignored the fact that these rocks do not consist of volcanic materials4 ,leonardo da ,vinci2 ,fracastorio2 ,falloppio4 ,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 remains1 geology as a science could not begin its existence4 ,the diluvialists 7those who relied on the hypothesis of the ,flood7 held the field during the #afth1 #agth and a great part of the #ahth century4 ,they were looked on as the champions of orthodoxy2 and1 on that account1 they doubtless wielded much more influence than would have been gained by them from the force of their arguments4 ,yet during those ages there were not wanting occasional observers who did good service in combating the prevalent misconceptions1 and in preparing the way for the ultimate triumph of truth4 ,it was more especially in ,italy1 where many of the more striking phenomena of geology are conspicuously displayed1 that the early pioneers of the science arose1 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 deduction4 ,one of the first of these leaders was ,leonardo da ,vinci 7#adeb-aeai71 who1 besides his achievements in painting1 sculpture1 architecture and engineering1 contributed some notable observations regarding the great problem of the origin of fossil shells4 ,he ridiculed the notion that these objects could have been formed by the influence of the stars1 and maintained that they had once belonged to living organisms1 and therefore that what is now land was formerly covered by the sea4 ,girolamo ,fracastorio 7#adhc-aeec7 claimed that the shells could never have been left by the ,flood1 which was a mere temporary inundation1 but that they proved the mountains1 in which they occur1 to have been successively uplifted out of the sea4 ,on the other hand1 even an accomplished anatomist like ,gabriello ,falloppio 7#aebc-aefb7 found it easier to believe that the bones of elephants1 teeth of sharks1 shells and other fossils were mere earthy inorganic concretions1 than that the waters of ,noah's ,flood could ever nave reached as far as ,italy4 ,nicolas ,steno4 ,by much the most important member of this early band of ,italian writers was undoubtedly ,nicolas ,steno 7#afca-afhg71 who1 though born in ,copenhagen1 ultimately settled in ,florence4 ,having made a ,european reputation as an anatomist1 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-fish1 of which he had published the anatomy4 ,cautiously at first1 for fear of offending orthodox opinions1 but afterwards more boldly1 he proclaimed his conviction that those objects had once been part of living animals1 and that they threw light on some of the past history of the earth4 ,he published in #affi a small tract1 ,de solido intra solidum naturaliter contento1 in which he developed the ideas he had formed of this history from an attentive study of the rocks4 ,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 water2 that where they contain marine productions this water is proved to have been the sea2 that diversities in their composition point to commingling of currents1 carrying different kinds of sediment of which the heaviest would first sink to the bottom4 ,he made original and important observations on stratification1 and laid down some of the fundamental axioms in stratigraphy4 ,he reasoned that as the original position of strata was approximately horizontal1 when they are found to be steeply inclined or vertical1 or bent into arches1 they have been disrupted by subterranean exhalations1 or by the falling in of the roofs of underground cavernous spaces4 ,it is to this alteration of the original position of the strata that the inequalities of the earth's surface1 such as mountains1 are to be ascribed1 though some have been formed by the outburst of fire1 ashes and stones from inside the earth4 ,another effect of the dislocation has been to provide fissures1 which serve as outlets for springs4 ,steno's anatomical training peculiarly fitted him for dealing authoritatively with the question of the nature and origin of the fossils contained in the rocks4 ,he had no hesitation in affirming that1 even if no shells had ever been found living in the sea1 the internal structure of these fossils would demonstrate that they once formed parts of living animals4 ,and not only shells1 but teeth1 bones and skeletons of many kinds of fishes had been quarried out of the rocks1 while some of the strata had skulls1 horns and teeth of land-animals4 ,illustrating his general principles by a sketch of what he supposed to have been the past history of ,tuscany1 he added a series of diagrams which show how clearly he had conceived the essential elements of stratigraphy4 ,he thought he could perceive the records of six successive phases in the evolution of the framework of that country1 and was inclined to believe that a similar chronological sequence would be found all over the world4 ,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 required4 ,he thought that many of the fossils must be as old as the time of the general deluge1 but he was careful not to indulge in any speculation as to the antiquity of the earth4 ,lazzaro ,moro4 ,to the ,italian school1 as especially typified in ,steno1 must be assigned the honour of having thus begun to lay firmly and truly the first foundation stones of the modern science of geology4 ,the same school included ,antonio ,vallisneri 7#affa-agcj71 who surpassed his predecessors in his wider and more exact knowledge of the fossiliferous rocks that form the backbone of the ,italian peninsula1 which he contended were formed during a wide and prolonged submergence of the region1 altogether different from the brief deluge of ,noah4 ,there was likewise ,lazzaro ,moro 7#afhg-agdj71 who did good service against the diluvialists1 but the fundamental feature of his system of nature lay in the preponderant part which1 unaware of the great difference between volcanic materials and ordinary sediment1 he assigned to volcanic action in the production of the sedimentary rocks of the earth's crust4 ,he supposed that in the beginning the globe was completely surrounded with water1 beneath which the solid earth lay as a smooth ball4 ,on the third day of creation1 however1 vast fires were kindled inside the globe1 whereby the smooth surface of stone was broken up1 and portions of it1 appearing above the water1 formed the earliest land4 ,from that time onward1 volcanic eruptions succeeded each other1 not only on the emerged land1 but on the sea-floor1 over which the ejected material spread in an ever augmenting thickness of sedimentary strata4 ,in this way ,moro carried the history of the stratified rocks beyond the time of the ,flood back to the ,creation1 which was supposed to have been some #afjj years earlier2 and he brought it down to the present day1 when fresh sedimentary deposits are continually accumulating4 ,he thus incurred no censure from the ecclesiastical guardians of the faith1 and he succeeded in attracting increased public attention to the problems of geology4 ,the influence of his teaching1 however1 was subsequently in great part due to the ,carmelite friar ,generelli1 who published an eloquent exposition of ,moro's views4 ,the ,cosmogonists and ,theories of the ,earth4--,while in ,italy substantial progress was made in collecting information regarding the fossiliferous formations of that country1 and in forming conclusions concerning them based upon more or less accurate observations1 the tendency to mere fanciful speculation1 which could not be wholly repressed in any country1 reached a remarkable extravagance in ,england4 ,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 church1 imagination supplied the place of ascertained fact1 and there appeared during the last twenty years of the #ahth century a group of ,english cosmogonists1 who1 by the sensational character of their speculations1 aroused general attention both in ,britain and on the continent4 ,it may be doubted1 however1 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 unrealities4 ,it is not needful here to do more than mention the names of ,thomas ,burnet1 whose ,sacred ,theory of the ,earth appeared in #afha1 and ,william ,whiston1 whose ,new ,theory of the ,earth was published in #afif4 ,hardly less fanciful than these writers1 though his practical acquaintance with rocks and fossils was infinitely greater1 was ,john ,woodward1 whose ,essay towards a ,natural ,history of the ,earth dates from #afie4 ,more important as a contribution to science was the catalogue of the large collection of fossils1 which he had made from the rocks of ,england and which he bequeathed to the university of ,cambridge4 ,this catalogue appeared in #agbh-agbi with the title of ,an attempt towards a ,natural ,history of the ,fossils of ,england4 ,descartes4 ,a striking contrast to these cosmogonists is furnished by another group1 which arose in ,france and ,germany1 and gave to the world the first rational ideas concerning the probable primeval evolution of our globe4 ,the earliest of these pioneers was the illustrious philosopher ,rene ,descartes 7#aeif-afej74 ,he propounded a scheme of cosmical development in which he represented the earth1 like the other planets1 to have been originally a mass of glowing material like the sun1 and to have gradually cooled on the outside1 while still retaining an incandescent1 self-luminous nucleus4 ,yet with this noble conception1 which modern science has accepted1 ,descartes could not shake himself free from the time-honoured error in regard to the origin of volcanic action4 ,he thought that certain exhalations within the earth condense into oil1 which1 when in violent motion1 enters into the subterranean cavities1 where it passes into a kind of smoke4 ,this smoke is from time to time ignited by a spark of fire and1 pressing violently against its containing walls1 gives rise to earthquakes4 ,if the flame breaks through to the surface at the top of a mountain1 it may escape with enormous energy1 hurling forth much earth mingled with sulphur or bitumen1 and thus producing a volcano4 ,the mountain might burn for a long time until at last its store of fuel in the shape of sulphur or bitumen would be exhausted4 ,not only did the philosopher refrain from availing himself of the high internal temperature of the globe as the source of volcanic energy1 he even did not make use of it as the cause of the ignition of his supposed internal fuel1 but speculated on the kindling of the subterranean fires by the spirits or gases setting fire to the exhalations1 or by the fall of masses of rock and the sparks produced by their friction or percussion4 ,leibnitz4 ,the ideas of ,descartes regarding planetary evolution were enlarged and made more definite by ,wilhelm ,gottfried ,leibnitz 7#afdf-agaf71 whose teaching has largely influenced all subsequent speculation on the subject4 ,in his great tract1 the ,protogaea 7published in #agdi1 thirty-three years after his death71 he traced the probable passage of our earth from an original condition of incandescent vapour into that of a smooth molten globe1 which1 by continuous cooling1 acquired an external solid crust and rugose surface4 ,he thought that the more ancient rocks1 such as granite and gneiss1 might be portions of the earliest outer crust2 and that as the external solidification advanced1 immense subterranean cavities were left which were filled with air and water4 ,by the collapse of the roofs of these caverns1 valleys might be originated at the surface1 while the solid intervening walls would remain in place and form mountains4 ,by the disruption of the crust1 enormous bodies of water were launched over the surface of the earth1 which swept vast quantities of sediment together1 and thus gave rise to sedimentary deposits4 ,after many vicissitudes of this kind1 the terrestrial forces calmed down1 and a more stable condition of things was established4 ,an important feature in the cosmogony of ,leibnitz is the prominent place which he assigned to organic remains in the stratified rocks of the crust4 ,ridiculing the foolish attempts to account for the presence of these objects by calling them sports of nature1 he showed that they are to be regarded as historical monuments2 and he adduced a number of instances wherein successive platforms of strata1 containing organic remains1 bear witness to a series of advances and retreats of the sea4 ,he recognized that some of the fossils appeared to have nothing like them in the living world of to-day1 but some analogous forms might yet be found1 he thought1 in still unexplored parts of the earth2 and even if no living representatives should ever be discovered1 many types of animals might have undergone transformation during the great changes which had affected the surface of the earth4 ,in spite of his clear realization of the vast store of potential energy residing within the highly heated interior of the earth1 ,leibnitz continued to regard volcanic action as due to the combustion of inflammable substances enclosed within the terrestrial crust1 such as stone-coal1 naphtha and sulphur4 ,buffon4 ,appealing to a much wider public than ,descartes or ,leibnitz1 and basing his speculations on a wider acquaintance with the organic and inorganic realms of nature1 ,g4,l4,l4 de ,buffon 7#agjg-aghh7 was undoubtedly one of the most influential forces that in ,europe guided the growth of geological ideas during the #ahth century4 ,he published in #agdi a ,theory of the ,earth1 in which he adopted views similar to those of ,descartes and ,leibnitz as to planetary evolution2 but though he realized the importance of fossils as records of former conditions of the earth's surface1 he accounted for them by supposing that they had been deposited from a universal ocean1 a large part of which had subsequently been engulfed into caverns in the interior of the globe4 ,thirty years later1 after having laboured with skill and enthusiasm in all branches of natural history1 he published another work1 his famous ,epoques de la nature 7#aggh71 which is specially remarkable as the first attempt to deal with the history of the earth in a chronological manner1 and to compute1 on a basis of experiment1 the antiquity of the several stages of this history4 ,his experiments were made with globes of cast iron1 and could not have yielded results of any value for his purpose2 but in so far as his calculations were not mere random guesses but had some kind of foundation on experiment1 they deserve respectful recognition4 ,he divided the history of our earth into six periods of unequal duration1 the whole comprising a period of some #gj1jjj or #ge1jjj years4 ,he supposed that the stage of incandescence1 before the globe had consolidated to the centre1 lasted #bicf years1 and that about #ce1jjj years elapsed before the surface had cooled sufficiently to be touched1 and therefore to be capable of supporting living things4 ,terrestrial animal life1 however1 was not introduced until #ee1jjj or #fj1jjj years after the beginning of the world or about #ae1jjj years before our time4 ,looking into the future1 he foresaw that1 by continued refrigeration1 our globe will eventually become colder than ice1 and this fair face of nature1 with its manifold varieties of plant and animal life1 will perish after having existed for #acb1jjj years4 ,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 treatises4 ,he still continued to believe in the lowering of the ocean by subsidence into vast subterranean cavities1 with a consequent emergence of land4 ,he still looked on volcanoes as due to the burning of pyritous and combustible stones1 though he now called in the co-operation of electricity4 ,he calculated that the first volcanoes could not arise until some #ej1jjj years after the beginning of the world1 by which time a sufficient extent of dense vegetation had been buried in the earth to supply them with fuel4 ,he appears to have had but an imperfect acquaintance with the literature of his own time4 ,at least there can be little doubt that had he availed himself of the labours of his own countryman1 ,jean ,etienne ,guettard 7#agae-aghf71 of ,giovanni ,arduino 7#agad-agie7 in ,italy1 and of ,johann ,gottlob ,lehmann 7d4 #agfg7 and ,george ,christian ,fuchsel 7#agbb-aggc7 in ,germany1 he would have been able to give to his epochs a more definite succession of events and a greater correspondence with the facts of nature4 ,james ,hutton4 ,among the writers of the #ahth century1 who formed philosophical conceptions of the system of processes by which the life of our earth as a habitable globe is carried on1 a foremost place must be assigned to ,james ,hutton 7#agbf-agig74 ,educated for the medical profession1 he studied at ,edinburgh and at ,paris1 and took his doctor's degree at ,leiden4 ,but having inherited a small landed property in ,berwickshire1 he took to agriculture1 and after putting his land into excellent order1 let his farm and betook himself to ,edinburgh1 there to gratify the scientific tastes which he had developed early in life4 ,he had been more especially led to study minerals and rocks1 and to meditate on the problems which they suggest as to the constitution and history of the earth4 ,his journeys in ,britain and on the continent of ,europe had furnished him with material for reflection2 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 world4 ,he used to discuss his views with one or two of his friends1 but refrained from publishing them to the world until1 on the foundation of the ,royal ,society of ,edinburgh1 he communicated an outline of his doctrine to that learned body in #aghe4 ,some years later he expanded this first essay into a larger work in two volumes1 which were published in #agie with the title of ,theory of the ,earth1 with ,proofs and ,illustrations4 ,john ,playfair4 ,hutton's teaching has exercised a profound influence on modern geology4 ,this influence1 however1 has arisen less from his own writings than from the account of his doctrines given by his friend ,john ,playfair in the classic work entitled ,illustrations of the ,huttonian ,theory1 published in #ahjb4 ,hutton wrote in so prolix and obscure a style as rather to repel than attract readers4 ,playfair1 on the other hand1 expressed himself in such clear and graceful language as to command general attention1 and to gain wide acceptance for his master's views4 ,unlike the older cosmogonists1 ,hutton refrained from trying to explain the origin of things1 and from speculations as to what might possibly have been the early history of our globe4 ,he determined from the outset to interpret the past by what can be seen to be the present order of nature2 and he refused to admit the operation of causes which cannot be shown to be part of the actual terrestrial system4 ,like other observers who had preceded him1 he recognized in the various rocks composing the dry land evidence of former geographical conditions very different from those which now prevail4 ,he saw that the vast majority of rocks consist of hardened sediments and must have been deposited in the sea4 ,he could distinguish among them an older or ,primary series1 and a younger or ,secondary series2 and did not dispute the existence of a ,tertiary series claimed by ,peter ,simon ,pallas 7#agda-ahaa74 ,he believed that these various aqueous accumulations had been consolidated by subterranean heat1 that the oldest and lowest rocks had suffered most from this action1 that into these more deep-seated masses subsequent veins and larger bodies of molten matter were injected from below1 and thus that what was originally loose detritus eventually became changed in such crystalline schists as are now found in mountain-chains4 ,in the course of these terrestrial revolutions sedimentary strata1 originally more or less nearly horizontal1 have been pushed upward1 dislocated1 crumpled1 placed on end1 and even elevated to form ranges of lofty mountains4 ,hutton looked upon these disturbances as due to the expansive power of subterranean heat2 but he did not attempt to sketch the mechanism of the process1 and he expressly declined to offer any conjecture as to how the land so elevated remains in that position4 ,he thought that the interior of our planet may be a fluid mass1 melted1 but unchanged by the action of heat2 and1 far from connecting volcanoes with the combustion of inflammable substances1 as had been the prevalent belief for so many centuries1 he looked upon them as a beneficent provision of spiracles to the subterranean furnace1 in order to prevent the unnecessary elevation of land and fatal effects of earthquakes4 ,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 globe4 ,hutton saw that the land is undergoing a ceaseless process of degradation1 through the influence of the air1 frost1 rain1 rivers and the sea1 and that in course of time1 if no countervailing agency should intervene1 the whole of the dry land will be washed away into the sea4 ,but he also perceived that this universal erosion is not everywhere carried on at the same rate2 that it is specially active along the channels of torrents and rivers1 and that1 owing to this difference these channels are gradually deepened and widened1 until the complicated valley-system of a country is carved out4 ,he recognized that the detritus worn away from the land must be spread out over the floor of the sea1 so as to form there strata similar to those that compose most of the dry land4 ,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-day1 he could look forward to future ages1 when the same subterranean agency which had raised up the present land would again be employed to uplift the bed of the existing ocean1 thus to renew the surface of our earth as a habitable globe1 and to start a fresh cycle of erosion and deposition4 ,lamarck4 ,though ,hutton was not unaware that organic remains abound in many of the stratified rocks1 he left them out of consideration in the elaboration of his theory4 ,it was otherwise with one of his ,french contemporaries1 the illustrious ,j4,b4 ,lamarck 7#agdd-ahbi71 who1 after having attained great eminence as a botanist1 turned to zoology when he was nearly fifty years of age1 and before long rose to even greater distinction in that department of science4 ,his share in the classification and description of the mollusca and in founding invertebrate palaeontology1 his theory of organic evolution and his philosophical treatment of many biological questions have been tardily recognized1 but his contributions to geology have been less generally acknowledged4 ,when he accepted the professorship of zoology2 of insects1 of worms and of microscopic animals at the ,museum of ,natural ,history1 ,paris1 in #agic1 he at once entered with characteristic ardour and capacity into the new field of research then opened to him4 ,in dealing with the mollusca he considered not merely the living but also the extinct forms1 especially the abundant1 varied and well-preserved genera and species furnished by the ,tertiary deposits of the ,paris basin1 of which he published descriptions and plates that proved of essential service in the stratigraphical work of ,cuvier and ,alexandre ,brongniart 7#aggj-ahdg74 ,his labours among these relics of ancient seas and lakes led him to ponder over the past history of the globe1 and as he was seldom dilatory in making known the opinions he had formed1 he communicated some of his conclusions to the ,national ,institute in #agii4 ,these1 including a further elaboration of his views1 he published in #ahjb in a small volume entitled ,hydrogeologie4 ,this treatise1 though it did not reach a second edition and has never been reprinted1 deserves an honourable place in geological literature4 ,its object1 the author states1 was to present some important and novel considerations1 which he thought should form the basis of a true theory of the earth4 ,he entirely agreed with the doctrine of the subaerial degradation of the land and the erosion of valleys by running water4 ,not even ,playfair could have stated this doctrine more emphatically1 and it is worthy of notice that ,playfair's ,illustrations of the ,huttonian ,theory appeared in the same year with ,lamarck's book4 ,the ,french naturalist1 however1 carried his conclusions so far as to take no account of any great movements of the terrestrial crust1 which might have produced or modified the main physical features of the surface of the globe4 ,he thought that all mountains1 except such as were thrown up by volcanic agency or local accidents1 have been cut out of plains1 the original surfaces of which are indicated by the crests and summits of these elevations4 ,lamarck1 in reflecting upon the wide diffusion of fossil shells and the great height above the sea at which they are found1 conceived the extraordinary idea that the ocean basin has been scoured out by the sea1 and that1 by an impulse communicated to the waters through the influence chiefly of the moon1 the sea is slowly eating away the eastern margins of the continents1 and throwing up detritus on their western coasts1 and is thus gradually shifting its basin round the globe4 ,he would not admit the operation of cataclysms2 but insisted as strongly as ,hutton on the continuity of natural processes1 and on the necessity of explaining former changes of the earth's surface by causes which can still be seen to be in operation4 ,as might be anticipated from his previous studies1 he brought living things and their remains into the forefront of his theory of the earth4 ,he looked upon fossils as one of the chief means of comprehending the revolutions which the surface of the earth has undergone2 and in his little volume he again and again dwells on the vast antiquity to which these revolutions bear witness4 ,he acutely argues1 from the condition of fossil shells1 that they must have lived and died where their remains are now found4 ,in the last part of his treatise ,lamarck advances some peculiar opinions in physics and chemistry1 which he had broached eighteen years before1 but which had met with no acceptance among the scientific men of his time4 ,he believed that the tendency of all compound substances is to decay1 and thereby to be resolved into their component constituents4 ,yet he saw that the visible crust of the earth consists almost wholly of compound bodies4 ,he therefore set himself to solve the problem thus presented4 ,perceiving that the biological action of living organisms is constantly forming combinations of matter1 which would never have otherwise come into existence1 he proceeded to draw the extraordinary conclusion that the action of plant and animal life 7the ,pouvoir de la vie7 upon the inorganic world is so universal and so potent1 that the rocks and minerals which form the outer part of the earth's crust are all1 without exception1 the result of the operations of once living bodies4 ,though this sweeping deduction must be allowed to detract from the value of ,lamarck's work1 there can be no doubt that he realized1 more fully than any one had done before him1 the efficacy of plants and animals as agents of geological change4 ,cuvier4 ,the last notable contributor to the cosmological literature of geology was another illustrious ,frenchman1 the comparative anatomist ,cuvier 7#agfi-ahcb74 ,he was contemporary with ,lamarck1 but of a very different type of mind4 ,the brilliance of his speculations1 and the charm with which he expounded them1 early gained for him a prominent place in the society of ,paris4 ,he too was drawn by his zoological studies to investigate fossil organic remains1 and to consider the former conditions of the earth's surface1 of which they are memorials4 ,it was among the vertebrate organisms of the ,paris basin that he found his chief material1 and from them that he prepared the memoirs which led to him being regarded as the founder of vertebrate palaeontology4 ,but beyond their biological interest1 they awakened in him a keen desire to ascertain the character and sequence of the geographical revolutions to which they bear witness4 ,he approached the subject from an opposite and less philosophical point of view than that of ,lamarck1 coming to it with certain preconceived notions1 which affected all his subsequent writings4 ,while ,lamarck was by instinct an evolutionist1 who sought to trace in the history of the past the operation of the same natural processes as are still at work1 ,cuvier1 on the other hand1 was a catastrophist1 who invoked a succession of vast cataclysms to account for the interruptions in the continuity of the geological record4 ,in a preliminary ,discourse prefixed to his ,recherches sur les ossemens fossiles 7#ahba7 ,cuvier gave an outline of what he conceived to have been the past history of our globe1 so far as he had been able to comprehend it from his investigations of the ,tertiary formations of ,france4 ,he believed that in that history evidence can be recognized of the occurrence of many sudden and disastrous revolutions1 which1 to judge from their effects on the animal life of the time1 must have exceeded in violence anything we can conceive at the present day1 and must have been brought about by other agencies than those which are now in operation4 ,yet1 in spite of these catastrophes1 he saw that there has been an upward progress in the animal forms inhabiting the globe1 until the series ended in the advent of man4 ,he could not1 however1 find any evidence that one species has been developed from another1 for in that case there should have been traces of intermediate forms among the stratified formations1 where he affirmed that they had never been found4 ,a prominent position in the ,discourse is given to a strenuous argument to disprove the alleged antiquity of some nations1 and to show that the last great catastrophe occurred not more than some #ejjj or #fjjj years ago4 ,cuvier thus linked himself with those who in previous generations had contended for the efficacy of the ,deluge4 ,but his researches among fossil animals had given him a far wider outlook into the geological past1 and had opened up to him a succession of deeply interesting problems in the history of life upon the earth1 which1 though he had not himself material for their solution1 he could foresee would be cleared up in the future4 ,gradual ,shaping of ,geology into a ,distinct ,branch of ,science4--,it will be seen from the foregoing historical sketch that it was only after the lapse of long centuries1 and from the labours of many successive generations of observers and writers1 that what we now know as the science of geology came to be recognized as a distinct department of natural knowledge1 founded upon careful and extended study of the structure of the earth1 and upon observation of the natural processes1 which are now at work in changing the earth's surface4 ,the term geology1#a descriptive of this branch of the investigation of nature1 was not proposed until the last quarter of the #ahth century by ,jean ,andre ,de ,luc 7#agbg-ahag7 and ,horace ,benedict ,de ,saussure 7#agdj-agdi74 ,but the science was then in a markedly half-formed condition1 theoretical speculation still in large part supplying the place of deductions from a detailed examination of actual fact4 ,in #ahjg 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 earth4 ,the cosmogonists and framers of ,theories of the ,earth were succeeded by other schools of thought4 ,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 are1 and that they had from time to time devastated the earth's surface2 extirpating the races of plants and animals1 and preparing the ground for new creations of organized life4 ,then came the ,uniformitarians1 who1 pushing the doctrines of ,hutton to an extreme which he did not propose1 saw no evidence that the activity of the various geological causes has ever seriously differed from what it is at present4 ,they were inclined to disbelieve that the stratified formations of the earth's crust furnish conclusive evidence of a gradual progression1 from simple types of life in the oldest strata to the most highly developed forms in the youngest2 and saw no reason why remains of the higher vertebrates should not be met with among the ,palaeozoic formations4 ,sir ,charles ,lyell 7#agig-ahge7 was the great leader of this school4 ,his admirably clear and philosophical presentations of geological facts which1 with unwearied industry1 he collected from the writings of observers in all parts of the world1 impressed his views upon the whole ,english-speaking world1 and gave to geological science a coherence and interest which largely accelerated its progress4 ,in his later years1 however1 he frankly accepted the views of ,darwin in regard to the progressive character of the geological record4 ,the youngest of the schools of geological thought is that of the ,evolutionists4 ,pointing to the whole body of evidence from inorganic and organic nature1 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 time1 and that the crust of the earth contains an abundant1 though incomplete1 record of the successive stages through which the plant and animal kingdoms have reached their existing organization4 ,the publication of ,darwin's ,origin of ,species in #ahei1 in which evolution was made the key to the history of the animal and vegetable kingdoms1 produced an extraordinary revolution in geological opinion4 ,the older schools of thought rapidly died out1 and evolution became the recognized creed of geologists all over the world4 ,werner4 ,development of ,opinion regarding ,igneous ,rocks4--,so long as the idea prevailed that volcanoes are caused by the combustion of inflammable substances underground1 there could be no rational conception of volcanic action and its products4 ,even so late as the middle of the #ahth century1 as above remarked1 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 vents4 ,after his time the notion continued to prevail that all the rocks which form the dry land were laid down under water4 ,even streams of lava1 which were seen to flow from an active crater1 were regarded only as portions of sedimentary or other rocks1 which had been melted by the fervent heat of the burning inflammable materials that had been kindled underground4 ,in spite of the speculations of ,descartes and ,leibnitz1 it was not yet generally comprehended that there exists beneath the terrestrial crust a molten magma1 which1 from time to time1 has been injected into that crust1 and has pierced through it1 so as to escape at the surface with all the energy of an active volcano4 ,what we now recognize to be memorials of these former injections and propulsions were all confounded with the rocks of unquestionably aqueous origin4 ,the last great teacher by whom these antiquated doctrines were formulated into a system and promulgated to the world was ,abraham ,gottlob ,werner 7#agdi-ahae71 the most illustrious ,german mineralogist and geognost of the second half of the #ahth century4 ,while still under twenty-six years of age1 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 life4 ,possessed of great enthusiasm for his subject1 clear1 methodical and eloquent in his exposition of it1 he soon drew around him men from all parts of the world1 who repaired to study under the great oracle of what he called geognosy 7,gr4 ,greek3 ge1 the earth1 ,greek3 gnosis1 knowledge7 or earth-knowledge4 ,reviving doctrines that had been current long before his time1 he taught that the globe was once completely surrounded with an ocean1 from which the rocks of the earth's crust were deposited as chemical precipitates1 in a certain definite order over the whole planet4 ,among these universal formations of aqueous origin were included many rocks1 which have long been recognized to have been once molten1 and to have risen from below into the upper parts of the terrestrial crust4 ,werner1 following the old tradition1 looked upon volcanoes as modern features in the history of the planet1 which could not have come into existence until a sufficient amount of vegetation had been buried to furnish fuel for their maintenance4 ,hence he attached but little importance to them1 and did not include in his system of rocks any division of volcanic or igneous materials4 ,from the predominant part assigned by him to the sea in the accumulation of the materials of the visible part of the earth1 ,werner and his school were known as ,neptunists4 ,origin of basalt4 ,but many years before the ,saxon professor began to teach1 clear evidence had been produced from central ,france that basalt1 one of the rocks claimed by him as a chemical precipitate and a universal formation1 is a lava which has been poured out in a molten state at various widely separated periods of time and at many different places4 ,so far back as #ageb ,j4,e4 ,guettard 7#agae-aghf7 had shown that the basaltic rocks of ,auvergne are true lavas1 which have flowed out in streams from groups of once active cones4 ,eleven years later the observation was confirmed and greatly extended by ,nicholas ,desmarest 7#agbe-ahae71 who1 during a long course of years1 worked out and mapped the complicated volcanic records of that interesting region1 and demonstrated to all who were willing impartially to examine the evidence the true volcanic nature of basalt4 ,these views found acceptance from some observers1 but they were vehemently opposed by the followers of ,werner1 who1 by the force of his genius1 made his theoretical conceptions predominate all over ,europe4 ,the controversy as to the origin of basalt was waged with great vigour during the later decades of the #ahth century4 ,desmarest took no part in it4 ,he had accumulated such conclusive proof of the correctness of his deductions1 and had so fully expounded the clearness of the evidence in their favour furnished by the region of ,auvergne1 that1 when any one came to consult him on the subject1 he contented himself with giving the advice to go and see4 ,while the debate was in progress on the continent1 the subject was approached from a new and independent point of view by ,hutton in ,scotland4 ,this illustrious philosopher1 as already stated1 realized the importance of the internal heat of the globe in consolidating the sedimentary rocks1 and believed that molten material from the earth's interior has been protruded from below into the overlying crust4 ,some of the material thus injected could be recognized1 he thought1 in granite and in the various dark massive rocks which1 known in ,scotland under the name of whinstone1 were afterwards called ,trap1 and are now grouped under various names1 such as basalt1 dolerite and diorite4 ,so important a share did ,hutton thus assign to the internal heat in the geological evolution of the planet1 that he and those who adopted the same opinions were styled ,plutonists1 or1 especially where they concerned themselves with the volcanic origin of basalt1 ,vulcanists4 ,the geological world was thus divided into two hostile camps1 that of the ,neptunists or ,wernerians1 and that of the ,plutonists1 ,vulcanists or ,huttonians4 ,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 pupils4 ,in particular ,jean ,francois ,d',aubuisson de ,voisins 7#agfi-ahai71 who had written a treatise on the aqueous origin of the basalts of ,saxony1 went afterwards to ,auvergne1 where he was speedily a convert to the views expounded by ,desmarest as to the volcanic nature of basalt4 ,having thus to relinquish one of the fundamental articles of the ,freiberg faith1 he was subsequently led to modify his adherence to others until1 as he himself confessed1 his views came almost wholly to agree with those of ,hutton4 ,not less complete1 and even more important1 was the conversion of the great ,leopold von ,buch 7#aggd-ahec74 ,he1 too1 was trained by ,werner himself1 and proved to be the most illustrious pupil of the ,saxon professor4 ,full of admiration for the ,neptunism in which he had been reared1 he1 in his earliest separate work1 maintained the aqueous origin of basalt1 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 theory4 ,but a little further acquaintance with the facts of nature led ,von ,buch also to abandon his earlier prepossessions4 ,it was a personal visit to the volcanic region of ,auvergne that first opened his eyes1 and led him to recant what he had believed and written about basalt4 ,but the abandonment of so essential a portion of the ,wernerian creed prepared the way for further relinquishments4 ,when a few years later he went to ,norway and found to his astonishment that granite1 which he had been taught to regard as the oldest chemical precipitate from the universal ocean1 could there be seen to have broken through and metamorphosed fossiliferous limestones1 and to have sent veins into them1 his faith in ,werner's order of the succession of the rocks in the earth's crust received a further momentous shock4 ,while one after another of the ,freiberg doctrines crumbled away before him1 he was now able to interrogate nature on a wider field than the narrow limits of ,saxony1 and he was thus gradually led to embrace the tenets of the opposite school4 ,his commanding position1 as the most accomplished geologist on the continent1 gave great importance to his recantation of the ,neptunist creed4 ,his defection indeed was the severest blow that this creed had yet sustained4 ,it may be said to have rung the knell of ,wernerianism1 which thereafter rapidly declined in influence1 while ,plutonism came steadily to the front1 where it has ever since remained4 ,although ,desmarest had traced in ,auvergne a long succession of volcanic eruptions1 of which the oldest went back to a remote period of time1 and although he had shown that this succession1 coupled with the records of contemporaneous denudation1 might be used in defining epochs of geological history1 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 globe1 which had been in operation from the remotest past1 and which had left numerous records among the rocks of the terrestrial crust4 ,during the progress of the controversy between the two great opposing factions in the later portion of the #ahth and the first three decades of the #aith century1 those who espoused the ,vulcanist cause were intent on proving that certain rocks1 which are intercalated among the stratified formations and which were claimed by the ,neptunists as obviously formed by water1 are nevertheless of truly igneous origin4 ,these observers fixed their eyes on the evidence that the material of such rocks1 instead of having been deposited from aqueous solution1 had once been actually molten1 and had in that condition been thrust between the strata1 had enveloped portions of them1 and had indurated or otherwise altered them4 ,they spoke of these masses as unerupted lavas2 and undoubtedly in innumerable instances they were right4 ,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 strata1 but had been poured out at the surface as truly volcanic discharges1 and therefore belonged to the ancient periods represented by the strata between which they are interposed4 ,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 theory4 ,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 ,vulcanists1 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 planet4 ,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 them4 ,in no part of the world has an ampler record of such episodes been preserved than in the ,british ,isles4 ,it was natural1 therefore1 that the subject should there receive most attention4 ,as far back as #ahbj ,ami ,boue 7#agid-ahha7 showed that the ,old ,red ,sandstone of ,scotland includes a great series of volcanic rocks1 and that other rocks of volcanic origin are associated with the ,carboniferous formations4 ,h4,t4 de la ,beche 7#agif-ahee7 afterwards traced proofs of contemporaneous eruptions among the ,devonian rocks of the south-west of ,england4 ,adam ,sedgwick 7#aghe-ahgc7 showed1 first in the ,lake ,district1 and afterwards in ,north ,wales1 the presence of abundant volcanic sheets among the oldest divisions of the ,palaeozoic series2 while ,roderick ,impey ,murchison 7#agib-ahga7 made similar discoveries among the ,lower ,silurian rocks4 ,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 region4 ,growth of ,opinion regarding ,earthquakes4--,we have seen how crude were the conceptions of the ancients regarding the causes of volcanic action1 and that they connected volcanoes and earthquakes as results of the commotion of wind imprisoned within subterranean caverns and passages4 ,one of the earliest treatises1 in which the phenomena of terrestrial movements were discussed in the spirit of modern science1 was the posthumous collection of papers by ,robert ,hooke 7#afce-agjc71 entitled ,lectures and ,discourses of ,earthquakes and ,subterranean ,eruptions1 where the probable agency of earthquakes in upheaving and depressing land is fully considered1 but without any definite pronouncement as to the author's conception of its origin4 ,hooke still associated earthquakes with volcanic action1 and connected both with what he called the general congregation of sulphurous subterraneous vapours4 ,he conceived that some kind of fermentation takes place within the earth1 and that the materials which catch fire and give rise to eruptions or earthquakes are analogous to those that constitute gunpowder4 ,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 ,rev4 ,john ,michell1 and communicated to the ,royal ,society in the year #agfj4 ,still under the old misconception that volcanoes are due to the combustion of inflammable materials1 which he thought might be set on fire by the spontaneous combustion of pyritous strata1 he supposed that1 by the sudden access of large bodies of water to these subterranean fires1 vapour is produced in such quantity and with such force as to give rise to the shock4 ,from the centre of origin of this shock waves1 he thought1 are propagated through the earth1 which are largest at the start and gradually diminish as they travel outwards4 ,by drawing lines at different places in the direction of the track of these waves1 he believed that the place of common intersection of these lines would be nearly the centre of the disturbance4 ,in this way he showed that the great ,lisbon earthquake of #agee had its focus under the ,atlantic1 somewhere between the latitudes of ,lisbon and ,oporto1 and he estimated that the depth at which it originated could not be much less than #a m41 and probably did not exceed #c m4 ,michell1 however1 misconceived the character of the waves which he described1 seeing that he believed them to be due to the actual propagation of the vapour itself underneath the surface of the earth4 ,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 phenomena4 ,in #ahdf ,robert ,mallet 7#ahaj-ahha7 published an important paper ,on the ,dynamics of ,earthquakes in the ,transactions of the ,royal ,irish ,academy4 ,from that time onward he continued to devote his energies to the investigation1 studying the effects of the ,calabrian earthquake of #aheg1 experimenting on the transmission of waves of shock through various materials1 caused by exploding charges of gunpowder1 and collecting all the information to be obtained on the subject4 ,his writings1 and especially his work in two volumes on ,the ,first ,principles of ,observational ,seismology1 must be regarded as having laid the foundations of this branch of modern geology 7see ,,earthquake2 ,,seismometer74 ,history of the ,evolution of ,stratigraphical ,geology4--,men had long been familiar with the evidence that the present dry land once lay under the sea1 before they began to realize that the rocks1 of which the land consists1 contain a record of many alternations of land and sea1 and relics of a long succession of plants and animals from early and simple types up to the manifold and complex forms of to-day4 ,in countries where coal-mining had been prosecuted for generations1 it had been recognized that the rocks consist of strata superposed on each other in a definite order1 which was found to extend over the whole of a district4 ,as far back as #agai ,john ,strachey drew attention to this fact in a communication published in the ,philosophical ,transactions4 ,john ,michell 7#agfj71 in the paper on earthquakes already cited1 showed that he had acquired a clear understanding of the order of succession among stratified formations1 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 mountains1 while the younger and more horizontal strata are spread over the plains4 ,in ,italy ,g4 ,arduino 7#agac-agie7 classified the rocks in the north of the peninsula as ,primitive1 ,secondary1 ,tertiary and ,volcanic4 ,a similar threefold order was announced for the ,harz and ,erzgebirge by ,j4,g4 ,lehmann in #agef4 ,he recognized in that region an ancient series of rocks in inclined or vertical strata1 which rise to the tops of the hills and descend to an unknown depth into the interior4 ,these masses1 he thought1 were contemporaneous with the making of the world4 ,next came the ,flotzgebirge1 consisting of younger sediments1 disposed in flat or gently inclined sheets which overlie the first and more disturbed series1 and are full of petrified remains of plants and animals4 ,lastly he included the mountains which have from time to time been formed by local accidents4 ,still more advanced were the conceptions of ,g4,c4 ,fuchsel1 who in the year #agfb published in ,latin ,a ,history of the ,earth and the ,sea1 based on a ,history of the ,mountains of ,thuringia2 and in #aggc1 in ,german1 a ,sketch of the most ,ancient ,history of the ,earth and ,man4 ,in these works he described the stratigraphical relations and general characters of the various geological formations in his little principality2 and taking them as indicative of a general order of succession1 he traced what he believed to have been a series of revolutions through which the earth has passed4 ,in interpreting this geological history1 he laid great stress on the evidence of the fossils contained in the rocks4 ,he recognized that the various formations differ from each other in their enclosed organic remains1 and that from these differences the existence of former sea-bottoms and land surfaces can be determined4 ,the labours of these pioneers paved the way for the advent of ,werner4 ,though the system evolved by this teacher claimed to discard theory and to be established on a basis of observed facts1 it rested on a succession of hypotheses1 for which no better foundation could be shown than the belief of their author in their validity4 ,starting from the extremely limited stratigraphical range displayed in the geological structure of ,saxony1 he took it as a type for the rest of the globe1 persuading himself and impressing upon his followers that the rocks of that small kingdom were to be taken as examples of his universal formations4 ,the oldest portion of the series1 classed by him as ,primitive1 consisted of rocks which he maintained had been deposited from chemical solution4 ,yet they included granite1 gneiss1 basalt1 porphyry and serpentine1 which1 even in his own day1 were by many observers correctly regarded as of igneous origin4 ,a later group of rocks1 to which he gave the name of ,transition1 comprised1 in his belief1 partly chemical1 partly mechanical sediments1 and contained the earliest fossil organic remains4 ,a third group1 for which he reserved ,lehmann's name ,flotz1 was made up chiefly of mechanical detritus1 while youngest of all came the ,alluvial series of loams1 clays1 sands1 gravels and peat4 ,it was by the gradual subsidence of the ocean that1 as he believed1 the general mass of the dry land emerged1 the first-formed rocks being left standing up1 sometimes on end1 to form the mountains1 while those of later date1 less steeply inclined1 occupied successively lower levels down to the flat alluvial accumulations of the plains4 ,neither ,werner1 nor any of his followers1 ventured to account for what became of the water as the sea-level subsided1 though1 in despite of their antipathy to anything like speculation1 they could not help suggesting1 as an answer to the cogent arguments of their opponents1 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 ocean4 ,nor was any attempt made to explain the extraordinary nature of the supposed chemical precipitates of the universal ocean4 ,the progress of inquiry even in ,werner's lifetime disproved some of the fundamental portions of his system4 ,many of the chemical precipitates were shown to be masses that had been erupted in a molten state from below4 ,his order of succession was found not to hold good2 and though he tried to readjust his sequence and to introduce into it modifications to suit new facts1 its inherent artificiality led to its speedy decline after his death4 ,it must be conceded1 however1 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 subject1 and in preparing the way for a more natural system1 based not on mere mineralogical characters1 but having regard to the organic remains1 which were now being gathered in ever-increasing numbers and variety from stratified formations of many different ages and from all parts of the globe4 ,it was in ,france and in ,england that the foundations of stratigraphy1 based upon a knowledge of organic remains1 were first successfully laid4 ,abbe ,j4,l4 ,giraud-,soulavie 7#ageb-ahac71 in his ,histoire naturelle de la ,france meridionale1 which appeared in seven volumes1 subdivided the limestones of ,vivarais into five ages1 each marked by a distinct assemblage of shells4 ,in the lowest strata1 representing the first age1 none of the fossils were believed by him to have any living representatives1 and he called these rocks ,primordial4 ,in the next group a mingling of living with extinct forms was observable4 ,the third age was marked by the presence of shells of still existing species4 ,the strata of the fourth series were characterized by carbonaceous shales or slates1 containing remains of primordial vegetation1 and perhaps equivalents of the first three calcareous series4 ,the fifth age was marked by recent deposits containing remains of terrestrial vegetation and of land animals4 ,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 basalt1 or in disputes about the character and stratigraphical position of the supposed universal formations1 and when the interest and importance of fossil organic remains still remained unrecognized by the vast majority of the combatants4 ,the rocks of the ,paris basin display so clearly an orderly arrangement1 and are so distinguished for the variety and perfect preservation of their enclosed organic remains1 that they could not fail to attract the early notice of observers4 ,j4 ,e4 ,guettard1 ,g4,f4 ,rouelle 7#agjc-aggj71 ,n4 ,desmarest1 ,a4,l4 ,lavoisier 7#agdc-agid7 and others made observations in this interesting district4 ,but it was reserved for ,cuvier 7#agfi-ahcb7 and ,a4 ,brongniart 7#aggj-ahdg7 to work out the detailed succession of the ,tertiary formations1 and to show how each of these is characterized by its own peculiar assemblage of organic remains4 ,the later progress of investigation has slightly corrected and greatly amplified the tabular arrangement established by these authors in #ahjh1 but the broad outlines of the ,tertiary stratigraphy of the ,paris basin remain still as ,cuvier and ,brongniart left them4 ,the most important subsequent change in the classification of the ,tertiary formations was made by ,sir ,charles ,lyell1 who1 conceiving in #ahbh the idea of a classification of these rocks by reference to their relative proportions of living and extinct species of shells1 established1 in collaboration with ,g4,p4 ,deshayes1 the now universally accepted divisions ,eocene1 ,miocene and ,pliocene4 ,long before ,cuvier and ,brongniart published an account of their researches1 another observer had been at work among the ,secondary formations of the west of ,england1 and had independently discovered that the component members of these formations were each distinguished by a peculiar group of organic remains2 and that this distinction could be used to discriminate them over all the region through which he had traced them4 ,the remarkable man who arrived at this far-reaching generalization was ,william ,smith 7#agfi-ahci71 a land surveyor who1 in the prosecution of his professional business1 found opportunities of traversing a great part of ,england1 and of putting his deductions to the test4 ,as the result of these journeys he accumulated materials enough to enable him to produce a geological map of the country1 on which the distribution and succession of the rocks were for the first time delineated4 ,smith's labours laid the foundation of stratigraphical geology in ,england and he was styled even in his lifetime the ,father of ,english geology4 ,from his day onward the significance of fossil organic remains gained rapidly increasing recognition4 ,thus in ,england the outlines traced by him among the ,secondary and ,tertiary formations were admirably filled in by ,thomas ,webster 7#aggc-ahdd72 while the ,cretaceous series was worked out in still greater detail in the classic memoirs of ,william ,henry ,fitton 7#aghj-ahfa74 ,there was one stratigraphical domain1 however1 into which ,william ,smith did not enter4 ,he traced his sequence of rocks down into the ,coal ,measures1 but contented himself with only a vague reference to what lay underneath that formation4 ,though some of these underlying rocks had in various countries yielded abundant fossils1 they had generally suffered so much from terrestrial disturbances1 and their order of succession was consequently often so much obscured throughout western ,europe1 that they remained but little known for many years after the stratigraphy of the ,secondary and ,tertiary series had been established4 ,at last in #ahca ,murchison began to attack this terra incognita on the borders of ,south ,wales1 working into it from the ,old ,red ,sandstone1 the stratigraphical position of which was well known4 ,in a few years he succeeded in demonstrating the existence of a succession of formations1 each distinguished by its own peculiar assemblage of organic remains which were distinct from those in any of the overlying strata4 ,to these formations he gave the name of ,silurian 7q4v474 ,from the key which his researches supplied1 it was possible to recognize in other countries the same order of formations and the same sequence of fossils1 so that1 in the course of a few years1 representatives of the ,silurian system were found far and wide over the globe4 ,while ,murchison was thus engaged1 ,sedgwick devoted himself to the more difficult task of unravelling the complicated structure of ,north ,wales4 ,he eventually made out the order of the several formations there1 with their vast intercalations of volcanic material4 ,he named them the ,cambrian system 7q4v471 and found them to contain fossils1 which1 however1 lay for some time unexamined by him4 ,he at first believed1 as ,murchison also did1 that his rocks were all older than any part of the ,silurian series4 ,it was eventually discovered that a portion of them was equivalent to the lower part of that series4 ,the oldest of ,sedgwick's groups1 containing distinctive fossils1 retain the name ,cambrian1 and are of high interest1 as they enclose the remains of the earliest faunas which are yet well known4 ,sedgwick and ,murchison rendered yet another signal service to stratigraphical geology by establishing1 in #ahci1 on a basis of palaeontological evidence supplied by ,w4 ,lonsdale1 the independence of the ,devonian system 7q4v474 ,for many years the rocks below the oldest fossiliferous deposits received comparatively little attention4 ,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 for4 ,w4,e4 ,logan 7#agih-ahge7 led the way1 in ,canada1 by establishing there several vast series of rocks1 partly of crystalline schists and gneisses 7,laurentian7 and partly of slates and conglomerates 7,huronian74 ,later observers1 both in ,canada and the ,united ,states1 have greatly increased our knowledge of these rocks1 and have shown their structure to be much more complex than was at first supposed 7see ,,archean ,,system74 ,during the latter half of the #aith 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 is1 the determination of the sequence and distribution of organic remains in these formations1 and the arrangement of the strata into zones1 each of which is distinguished by a peculiar assemblage of fossil species 7see under ,part ,,vi474 ,the zones are usually named after one especially characteristic species4 ,this system of classification was begun in ,germany with reference to the members of the ,jurassic system 7q4v47 by ,a4 ,oppel 7#ahef-aheh7 and ,f4,a4 von ,quenstedt 7#aheh71 and it has since been extended through the other ,mesozoic formations4 ,it has even been found to be applicable to the ,palaeozoic rocks1 which are now subdivided into palaeontological zones4 ,in the ,silurian system1 for example1 the graptolites have been shown by ,c4 ,lapworth to furnish a useful basis for zonal subdivisions4 ,the lowest fossiliferous horizon in the ,cambrian rocks of ,europe and ,north ,america is known as the ,olenellus zone1 from the prominence in it of that genus of trilobite4 ,another conspicuous feature in the progress of stratigraphy during the second half of the #aith century was displayed by the rise and rapid development of what is known as ,glacial geology4 ,the various deposits of drift spread over northern ,europe1 and the boulders scattered across the surface of the plains had long attracted notice1 and had even found a place in popular legend and superstition4 ,when men began to examine them with a view to ascertain their origin1 they were naturally regarded as evidences of the ,noachian deluge4 ,the first observer who drew attention to the smoothed and striated surfaces of rock that underlie the ,drifts was ,hutton's friend1 ,sir ,james ,hall1 who studied them in the lowlands of ,scotland and referred them to the action of great debacles of water1 which1 in the course of some ancient terrestrial convulsion1 had been launched across the face of the country4 ,playfair1 however1 pointed out that the most potent geological agents for the transportation of large blocks of stone are the glaciers4 ,but no one was then bold enough to connect the travelled boulders with glaciers on the plains of ,germany and of ,britain4 ,yet the transporting agency of ice was invoked in explanation of their diffusion4 ,it came to be the prevalent belief among the geologists of the first half of the #aith century1 that the fall of temperature1 indicated by the gradual increase in the number of northern species of shells in the ,english ,crag deposits1 reached its climax during the time of the ,drift1 and that much of the north and centre of ,europe was then submerged beneath a sea1 across which floating icebergs and floes transported the materials of the ,drift and dropped the scattered boulders4 ,as the phenomena are well developed around the ,alps1 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 support4 ,at last ,louis ,agassiz 7#ahjg-ahgc71 who had varied his palaeontological studies at ,neuchatel by excursions into the ,alps1 was so much struck by the proofs of the former far greater extension of the ,swiss glaciers1 that he pursued the investigation and satisfied himself that the ice had formerly extended from the ,alpine valleys right across the great plain of ,switzerland1 and had transported huge boulders from the central mountains to the flanks of the ,jura4 ,in the year #ahdj he visited ,britain and soon found evidence of similar conditions there4 ,he showed that it was not by submergence in a sea cumbered with floating ice1 but by the former presence of vast glaciers or sheets of ice that the ,drift and erratic blocks had been distributed4 ,the idea thus propounded by him did not at once command complete approval1 though traces of ancient glaciers in ,scotland and ,wales were soon detected by native geologists1 particularly by ,w4 ,buckland1 ,lyell1 ,j4,d4 ,forbes and ,charles ,maclaren4 ,robert ,chambers 7#ahjb-ahga7 did good service in gathering additional evidence from ,scotland and ,norway in favour of ,agassiz's views1 which steadily gained adherents until1 after some quarter of a century1 they were adopted by the great majority of geologists in ,britain1 and subsequently in other countries4 ,since that time the literature of geology has been swollen by a vast number of contributions in which the history of the ,glacial period1 and its records both in the ,old and ,new ,world1 have been fully discussed4 ,rise and ,progress of ,palaeontological ,geology4--,as this branch of the science deals with the evidence furnished by fossil organic remains as to former geographical conditions1 it early attracted observers who1 in the superficial beds of marine shells found at some distance from the coast1 saw proofs of the former submergence of the land under the sea4 ,but the occurrence of fossils embedded in the heart of the solid rocks of the mountains offered much greater difficulties of explanation1 and further progress was consequently slow4 ,especially baneful was the belief that these objects were mere sports of nature1 and had no connexion with any once living organisms4 ,so long as the true organic origin of the fossil plants and animals contained in the rocks was in dispute1 it was hardly possible that much advance could be made in their systematic study1 or in the geological deductions to be drawn from them4 ,one good result of the controversy1 however1 was to be seen in the large collections of these formed stones that were gathered together in the cabinets and museums of the #agth and #ahth centuries4 ,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 engravings4 ,switzerland was more particularly noted for the number and merit of its works of this kind1 such as that of ,k4,n4 ,lang 7,historia lapidum figuratorum ,helvetiae1 #agjh7 and those of ,johann ,jacob ,scheuchzer 7#afgb-agcc74 ,in ,england1 also1 illustrated treatises were published both by men who looked on fossils as mere freaks of nature1 and by those who regarded them as proofs of ,noah's flood4 ,of the former type were the works of ,martin ,lister 7#afch-agab7 and ,robert ,plot 7,natural ,history of ,oxfordshire1 #afgg74 ,the ,celtic scholar ,edward ,llwyd 7#affj-agji7 wrote a ,latin treatise containing good plates of a thousand fossils in the ,ashmolean ,museum1 ,oxford1 and ,j4 ,woodward1 in #agbh-agbi1 published his ,natural ,history of the ,fossils of ,england1 already mentioned1 wherein he described his own extensive collection1 which he bequeathed to the ,university of ,cambridge1 where it is still carefully preserved4 ,the most voluminous and important of all these works1 however1 appeared at a later date at ,nuremberg4 ,it was begun by ,g4,w4 ,knorr 7#agje-agfa71 who himself engraved for it a series of plates1 which for beauty and accuracy have seldom been surpassed4 ,after his death the work was continued by ,j4,e4,i4 ,walch 7#agbe-aggh71 and ultimately consisted of four massive folio volumes and nearly #cjj plates under the title of ,lapides diluvii universalis testes4 ,although the authors supposed their fossils to be relics of ,noah's flood1 their work must be acknowledged to mark a distinct onward stage in the palaeontological department of geology4 ,it was in ,france that palaeontological geology began to be cultivated in a scientific spirit4 ,the potter ,bernard ,palissy1 as far back as #aehj1 had dwelt on the importance of fossil shells as monuments of revolutions of the earth's surface2 but the observer who first undertook the detailed study of the subject was ,jean ,etienne ,guettard1 who began in #agea to publish his descriptions of fossils in the form of memoirs presented to the ,academy of ,sciences of ,paris4 ,to him they were not only of deep interest as monuments of former types of existence1 but they had an especial value as records of the changes which the country had undergone from sea to land and from land to sea4 ,more especially noteworthy was a monograph by him which appeared in #agfe 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 ,sea4 ,in this treatise he showed that the fossils have been encrusted with barnacles and serpulae1 have been bored into by other organisms1 and have often been rounded or broken before final entombment2 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-day4 ,his argument was the most triumphant that had ever been brought against the doctrine of lusus naturae1 and that of the efficacy of ,noah's flood--doctrines which still held their ground in ,guettard's day4 ,when ,soulavie1 ,cuvier and ,brongniart in ,france1 and ,william ,smith in ,england1 showed that the rock formations of the earth's crust could be arranged in chronological order1 and could be recognized far and wide by means of their enclosed organic remains1 the vast significance of these remains in geological research was speedily realized1 and palaeontological geology at once entered on a new and enlarged phase of development4 ,but apart from their value as chronological monuments1 and as witnesses of former conditions of geography1 fossils presented in themselves a wide field of investigation as types of life that had formerly existed1 but had now passed away4 ,it was in ,france that this subject first took definite shape as an important branch of science4 ,the mollusca of the ,tertiary deposits of the ,paris basin became1 in the hands of ,lamarck1 the basis on which invertebrate palaeontology was founded4 ,the same series of strata furnished to ,cuvier the remains of extinct land animals1 of which1 by critical study of their fragmentary bones and skeletons1 he worked out restorations that may be looked on as the starting-point of vertebrate palaeontology4 ,these brilliant researches1 rousing widespread interest in such studies1 showed how great a flood of light could be thrown on the past history of the earth and its inhabitants4 ,but the full significance of these extinct types of life could not be understood so long as the doctrine of the immutability of species1 so strenuously upheld by ,cuvier1 maintained its sway among naturalists4 ,lamarck1 as far back as the year #ahjj1 had begun to propound his theory of evolution and the transformation of species2 but his views1 strongly opposed by ,cuvier and the great body of naturalists of the day1 fell into neglect4 ,not until after the publication in #ahei of the ,origin of ,species by ,charles ,darwin were the barriers of old prejudice in this matter finally broken down4 ,the possibility of tracing the ancestry of living forms back into the remotest ages was then perceived2 the time-honoured fiction that the stratified formations record a series of catastrophes and re-creations was finally dissipated2 and the earth's crust was seen to contain a noble1 though imperfect1 record of the grand evolution of organic types of which our planet has been the theatre4 ,development of ,petrographical ,geology4--,theophrastus1 the favourite pupil of ,aristotle1 wrote a treatise ,on ,stones1 which has come down to our own day1 and may be regarded as the earliest work on petrography4 ,at a subsequent period ,pliny1 in his ,natural ,history1 collected all that was known in his day regarding the occurrence and uses of minerals and rocks4 ,but neither of these works is of great scientific importance1 though containing much interesting information4 ,minerals from their beauty and value attracted notice before much attention was paid to rocks1 and their study gave rise to the science of mineralogy long before geology came into existence4 ,when rocks began to be more particularly scrutinized1 it was chiefly from the side of their usefulness for building and other economic purposes4 ,the occurrence of marine shells in many of them had early attracted attention to them4 ,but their varieties of composition and origin did not become the subject of serious study until after ,linnaeus and ,j4,g4 ,wallerius in the #ahth century had made a beginning4 ,the first important contribution to this department of the science was that of ,werner1 who in #aghf published a classification and description of rocks in which he arranged them in two divisions1 simple and compound1 and further distinguished them by various external characters and by their relative age4 ,the publication of this scheme may be said to mark the beginning of scientific petrography4 ,werner's system1 however1 had the serious defect that the chronological order in which he grouped the rocks1 and the hypothesis by which he accounted for them as chemical precipitates from the original ocean1 were both alike contrary to nature4 ,it was hardly possible indeed that much progress could be made in this branch of geology until chemistry and mineralogy had made greater advances2 and especially until it was possible to ascertain the intimate chemical and mineralogical composition1 and the minute structure of rocks4 ,the study1 however1 continued to be pursued in ,germany1 where the influence of ,werner's enthusiasm still led men to enter the petrographical rather than the palaeontological domain4 ,the resources of modern chemistry were pressed into the service1 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 revealed4 ,the condition of the science in the middle of the #aith century was well shown by ,j4,l4,a4 ,roth1 who in #ahfa collected about #ajjj trustworthy analyses which up to that time had been made4 ,but though the chemical elements of the rocks had been fairly well determined1 the manner in which they were combined in the compound rocks could for the most part be only more or less plausibly conjectured4 ,as far back as #ahca an account was published of a process devised by ,william ,nicol of ,edinburgh1 whereby sections of fossil wood could be cut1 mounted on glass1 and reduced to such a degree of transparency as to be easily examined under a microscope4 ,henry ,sorby1 of ,sheffield1 having seen ,nicol's preparations1 perceived how admirably adapted the process was for the study of the minute structure and composition of rocks4 ,in #aheh he published in the ,quarterly ,journal of the ,geological ,society a paper ,on the ,microscopical ,structure of ,crystals4 ,this essay led to a complete revolution of petrographical methods and gave a vast impetus to the study of rocks4 ,petrology entered upon a new and wider field of investigation4 ,not only were the mineralogical constituents of the rocks detected1 but minute structures were revealed which shed new light on the origin and history of these mineral masses1 and opened up new paths in theoretical geology4 ,in the hands of ,h4 ,vogelsang1 ,f4 ,zirkel1 ,h4 ,rosenbusch1 and a host of other workers in all civilized countries1 the literature of this department of the science has grown to a remarkable extent4 ,armed with the powerful aid of modern optical instruments1 geologists are now able with far more prospect of success to resume the experiments begun a century before by de ,saussure and ,hall4 ,g4,a4 ,daubree1 ,c4 ,friedel1 ,e4 ,sarasin1 ,f4 ,fouque and ,a4 ,michel ,levy in ,france1 ,c4 ,doelter y ,cisterich and ,e4 ,hussak of ,gratz1 ,j4 ,morozewicz of ,warsaw and others1 have greatly advanced our knowledge by their synthetical analyses1 and there is every reason to hope that further advances will be made in this field of research4 ,rise of ,physiographical ,geology4--,until stratigraphical geology had advanced so far as to show of what a vast succession of rocks the crust of the earth is built up1 by what a long and complicated series of revolutions these rocks have come to assume their present positions1 and how enormous has been the lapse of time which all these changes represent1 it was not possible to make a scientific study of the surface features of our globe4 ,from ancient times it had been known that many parts of the land had once been under the sea2 but down even to the beginning of the #aith century the vaguest conceptions continued to prevail as to the operations concerned in the submergence and elevation of land1 and as to the processes whereby the present outlines of terrestrial topography were determined4 ,we have seen1 for instance1 that according to the teaching of ,werner the oldest rocks were first precipitated from solution in the universal ocean to form the mountains1 that the vertical position of their strata was original1 that as the waters subsided successive formations were deposited and laid bare1 and that finally the superfluous portion of the ocean was whisked away into space by some unexplained co-operation of another planetary body4 ,desmarest1 in his investigation of the volcanic history of ,auvergne1 was the first observer to perceive by what a long process of sculpture the present configuration of the land has been brought about4 ,he showed conclusively that the valleys have been carved out by the streams that flow in them1 and that while they have sunk deeper and deeper into the framework of the land1 the spaces of ground between them have been left as intervening ridges and hills4 ,de ,saussure learnt a similar lesson from his studies of the ,alps1 and ,hutton and ,playfair made it a cardinal feature in their theory of the earth4 ,nevertheless the idea encountered so much opposition that it made but little way until after the middle of the #aith century4 ,geologists preferred to believe in convulsions of nature1 whereby valleys were opened and mountains were upheaved4 ,that the main features of the land1 such as the great mountain-chains1 had been produced by gigantic plication of the terrestrial crust was now generally admitted1 and also that minor fractures and folds had probably initiated many of the valleys4 ,but those who realized most vividly the momentous results achieved by ages of subaerial denudation perceived that1 as ,hutton showed1 even without the aid of underground agency1 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 seen4 ,it was ,j4,b4 ,jukes who1 in #ahfb1 first revived the ,huttonian doctrine1 and showed how completely it explained the drainage-lines in the south of ,ireland4 ,other writers followed in quick succession until1 in a few years1 the doctrine came to be widely recognized as one of the established principles of modern geology4 ,much help was derived from the admirable illustrations of land-sculpture and river-erosion supplied from the ,western ,territories and ,states of the ,american ,union4 ,another branch of physiographical geology which could only come into existence after most of the other departments of the science had made large progress1 deals with the evolution of the framework of each country and of the several continents and oceans of the globe4 ,it is now possible1 with more or less confidence1 to trace backward the history of every terrestrial area1 to see how sea and land have there succeeded each other1 how rivers and lakes have come and gone1 how the crust of the earth has been ridged up at widely separated intervals1 each movement determining some line of mountains or plains1 how the boundaries of the oceans have shifted again and again in the past1 and thus how1 after so prolonged a series of revolutions1 the present topography of each country1 and of the globe as a whole1 has been produced4 ,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 world1 and thus to indicate the successive stages through which the architecture of the land has been gradually evolved4 ,the most noteworthy contribution to this department of the science is the ,antlitz der ,erde of ,professor ,suess of ,vienna4 ,this important and suggestive work has been translated into ,french and ,english4 ,,part ,,ii4--,,cosmical ,,aspects ,before geology had attained to the position of an inductive science1 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 planet1 or even of the universe4 ,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 together4 ,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 things4 ,he taught that in the materials from which geological evidence is to be compiled there can be found no traces of a beginning1 no prospect of an end4 ,in ,england1 mainly to the influence of the school which he founded1 and to the subsequent rise of the ,geological ,society of ,london1 which resolved to collect facts instead of fighting over hypotheses1 is due the disappearance of the crude and unscientific cosmologies by which the writings of the earlier geologists were distinguished4 ,but there can now be little doubt that in the reaction against those visionary and often grotesque speculations1 geologists were carried too far in an opposite direction4 ,in allowing themselves to believe that geology had nothing to do with questions of cosmogony1 they gradually grew up in the conviction that such questions could never be other than mere speculation1 interesting or amusing as a theme for the employment of the fancy1 but hardly coming within the domain of sober and inductive science4 ,nor would they soon have been awakened out of this belief by anything in their own science4 ,it is still true that in the data with which they are accustomed to deal1 as comprising the sum of geological evidence1 there can be found no trace of a beginning1 though the evidence furnished by the terrestrial crust shows a general evolution of organic forms from some starting-point which cannot be seen4 ,the oldest rocks which have been discovered on any part of the globe have probably been derived from other rocks older than themselves4 ,geology by itself has not yet revealed1 and is little likely ever to reveal1 a trace of the first solid crust of our globe4 ,if1 then1 geological history is to be compiled from direct evidence furnished by the rocks of the earth1 it cannot begin at the beginning of things1 but must be content to date its first chapter from the earliest period of which any record has been preserved among the rocks4 ,nevertheless1 though geology in its usual restricted sense has been1 and must ever be1 unable to reveal the earliest history of our planet1 it no longer ignores1 as mere speculation1 what is attempted in this subject by its sister sciences4 ,astronomy1 physics and chemistry have in late years all contributed to cast light on the earlier stages of the earth's existence1 previous to the beginning of what is commonly regarded as geological history4 ,but whatever extends our knowledge of the former conditions of our globe may be legitimately claimed as part of the domain of geology4 ,if this branch of inquiry1 therefore1 is to continue worthy of its name as the science of the earth1 it must take cognizance of these recent contributions from other sciences4 ,it must no longer be content to begin its annals with the records of the oldest rocks1 but must endeavour to grope its way through the ages which preceded the formation of any rocks4 ,thanks to the results achieved with the telescope1 the spectroscope and the chemical laboratory1 the story of these earliest ages of our earth is every year becoming more definite and intelligible4 ,up to the present time no definite light has been thrown by physics on the origin and earliest condition of our globe4 ,the famous nebular theory 7q4v47 of ,kant and ,laplace sketched the supposed evolution of the solar system from a gaseous nebula1 slowly rotating round a more condensed central portion of its mass1 which eventually became the sun4 ,as a consequence of increased rapidity of rotation resulting from cooling and contraction1 the nebula acquired a more and more lenticular form1 until at last it threw off from its equatorial protuberance a ring of matter4 ,subsequently the same process was repeated1 and other similar rings successively separated from the parent mass4 ,each ring went through a corresponding series of changes until it ultimately became a planet1 with or without one or more attendant satellites4 ,the intimate relationship of our earth to the sun and the other planets was1 in this way1 shown4 ,but there are some serious physical difficulties in the way of the acceptance of the nebular hypothesis4 ,another explanation is given by the meteoritic hypothesis1 according to which1 out of the swarms of meteorites with which the regions of space are crowded1 the sun and planets have been formed by gradual accretion4 ,according to these theoretical views we should expect to find a general uniformity of composition in the constituent matter of the solar system4 ,for many years the only available evidence on this point was derived from the meteorites 7q4v47 which so constantly fall from outer space upon the surface of the earth4 ,these bodies were found to consist of elements1 all of which had been recognized as entering into the constitution of the earth4 ,but the discoveries of spectroscopic research have made known a far more widely serviceable method of investigation1 which can be applied even to the luminous stars and nebulae that lie far beyond the bounds of the solar system4 ,by this method information has been obtained regarding the constitution of the sun1 and many of our terrestrial metals1 such as iron1 nickel and magnesium1 have been ascertained to exist in the form of incandescent vapour in the solar atmosphere4 ,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 space4 ,if our globe was at first1 like its parent sun1 an incandescent mass of probably gaseous matter1 occupying much more space than it now fills1 we can conceive that it has ever since been cooling and contracting until it has reached its present form and dimensions1 and that it still retains a high internal temperature4 ,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 solidity4 ,but it has been claimed that even a solid spherical globe might develop1 under the influence of protracted rotation1 such a shape as the earth at present possesses4 ,the observed increase of temperature downwards in our planet has hitherto been generally accepted as a relic and proof of an original high temperature and mobility of substance4 ,recently1 however1 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 temperature4 ,too little1 however1 is known of the history and properties of what is called radium to afford a satisfactory ground on which to discard what has been1 and still remains1 the prevalent belief on this subject4 ,an important epoch in the geological history of the earth was marked by the separation of the moon from its mass 7see ,,tide74 ,whether the severance arose from the rupture of a surrounding ring or the gradual condensation of matter in such a ring1 or from the ejection of a single mass of matter from the rapidly rotating planet1 it has been shown that our satellite was only a few thousand miles from the earth's surface1 since when it has retreated to its present distance of #bdj1jjj m4 ,hence the influence of the moon's attraction1 and all the geological effects to which it gives rise1 attained their maximum far back in the development of the globe1 and have been slowly diminishing throughout geological history4 ,the sun by virtue of its vast size has not yet passed out of the condition of glowing gas1 and still continues to radiate heat beyond the farthest planet of the solar system4 ,the earth1 however1 being so small a body in comparison1 would cool down much more quickly4 ,underneath its hot atmosphere a crust would conceivably begin to form over its molten surface1 though the interior might still possess a high temperature and1 owing to the feeble conducting power of rocks1 would remain intensely hot for a protracted series of ages4 ,full information regarding the form and size of the earth1 and its relations to the other planetary members of the solar system1 will be found in the articles ,,planet and ,,solar ,,system4 ,for the purposes of geological inquiry the reader will bear in mind that the equatorial diameter of our globe is estimated to be about #gibe m41 and the polar diameter about #ghii m42 the difference between these two sums representing the amount of flattening at the poles 7about #bf-a_/b m474 ,the planet has been compared in shape to an orange1 but it resembles an orange which has been somewhat squeezed1 for its equatorial circumference is not a regular circle but an ellipse1 of which the major axis lies in long4 #h deg4 #ae' ,w4--on a meridian which cuts the north-west corner of ,america1 passing through ,portugal and ,ireland1 and the north-east corner of ,asia in the opposite hemisphere4 ,the rotation of the earth on its axis exerts an important influence on the movements of the atmosphere1 and thereby affects the geological operations connected with these movements4 ,the influence of rotation is most marked in the great aerial circulation between the poles and the equator4 ,currents of air1 which set out in a meridional direction from high latitudes towards the equator1 come from regions where the velocity due to rotation is small to where it is greater1 and they consequently fall behind4 ,thus1 in the northern hemisphere a north wind1 as it moves away from its northern source of origin1 is gradually deflected more and more towards the west and becomes a north-east current2 while in the opposite hemisphere a wind making from high southern latitudes towards the equator becomes1 from the same cause1 a south-east current4 ,where1 on the other hand1 the air moves from the equatorial to the polar regions its higher velocity of rotation carries it eastward1 so that on the south side of the equator it becomes a north-west current and on the north side a south-west current4 ,it is to this cause that the easting and westing of the great atmospheric currents are to be attributed1 as is familiarly exemplified in the trade winds4 ,the atmospheric circulation thus deflected influences the circulation of the ocean4 ,the winds which persistently blow from the north-east on the north side of the equator1 and from the south-east on the south side1 drive the superficial waters onwards1 and give rise to converging oceanic currents which unite to form the great westerly equatorial current4 ,a more direct effect of terrestrial rotation has been claimed in the case of rivers which flow in a meridional direction4 ,it has been asserted that those1 which in the northern hemisphere flow from north to south1 like the ,volga1 by continually passing into regions where the velocity of rotation is increasingly greater1 are thrown more against their western than their eastern banks1 while those whose general course is in an opposite direction1 like the ,irtisch and ,yenesei1 press more upon their eastern sides4 ,there cannot be any doubt that the tendency of the streams must be in the directions indicated4 ,but when the comparatively slow current and constantly meandering course of most rivers are taken into consideration1 it may be doubted whether the influence of rotation is of much practical account so far as river-erosion is concerned4 ,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 rotation4 ,abundant evidence has now been obtained to prove that at a comparatively late geological period a rich flora1 resembling that of warm climates at the present day1 existed in high latitudes even within less than #i deg4 of the north pole1 where1 with an extremely low temperature and darkness lasting for half of the year1 no such vegetation could possibly now exist4 ,it has accordingly been maintained by many geologists that the axis of rotation must have shifted1 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 time4 ,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 astronomers1 and their arguments have been generally1 but somewhat reluctantly1 accepted by geologists1 who find themselves confronted with a problem which has hitherto seemed insoluble4 ,that the axis is not rigidly stable1 however1 has been postulated by some physicists1 and has now been demonstrated by actual observation and measurement4 ,it is admitted that by the movement of large bodies of water the air over the surface of the globe1 and more particularly by the accumulation of vast masses of snow and ice in different regions1 the position of the axis might be to some extent shifted2 more serious effects might follow from widespread upheavals or depressions of the surface of the lithosphere4 ,on the assumption of the extreme rigidity of the earth's interior1 however1 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 record1 that is1 since the time of the oldest known sedimentary formations1 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 latitudes4 ,if1 however1 the inside of the globe possesses a great plasticity than has been allowed1 the shifting of the axis might not be impossible1 even to such an extent as would satisfy the geological requirements4 ,this question is one on which the last word has not been said1 and regarding which judgment must remain in suspense4 ,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 ,america4 ,it has thus been ascertained that the pole wanders with a curiously irregular but somewhat spiral movement1 within an amplitude of between #dj and #ej ft41 and completes its erratic circuit in about #dbh days4 ,it was not supposed that its movement had any geological interest1 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 earthquakes1 and has suggested that1 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 axis1 both effects may not improbably arise from the same redistribution of surface material by ocean currents and meteorological causes4 ,if for any reason the earth's centre of gravity were sensibly displaced1 momentous geological changes would necessarily ensue4 ,that the centre of gravity does not coincide with the centre of figure of the globe1 but lies to the south of it1 has long been known4 ,this greater aggregation of dense material in the southern hemisphere probably dates from the early ages of the earth's consolidation1 and it is difficult to believe that any readjustment of the distribution of this material in the earth's interior is now possible4 ,but certain rearrangements of the hydrosphere on the surface of the globe may1 from time to time1 cause a shifting of the centre of gravity1 which will affect the level of the ocean4 ,the accumulation of enormous masses of ice around the pole will give rise to such a displacement1 and will thus increase the body of oceanic water in the glaciated hemisphere4 ,various calculations have been made of the effect of the transference of the ice-cap from one pole to the other1 a revolution which may possibly have occurred more than once in the past history of the globe4 ,james ,croll estimated that if the mass of ice in the southern hemisphere be assumed to be #ajjj ft4 thick down to lat4 #fj deg41 its removal to the opposite hemisphere would raise the level of the sea #hj ft4 at the north pole1 while the ,rev4 ,osmond ,fisher made the rise as much as #dji ft4 ,the melting of the ice would still further raise the sea-level by the addition of so large a volume of water to the ocean4 ,to what extent superficial changes of this kind have operated in geological history remains an unsolved problem1 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 surface4 ,the ,age of the ,earth4--,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 importance1 regarding which much discussion has taken place since the middle of the #aith century4 ,though an account of this discussion necessarily involves allusion to departments of geology which are more appropriately referred to in later parts of this article1 it may perhaps be most conveniently included here4 ,geologists were for many years in the habit of believing that no limit could be assigned to the antiquity of the planet1 and that they were at liberty to make unlimited drafts on the ages of the past4 ,in #ahfb and subsequent years1 however1 ,lord ,kelvin 7then ,sir ,william ,thomson7 pointed out that these demands were opposed to known physical facts1 and that the amount of time required for geological history was not only limited1 but must have been comprised within a comparatively narrow compass4 ,his argument rested on three kinds of evidence3 7#a7 the internal heat and rate of cooling of the earth2 7#b7 the tidal retardation of the earth's rotation2 and 7#c7 the origin and age of the sun's heat4 #a4 ,applying ,fourier's theory of thermal conductivity1 ,lord ,kelvin contended that in the known rate of increase of temperature downward and beneath the surface1 and the rate of loss of heat from the earth1 we have a limit to the antiquity of the planet4 ,he showed1 from the data available at the time1 that the superficial consolidation of the globe could not have occurred less than #bj million years ago1 or the underground heat would have been greater than it is2 nor more than #djj million years ago1 otherwise the underground temperature would have shown no sensible increase downwards4 ,he admitted that very wide limits were necessary4 ,in subsequently discussing the subject1 he inclined rather towards the lower than the higher antiquity1 but concluded that the limit1 from a consideration of all the evidence1 must be placed within some such period of past time as #ajj millions of years4 #b4 ,the argument from tidal retardation proceeds on the admitted fact that1 owing to the friction of the tide-wave1 the rotation of the earth is retarded1 and is1 therefore1 much slower now than it must have been at one time4 ,lord ,kelvin affirmed that had the globe become solid some #aj1jjj million years ago1 or indeed any high antiquity beyond #ajj million years1 the centrifugal force due to the more rapid rotation must have given the planet a very much greater polar flattening than it actually possesses4 ,he admitted1 however1 that1 though #ajj million years ago that force must have been about #c greater than now1 yet nothing we know regarding the figure of the earth1 and the disposition of land and water1 would justify us in saying that a body consolidated when there was more centrifugal force by #c than now1 might not now be in all respects like the earth1 so far as we know it at present4 #c4 ,the third argument1 based upon the age of the sun's heat1 is confessedly less to be relied on than the two previous ones4 ,it proceeds upon calculations as to the amount of heat which would be available by the falling together of masses from space1 which gave rise by their impact to our sun4 ,the vagueness of the data on which this argument rests may be inferred from the fact that in one passage ,p4,g4 ,tait placed the limit of time during which the sun has been illuminating the earth as1 on the very highest computation1 not more than about #ae or #bj millions of years2 while1 in another sentence of the same volume1 he admitted that1 by calculations in which there is no possibility of large error1 this hypothesis of the origin of the sun's heat by the falling together of masses of matter is thoroughly competent to explain #ajj millions of years' solar radiation at the present rate1 perhaps more4 ,in more recently reviewing his argument1 ,lord ,kelvin expressed himself in favour of more strictly limiting geological time than he had at first been disposed to do4 ,he insists that the time was more than #bj and less than #dj millions of years and probably much nearer #bj than #dj4 ,geologists appear to have reluctantly brought themselves to believe that perhaps1 after all1 #ajj millions of years might suffice for the evolution of geological history4 ,but when the time was cut down to #ae or #bj millions they protested that such a restricted period was insufficient for that evolution1 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 based4 ,by degrees1 however1 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 founded1 and at the same time greater appreciation has been shown of the signification and strength of the geological proofs of the high antiquity of our planet4 ,in an address from the chair of the ,mathematical ,section of the ,british ,association in #ahhf1 ,professor 7afterwards ,sir7 ,george ,darwin reviewed the controversy1 and pronounced the following deliberate judgment in regard to it3 ,in considering these three arguments ,i have adduced some reasons against the validity of the first tidal friction1 and have endeavoured to show that there are elements of uncertainty surrounding the second secular cooling of the earth2 nevertheless1 they undoubtedly constitute a contribution of the first importance to physical geology4 ,whilst1 then1 we may protest against the precision with which ,professor ,tait seeks to deduce results from them1 we are fully justified in following ,sir ,william ,thomson1 who says that 'the existing state of things on the earth1 life on the earth--all geological history showing continuity of life--must be limited within some such period of past time as #ajj million years'4 ,lord ,kelvin has never dealt with the geological and palaeontological objections against the limitation of geological time to a few millions of years4 ,but ,professor ,darwin1 in the address just cited1 uttered the memorable warning3 ,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 allowable4 ,in his presidential address to the ,british ,association at ,cape ,town in #aije he returned to the subject1 remarking that the argument derived from the increase of underground temperature seems to be entirely destroyed by the discovery of the properties of radium4 ,he thinks that it does not seem extravagant to suppose that #ejj to #ajjj million years may have elapsed since the birth of the moon4 ,he has always believed that the geologists were more nearly correct than the physicists1 notwithstanding the fact that appearances were so strongly against them1 and he concludes thus3 ,it appears1 then1 that the physical argument is not susceptible of a greater degree of certainty than that of the geologists1 and the scale of geological time remains in great measure unknown 7see also ,tide1 chap4 viii474 ,in an address to the mathematical section of the ,american ,association for the ,advancement of ,science in #ahhi1 the vice-president of the section1 ,r4,s4 ,woodward1 thus expressed himself with regard to the physical arguments brought forward by ,lord ,kelvin and ,professor ,tait in limitation of geological time3 ,having been at some pains to look into this matter1 ,i feel bound to state that1 although the hypothesis appears to be the best which can be formulated at present1 the odds are against its correctness4 ,its weak links are the unverified assumptions of an initial uniform temperature and a constant diffusivity4 ,very likely these are approximations1 but of what order we cannot decide4 ,furthermore1 if we accept the hypothesis1 the odds appear to be against the present attainment of trustworthy numerical results1 since the data for calculation1 obtained mostly from observations on continental areas1 are far too meagre to give satisfactory average values for the entire mass of the earth4 ,still more emphatic is the protest made from the physical side by ,professor ,john ,perry4 ,he has attacked each of the three lines of argument of ,lord ,kelvin1 and has impugned the validity of the conclusions drawn from them4 ,the argument from tidal retardation he dismisses as fallacious1 following in this contention the previous criticism of the ,rev4 ,maxwell ,close and ,sir ,george ,darwin4 ,in dealing with the argument based on the secular cooling of the earth1 he holds it to be perfectly allowable to assume a much higher conductivity for the interior of the globe1 and that such a reasonable assumption would enable us greatly to increase our estimate of the earth's antiquity4 ,as for the third argument1 from the age of the sun's heat1 he points out that the sun may have been repeatedly fed by a supply of meteorites from outside1 while the earth may have been protected from radiation1 and been able to retain much of its heat by being enveloped in a dense atmosphere4 ,remarking that almost anything is possible as to the present internal state of the earth1 he concludes thus3 ,to sum up1 we can find no published record of any lower maximum age of life on the earth1 as calculated by physicists1 than #djj millions of years4 ,from the three physical arguments ,lord ,kelvin's higher limits are #ajjj1 #djj and #ejj million years4 ,i have shown that we have reasons for believing that the age1 from all these1 may be very considerably underestimated4 ,it is to be observed that if we exclude everything but the arguments from mere physics1 the probable age of life on the earth is much less than any of the above estimates2 but if the palaeontologists have good reasons for demanding much greater times1 ,i see nothing from the physicists' point of view which denies them four times the greatest of these estimates4 ,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-activity4 ,from the ascertained properties of radium it appears to be possible that our estimates of solar heat1 as derived from the theory of gravitation1 may have to be augmented ten or twenty times2 that stores of radium and similar bodies within the earth may have indefinitely deferred the establishment of the present temperature gradient from the surface inward2 that consequently the earth may have remained for long ages at a temperature not greatly different from that which it now possesses1 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 7see ,,radioactivity74 ,the arguments from the geological side against the physical contention that would limit the age of our globe to some #aj or #bj millions of years are mainly based on the observed rates of geological and biological changes at the present time upon land and sea1 and on the nature1 physical history and organic contents of the stratified crust of the earth4 ,unfortunately1 actual numerical data are not obtainable in many departments of geological activity1 and even where they can be procured they do not yet rest on a sufficiently wide collection of accurate and co-ordinated observations4 ,but in some branches of dynamical geology1 material exists for1 at least1 a preliminary computation of the rate of change4 ,this is more especially the case in respect of the wide domain of denudation4 ,the observational records of the action of the sea1 of springs1 rivers and glaciers are becoming gradually fuller and more trustworthy4 ,a method of making use of these records for estimating the rate of denudation of the land has been devised4 ,taking the ,mississippi as a general type of river action1 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 #a_/fjjj of a foot4 ,this would amount to one foot in #fjjj years and #ajjj ft4 in #f million years4 ,so that at the present rate of waste in the ,mississippi basin a whole continent might be worn away in a few millions of years4 ,it is evident that as deposition and denudation are simultaneous processes1 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 sea1 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 globe4 ,if the thickness of these rocks be assumed to be #ajj1jjj ft41 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 derived1 then at the present rate of denudation their accumulation would require some #fjj millions of years4 ,but1 as ,dr ,a4,r4 ,wallace has justly pointed out1 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 away4 ,we have no means1 however1 of determining what may have been the ratio between the two areas in past time4 ,certainly ancient marine sedimentary rocks cover at the present day a much more extensive area than that in which they are now being elaborated4 ,if we take the ratio postulated by ,dr ,wallace--#a to #ai--the #ajj1jjj ft4 of sedimentary strata would require #ca millions of years for their accumulation4 ,it is quite possible1 however1 that this ratio may be much too high4 ,there are reasons for believing that the proportion of coast-line to land area has been diminishing during geological time2 in other words1 that in early times the land was more insular and is now more continental4 ,so that the #ca millions of years may be much less than the period that would be required1 even on the supposition of continuous uninterrupted denudation and sedimentation1 during the whole of the time represented by the stratified formations4 ,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 vague1 imperfect and misleading are the data on which such computations are founded4 ,it requires no prolonged acquaintance with the earth's crust to impress upon the mind that one all-important element is omitted1 and indeed can hardly be allowed for from want of sufficiently precise data1 but the neglect of which must needs seriously impair the value of all numerical calculations made without it4 ,the assumption that the stratified formations can be treated as if they consisted of a continuous unbroken sequence of sediments1 indicating a vast and uninterrupted process of waste and deposition1 is one that is belied on every hand by the actual structure of these formations4 ,it can only give us a minimum of the time required2 for1 instead of an unbroken series1 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 work4 ,it can often be shown that these breaks of continuity must have been of vast duration1 and actually exceeded in chronological importance thick groups of strata lying below and above them 7see ,part ,,vi474 ,moreover1 even among the uninterrupted strata1 where no such unconformabilities exist1 but where the sediments follow each other in apparently uninterrupted sequence1 and might be thought to have been deposited continuously at the same general rate1 and without the intervention of any pause1 it can be demonstrated that sometimes an inch or two of sediment might1 on certain horizons1 represent the deposit of an enormously longer period than a hundred or a thousand times the same amount of sediment on other horizons4 ,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 represented4 ,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 time1 because the total sum of terrestrial energy has been steadily diminishing1 and geological processes must consequently have been more vigorous in former ages than they are now4 ,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 study4 ,what they do insist on is that the present rate of change is the only one which we can watch and measure1 and which will thus supply a statistical basis for any computations on the subject4 ,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 now2 that there were far more abundant and more stupendous volcanoes1 more frequent and more destructive earthquakes1 more gigantic upheavals and subsidences1 more powerful oceanic waves and tides1 more violent atmospheric disturbances with heavier rainfall and more active denudation4 ,it is easy to make these assertions1 and they look plausible2 but1 after all1 they rest on nothing stronger than assumption4 ,they can be tested by an appeal to the crust of the earth1 in which the geological history of our planet has been so fully recorded4 ,had such portentous manifestations of geological activity ever been the normal condition of things since the beginning of that history1 there ought to be a record of them in the rocks4 ,but no evidence for them has been found there1 though it has been diligently sought for in all quarters of the globe4 ,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 existence1 of which no geological record remains1 there is no proof that they have ever done so since the time when the very oldest of the stratified formations were deposited4 ,there is no need to maintain that they have always been conducted precisely on the same scale as now1 or to deny that they may have gradually become less vigorous as the general sum of terrestrial energy has diminished4 ,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 earth3 that1 on the contrary1 no appearances have been detected there which necessarily demand the assumption of those more powerful operations postulated by physicists1 or which are not satisfactorily explicable by reference to the existing scale of nature's processes4 ,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 energy4 ,take1 for example1 the proofs of gigantic plication1 fracture and displacement within the terrestrial crust4 ,these1 as they have affected the most ancient rocks of ,europe1 have been worked out in great detail in the north-west of ,scotland4 ,but they are not essentially different from or on a greater scale than those which have been proved to have affected the ,alps1 and to have involved strata of so recent a date as the older ,tertiary formations4 ,on the contrary1 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 globe4 ,it may1 indeed1 quite well have been the rule that instead of diminishing in intensity of effect1 the consequences of terrestrial contraction have increased in magnitude1 the augmenting thickness of the crust offering greater resistance to the stresses1 and giving rise to vaster plications1 faults1 thrust-planes and metamorphism1 as this growing resistance had to be overcome4 ,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 support4 ,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 now1 and that traces of copious eruptions of volcanic material can be followed back into some of the oldest parts of the geological record4 ,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 periods4 ,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 studied4 ,but most assuredly1 as far as geological investigation has yet gone1 there is an overwhelming body of evidence to show that from the earliest epochs in geological history1 as registered in the stratified rocks1 volcanic action has manifested itself very much as it does now1 but on a less rather than on a greater scale4 ,nowhere can this subject be more exhaustively studied than in the ,british ,isles1 where a remarkably complete series of volcanic eruptions has been chronicled ranging from the earliest ,palaeozoic down to older ,tertiary time4 ,the result of a prolonged study of ,british volcanic geology has demonstrated that1 even to minute points of detail1 there has been a singular uniformity in the phenomena from beginning to end4 ,the oldest lavas and ashes differ in no essential respect from the youngest4 ,nor have they been erupted more copiously or more frequently4 ,many successive volcanic periods have followed each other after prolonged intervals of repose1 each displaying the same general sequence of phenomena and similar evidence of gradual diminution and extinction4 ,the youngest1 instead of being the feeblest1 were the most extensive outbursts in the whole of this prolonged series4 ,if now we turn for evidence of the alleged greater activity of all the epigene or superficial forces1 and especially for proofs of more rapid denudation and deposition on the earth's surface1 we search for it in vain among the stratified formations of the terrestrial crust4 ,had the oldest of these rocks been accumulated in a time of great atmospheric perturbation1 of torrential rains1 colossal tides and violent storms1 we might surely expect to find among the sediments some proof of such disturbed meteorological and geographical conditions4 ,we should look1 on the one hand1 for tumultuous accumulations of coarse unworn detritus1 rapidly swept by rains1 floods and waves from land to sea1 and on the other hand1 for an absence of any evidence of the tranquil and continuous deposit of such fine laminated silt as could only settle in quiet water4 ,but an appeal to the geological record is made in vain for any such proofs4 ,the oldest sediments1 like the youngest1 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 deposits4 ,if1 for instance1 we search the most ancient thick sedimentary formation in ,britain--the ,torridon ,sandstone of north-west ,scotland1 which is older than the oldest fossiliferous deposits--we meet with nothing which might not be found in any ,palaeozoic1 ,mesozoic or ,cainozoic group of similar sediments4 ,we see an accumulation1 at least #hjjj or #aj1jjj ft4 thick1 of consolidated sand1 gravel and mud1 such as may be gathering now on the floor of any large mountain-girdled lake4 ,the conglomerates of this ancient series are not pell-mell heaps of angular detritus1 violently swept away from the land and huddled promiscuously on the sea-floor4 ,they are1 in general1 built up of pebbles that have been worn smooth1 rounded and polished by prolonged attrition in running water1 and they follow each other on successive platforms with intervening layers of finer sediment4 ,the sandstones are composed of well water-worn sand1 some of which has been laid down so tranquilly that its component grains have been separated out in layers according to their specific gravity1 in such manner that they now present dark laminae in which particles of magnetic iron1 zircon and other heavy minerals have been sifted out together1 just as iron-sand may be seen gathered into thin sheets on sandy beaches at the present day4 ,again1 the same series of primeval sediments includes intercalations of fine silt1 which has been deposited as regularly and intermittently there as it has been among the most recent formations4 ,these bands of shale have been diligently searched for fossils1 as yet without success2 but they may eventually disclose organic remains older than any hitherto found in ,europe4 ,we now come to the consideration of the palaeontological evidence as to the value of geological time4 ,here the conclusions derived from a study of the structure of the sedimentary formations are vastly strengthened and extended4 ,in the first place1 the organization of the most ancient plants and animals furnishes no indication that they had to contend with any greater violence of storm1 flood1 wave or ocean-current than is familiar to their modern descendants4 ,the oldest trees1 shrubs1 ferns and club-mosses display no special structures that suggest a difference in the general conditions of their environment4 ,the most ancient crinoids1 sponges1 crustaceans1 arachnids and molluscs were as delicately constructed as those of to-day1 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 experience4 ,of much more cogency1 however1 is the evidence supplied by the grand upward succession of organic forms1 from the most ancient stratified rocks up to the present day4 ,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 organization4 ,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 advanced1 but no one who is conversant with the facts will now venture to deny that it has taken place1 and that1 on any possible explanation of its progress1 it must have demanded an enormous lapse of time4 ,in the ,cambrian or oldest fossiliferous formations there is already a large and varied fauna1 in which the leading groups of invertebrate life are represented4 ,on no tenable hypothesis can these be regarded as the first organisms that came into being on our planet4 ,they must have had a long ancestry1 and as ,darwin first maintained1 the time required for their evolution may have been as long as1 or probably far longer than1 the whole interval from the ,silurian ,cambrian age to the present day4 ,the records of these earliest eras of organic development have unfortunately not survived the geological revolutions of the past2 at least1 they have not yet been recovered4 ,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 destruction4 ,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 subject4 ,professor ,e4,b4 ,poulton1 in an address to the zoological section of the ,british ,association at the ,liverpool ,meeting in #ahif1 showed how overwhelming are the demands which this evidence makes for long periods of time1 and how impossible it is of comprehension unless these demands be conceded4 ,the history of life upon the earth1 though it will probably always be surrounded with great and even insuperable difficulties1 becomes broadly comprehensible in its general progress when sufficient time is granted for the evolution which it records2 but it remains unintelligible on any other conditions4 ,taken then as a whole1 the body of evidence1 geological and palaeontological1 in favour of the high antiquity of our globe is so great1 so manifold1 and based on such an ever-increasing breadth of observation and reflection1 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 years4 ,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 earth4 ,but we cannot assume it to be much less1 and it may possibly have been much more1 than the #ajj millions of years which ,lord ,kelvin was at one time willing to concede4#b ,,part ,,iii4--,,geognosy4 ,,the ,,investigation ,,of ,,the ,,nature ,,and ,,composition ,,of ,,the ,,materials ,,of ,,which ,,the ,,earth ,,consists ,this division of the science is devoted to a description of the parts of the earth--of the atmosphere and ocean that surround the planet1 and more especially of the solid materials that underlie these envelopes and extend downwards to an unknown distance into the interior4 ,these various constituents of the globe are here considered as forms of matter capable of being analysed1 and arranged according to their composition and the place they take in the general composition of the globe4 ,viewed in the simplest way the earth may be regarded as made up of three distinct parts1 each of which ever since an early period of planetary history has been the theatre of important geological operations4 7#a7 ,an envelope of air1 termed the atmosphere1 which surrounds the whole globe2 7#b7 ,a lower and less extensive envelope of water1 known as the hydrosphere 7,gr4 ,greek3 hydor1 water7 which1 constituting the oceans and seas1 covers nearly three-fourths of the underlying solid surface of the planet2 7#c7 ,a globe1 called the lithosphere 7,gr4 ,greek3 lithos1 stone71 the external part of which1 consisting of solid stone1 forms the crust1 while underneath1 and forming the vast mass of the interior1 lies the nucleus1 regarding the true constitution of which we are still ignorant4 #a4 ,the ,atmosphere4--,the general characters of the atmosphere are described in separate articles 7see especially ,,atmosphere2 ,,meteorology74 ,only its relations to geology have here to be considered4 ,as this gaseous envelope encircles the whole globe it is the most universally present and active of all the agents of geological change4 ,its efficacy in this respect arises partly from its composition1 and the chemical reactions which it effects upon the surface of the land1 partly from its great variations in temperature and moisture1 and partly from its movements4 ,many speculations have been made regarding the chemical composition of the atmosphere during former geological periods4 ,there can indeed be little doubt that it must originally have differed greatly from its present condition4 ,if the whole mass of the planet originally existed in a gaseous state1 there would be practically no atmosphere4 ,the present outer envelope of air may be considered to be the surviving relic of this condition1 after all the other constituents have been incorporated into the hydrosphere and lithosphere4 ,the oxygen1 which now forms fully a half of the outer crust of the earth1 was doubtless originally1 whether free or in combination1 part of the atmosphere4 ,so1 too1 the vast beds of coal found all over the world1 in geological formations of many different ages1 represent so much carbonic acid once present in the air4 ,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 vapour1 though they have been continually increased ever since by contributions from the drainage of the land4 ,it has often been suggested that1 during the ,carboniferous period1 the atmosphere must have been warmer and more charged with aqueous vapour and carbon dioxide than at the present day1 to admit of so luxuriant a flora as that from which the coal-seams were formed4 ,there seems1 however1 to be at present no method of arriving at any certainty on this subject4 ,lastly1 the amount of carbonic acid absorbed in the weathering of rocks at the surface1 and the consequent production of carbonates1 represents an enormous abstraction of this gas4 ,as at present constituted1 the atmosphere is regarded as a mechanical mixture of nearly four volumes of nitrogen and one of oxygen1 together with an average of #c4e parts of carbon dioxide in every #aj1jjj parts of air1 and minute quantities of various other gases and solid particles4 ,of the vapours contained in it by far the most important is that of water which1 although always present1 varies greatly in amount according to variations in temperature4 ,by condensation the water vapour appears in visible form as dew1 mist1 cloud1 rain1 hail1 snow and ice1 and in these forms includes and carries down some of the other vapours1 gases and solid particles present in the air4 ,the circulation of water from the atmosphere to the land1 from the land to the sea1 and again from the sea to the land1 forms the great geological process whereby the habitable condition of the planet is maintained and the surface of the land is sculptured 7,part ,,iv474 #b4 ,the ,hydrosphere4--,the water envelope covers nearly three-fourths of the surface of the earth1 and forms the various oceans and seas which1 though for convenience of reference distinguished by separate names1 are all linked together in one great body4 ,the physical characters of this vast envelope are discussed in separate articles 7see ,,ocean and ,,oceanography74 ,viewed from the geological standpoint1 the features of the sea that specially deserve attention are first the composition of its waters1 and secondly its movements4 ,sea-water is distinguished from that of ordinary lakes and rivers by its greater specific gravity and its saline taste4 ,its average density is about #a4jbf1 but it varies even within the same ocean1 being least where large quantities of fresh water are added from rain or melting snow and ice1 and greatest where evaporation is most active4 ,that sea-water is heavier than fresh arises from the greater proportion of salts which it contains in solution4 ,these salts constitute about three and a half parts in every hundred of water4 ,they consist mainly of chlorides of sodium and magnesium1 the sulphates of magnesium1 calcium and potassium1 with minuter quantities of magnesium bromide and calcium carbonate4 ,still smaller proportions of other substances have been detected1 gold for example having been found in the proportion of #a part in #ae1ahj1jjj4 ,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 probable4 ,it is manifest1 however1 that1 whatever may have been the original composition of the oceans1 they have for a vast section of geological time been constantly receiving mineral matter in solution from the land4 ,every spring1 brook and river removes various salts from the rocks over which it moves1 and these substances1 thus dissolved1 eventually find their way into the sea4 ,consequently sea-water ought to contain more or less traceable proportions of every substance which the terrestrial waters can remove from the land1 in short1 of probably every element present in the outer shell of the globe1 for there seems to be no constituent of this earth which may not1 under certain circumstances1 be held in solution in water4 ,moreover1 unless there be some counteracting process to remove these mineral ingredients1 the ocean water ought to be growing1 insensibly perhaps1 but still assuredly1 saltier1 for the supply of saline matter from the land is incessant4 ,to the geologist the presence of mineral solutions in sea-water is a fact of much importance1 for it explains the origin of a considerable part of the stratified rocks of the earth's crust4 ,by evaporation the water has given rise to deposits of rock-salt1 gypsum and other materials4 ,the lime contained in solution1 whether as sulphate or carbonate1 has been extracted by many tribes of marine animals1 which have thus built up out of their remains vast masses of solid limestone1 of which many mountain-chains largely consist4 ,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 land4 ,besides the limestones1 the visible parts of the terrestrial crust are1 in large measure1 composed of sedimentary rocks which were originally laid down on the sea-bottom4 ,moreover1 by its various movements1 the sea occupies a prominent place among the epigene or superficial agents which produce geological changes on the surface of the globe4 #c4 ,the ,lithosphere4--,beneath the gaseous and liquid envelopes lies the solid part of the planet1 which is conveniently regarded as consisting of two parts1--7a7 the crust1 and 7b7 the interior or nucleus4 ,the crust4 ,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 cooling4 ,hence the term crust was applied to this external solid envelope1 which was variously computed to be #aj1 #bj1 or more miles in thickness4 ,the portion of this crust accessible to human observation was seen to afford abundant evidence of vast plications and corrugations of its substance1 which were regarded as only explicable on the supposition of a thin solid collapsible shell floating on a denser liquid interior4 ,when1 however1 physical arguments were adduced to show the great rigidity of the earth as a whole1 the idea of a thin crust enclosing a molten nucleus was reluctantly abandoned by geologists1 who found the problem of the earth's interior to be incapable of solution by any evidence which their science could produce4 ,they continued1 however1 to use the term crust as a convenient word to denote the cool outer layer of the earth's mass1 the structure and history of which form the main subjects of geological investigation4 ,more recently1 however1 various lines of research have concurred in suggesting that1 whatever may be the condition of the interior1 its substance must differ greatly from that of the outer shell1 and that there may be more reason than appeared for the retention of the name of crust4 ,observations on earthquake motion by ,dr ,john ,milne and others1 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 crust4 ,this difference indicates that rocky material1 such as we know at the surface1 may extend inwards for some #cj m41 below which the earth's interior rapidly becomes fairly homogeneous and possesses a high rigidity4 ,from measurements of the force of gravity in ,india by ,colonel ,s4,g4 ,burrard1 it has been inferred that the variations in density of the outer parts of the earth do not descend farther than #cj or #dj m41 which might be assumed to be the limit of the thickness of the crust4 ,recent researches in regard to the radio-active substances present in rocks suggest that the crust is not more than #ej m4 thick1 and that the interior differs from it in possessing little or no radio-active material4 ,the interior4 ,though we cannot hope ever to have direct acquaintance with more than the mere outside skin of our planet1 we may be led to infer the irregular distribution of materials within the crust from the present distribution of land and water1 and the observed differences in the amount of deflection of the plumb-line near the sea and near mountain-chains4 ,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 planet4 ,the existence of such a vast sheet of water as that of the ,pacific ,ocean is to be accounted for1 as ,archdeacon ,j4,h4 ,pratt pointed out1 by the presence of some excess of matter in the solid parts of the earth between the ,pacific ,ocean and the earth's centre1 which retains the water in its place1 otherwise the ocean would flow away to the other parts of the earth4 ,a deflection of the plumb-line towards the sea1 which has in a number of cases been observed1 indicates that the density of the crust beneath the mountains must be less than that below the plains1 and still less than that below the ocean-bed4 ,apart therefore from the depression of the earth's surface in which the oceans lie1 we must regard the internal density1 whether of crust or nucleus1 to be somewhat irregularly arranged1 there being an excess of heavy materials in the water hemisphere1 and beneath the ocean-beds1 as compared with the continental masses4 ,in our ignorance regarding the chemical constitution of the nucleus of our planet1 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 crust4 ,this has been held by some writers to prove that the interior must consist of much heavier material and is therefore probably metallic4 ,but the effect of pressure ought to make the density of the nucleus much higher1 even if the interior consisted of matter no heavier than the crust4 ,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 pressure4 ,the only force we can suppose capable of so acting is heat4 ,but comparatively little is yet known regarding the compression of gases1 liquids and solids under such vast pressures as must exist within the nucleus4 ,that the interior of the earth possesses a high temperature is inferred from the evidence of various sources4 7#a7 ,volcanoes1 which are openings that constantly1 or intermittently1 give out hot vapours and molten lava from reservoirs beneath the crust4 ,besides active volcanoes1 it is known that former eruptive vents have been abundantly and widely distributed over the globe from the earliest geological periods down to our own day4 7#b7 ,hot springs are found in many parts of the globe1 with temperatures varying up to the boiling point of water4 7#c7 ,from mines1 tunnels and deep borings into the earth it has been ascertained that in all quarters of the globe below the superficial zone of invariable temperature1 there is a progressive increase of heat towards the interior4 ,the rate of this increase varies1 being influenced1 among other causes1 by the varying conductivity of the rocks4 ,but the average appears to be about #a deg4 ,fahr4 for every #ej or #fj ft4 of descent1 as far down as observations have extended4 ,though the increase may not advance in the same proportion at great depths1 the inference has been confidently drawn that the temperature of the nucleus must be exceedingly high4 ,the probable condition of the earth's interior has been a fruitful source of speculation ever since geology came into existence2 but no general agreement has been arrived at on the subject4 ,three chief hypotheses have been propounded3 7#a7 that the nucleus is a molten mass enclosed within a solid shell2 7#b7 that1 save in local vesicular spaces which may be filled with molten or gaseous material1 the globe is solid and rigid to the centre2 7#c7 that the great body of the nucleus consists of incandescent vapours and gases1 especially vaporous iron1 which under the gigantic pressure within the earth are so compressed as to confer practical rigidity on the globe as a whole1 and that outside this main part of the nucleus the gases pass into a shell of molten magma1 which1 in turn1 shades off outwards into the comparatively thin1 cool solidified crust4 ,recent seismological observations have led to the inference that the outer crust1 some #cj to #de m4 thick1 must rapidly merge into a fairly homogeneous nucleus which1 whatever be its constitution1 transmits undulatory movements through its substance with uniform velocity and is believed to possess a high rigidity4 ,the origin of the earth's high internal temperature has been variously accounted for4 ,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 globe4 ,according to another supposition the effects of the gradual gravitational compression of the earth's mass have been the main source of the high temperature4 ,recent researches in radio-activity1 to which reference has already been made1 have indicated another possible source of the internal heat in the presence of radium in the rocks of the crust4 ,this substance has been detected in all igneous rocks1 especially among the granites1 in quantity sufficient1 according to the ,hon4 ,r4,j4 ,strutt1 to account for the observed temperature-gradient in the crust1 and to indicate that this crust cannot be more than #de m4 thick1 otherwise the outflow of heat would be greater than the amount actually ascertained4 ,inside this external crust containing radio-active substances1 it is supposed1 as already stated1 that the nucleus consists of some totally different matter containing little or no radium4 ,constitution of the ,earth's ,crust4--,as the crust of the earth contains the geological record1 or stony chronicle from which geology interprets the history of our globe1 it forms the main subject of study to the geologist4 ,the materials of which this crust consists are known as minerals and rocks4 ,from many chemical analyses1 which have been made of these materials1 the general chemical constitution of1 at least1 the accessible portion of the crust has been satisfactorily ascertained4 ,this information becomes of much importance in speculations regarding the early history of the globe4 ,of the elements known to the chemist the great majority form but a small proportion of the composition of the crust1 which is mainly built up of about twenty of them4 ,of these by far the most important are the non-metallic elements oxygen and silicon4 ,the former forms about #dg and the latter rather more than #bh of the original crust1 so that these two elements make up about three-fourths of the whole4 ,next after them come the metals aluminium 7#h4af71 iron 7#d4fd71 calcium 7#c4ej71 magnesium 7#b4fb71 sodium 7#b4fc71 and potassium 7#b4ce74 ,the other twelve elements included in the twenty vary in amount from a proportion of #j4da in the case of titanium1 to not more than #j4ja of chlorine1 fluorine1 chromium1 nickel and lithium4 ,the other fifty or more elements exist in such minute proportions in the crust that1 probably1 not one of them amounts to as much as #j4ja1 though they include the useful metals1 except iron4 ,taking the crust1 and the external envelopes of the ocean and the air1 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 metals4 ,the combinations of the elements which are of most importance in the constitution of the terrestrial crust consist of oxides4 ,from the mean of a large number of analyses of the rocks of the lower or primitive portion of the crust1 it has been ascertained that silica 7,si,o#b7 forms almost #fj and alumina 7,al#b,o#c7 upwards of #ae of the whole4 ,the other combinations in order of importance are lime 7,ca,o7 #d4ij1 magnesia 7,mg,o7 #d4cf1 soda 7,na#b,o7 #c4ee1 ferrous oxide 7,fe,o7 #c4eb1 potash 7,k#b,o7 #b4hj1 ferric oxide 7,fe#b,o#c7 #b4fc1 water 7,h#b,o7 #a4eb1 titanium oxide 7,ti,o#b7 #j4fj1 phosphoric acid 7,p#b,o#e7 #j4bb2 the other combinations of elements thus form less than #a of the crust4 ,these different combinations of the elements enter into further combinations with each other so as to produce the wide assortment of simple minerals 7see ,,mineralogy74 ,thus1 silica and alumina are combined to form the aluminous silicates1 which enter so largely into the composition of the crust of the earth4 ,the silicates of magnesia1 potash and soda constitute other important families of minerals4 ,a mass of material composed of one1 but more usually of more than one mineral1 is known as a rock4 ,under this term geologists are accustomed to class not only solid stone1 such as granite and limestone1 but also less coherent materials such as clay1 peat and even loose sand4 ,the accessible portion of the earth's crust consists of various kinds of rocks1 which differ from each other in structure1 composition and origin1 and are therefore susceptible of diverse classifications according to the point of view from which they are considered4 ,the details of this subject will be found in the article ,,petrology4 ,classification of ,rocks4--,various systems of classification of rocks have been proposed1 but none of them is wholly satisfactory4 ,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 origin4 ,from this point of view they may be ranged in three divisions3 #a4 ,in the first place1 a large number of rocks may be described as original or underived1 for it is not possible to trace them back to any earlier source4 ,they belong to the primitive constitution of the planet1 and1 as they have all come up from below through the crust1 they serve to show the nature of the material which lies immediately below the outer parts of that crust4 ,they include the numerous varieties of lava1 which have been poured out in a molten state from volcanic vents1 also a great series of other rocks which1 though they may never have been erupted to the surface1 have been forced upward in a melted condition into the other rocks of the crust and have solidified there4 ,from their mode of origin this great class of rocks has been called igneous or eruptive4 ,as they generally show no definite internal structure save such as may result from joints1 they have been termed massive or unstratified1 to distinguish them from those of the second division which are strongly marked out by the presence of a stratified structure4 ,the igneous rocks present a considerable range of composition4 ,for the most part they consist mainly of aluminous silicates1 some of them being highly acid compounds with #ge or more of silica4 ,but they also include highly basic varieties wherein the proportion of silica sinks to #dj1 and where magnesia greatly predominates over alumina4 ,the textures of igneous rocks likewise comprise a wide series of varieties4 ,on the one hand1 some are completely vitreous1 like obsidian1 which is a natural glass4 ,from this extreme every gradation may be traced through gradual increase of the products of devitrification1 until the mass may become completely crystalline4 ,again1 some crystalline igneous rocks are so fine in grain as not to show their component crystals save under the microscope1 while in others the texture is so coarse as to present the component minerals in separate crystals an inch or more in length4 ,these differences indicate that1 at first1 the materials of the rock may have been as completely molten as artificial glass1 and that the crystalline condition has been subsequently developed by cooling1 and the separation of the chemical constituents into definite crystalline minerals4 ,many of the characters of igneous rocks have been reproduced experimentally by fusing together their minerals1 or the constituents of their minerals1 in the proper proportion4 ,but it has not yet been found possible to imitate the structure of such rocks as granite4 ,doubtless these rocks consolidated with extreme slowness at great depths below the surface1 under vast pressures and probably in the presence of water or water-vapour--conditions which cannot be adequately imitated in a laboratory4 ,though the igneous rocks occupy extensive areas in some countries1 they nevertheless cover a much smaller part of the whole surface of the land than is taken up by the second division or stratified rocks4 ,but they increase in quantity downwards and probably extend continuously round the globe below the other rocks4 ,this important series brings before us the relations of the molten magma within the earth to the overlying crust and to the outer surface4 ,on the one hand1 it includes the oldest and most deep-seated extravasations of that magma1 which have been brought to light by ruptures and upheavals of the crust and prolonged denudation4 ,on the other1 it presents to our study the varied outpourings of molten and fragmentary materials in the discharges of modern and ancient volcanoes4 ,between these two extremes of position and age1 we find that the crust has been1 as it were1 riddled with injections of the magma from below4 ,these features will be further noticed in ,part ,v4 of this article4 #b4 ,the sedimentary or stratified rocks form by much the larger part of the dry land of the globe1 and they are prolonged to an unknown distance from the shores under the bed of the sea4 ,they include those masses of mineral matter which1 unlike the igneous rocks1 can be traced back to a definite origin on the surface of the earth4 ,three distinct types may be recognized among them3 7a7 ,by far the largest proportion of them consists of different kinds of sediment derived from the disintegration of pre-existing rocks4 ,in this fragmental group are placed all the varieties of shingle1 gravel1 sand1 clay and mud1 whether these materials remain in a loose incoherent condition1 or have been compacted into solid stone4 7b7 ,another group consists of materials that have been deposited by chemical precipitation from solution in water4 ,the white sinter laid down by calcareous springs is a familiar example on a small scale4 ,beds of rock-salt1 gypsum and dolomite have1 in some regions1 been accumulated to a thickness of many thousand feet1 by successive precipitations of the salt contained in the water of inland seas4 7c7 ,an abundant and highly important series of sedimentary formations has been formed from the remains of plants and animals4 ,such accumulations may arise either from the transport and deposit of these remains1 as in the case of sheets of drift-wood1 and banks of drifted sea-shells1 or from the growth and decay of the organisms on the spot1 as happens in peat bogs and in coral-reefs4 ,as the sedimentary rocks have for the most part been laid down under water1 and more especially on the sea-floor1 they are often spoken of as aqueous1 in contradistinction to the igneous rocks4 ,some of them1 however1 are accumulated by the drifting action of wind upon loose materials1 and are known as aeolian formations4 ,familiar instances of such wind-formed deposits are the sand-dunes along many parts of the sea coast4 ,much more extensive in area are the sands of the great deserts in the arid regions of the globe4 ,it is from the sedimentary rocks that the main portion of geological history is derived4 ,they have been deposited one over another in successive strata from a remote period in the development of the globe down to the present time4 ,from this arrangement they have been termed stratified1 in contrast to the unstratified or igneous series4 ,they have preserved memorials of the geographical revolutions which the surface of the earth has undergone2 and above all1 in the abundant fossils which they have enclosed1 they furnish a momentous record of the various tribes of plants and animals which have successively flourished on land and sea4 ,their investigation is thus the most important task which devolves upon the geologist4 #c4 ,in the third place comes a series of rocks which are not now in their original condition1 but have undergone such alteration as to have acquired new characters that more or less conceal their first structures4 ,some of them can be readily recognized as altered igneous masses2 others are as manifestly of sedimentary origin2 while of many it is difficult to decide what may have been their pristine character4 ,to this series the term metamorphic has been applied4 ,its members are specially distinguished by a prevailing fissile1 or schistose1 structure which they did not at first possess1 and which differs from anything found in unaltered igneous or sedimentary rocks4 ,this fissility is combined with a more or less pronounced crystalline structure4 ,these changes are believed to be the result of movements within the crust of the earth1 whereby the most solid rocks were crushed and sheared1 while1 at the same time1 under the influence of a high temperature and the presence of water1 they underwent internal chemical reactions1 which led to a rearrangement and recomposition of their mineral constituents and the production of a crystalline structure 7see ,,metamorphism74 ,among the less altered metamorphic rocks of sedimentary origin1 the successive laminae of deposit of the original sediment can be easily observed2 but they are also traversed by a new set of divisional planes1 along which they split across the original bedding4 ,together with this superinduced cleavage there have been developed in them minute hairs1 scales and rudimentary crystals4 ,further stages of alteration are marked by the increase of micaceous scales1 garnets and other minerals1 especially along the planes of cleavage1 until the whole rock becomes crystalline1 and displays its chief component minerals in successive discontinuous folia which merge into each other1 and are often crumpled and puckered4 ,massive igneous rocks can be observed to have undergone intense crushing and cleavage1 and to have ultimately assumed a crystalline foliated character4 ,rocks which present this aspect are known as schists 7q4v474 ,they range from the finest silky slates1 or phyllites1 up to the coarsest gneisses1 which in hand-specimens can hardly be distinguished from granites4 ,there is indeed every reason to believe that such gneisses were probably originally true granites1 and that their foliation and recrystallization have been the result of metamorphism4 ,the schists are more especially to be found in the heart of mountain-chains1 and in regions where the lowest and oldest parts of the earth's crust have1 in the course of geological revolutions1 been exposed to the light of day4 ,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 nucleus4 ,but the progress of investigation all over the world has shown that this supposition cannot be sustained4 ,the oldest known rocks present none of the characters of molten material that has cooled and hardened in the air1 like the various forms of recent lava4 ,on the contrary1 they possess many of the features characteristic of bodies of eruptive material that have been injected into the crust at some depth underground1 and are now visible at the surface1 owing to the removal by denudation of the rocks under which they consolidated4 ,in their less foliated portions they can be recognized as true eruptive rocks4 ,in many places gneisses that possess a thoroughly typical foliation have been found to pierce ancient sedimentary formations as intrusive bosses and veins4 ,,part ,,iv4--,,dynamical ,,geology ,this section of the science includes the investigation of those processes of change which are at present in progress upon the earth1 whereby modifications are made on the structure and composition of the crust1 on the relations between the interior and the surface1 as shown by volcanoes1 earthquakes and other terrestrial disturbances1 on the distribution of oceans and continents1 on the outlines of the land1 on the form and depth of the sea-bottom1 on climate1 and on the races of plants and animals by which the earth is tenanted4 ,it brings before us1 in short1 the whole range of activities which it is the province of geology to study1 and leads us to precise notions regarding their relations to each other and the results which they achieve4 ,a knowledge of this branch of the subject is thus the essential groundwork of a true and fruitful acquaintance with the principles of geology1 seeing that it necessitates a study of the present order of nature1 and thus provides a key for the interpretation of the past4 ,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 change1 into which we may break at any point1 and round which we may travel1 only to find ourselves brought back to our starting-point4 ,it is a matter of comparatively small moment at what part of the cycle we begin our inquiries4 ,we shall always find that the changes we see in action have resulted from some that preceded1 and give place to others which follow them4 ,at an early time in the earth's history1 anterior to any of the periods of which a record remains in the visible rocks1 the chief sources of geological action probably lay within the earth itself4 ,if1 as is generally supposed1 the planet still retained a great store of its initial heat1 it was doubtless the theatre of great chemical changes1 giving rise1 perhaps1 to manifestations of volcanic energy somewhat like those which have so marvellously roughened the surface of the moon4 ,as the outer layers of the globe cooled1 and the disturbances due to internal heat and chemical action became less marked1 the conditions would arise in which the materials for geological history were accumulated4 ,the influence of the sun1 which must always have operated1 would then stand out more clearly1 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 play4 ,in the pursuit of his inquiries into the past history and into the present regime of the earth1 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 imagined4 ,human experience has been too short to allow him to assume that all the causes and modes of geological change have been definitively ascertained4 ,on the earth itself there may remain for future discovery evidence of former operations by heat1 magnetism1 chemical change or otherwise1 which may explain many of the phenomena with which geology has to deal4 ,of the influences1 so many and profound1 which the sun exerts upon our planet1 we can as yet only perceive a little4 ,nor can we tell what other cosmical influences may have lent their aid in the evolution of geological changes4 ,much useful information regarding many geological processes has been obtained from experimental research in laboratories and elsewhere1 and much more may be confidently looked for from future extensions of this method of inquiry4 ,the early experiments of ,sir ,james ,hall1 already noticed1 formed the starting-point for numerous subsequent researches1 which have elucidated many points in the origin and history of rocks4 ,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 time4 ,but experience has shown that in regard to a large number of processes1 it is possible to imitate nature's working with sufficient accuracy to enable us to understand them1 and so to modify and control the results as to obtain a satisfactory solution of some geological problems4 ,in the present state of our knowledge1 all the geological energy upon and within the earth must ultimately be traced back to the primeval energy of the parent nebula or sun4 ,there is1 however1 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 sun4 ,in the former case we have to deal with the interior of the earth1 and its reaction upon the surface2 in the latter1 we deal with the surface of the earth and to some extent with its reaction on the interior4 ,this distinction allows of a broad treatment of the subject under two divisions3 ,i4 ,hypogene or ,plutonic ,action3 ,the changes within the earth caused by internal heat1 mechanical movement and chemical rearrangements4 ,,ii4 ,epigene or ,surface ,action3 ,the changes produced on the superficial parts of the earth1 chiefly by the circulation of air and water set in motion by the sun's heat4 ,,division ,i4--,,hypogene ,,or ,,plutonic ,,action ,in the discussion of this branch of the subject we must carry in our minds the conception of a globe still possessing a high internal temperature1 radiating heat into space and consequently contracting in bulk4 ,portions of molten rocks from inside are from time to time poured out at the surface4 ,sudden shocks are generated by which destructive earthquakes are propagated through the diameter of the globe as well as to and along its surface4 ,wide geographical areas are pushed up or sink down4 ,in the midst of these movements remarkable changes are produced upon the rocks of the crust2 they are plicated1 fractured1 crushed1 rendered crystalline and even fused4 7,a7 ,volcanoes and ,volcanic ,action4 ,this subject is discussed in the article ,,volcano1 and only a general view of its main features will be given here4 ,under the term volcanic action 7vulcanism1 vulcanicity7 are embraced all the phenomena connected with the expulsion of heated materials from the interior of the earth to the surface4 ,a volcano may be defined as a conical hill or mountain1 built up wholly or mainly of materials which have been ejected from below1 and which have accumulated around the central vent of eruption4 ,as a rule its truncated summit presents a cup-shaped cavity1 termed the crater1 at the bottom of which is the opening of the main funnel or pipe whereby communication is maintained with the heated interior4 ,from time to time1 however1 in large volcanoes rents are formed on the sides of the cone1 whence steam and other hot vapours and also streams of molten lava are poured forth4 ,on such rents smaller or parasitic cones are often formed1 which imitate the operations of the parent cone and1 after repeated eruptions1 may rise to hills hundreds of feet in height4 ,in course of centuries the result of the constant outpouring of volcanic materials may be to build up a large mountain like ,etna1 which towers above the sea to a height of #aj1hdj feet1 and has some #bjj minor cones along its flanks4 ,but all volcanic eruptions do not proceed from central orifices4 ,in ,iceland it has been observed that1 from fissures opened in the ground and extending for long distances1 molten material has issued in such abundance as to be spread over the surrounding country for many miles1 while along the lines of fissure small cones or hillocks of fragmentary material have accumulated round more active parts of the rent4 ,there is reason to believe that in the geological past this fissure-type of eruption has repeatedly been developed1 as well as the more common form of central cones like ,vesuvius or ,etna4 ,in the operations of existing volcanoes only the superficial manifestations of volcanic action are observable4 ,but when the rocks of the earth's crust are studied1 they are found to enclose the relics of former volcanic eruptions4 ,the roots of ancient volcanoes have thus been laid bare by geological revolutions2 and some of the subterranean phases of volcanic action are thereby revealed which are wholly concealed in an active volcano4 ,hence to obtain as complete a conception as possible of the nature and history of volcanic action1 regard must be had1 not merely to modern volcanoes1 but to the records of ancient eruptions which have been preserved within the crust4 ,the substances discharged from volcanic vents consist of--7#a7 ,gases and vapours3 which1 dissolved in the molten magma of the interior1 take the chief share in volcanic activity4 ,they include in greatest abundance water-gas1 which condenses into the clouds of steam so conspicuous in volcanic eruptions4 ,hydrochloric acid and sulphuretted hydrogen are likewise plentiful1 together with many other substances which1 sublimed by the high internal temperature1 take a solid form on cooling at the surface4 7#b7 ,molten rock or lava3 which ranges from the extremely acid type of the obsidians and rhyolites with #gj or more of silica1 to the more basic and heavy varieties such as basalts and leucite-lavas with much iron1 and sometimes no more than #de of silica4 ,the specific gravity of lavas varies between #b4cg and #c4bb1 and the texture ranges from nearly pure glass1 like obsidian1 to a coarse granitoid compound1 as in some rhyolites4 7#c7 ,fragmentary materials1 which are sometimes discharged in enormous quantity and dispersed over a wide extent of country1 the finer particles being transported by upper air-currents for hundreds of miles4 ,these materials arise either from the explosion of lava by the sudden expansion of the dissolved vapours and gases1 as the molten rock rises to the surface1 or from the breaking up and expulsion of portions of the walls of the vent1 or of the lava1 which happens to have solidified within these walls4 ,they vary from the finest impalpable dust and ashes1 through increasing stages of coarseness up to huge bombs torn from the upper surface of the molten rock in the vent1 and large blocks of already solidified lava1 or of non-volcanic rock detached from the sides of the pipe up which the eruptions take place4 ,nothing is yet known as to the determining cause of any particular volcanic eruption4 ,some vents1 like that of ,stromboli1 in the ,mediterranean1 are continually active1 and have been so ever since man has observed them4 ,others again have been only intermittently in eruption1 with intervals of centuries between their periods of activity4 ,we are equally in the dark as to what has determined the sites on which volcanic action has manifested itself4 ,there is reason1 indeed1 to believe that extensive fractures of the terrestrial crust have often provided passages up which the vapours1 imprisoned in the internal magma1 have been able to make their way1 accompanied by other products4 ,where chains of volcanoes rise along definite lines1 like those of ,sumatra1 ,java1 and many other tracts both in the ,old and the ,new ,world1 there appears to be little doubt that their linear distribution should be attributed to this cause4 ,but where a volcano has appeared by itself1 in a region previously exempt from volcanic action1 the existence of a contributing fissure cannot be so confidently presumed4 ,the study of certain ancient volcanoes1 the roots of which have been exposed by long denudation1 has shown an absence of any visible trace of their having availed themselves of fractures in the crust4 ,the inference has been drawn that volcanic energy is capable of itself drilling an orifice through the crust1 probably at some weaker part1 and ejecting its products at the surface4 ,the source of this energy is to be sought in the enormous expansive force of the vapours and gases dissolved in the magma4 ,they are kept in solution by the enormous pressure within the earth2 but as the lava approaches the surface and this pressure is relieved these dissolved vapours and gases rush out with explosive violence1 blowing the upper part of the lava column into dust1 and allowing portions of the liquid mass below to rise and escape1 either from the crater or from some fissure which the vigour of explosion has opened on the side of the cone4 ,so gigantic is the energy of these pent-up vapours1 that1 after a long period of volcanic quiescence1 they sometimes burst forth with such violence as to blow off the whole of the upper part or even one side of a large cone4 ,the history of ,vesuvius1 and the great eruptions of ,krakatoa in #ahhc and of ,bandaizan in #ahhh furnish memorable examples of great volcanic convulsions4 ,it has been observed that such stupendous discharges of aeriform and fragmentary matter may be attended with the emission of little or no lava4 ,on the other hand1 some of the largest outflows of lava have been accompanied by comparatively little fragmentary material4 ,thus1 the great lava-floods of ,iceland in #aghc spread for #dj m4 away from their parent fissure1 which was marked only by a line of little cones of slag4 ,the temperature of lava as it issues from underground has been measured more or less satisfactorily1 and affords an indication of that existing within the earth4 ,at ,vesuvius it has been ascertained to be more than #bjjj deg4 ,fahr4 ,at first the molten rock glows with a white light1 which rapidly reddens1 and disappears under the rugged brown and black crust that forms on the surface4 ,underneath this badly conducting crust1 the lava cools so slowly that columns of steam have been noticed rising from its surface more than #hj years after its eruption4 ,considerable alteration in the topography of volcanic regions may be produced by successive eruptions4 ,the fragmentary materials are sometimes discharged in such abundance as to cover the ground for many miles around with a deposit of loose ashes1 cinders and slag4 ,such a deposit accumulating to a depth of many feet may completely bury valleys and water-courses1 and thus greatly affect the drainage4 ,the coarsest materials accumulate nearest to the vent that emits them4 ,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-currents1 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 volcano4 ,outflows of lava1 from their greater solidity and durability1 produce still more serious and lasting changes in the external features of the ground over which they flow4 ,as they naturally seek the lowest levels1 they find their way into the channels of streams4 ,if they keep along the channels1 they seal them up under a mass of compact stone which the running water1 if not wholly diverted elsewhere1 will take many long centuries to cut through4 ,if1 on the other hand1 the lava crosses a stream1 it forms a massive dam1 above which the water is ponded back so as to form a lake4 ,as the result of prolonged activity a volcanic cone is gradually built up by successive outflows of lava and showers of dust and stones4 ,these materials are arranged in beds1 or sheets1 inclined outwards from the central vent4 ,on surrounding level ground the alternating beds are flat4 ,in course of time1 deep gullies are cut on the outer slopes of the cone by rain1 and by the heavy showers that arise from the condensation of the copious discharges of steam during eruptions4 ,along the sides of these ravines instructive sections may be studied of the volcanic strata4 ,the larger rivers of some volcanic regions have likewise eroded vast gorges in the more horizontal lavas and ashes of the flatter country1 and have thus laid bare stupendous cliffs1 along which the successive volcanic sheets can be seen piled above each other for many hundred feet4 ,on a small scale1 some of these features are well displayed among the rivers that drain the volcanic tracts of central ,france2 on a great scale1 they are presented in the course of the ,snake river1 and other streams that traverse the great volcanic country of western ,north ,america4 ,similar volcanic scenery has been produced in western ,europe by the action of denudation in dissecting the flat ,tertiary lavas of ,scotland1 the ,faeroe ,isles and ,iceland4 ,of special interest to the geologist are those volcanoes which have taken their rise on the sea-bottom2 for the volcanic intercalations among the stratified formations of the earth's crust are almost entirely of submarine origin4 ,many active volcanoes situated on islands have begun their eruptions below sea-level4 ,both ,vesuvius and ,etna sprang up on the floor of the ,mediterranean sea1 and have gradually built up their cones into conspicuous parts of the dry land4 ,examples of a similar history are to be found among the volcanic islands of the ,pacific ,ocean4 ,in some of these cases a movement of elevation has carried the submarine lavas1 tuffs and agglomerates above sea-level1 and has furnished opportunities of comparing these materials with those of recent subaerial origin1 and also with the ancient records of submarine eruptions which have been preserved among the stratified formations4 ,from the evidence thus supplied1 it can be shown that the materials ejected from modern submarine volcanic vents closely resemble those accumulated by subaerial volcanoes2 that the dust1 ashes and stones become intermingled or interstratified with coral-mud1 or other non-volcanic deposit of the sea-bottom1 that vesicular lavas may be intercalated among them as on land1 and that between the successive sheets of volcanic origin1 layers of limestone may be laid down which are composed chiefly1 or wholly1 of the remains of calcareous marine organisms4 ,though active volcanoes are widely distributed over the globe1 and are especially abundant around the vast basin of the ,pacific ,ocean1 they afford an incomplete picture of the extent to which volcanic action has displayed itself on the surface of our planet4 ,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 it4 ,not only so1 but they reveal that volcanoes have been in eruption in some of these districts during many different periods of the past1 back to the beginnings of geological history4 ,the ,british ,islands furnish a remarkable example of such a series of ancient eruptions4 ,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 discharged4 ,again in older ,tertiary times the same region witnessed a stupendous outpouring of basalt1 the surviving relics of which are more than #cjjj ft4 thick1 and cover many hundreds of square miles4 ,similar evidence is supplied in other countries both in the ,old and the ,new world4 ,hence it is proved that1 in the geological past1 volcanic action has been vigorous at long intervals on the same sites during a vast series of ages1 though no active vents are to be seen there now4 ,the volcanoes now active form but a small proportion of the total number which has appeared on the surface of the earth4 ,with regard to the cause of volcanic action much has been speculated1 but little can be confidently affirmed4 ,that water in the form of occluded gas plays the chief part in forcing the lava column up a volcanic chimney1 and in the violent explosions that accompany the rise of the molten material1 is generally admitted4 ,but opinions differ as to the source of this water4 ,according to some investigators1 it should be regarded as in large measure of meteoric origin1 derived from the descent of rain into the earth1 and its absorption by the molten magma in the interior4 ,others1 contending that the supply so furnished1 even if it could reach and be dissolved in the magma1 would yet be insufficient to furnish the prodigious quantity of aqueous vapour discharged during an eruption1 maintain that the water belongs to the magma itself4 ,they point to the admitted fact that many substances1 particularly metals in a state of fusion1 can absorb large quantities of vapours and gases without chemical combination1 and on cooling discharge them with eruptive phenomena somewhat like those of volcanoes4 ,this question must be regarded as one of the still unsolved problems of geology4 7,b7 ,movements of the ,earth's ,crust4 ,among the hypogene forces in geological dynamics an important place must be assigned to movements of the terrestrial crust4 ,though the expression the solid earth has become proverbial1 it appears singularly inappropriate in the light of the results obtained in recent years by the use of delicate instruments of observation4 ,with the facilities supplied by these instruments 7see ,,seismometer71 it has been ascertained that the ground beneath our feet is subject to continual slight tremors1 and feeble pulsations of longer duration1 some of which may be due to daily or seasonal variations of temperature1 atmospheric pressure or other meteorological causes4 ,the establishment of self-recording seismometers all over the world has led to the detection of many otherwise imperceptible shocks1 over and above the appreciable earth-waves propagated from earthquake centres of disturbance4 ,moreover1 it has been ascertained that some parts of the surface of the land are slowly rising1 while others are falling with reference to the sea-level4 ,from time to time the surface suffers calamitous devastation from earthquakes1 when portions of the crust under great strain suddenly give way4 ,lastly1 at intervals1 probably separated from each other by vast periods of time1 the terrestrial crust undergoes intense plication and fracture1 and is consequently ridged up into mountain-chains4 ,no event of this kind has been witnessed since man began to record his experiences4 ,but from the structure of mountains1 as laid open by prolonged denudation1 it is possible to form a vivid conception of the nature and effects of these most stupendous of all geological revolutions4 ,in considering this department of geological inquiry it will be convenient to treat it under the following heads3 7#a7 ,slow depression and upheaval2 7#b7 ,earthquakes2 7#c7 ,mountain-making2 7#d7 ,metamorphism of rocks4 #a4 ,slow ,depression and ,upheaval4--,on the west side of ,japan the land is believed to be sinking below the sea1 for fields are replaced by beaches of sand or shingle1 while the depth of the sea off shore has perceptibly increased4 ,a subsidence of the south of ,sweden has taken place in comparatively recent times1 for streets and foundations of houses at successive levels are found below high-water mark4 ,the west coast of ,greenland over an extent of more than #fjj m4 is sinking1 and old settlements are now submerged4 ,proofs of submergence of land are furnished by submerged forests1 and beds of terrestrial peat now lying at various depths below the level of the sea1 of which many examples have been collected along the shores of the ,british ,isles1 ,holland and ,france4 ,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 ,atlantic1 which show that the valleys of the land are prolonged under the sea4 ,these valleys have been eroded out of the rocks by the streams which flow in them1 and the depth of their submerged portions below the sea level affords an indication of the extent of the subsidence4 ,the uprise of land has been detected in various parts of the world4 ,one of the most celebrated instances is that of the shores of the ,gulf of ,bothnia1 where1 at ,stockholm1 the elevation1 between the years #aggd and #ahge1 appears to have been #dh centimetres 7#ah-a_/b in47 in a century4 ,but on the west side of ,sweden1 fronting the ,skager ,rak1 the coast1 between the years #ahbj and #ahgj1 rose #cj centimetres1 which is at the rate of #fj centimetres1 or nearly #b ft4 in a century4 ,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-west1 of which the mean rate appears to be rather less than #f in4 in a century4 ,if this rate of change should continue the waters of ,lake ,michigan1 owing to the progress of the tilt1 will1 in some #ejj or #fjj years1 submerge the city of ,chicago1 and eventually the drainage of the lakes will be diverted into the basin of the ,mississippi4 ,proof of recent emergence of land is supplied by what are called raised beaches or strand-lines1 that is1 lines of former shores marked by sheets of littoral deposits1 or platforms cut by shore-waves in rock and flanked by old sea-cliffs and lines of sea-worn caves4 ,admirable examples of these features are to be seen along the west coast of ,europe from the south of ,england to the north of ,norway4 ,these lines of old shores become fainter in proportion to their antiquity4 ,in ,britain they occur at various heights1 the platforms at #be1 #ej and #ajj ft4 being well marked4 ,the cause of these slow upward and downward movements of the crust of the earth is still imperfectly understood4 ,upheaval might conceivably be produced by an ascent of the internal magma1 and the consequent expansion of the overlying crust by heat2 while depression might follow any subsidence of the magma1 or its displacement to another district4 ,if1 as is generally believed1 the globe is still contracting1 the shrinkage of the surface may cause both these movements4 ,subsidence will be in excess1 but between subsiding tracts lateral thrust may suffice to push upward intervening more solid and stable ground2 but no solution of the problem yet proposed is wholly satisfactory4 #b4 ,earthquakes4--,as this subject is discussed in a separate article it will be sufficient here to take note of its more important geological bearings4 ,it was for many centuries taken for granted that earthquakes and volcanoes are due to a common cause4 ,we have seen that in classical antiquity they were looked on as the results of the movements of wind imprisoned within the earth4 ,long after this notion was discarded1 and a more scientific appreciation of volcanic action was reached1 it was still thought that earthquakes should be regarded as manifestations of the same source of energy as that which displays itself in volcanic eruptions4 ,it is true that earthquakes are frequent in districts of active volcanoes1 and they may undoubtedly be often due there to the explosions of the magma1 or to the rupture of rocks caused by its ascent towards the surface4 ,but such shocks are comparatively local in their range and feeble in their effects4 ,there is now a general agreement that between the great world-shaking earthquakes and volcanic phenomena1 no immediate and intimate relationship can be traced1 though they may be connected in ways which are not yet perceived4 ,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 strain1 either along lines of previous fracture or of new rents in the terrestrial crust2 and that such ruptures may occur at a remote distance from any volcano4 ,thus the recent disastrous ,san ,francisco earthquake has been recognized to have resulted from a slipping of ground along the line of an old fault1 which has been traced for a long distance in ,california generally parallel to the coast4 ,the position of this fault at the surface has long been clearly followed by its characteristic topography4 ,after the earthquake these superficial features were found to have been removed by the same cause that had originated them4 ,for some #cjj m4 on the track of this old fault-line a renewed slipping was seen to have taken place along one or both sides1 and the ground at the surface was ruptured as well as displaced horizontally4 ,obviously1 the jar occasioned by the sudden and simultaneous subsidence of a portion of the earth's crust several hundred miles long1 must be far more serious than could be produced by an earthquake radiating from a single local volcanic focus4 ,from their disastrous effects on buildings and human lives1 an exaggerated importance has been imputed to earthquakes as agents of geological change4 ,experience shows that even after a severe shock which may have destroyed numerous towns and villages1 together with thousands of their inhabitants1 the face of the country has suffered scarcely any perceptible change1 and that1 in the course of a year or two1 when the ruined houses and prostrate trees have been cleared away1 little or no obvious trace of the catastrophe may remain4 ,among the more enduring records of a great earthquake may be enumerated 7a7 landslips1 which lay bare hillsides1 and sometimes pond back the drainage of valleys so as to give rise to lakes2 7b7 alterations of the topography1 as in fissuring of the ground1 or in the production of inequalities whereby the drainage is affected2 new valleys and new lakes may thus be formed1 while previously existing lakes may be emptied2 7c7 permanent changes of level1 either in an upward or downward direction4 #c4 ,mountain-making4--,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 processes4 ,the structure of a great mountain-chain such as the ,alps proves that the crust of the earth has been intensely plicated1 crumpled and fractured4 ,vast piles of sedimentary strata have been folded to such an extent as to occupy now only half of their original horizontal extent4 ,this compression in the case of the ,alps has been computed to amount to as much as #abj1jjj metres or #gd ,english miles1 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 movements4 ,besides such intense plication1 extensive rupturing of the crust has taken place in the same range of mountains4 ,not only have the most ancient rocks been squeezed up into the central axis of the chain1 but huge slices of them have been torn away from the main body1 and thrust forward for many miles1 so as now actually to form the summits of mountains1 which are almost entirely composed of much younger formations4 ,if these colossal disturbances occurred rapidly1 they would give rise to cataclysms of inconceivable magnitude over the surface of the globe4 ,no record has been discovered of such accompanying devastation4 ,but whether sudden and violent1 or prolonged and gradual1 such stupendous upturnings of the crust did undoubtedly take place1 as is clearly revealed in innumerable natural sections1 which have been laid open by the denudation of the crests and sides of the mountains4 #d4 ,metamorphism of ,rocks 7see ,,metamorphism74--,during the movements to which the crust of the earth has been subject1 not only have the rocks been folded and fractured1 but they have likewise1 in many regions1 acquired new internal structures1 and have thus undergone a process of regional metamorphism4 ,this rearrangement of their substance has been governed by conditions which are probably not yet all recognized1 but among them we should doubtless include a high temperature1 intense pressure1 mechanical movement resulting in crushing1 shearing and foliation1 and the presence of water in their pores4 ,it is among igneous rocks that the progressive stages of metamorphism can be most easily traced4 ,their definite original structure and mineral composition afford a starting-point from which the investigation may be begun and pursued4 ,where an igneous rock has been invaded by metamorphic changes1 it may be observed to have been first broken down into separate lenticles1 the cores of which may still retain1 with little or no alteration1 the original characteristic minerals and crystalline structure of the rock4 ,between these lenticles1 the intervening portions have been crushed down into a powder or paste1 which seems to have been squeezed round and past them1 and shows a laminated arrangement that resembles the flow-structure in lavas4 ,as the degree of metamorphism increases1 the lenticles diminish in size1 and the intervening crushed and foliated matrix increases in amount1 until at last it may form the entire mass of the rock4 ,while the original minerals are thus broken down1 new varieties make their appearance4 ,of these1 among the earliest to present themselves are usually the micas1 that impart their characteristic silvery sheen to the surfaces of the folia along which they spread4 ,younger felspars1 as well as mica1 are developed1 and there arise also sillimanite1 garnet1 andalusite and many others4 ,the texture becomes more coarsely crystalline1 and the segregation of the constituent minerals more definite along the lines of foliation4 ,from the finest silky phyllites a graduation may be traced through successively coarser mica-schists1 until we reach the almost granitic texture of the coarsest gneisses4 ,regional metamorphism has arisen in the heart of mountain-chains1 and in any other district where the deformation of the crust has been sufficiently intense4 ,there is another type of alteration termed contact-metamorphism1 which is developed around masses of igneous rock1 especially where these have been intruded in large bosses among stratified formations4 ,it is particularly displayed around masses of granite1 where sandstones are found altered into quartzite1 shales and grits into schistose compounds1 and where sometimes fossils are still recognizable among the metamorphic minerals4 ,,division ,,ii4--,,epigene ,,or ,,superficial ,,action ,it is on the surface of the globe1 and by the operation of agents working there1 that at present the chief amount of visible geological change is effected4 ,in considering this branch of inquiry1 we are not involved in a preliminary difficulty regarding the very nature of the agencies as is the case in the investigation of plutonic action4 ,on the contrary1 the surface agents are carrying on their work under our very eyes4 ,we can watch it in all its stages1 measure its progress1 and mark in many ways how accurately it represents similar changes which1 for long ages previously1 must have been effected by the same means4 ,but in the systematic treatment of this subject we encounter a difficulty of another kind4 ,we discover that while the operations to be discussed are numerous and readily observable1 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 impression4 ,while1 therefore1 under the unavoidable necessity of making use of such a classification of subjects1 we must always bear in mind that it is employed merely for convenience1 and that in nature superficial geological action must be continually viewed as a whole1 since the work of each agent has constant reference to that of the others1 and is not properly intelligible unless that connexion be kept in view4 ,the movements of the air2 the evaporation from land and sea2 the fall of rain1 hail and snow2 the flow of rivers and glaciers2 the tides1 currents and waves of the ocean2 the growth and decay of organized existence1 alike on land and in the depths of the sea2--in short1 the whole circle of movement1 which is continually in progress upon the surface of our planet1 are the subjects now to be examined4 ,it is desirable to adopt some general term to embrace the whole of this range of inquiry4 ,for this end the word epigene 7,gr4 ,greek3 epi1 upon7 has been suggested as a convenient term1 and antithetical to hypogene 7,gr4 ,greek3 hypo1 under71 or subterranean action4 ,a simple arrangement of this part of ,geological ,dynamics is in three sections3 ,a4 ,air4--,the influence of the atmosphere in destroying and forming rocks4 ,b4 ,water4--,the geological functions of the circulation of water through the air and between sea and land1 and the action of the sea4 ,c4 ,life4--,the part taken by plants and animals in preserving1 destroying or reproducing geological formations4 ,the words destructive1 reproductive and conservative1 employed in describing the operations of the epigene agents1 do not necessarily imply that anything useful to man is destroyed1 reproduced or preserved4 ,on the contrary1 the destructive action of the atmosphere may turn barren rock into rich soil1 while its reproductive effects sometimes turn rich land into barren desert4 ,again1 the conservative influence of vegetation has sometimes for centuries retained as barren morass what might otherwise have become rich meadow or luxuriant woodland4 ,the terms1 therefore1 are used in a strictly geological sense1 to denote the removal and re-deposition of material1 and its agency in preserving what lies beneath it4 7,a7 ,the ,air4 ,as a geological agent1 the air brings about changes partly by its component gases and partly by its movements4 ,its destructive action is both chemical and mechanical4 ,the chemical changes are probably mainly1 if not entirely1 due to the moisture of the air1 and particularly to the gases1 vapours and organic matter which the moisture contains4 ,dry air seems to have little or no appreciable influence in promoting these reactions4 ,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 rain4 ,among the more recognizable mechanical changes effected in the atmosphere1 one of considerable importance is to be seen in the result of great and rapid changes of temperature4 ,heat expands rocks1 while cold contracts them4 ,in countries with a great annual range of temperature1 considerable difficulty is sometimes experienced in selecting building materials liable to be little affected by the alternate expansion and contraction1 which prevents the joints of masonry from remaining close and tight4 ,in dry tropical climates1 where the days are intensely hot and the nights extremely cold1 the rapid nocturnal contraction produces a strain so great as to rival frost in its influence upon the surface of exposed rocks1 disintegrating them into sand1 or causing them to crack or peel off in skins or irregular pieces4 ,dr ,livingstone found in ,africa 7#ab deg4 ,s4 lat41 #cd deg4 ,e4 long47 that surfaces of rock which during the day were heated up to #acg deg4 ,fahr41 cooled so rapidly by radiation at night that1 unable to sustain the strain of contraction1 they split and threw off sharp angular fragments from a few ounces to #ajj or #bjj lb in weight4 ,in temperate regions this action1 though much less pronounced1 still makes itself felt4 ,in these climates1 however1 and still more in high latitudes1 somewhat similar results are brought about by frost4 ,by its motion in wind the air drives loose sand over rocks1 and in course of time abrades and smoothes them4 ,desert polish is the name given to the characteristic lustrous surface thus imparted4 ,holes are said to be drilled in window glass at ,cape ,cod by the same agency4 ,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 wind4 ,hurricanes form important geological agents upon land in uprooting trees1 and thus sometimes impeding the drainage of a country and giving rise to the formation of peat mosses4 ,the reproductive action of the air arises partly from the effect of the chemical and mechanical disintegration involved in the process of weathering1 and partly from the transporting power of wind and of aerial currents4 ,the layer of soil1 which covers so much of the surface of the land1 is the result of the decay of the underlying rocks1 mingled with mineral matter blown over the ground by wind1 or washed thither by rain1 and with the mouldering remains of plants and animals4 ,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 vegetation1 though even there1 in dry weather during spring1 clouds of dust may often be seen blown away by wind from bare ploughed fields4 ,intercepted by the leaves of plants and washed down to their roots by rain1 this dust goes to increase the soil below4 ,in arid climates1 where dust clouds are dense and frequent1 enormous quantities of fine mineral particles are thus borne along and accumulated4 ,the remarkable deposit of ,loess1 which is sometimes more than #aejj ft4 thick and covers extensive areas in ,china and other countries1 is regarded as due to the drifting of dust by wind4 ,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 action4 7,b7 ,water4 ,in treating of the epigene action of water in geological processes it will be convenient to deal first with its operations in traversing the land1 and then with those which it performs in the sea4 ,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 affected4 #a4 ,rain4--,rain effects two kinds of changes upon the surface of the land4 ,it acts chemically upon soils and stones1 and sinking under ground continues a great series of similar reactions there4 ,it acts mechanically1 by washing away loose materials1 and thus powerfully affecting the contours of the land4 ,its chemical action depends mainly upon the nature and proportion of the substances which1 in descending to the earth1 it abstracts from the atmosphere4 ,rain always absorbs a little air1 which1 in addition to its nitrogen and oxygen1 contains carbonic acid1 and in minute proportions1 sodium chloride1 sulphuric acid and other ingredients1 especially inorganic dust1 organic particles and living germs4 ,probably the most generally efficient of these constituents are oxygen1 carbonic acid and organic matter4 ,armed with these reagents1 rain effects a chemical decomposition of the rocks on which it falls1 and through which it sinks underground4 ,the principal changes thus produced are as follows3 7a7 ,oxidation4--,owing to the prominence of oxygen in rain-water1 and its readiness to unite with any substance which can contain more of it1 a thin oxidized pellicle is formed on the surface of many rocks on which rain falls1 and this oxidized layer if not at once washed off1 sinks deeper until a crust is formed over the stone4 ,a familiar illustration of this action is afforded by the rust1 or oxide1 which forms on iron when exposed to moisture1 though this iron may be kept long bright if allowed to remain screened from moist air and rain4 7b7 ,deoxidation4--,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 protoxides4 ,these changes are especially noticeable among the iron oxides so abundantly diffused among rocks4 ,hence rain-water1 in sinking through soil and obtaining such organic matter1 becomes thereby a reducing agent4 7c7 ,solution4--,this may take place either by the simple action of the water1 as in the solution of rock-salt1 or by the influence of the carbonic acid present in the rain4 7d7 ,formation of ,carbonates4--,a familiar example of the action of carbonic acid in rain is to be seen in the corrosion of exposed marble slabs4 ,the carbonic acid dissolves some of the lime1 which1 as a bicarbonate1 is held in solution in the carbonated water1 but is deposited again when the water loses its carbonic acid or evaporates4 ,it is not merely carbonates1 however1 which are liable to this kind of destruction4 ,even silicates of lime1 potash and soda1 combinations existing abundantly as constituents of rocks1 are attacked2 their silica is liberated1 and their alkalis or alkaline earths1 becoming carbonates1 are removed in solution4 7e7 ,hydration4--,some minerals1 containing little or no water1 and therefore called anhydrous1 when exposed to the action of the atmosphere1 absorb water1 or become hydrous1 and are then usually more prone to further change4 ,hence the rocks of which they form part become disintegrated4 ,besides the reactions here enumerated1 a considerable amount of decay may be observed as the result of the presence of sulphuric and nitric acid in the air1 especially in that of large towns and manufacturing districts1 where much coal is consumed4 ,metallic surfaces1 as well as various kinds of stone1 are there corroded1 while the mortar of walls may often be observed to be slowly swelling out and dropping off1 owing to the conversion of the lime into sulphate4 ,great injury is likewise done from a similar cause to marble monuments in exposed graveyards4 ,the general result of the disintegrating action of the air and of rain1 including also that of plants and animals1 to be noticed in the sequel1 is denoted by the term weathering4 ,the amount of decay depends partly on conditions of climate1 especially the range of temperature1 the abundance of moisture1 height above the sea and exposure to prevalent winds4 ,many rocks liable to be saturated with rain and rapidly dried under a warm sun are apt to disintegrate at the surface with comparative rapidity4 ,the nature and progress of the weathering are mainly governed by the composition and texture of the rocks exposed to it4 ,rocks composed of particles liable to little chemical change from the influence of moisture are best fitted to resist weathering1 provided they possess sufficient cohesion to withstand the mechanical processes of disintegration4 ,siliceous sandstones are excellent examples of this permanence4 ,consisting wholly or mainly of the durable mineral quartz1 they are sometimes able so to withstand decay that buildings made of them still retain1 after the lapse of centuries1 the chisel-marks of the builders4 ,some rocks1 which yield with comparative rapidity to the chemical attacks of moisture1 may show little or no mark of disintegration on their surface4 ,this is particularly the case with certain calcareous rocks4 ,limestone when pure is wholly soluble in acidulated water4 ,rain falling on such a rock removes some of it in solution1 and will continue to do so until the whole is dissolved away4 ,but where a limestone is full of impurities1 a weathered crust of more or less insoluble particles remains after the solution of the calcareous part of the stone4 ,hence the relative purity of limestones may be roughly determined by examining their weathered surfaces1 where1 if they contain much sand1 the grains will be seen projecting from the calcareous matrix1 and where1 should the rock be very ferruginous1 the yellow hydrous peroxide1 or ochre1 will be found as a powdery crust4 ,in limestones containing abundant encrinites1 shells1 or other organic remains1 the weathered surface commonly presents the fossils standing out in relief4 ,the crystalline arrangement of the lime in the organic structures enables them to resist disintegration better than the general mechanically aggregated matrix of the rock4 ,an experienced fossil collector will always search well such weathered surfaces1 for he often finds there1 delicately picked out by the weather1 minute and frail fossils which are wholly invisible on a freshly broken surface of the stone4 ,many rocks weather with a thick crust1 or even decay inwards for many feet or yards4 ,basalt1 for example1 often shows a yellowish-brown ferruginous layer on its surface1 formed by the conversion of its felspar into kaolin1 and the removal of its calcium silicate as carbonate1 by the hydration of its olivine and augite and their conversion into serpentine1 or some other hydrous magnesian silicate1 and by the conversion of its magnetite into limonite4 ,granite sometimes shows in a most remarkable way the distance to which weathering can reach4 ,it may occasionally be dug into for a depth of #bj or #cj ft41 the quartz crystals and veins retaining their original positions1 while the felspar is completely kaolinized4 ,it is to the endlessly varied effects of weathering that the abundant fantastic shapes assumed by crags and other rocky masses are due4 ,most varieties of rock have their own characteristic modes of weathering1 whereby they may be recognized even from a distance4 ,to some of these features reference will be made in ,part ,,viii4 ,the mechanical action of rain1 which is intimately bound up with its chemical action1 consists in washing off the fine superficial particles of rocks which have been corroded and loosened by the process of weathering1 and in thus laying open fresh portions to the same influences of decay4 ,the detritus so removed is partly carried down into the soil which is thereby enriched1 partly held in suspension in the little runnels into which the rain-drops gather as they begin to flow over the land1 partly pushed downwards along the surface of sloping ground4 ,a good deal of it finds its way into the nearest brooks and rivers1 which are consequently made muddy by heavy rain4 ,it is natural that a casual consideration of the subject should lead to an impression that1 though the general result of the fall of rain upon a land-surface must lead to some amount of disintegration and lowering of that surface1 the process must be so slow and slight as hardly to be considered of much importance among geological operations4 ,but further attention will show such an impression to be singularly erroneous4 ,it loses sight of the fact that a change which may be hardly appreciable within a human lifetime1 or even within the comparatively brief span of geological time embraced in the compass of human history1 may nevertheless become gigantic in its results in the course of immensely protracted periods4 ,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 cottage4 ,the fragments of stone and pebbles of gravel that form part of the soil can there be seen sticking out of the ground1 because being hard they resist the impetus of the falling drops1 protecting for a time the earth beneath them1 while that which surrounded and covered them is washed away4 ,from this familiar illustration the observer may advance through every stage in the disappearance of material which once covered the surface1 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 removed4 ,since the whole land surface over which rain falls is exposed to this waste1 the superficial covering of decayed rock or soil1 as ,hutton insisted1 is constantly1 though imperceptibly1 travelling outward and downward to the sea4 ,in this process of transport rain is an important carrying agent1 while at the same time it serves to connect the work of the other disintegrating forces1 and to make it conducive to the general degradation of the land4 ,though this decay is general and constant1 it is obviously not uniform4 ,in some places where1 from the nature of the rock1 from the flatness of the ground1 or from other causes1 rain works under great difficulties1 the rate of waste may be extremely slow4 ,in other places it may be rapid enough to be appreciable from year to year4 ,a survey of this department of geological activity shows how unequal wasting by rain1 combined with the operations of brooks and rivers1 has produced the details of the present relief of the land1 those tracts where the destruction has been greatest forming hollows and valleys1 others1 where it has been less1 rising into ridges and hills 7,part ,,viii474 ,rain-action is not merely destructive1 but is accompanied with reproductive effects1 chief of which is the formation of soil4 ,in favourable situations it has gathered together accumulations of loam and earth from neighbouring higher ground1 such as the brick-earth1 head1 and rain-wash of the south of ,england--earthy deposits1 sometimes full of angular stones1 derived from the subaerial waste of the rocks of the neighbourhood4 #b4 ,underground ,water4--,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 ocean2 the larger part1 however1 sinks into the ground and disappears4 ,it is this latter part which has now to be considered4 ,over and above the proportion of the rainfall which is absorbed by living vegetation and by the soil1 there is a continual filtering down of the water from the surface into the rocks that lie below1 where it partly lodges in pores and interstices1 and partly finds its way into subterranean joints and fissures1 in which it performs an underground circulation1 and ultimately issues once more at the surface in the form of springs 7q4v474 ,in the course of this circulation the water performs an important geological task4 ,not only carrying down with it the substances which the rain has abstracted from the air1 but obtaining more acids and organic matter from the soil1 it is enabled to effect chemical changes in the rocks underneath1 and especially to dissolve limestone and other calcareous formations4 ,so considerable is the extent of this solution in some places that the springs which come to the surface1 and begin there to evaporate and lose some of their carbonic acid1 contain more dissolved lime than they can hold4 ,they consequently deposit it in the form of calcareous tuff or sinter 7q4v474 ,other subterranean waters issue with a large proportion of iron-salts in solution which form deposits of ochre4 ,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 underground4 ,as the result of prolonged subterranean solution in limestone districts1 passages and caves 7q4v471 sometimes of great width and length1 are formed4 ,when these lie near the surface their roofs sometimes fall in and engulf brooks and rivers1 which then flow for some way underground until the tunnels conduct them back again to daylight on some lower ground4 ,besides its chemical activity water exerts among subterranean rocks a mechanical influence which leads to important changes in the topography of the surface4 ,in removing the mineral matter1 either in solution or as fine sediment1 it sometimes loosens the support of overlying masses of rock which may ultimately give way on sloping ground1 and rush down the declivities in the form of landslips4 ,these destructive effects are specially frequent on the sides of valleys in mountainous countries and on lines of sea-cliff4 #c4 ,brooks and ,rivers4--,as geological agents the running waters on the face of the land play an important part in epigene changes4 ,like rain and springs they have both a chemical and a mechanical action4 ,the latter receives most attention1 as it undoubtedly is the more important2 but the former ought not to be omitted in any survey of the general waste of the earth's surface4 ,the water of rivers must possess the powers of a chemical solvent like rain and springs1 though its actual work in this respect can be less easily measured1 seeing that river water is directly derived from rain and springs1 and necessarily contains in solution mineral substances supplied to it by them and not by its own operation4 ,nevertheless1 it is sometimes easy to prove that streams dissolve chemically the rocks of their channels4 ,thus1 in limestone districts the base of the cliffs of river ravines may be found eaten away into tunnels1 arches1 and overhanging projections1 presenting in their smooth surfaces a great contrast to the angular jointed faces of the same rock1 where now exposed to the influence only of the weather on the higher parts of the cliff4 ,the mechanical action of rivers consists 7a7 in transporting mud1 sand1 gravel and blocks of stone from higher to lower levels2 7b7 in using these loose materials to widen and deepen their channels by erosion2 7c7 in depositing their load of detritus wherever possible and thus to make new geological formations4 7a7 ,transporting ,power4--,river-water is distinguished from that of springs by being less transparent1 because it contains more or less mineral matter in suspension1 derived mainly from what is washed down by rain1 or carried in by brooks1 but partly also from the abrasion of the water-channels by the erosive action of the rivers themselves4 ,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 stream4 ,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 side4 ,traced downwards the blocks are seen to become gradually smaller and more rounded4 ,they are ground against each other1 and upon the rocky sides and bottom of the channel1 getting more and more reduced as they descend1 and at the same time abrading the rocks over or against which they are driven4 ,hence a great deal of debris is produced1 and is swept along by the onward and downward movement of the water4 ,the finer portions1 such as mud and fine sand1 are carried in suspension1 and impart the characteristic turbidity to river-water2 the coarser sand and gravel are driven along the river-bottom4 ,the proportion of suspended mineral matter has been ascertained with more or less precision for a number of rivers4 ,as an illustrative example of a river draining a vast area with different climates1 forms of surface and geological structure the ,mississippi may be cited4 ,the average proportion of sediment in its water was ascertained by ,humphreys and ,abbot to be #a_/aejj by weight or #a_/bijj by volume4 ,these engineers found that1 in addition to this suspended material1 coarse detritus is constantly being pushed forward along the bed of the river into the ,gulf of ,mexico1 to an amount which they estimated at about #gej1jjj1jjj cubic ft4 of sand1 earth and gravel2 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 #bfh ft4 in height4 7b7 ,excavating ,power4--,it is by means of the sand1 gravel and stones which they drive against the sides and bottoms of their channels that streams have hollowed out the beds in which they flow4 ,not only is the coarse detritus reduced in size by the friction of the stones against each other1 but1 at the same time1 these materials abrade the rocks against which they are driven by the current4 ,where1 owing to the shape of the bottom of the channel1 the stones are caught in eddies1 and are kept whirling round there1 they become more and more worn down themselves1 and at the same time scour out basin-shaped cavities1 or pot-holes1 in the solid rock below4 ,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 bed4 ,the steeper the channel1 other conditions being equal1 the more rapid will be the erosion4 ,geological structure also affects the character and rate of the excavation4 ,where the rocks are so arranged as to favour the formation and persistence of a waterfall1 a long chasm may be hollowed out like that of the ,niagara below the falls1 where a hard thick bed of nearly flat limestone lies on softer and more easily eroded shales4 ,the latter are scooped out from underneath the limestone1 which from time to time breaks off in large masses and the waterfall gradually retreats up stream1 while the ravine is proportionately lengthened4 ,to the excavating power of rivers the origin of the valley systems of the dry land must be mainly assigned 7see ,part ,,viii474 7c7 ,reproductive ,power4--,so long as a stream flows over a steep declivity its velocity suffices to keep the sediment in suspension1 but when from any cause1 such as a diminution of slope1 the velocity is checked1 the transporting power is lessened and the sediment begins to fall to the bottom and to remain there4 ,hence various river-formed or alluvial deposits are laid down4 ,these sometimes cover considerable spaces at the foot of mountains4 ,the floors of valleys are strewn with detritus1 and their level may thereby be sensibly raised4 ,in floods the ground inundated on either side of a stream intercepts some part of the detritus1 which is then spread over the flood-plain and gradually heightens it4 ,at the same time the stream continues to erode the channel1 and ultimately is unable to reach the old flood-plain4 ,it consequently forms a new plain at a lower level1 and thus1 by degrees1 it comes to be flanked on either side by a series of successive terraces or platforms1 each of which marks one of its former levels4 ,where a river enters a large body of water its current is checked4 ,some of its sediment is consequently dropped1 and by slow accumulation forms a delta 7q4v474 ,on land1 every lake in mountain districts furnishes instances of this kind of alluvium4 ,but the most important deltas are those formed in the sea at the mouths of the larger rivers of the globe4 ,off many coast-lines the detritus washed from the land gathers into bars1 which enclose long strips of water more or less completely separated from the sea outside and known as lagoons4 ,a chain of such lagoon-barriers stretches for hundreds of miles round the ,gulf of ,mexico and the eastern shores of the ,united ,states4 #d4 ,lakes4--,these sheets of water1 considered as a whole1 do not belong to the normal system of drainage on the land whereby valleys are excavated4 ,on the contrary they are exceptional to it2 for the constant tendency of running water is to fill them up1 or to drain them by wearing down the barriers that contain them at their outflow4 ,some of them are referable to movements of the terrestrial crust whereby depressions arise on the surface of the land1 as has been noted after earthquakes4 ,others have arisen from solution such as that of rock-salt or of limestone1 the removal of which by underground water causes a subsidence of the ground above4 ,a third type of lake-basin occurs in regions that are now or have once been subject to the erosive action of glaciers 7see under next subdivision1 ,terrestrial ,ice74 ,many small lakes or tarns have been caused by the deposit of debris across a valley as by landslips or moraines4 ,considered from a geological point of view1 lakes perform an important function in regulating the drainage of the ground below their outfall and diminishing the destructive effects of floods1 in filtering the water received from their affluent streams1 and in providing undisturbed areas of deposit in which thick and extensive lacustrine formations may be accumulated4 ,in the inland basins of some dry climates the lakes are salt1 owing to excess of evaporation1 and their bottoms become the sites of chemical deposits1 particularly of chlorides of sodium and magnesium1 and calcium sulphate and carbonate4 #e4 ,terrestrial ,ice4--,each of the forms assumed by frozen water has its own characteristic action in geological processes4 ,frost has a powerful influence in breaking up damp soils and surfaces of stone in the pores or cracks of which moisture has lodged4 ,the water in freezing expands1 and in so doing pushes asunder the component particles of soil or stone1 or widens the space between the walls of joints or crevices4 ,when the ice melts the loosened grains remain apart ready to be washed away by rain or blown off by wind1 while by the widening of joints large blocks of rock are detached from the faces of cliffs4 ,where rivers or lakes are frozen over the ice exerts a marked pressure on their banks2 and when it breaks up large sheets of it are driven ashore1 pushing up quantities of gravel and stones above the level of the water4 ,the piling up of the disrupted ice against obstructions in rivers ponds back the water1 and often leads to destructive floods when the ice barriers break4 ,where the ice has formed round boulders in shallow water1 or at the bottom 7anchor-ice71 it may lift these up when the frost gives way1 and may transport them for some distance4 ,ice formed in the atmosphere1 and descending to the ground in the form of hail1 often causes great destruction to vegetation and not infrequently to animal life4 ,where the frozen moisture reaches the earth as snow1 it serves to protect rock1 soil and vegetation from the effects of frost2 but on sloping ground it is apt to give rise to destructive avalanches or landslips1 while indirectly1 by its rapid melting1 it may cause serious floods in rivers4 ,but the most striking geological work performed by terrestrial ice is that achieved by glaciers 7q4v47 and ice-sheets4 ,these vast masses of moving ice1 when they descend from mountains where the steeper rocks are clear of snow1 receive on their surface the debris detached by frost from the declivities above1 and bear these materials to lower levels or to the sea4 ,enormous quantities of rock-rubbish are thus transported in the ,alps and other high mountain ranges4 ,when the ice retreats the boulders carried by it are dropped where it melts1 and left there as memorials of the former extension of the glaciers4 ,evidence of this nature proves the much wider extent of the ,alpine ice at a comparatively recent geological date4 ,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 ,europe4 ,the ice1 by means of grains of sand and pieces of stone which it drags along1 scores1 scratches and polishes the surfaces of rock underneath it1 and1 in this way1 produces the abundant fine sediment that gives the characteristic milky appearance to the rivers that issue from the lower ends of glaciers4 ,by such long-continued attrition the rocks are worn down1 portions of them of softer nature1 or where the ice acts with especial vigour1 are hollowed out into cavities which1 on the disappearance of the ice1 may be filled with water and become tarns or lakes4 ,rocks over which land-ice has passed are marked by a peculiar smooth1 flowing outline1 which forms a contrast to the more rugged surface produced by ordinary weathering4 ,they are covered with groovings1 which range from the finest striae left by sharp grains of sand to deep ruts ground out by blocks of stone4 ,the trend of these markings shows the direction in which the ice flowed4 ,by their evidence the position and movement of former glaciers in countries from which the ice has entirely vanished may be clearly determined 7see ,,glacial ,,period74 #f4 ,the ,sea4--,the physical features of the sea are discussed in separate articles 7see ,,ocean ,,and ,,oceanography74 ,the sea must be regarded as the great regulator of temperature and climate over the globe1 and as thus exerting a profound influence on the distribution of plant and animal life4 ,its distinctly geological work is partly erosive and partly reproductive4 ,as an eroding agent it must to some extent effect chemical decompositions in the rocks and sediments over which it spreads2 but these changes have not yet been satisfactorily studied4 ,undoubtedly1 its chief destructive power is of a mechanical kind1 and arises from the action of its waves in beating upon shore-cliffs4 ,by the alternate compression and expansion of the air in crevices of the rocks on which heavy breakers fall1 and by the hydraulic pressure which these masses of sea-water exert on the walls of the fissures into which they rush1 large masses of rock are loosened and detached1 and caves and tunnels are drilled along the base of sea-cliffs4 ,probably still more efficacious are the blows of the loose shingle1 which1 caught up and hurled forward by the waves1 falls with great force upon the shore rocks1 battering them as with a kind of artillery until they are worn away4 ,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 waste1 while the rounded forms of the boulders and shingle show that they too are being continually reduced in size4 ,thus the sea1 by its action on the coasts1 produces much sediment1 which is swept away by its waves and currents and strewn over its floor4 ,besides this material1 it is constantly receiving the fine silt and sand carried down by rivers4 ,as the floor of the ocean is thus the final receptacle for the waste of the land1 it becomes the chief era on the surface of the globe for the accumulation of new stratified formations4 ,and such has been one of its great functions since the beginning of geological time1 as is proved by the rocks that form the visible part of the earth's crust1 and consist in great part of marine deposits4 ,chemical precipitates take place more especially in enclosed parts of the sea1 where concentration of the water by evaporation can take place1 and where layers of sodium chloride1 calcium sulphate and carbonate1 and other salts are laid down4 ,but the chief marine accumulations are of detrital origin4 ,near the land and for a variable distance extending sometimes to #bjj or #cjj m4 from shore the deposits consist chiefly of sediments derived from the waste of the land1 the finer silts being transported farthest from their source4 ,at greater depths and distances the ocean floor receives a slow deposit of exceedingly fine clay1 which is believed to be derived from the decomposition of pumice and volcanic dust from insular or submarine volcanoes4 ,wide tracts of the bottom are covered with various forms of ooze derived from the accumulation of the remains of minute organisms4 7,c7 ,life4 ,among the agents by which geological changes are carried on upon the surface of the globe living organisms must be enumerated4 ,both plants and animals co-operate with the inorganic agents in promoting the degradation of the land4 ,in some cases1 on the other hand1 they protect rocks from decay1 while1 by the accumulation of their remains1 they give rise to extensive formations both upon the land and in the sea4 ,their operations may hence be described as alike destructive1 conservative and reproductive4 ,under this heading also the influence of ,man as a geological agent deserves notice4 7a7 ,plants4--,vegetation promotes the disintegration of rocks and soil in the following ways3 7#a7 ,by keeping the surfaces of stone moist1 and thus promoting both mechanical and chemical dissolution1 as is especially shown by liverworts1 mosses and other moisture-loving plants4 7#b7 ,by producing through their decay carbonic and other acids1 which1 together with decaying organic matter taken up by passing moisture1 become potent in effecting the chemical decomposition of rocks and in promoting the disintegration of soils4 7#c7 ,by inserting their roots or branches between joints of rock1 which are thereby loosened1 so that large slices may be eventually wedged off4 7#d7 ,by attracting rain1 as thick woods1 forests and peat-mosses do1 and thus accelerating the general waste of a country by running water4 7#e7 ,by promoting the decay of diseased and dead plants and animals1 as when fungi overspread a damp rotting tree or the carcase of a dead animal4 ,that plants also exert a conservative influence on the surface of the land is shown in various ways4 7#a7 ,the formation of a stratum of turf protects the soil and rocks underneath from being rapidly disintegrated and washed away by atmospheric action4 7#b7 ,many plants1 even without forming a layer of turf1 serve by their roots or branches to protect the loose sand or soil on which they grow from being removed by wind4 ,the common sand-carex and other arenaceous plants bind the loose sand-dunes of our coasts1 and give them a permanence1 which would at once be destroyed were the sand laid bare again to storms4 ,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 case1 but serves to arrest the sediment in floods1 filtering the water and thereby adding to the height of the flood plain4 7#c7 ,some marine plants1 like the calcareous nullipores1 afford protection to shore rocks by covering them with a hard incrustation4 ,the tangles and smaller ,fuci which grow abundantly on the littoral zone break the force of the waves or diminish the effects of ground swell4 7#d7 ,forests and brushwood protect the soil1 especially on slopes1 from being washed away by rain or ploughed up by avalanches4 ,plants contribute by the aggregation of their remains to the formation of stratified deposits4 ,some marine algae which secrete carbonate of lime not only encrust rocks but give rise to sheets of submarine limestone4 ,an analogous part is played in fresh-water lakes by various lime-secreting plants1 such as ,chara4 ,long-continued growth of vegetation has1 in some regions1 produced thick accumulations of a dark loam1 as in the black cotton soil 7regur7 of ,india1 and the black earth 7tchernozom7 of ,russia4 ,peat-mosses are formed in temperate and arctic climates by the growth of marsh-loving plants1 sometimes to a thickness of #dj or #ej ft4 ,in tropical regions the mangrove swamps on low moist shores form a dense jungle1 sometimes #bj m4 broad1 which protects these shores from the sea until1 by the arrest of sediment and the constant contribution of decayed vegetation1 the spongy ground is at last turned into firm soil4 ,some plants 7diatoms7 can abstract silica and build it into their framework1 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 extent4 7b7 ,animals4--,these exert a destructive influence in the following ways3 7#a7 ,by seriously affecting the composition and arrangement of the vegetable soil4 ,worms bring up the lower portions of the soil to the surface1 and while thus promoting its fertility increase its liability to be washed away by rain4 ,burrowing animals1 by throwing up the soil and subsoil1 expose these to be dried and blown away by the wind4 ,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 them4 ,in ,britain the mole and rabbit are familiar examples4 7#b7 ,by interfering with or even diverting the flow of streams4 ,thus beaver-dams check the current of water-courses1 intercept floating materials1 and sometimes turn streams into new channels4 ,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 country4 ,similar results have happened in ,europe from subterranean operations of rats4 7#c7 ,some mollusca bore into stone or wood and by the number of contiguous perforations greatly weaken the material4 7#d7 ,many animals exercise a ruinously destructive influence upon vegetation4 ,of the numerous plagues of this kind the locust1 phylloxera and ,colorado beetle may be cited4 ,the most important geological function performed by animals is the formation of new deposits out of their remains4 ,it is chiefly by the lower grades of the animal kingdom that this work is accomplished1 especially by molluscs1 corals and foraminifera4 ,shell-banks are formed abundantly in such comparatively shallow and enclosed basins as that of the ,north ,sea1 and on a much more extensive scale on the floor of the ,west ,indian seas4 ,by the coral polyps thick masses of limestones have been built up in the warmer seas of the globe 7see ,,coral ,,reefs74 ,the floor of the ,atlantic and other oceans is covered with a fine calcareous ooze derived mainly from the remains of foraminifera1 while in other regions the bottom shows a siliceous ooze formed almost entirely of radiolaria4 ,vertebrate animals give rise to phosphatic deposits formed sometimes of their excrement1 as in guano and coprolites1 sometimes of an accumulation of their bones4 7c7 ,man4--,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 geography1 for man has introduced1 as it were1 an element of antagonism to nature4 ,his interference shows itself in his relations to climate1 where he has affected the meteorological conditions of different countries3 7#a7 ,by removing forests1 and laying bare to the sun and winds areas which were previously kept cool and damp under trees1 or which1 lying on the lee side1 were protected from tempests4 ,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 districts4 7#b7 ,by drainage1 whereby the discharged rainfall is rapidly removed1 and the evaporation is lessened1 with a consequent diminution of rainfall and some increase in the general temperature of a country4 7#c7 ,by the other processes of agriculture1 such as the transformation of moor and bog into cultivated land1 and the clothing of bare hillsides with green crops or plantations of coniferous and hardwood trees4 ,still more obvious are the results of human interference with the flow of water3 7#a7 ,by increasing or diminishing the rainfall man directly affects the volume of rivers4 7#b7 ,by his drainage operations he makes the rain to run off more rapidly than before1 and thereby increases the magnitude of floods and of the destruction caused by them4 7#c7 ,by wells1 bores1 mines1 or other subterranean works he interferes with the underground waters1 and consequently with the discharge of springs4 7#d7 ,by embanking rivers he confines them to narrow channels1 sometimes increasing their scour1 and enabling them to carry their sediment further seaward1 sometimes causing them to deposit it over the plains and raise their level4 7#e7 ,by his engineering operations for water-supply he abstracts water from its natural basins and depletes the streams4 ,in many ways man alters the aspect of a country3 7#a7 ,by changing forest into bare mountain1 or clothing bare mountains with forest4 7#b7 ,by promoting the growth or causing the removal of peat-mosses4 7#c7 ,by heedlessly uncovering sand-dunes1 and thereby setting in motion a process of destruction which may convert hundreds of acres of fertile land into waste sand1 or by prudently planting the dunes with sand-loving vegetation and thus arresting their landward progress4 7#d7 ,by so guiding the course of rivers as to make them aid him in reclaiming waste land1 and bringing it under cultivation4 7#e7 ,by piers and bulwarks1 whereby the ravages of the sea are stayed1 or by the thoughtless removal from the beach of stones which the waves had themselves thrown up1 and which would have served for a time to protect the land4 7#f7 ,by forming new deposits either designedly or incidentally4 ,the roads1 bridges1 canals1 railways1 tunnels1 villages and towns with which man has covered the surface of the land will in many cases form a permanent record of his presence4 ,under his hand the whole surface of civilized countries is very slowly covered with a stratum1 either formed wholly by him or due in great measure to his operations and containing many relics of his presence4 ,the soil of ancient towns has been increased to a depth of many feet by their successive destructions and renovations4 ,perhaps the most subtle of human influences are to be seen in the distribution of plant and animal life upon the globe4 ,some of man's doings in this domain are indeed plain enough1 such as the extirpation of wild animals1 the diminution or destruction of some forms of vegetation1 the introduction of plants and animals useful to himself1 and especially the enormous predominance given by him to the cereals and to the spread of sheep and cattle4 ,but no such extensive disturbance of the normal conditions of the distribution of life can take place without carrying with it many secondary effects1 and setting in motion a wide cycle of change and of reaction in the animal and vegetable kingdoms4 ,for example1 the incessant warfare waged by man against birds and beasts of prey in districts given up to the chase leads sometimes to unforeseen results4 ,the weak game is allowed to live1 which would otherwise be killed off and give more room for the healthy remainder4 ,other animals which feed perhaps on the same materials as the game are by the same cause permitted to live unchecked1 and thereby to act as a further hindrance to the spread of the protected species4 ,but the indirect results of man's interference with the regime of plants and animals still require much prolonged observation4 ,,part ,v4--,,geotectonic ,,or ,,structural ,,geology ,from a study of the nature and composition of minerals and rocks1 and an investigation of the different agencies by which they are formed and modified1 the geologist proceeds to inquire how these materials have been put together so as to build up the visible part of the earth's crust4 ,he soon ascertains that they have not been thrown together wholly at random1 but that they show a recognizable order of arrangement4 ,some of them1 especially those of most recent growth1 remain in their original condition and position1 but1 in proportion to their antiquity1 they generally present increasing alteration1 until it may no longer be possible to tell what was their pristine state4 ,as by far the largest accessible portion of the terrestrial crust consists of stratified rocks1 and as these furnish clear evidence of most of the modifications to which they have been subjected in the long course of geological history1 it is convenient to take them into consideration first4 ,they possess a number of structures which belong to the original conditions in which they were accumulated4 ,they present in addition other structures which have been superinduced upon them1 and which they share with the unstratified or igneous rocks4 #a4 ,,original ,,structures 7a7 ,stratified ,rocks4--,this extensive and important series is above all distinguished by possessing a prevailing stratified arrangement4 ,their materials have been laid down in laminae1 layers and strata1 or beds1 pointing generally to the intermittent deposition of the sediments of which they consist4 ,as this stratification was1 as a rule1 originally nearly or quite horizontal1 it serves as a base from which to measure any subsequent disturbance which the rocks have undergone4 ,the occurrence of false-bedding1 i4e4 bands of inclined layers between the normal planes of stratification1 does not form any real exception2 but indicates the action of shifting currents whereby the sediment was transported and thrown down4 ,other important records of the original conditions of deposit are supplied by ripple-marks1 sun-cracks1 rain-prints and concretions4 ,from the nature of the material further light is cast on the geographical conditions in which the strata were accumulated4 ,thus1 conglomerates indicate the proximity of old shore-lines1 sandstones mark deposits in comparatively shallow water1 clays and shales point to the tranquil accumulation of fine silt at a greater depth and further from land1 while fossiliferous limestones bear witness to clearer water in which organisms flourished at some distance from deposits of sand and mud4 ,again1 the alternation of different kinds of sediment suggests a variability in the conditions of deposition1 such as a shifting of the sediment-bearing currents and of the areas of muddy and clear water4 ,a thick group of conformable strata1 that is1 a series of deposits which show no discordance in their stratification1 may usually be regarded as having been laid down on a sea-floor that was gently sinking4 ,here and there evidence is obtainable of the limits or of the progress of the subsidence by what is called overlap4 ,of the absolute length of time represented by any strata or groups of strata no satisfactory estimates can yet be formed4 ,certain general conclusions may indeed be drawn1 and comparisons may be made between different series of rocks4 ,sandstones full of false-bedding were probably accumulated more rapidly than finely-laminated shales or clays4 ,it is not uncommon in certain ,carboniferous formations to find coniferous and other trunks embedded in sandstone4 ,some of these trees seem to have been carried along and to have sunk1 their heavier or root end touching the bottom and their upper end slanting upward in the direction of the current1 exactly as in the case of the snags of the ,mississippi4 ,in other cases the trees have been submerged while still in their positions of growth4 ,the continuous deposit of sand at last rose above the level of the trunks and buried them4 ,it is clear then that the rate of deposit must have been sometimes sufficiently rapid to allow sand to accumulate to a depth of #cj ft4 or more before the decay of the wood4 ,modern instances are known where1 under certain circumstances1 submerged trees may last for some centuries1 but even the most durable must decay in what1 after all1 is a brief space of geological time4 ,since continuous layers of the same kind of deposit suggest a persistence of geological conditions1 while numerous alternations of different kinds of sedimentary matter point to vicissitudes or alternations of conditions1 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 strata1 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 completion4 ,but this conclusion may often be erroneous4 ,it will be best supported when1 from the very nature of the rocks1 wide variations in the character of the water-bottom can be established4 ,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 strata4 ,a thick mass of limestone1 made up of organic remains which lived and died upon the spot1 and whose remains are crowded together generation above generation1 must have demanded many years or centuries for its formation4 ,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 character4 ,the interval between the deposit of two successive laminae of shale may have been as long as1 or even longer than1 that required for the formation of one of the laminae4 ,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 side4 ,but the relative chronological importance of the bars or lines in the geological record can seldom be satisfactorily discussed merely on lithological grounds4 ,this must mainly be decided on the evidence of organic remains1 as shown in ,part ,,vi41 where the grouping of the stratified rocks into formations and systems is described4 7b7 ,igneous ,rocks4--,as part of the earth's crust these rocks present characters by which they are strongly differentiated from the stratified series4 ,while the broad petrographical distinctions of their several varieties remain persistent1 they present sufficient local variations of type to point to the existence of what have been called petrographic provinces1 in each of which the eruptive masses are connected by a general family relationship1 differing more or less from that of a neighbouring province4 ,in each region presenting a long chronological series of eruptive rocks a petrographical sequence can be traced1 which is observed to be not absolutely the same everywhere1 though its general features may be persistent4 ,the earliest manifestations of eruptive material in any district appear to have been most frequently of an intermediate type between acid and basic1 passing thence into a thoroughly acid series and concluding with an effusion of basic material4 ,considered as part of the architecture of the crust of the earth1 igneous rocks are conveniently divisible into two great series3 7#a7 those bodies of material which have been injected into the crust and have solidified there1 and 7#b7 those which have reached the surface and have been ejected there1 either in a molten state as lava or in a fragmental form as dust1 ashes and scoriae4 ,the first of these divisions represents the plutonic1 intrusive or subsequent phase of eruptivity2 the second marks the volcanic1 interstratified or contemporaneous phase4 #a4 ,the plutonic or intrusive rocks1 which have been forced into the crust and have consolidated there1 present a wide range of texture from the most coarse-grained granites to the most perfect natural glass4 ,seeing that they have usually cooled with extreme slowness underground1 they are as a general rule more largely crystalline than the volcanic series4 ,the form assumed by each individual body of intrusive material has depended upon the shape of the space into which it has been injected1 and where it has cooled and become solid4 ,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 other4 ,it offers a convenient basis for the classification of the intrusive rocks1 which1 as part of the framework of the crust1 may thus be grouped according to the shape of the cavity which received them1 as bosses1 sills1 dikes and necks4 ,bosses1 or stocks1 are the largest and most shapeless extravasations of erupted material4 ,they include the great bodies of granite which1 in most countries of the world1 have risen for many miles through the stratified formations and have altered the rocks around them by contact-metamorphism4 ,sills1 or intrusive sheets1 are bed-like masses which have been thrust between the planes of sedimentary or even of igneous rocks4 ,the term laccolite has been applied to sills which are connected with bosses4 ,intrusive sheets are distinguishable from true contemporaneously intercalated lavas by not keeping always to the same platform1 but breaking across and altering the contiguous strata1 and by the closeness of their texture where they come in contact with the contiguous rocks1 which1 being cold1 chilled the molten material and caused it to consolidate on its outer margins more rapidly than in its interior4 ,dikes or veins are vertical walls or ramifying branches of intrusive material which has consolidated in fissures or irregular clefts of the crust4 ,necks are volcanic chimneys which have been filled up with erupted material1 and have now been exposed at the surface after prolonged denudation has removed not only the superficial volcanic masses originally associated with them1 but also more or less of the upper part of the vents4 ,plutonic rocks do not present evidence of their precise geological age4 ,all that can be certainly affirmed from them is that they must be younger than the rocks into which they have been intruded4 ,from their internal structure1 however1 and from the evidence of the rocks associated with them1 some more or less definite conjectures may be made as to the limits of time within which they were probably injected4 #b4 ,the interstratified or volcanic series is of special importance in geology1 inasmuch as it contains the records of volcanic action during the past history of the globe4 ,it was pointed out in ,part ,i4 that while towards the end of the #ahth and in the beginning of the #aith 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 crust1 these observers lost sight of the possibility that some of these rocks might have been erupted at the surface1 and might thus be chronicles of volcanic action in former geological periods4 ,it is not always possible to satisfactorily discriminate between the two types of contemporaneously intercalated and subsequently injected material4 ,but rocks of the former type have not broken into or involved the overlying strata1 and they are usually marked by the characteristic structures of superficial lavas and by their association with volcanic tuffs4 ,by means of the evidence which they supply1 it has been ascertained that volcanic action has been manifested in the globe since the earliest geological periods4 ,in the ,british ,isles1 for example1 the volcanic record is remarkably full for the long series of ages from ,cambrian to ,permian time1 and again for the older ,tertiary period4 #b4 ,,subsequently ,,induced ,,structures ,after their accumulation1 whether as stratified or eruptive masses1 all kinds of rocks have been subject to various changes1 and have acquired in consequence a variety of superinduced structures4 ,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 earth4 ,some of these movements are so slight as to be only recognizable by means of delicate instruments2 but from this inferior limit they range up to gigantic convulsions by which mountain-chains are upheaved4 ,the crust must be regarded as in a perpetual state of strain1 and its component materials are therefore subject to all the effects which flow from that condition4 ,it is the one great object of the geotectonic division of geology to study the structures which have been developed in consequence of earth-movements1 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 them4 ,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 structures4 ,all that need be offered here is a general outline connecting the several portions of the subject together4 ,one of the most universal of these later structures is to be seen in the divisional planes1 usually vertical or highly inclined1 by which rocks are split into quadrangular or irregularly shaped blocks4 ,to these planes the name of joints has been given4 ,they are of prime importance from an industrial point of view1 seeing that the art of quarrying consists mainly in detecting and making proper use of them4 ,their abundance in all kinds of rocks1 from those of recent date up to those of the highest antiquity1 affords a remarkable testimony to the strains which the terrestrial crust has suffered4 ,they have arisen sometimes from tension1 such as that caused by contraction from the drying and consolidation of an aqueous sediment or from the cooling of a molten mass2 sometimes from torsion during movements of the crust4 ,although the stratified rocks were originally deposited in a more or less nearly horizontal position on the floor of the sea1 where now visible on the dry land they are seldom found to have retained their flatness4 ,on the contrary1 they are seen to have been generally tilted up at various angles1 sometimes even placed on end 7crop1 dip1 strike74 ,when a sufficiently large area of ground is examined1 the inclination into which the strata have been thrown may be observed not to continue far in the same direction1 but to turn over to the opposite or another quarter4 ,it can then be seen that in reality the rocks have been thrown into undulations4 ,from the lowest and flattest arches where the departure from horizontality may be only trifling1 every step may be followed up to intense curvature1 where the strata have been compressed and plicated as if they had been piles of soft carpets 7anticline1 syncline1 monocline1 geo-anticline1 geo-syncline1 isoclinal1 plication1 curvature1 quaquaversal74 ,it has further happened abundantly all over the surface of the globe that relief from internal strain in the crust has been obtained by fracture1 and the consequent subsidence or elevation of one or both sides of the fissure4 ,the differential movement between the two sides may be scarcely perceptible in the feeblest dislocation1 but in the extreme cases it may amount to many thousand feet 7fault1 fissure1 dislocation1 hade1 slickensides74 ,the great faults in a country are among its most important structural features1 and as they not infrequently continue to be lines of weakness in the crust along which sudden slipping may from time to time take place1 they become the lines of origin of earthquakes4 ,the ,san ,francisco earthquake of #aijf1 already cited1 affords a memorable illustration of this connexion4 ,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 beheld4 ,the combination of overturned folds with rupture has been already referred to as a characteristic feature in the ,alps 7,part ,,iv474 ,the gigantic folds have in many places been pushed over each other so as to lie almost flat1 while the upper limb has not infrequently been driven for many miles beyond the lower by a rupture along the axis4 ,in this way successive slices of a thick series of formations have been carried northwards on the northern slope of the ,alps1 and have been piled so abnormally above each other that some of their oldest members recur several times on different thrust-planes1 the whole being underlain by ,tertiary strata 7see ,,alps74 ,further proof of the colossal compression to which the rocks have been subjected is afforded by their intense crumpling and corrugation1 and by the abundantly faulted and crushed condition to which they have been reduced4 ,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 countries4 ,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 7cleavage74 ,closely connected with this internal rearrangement has been the development of microscopic microlites or crystals 7rutile1 mica1 c47 in argillaceous slates which were undoubtedly originally fine marine mud and silt4 ,from this incipient form of metamorphism successive stages may be traced through the various kinds of argillite and phyllite into mica-schist1 and thence into more crystalline gneissoid varieties 7foliation1 slate1 mica-schist1 gneiss74 ,the ,alps afford excellent illustrations of these transformations4 ,the fissures produced in the crust are sometimes clean1 sharply defined divisional planes1 like cracks across a pane of glass4 ,much more usually1 however1 the rocks on either side have been broken up by the friction of movement1 and the fault is marked by a variable breadth of this broken material4 ,sometimes the walls have separated and molten rock has risen from below and solidified between them as a dike4 ,occasionally the fissures have opened to the surface1 and have been filled in from above with detritus1 as in the sandstone-dikes of ,colorado and ,california4 ,in mineral districts the fissures have been filled with various spars and ores1 forming what are known as mineral veins4 ,where one series of rocks is covered by another without any break or discordance in the stratification they are said to be conformable4 ,but where the older series has been tilted up or visibly denuded before being overlain by the younger1 the latter is termed unconformable4 ,this relation is one of the greatest value in structural geology1 for it marks a gap in the geological record1 which may represent a vast lapse of time not there recorded by strata4 ,,part ,,vi4--,,paleontological ,,geology ,this division of the science deals with fossils1 or the traces of plants and animals preserved in the rocks of the earth's crust1 and endeavours to gather from them information as to the history of the globe and its inhabitants4 ,the term fossil 7,lat4 fossilis1 from fodere1 to dig up71 meaning literally anything dug up1 was formerly applied indiscriminately to any mineral substance taken out of the earth's crust1 whether organized or not4 ,since the time of ,lamarck1 however1 the meaning of the word has been restricted1 so as to include only the remains or traces of plants and animals preserved in any natural formation whether hard rock or superficial deposit4 ,it includes not merely the petrified structures of organisms1 but whatever was directly connected with or produced by these organisms4 ,thus the resin which was exuded from trees of long-perished forests is as much a fossil as any portion of the stem1 leaves1 flowers or fruit1 and in some respects is even more valuable to the geologist than more determinable remains of its parent trees1 because it has often preserved in admirable perfection the insects which flitted about in the woodlands4 ,the burrows and trails of a worm preserved in sandstone and shale claim recognition as fossils1 and indeed are commonly the only indications to be met with of the existence of annelid life among old geological formations4 ,the droppings of fishes and reptiles1 called coprolites1 are excellent fossils1 and tell their tale as to the presence and food of vertebrate life in ancient waters4 ,the little agglutinated cases of the caddis-worm remain as fossils in formations from which1 perchance1 most other traces of life may have passed away4 ,nay1 the very handiwork of man1 when preserved in any natural manner1 is entitled to rank among fossils2 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-bottoms4 ,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 if1 as is probable1 the same conditions existed in former geological periods1 we should regard the occurrence of organic remains among the stratified formations of the earth's crust as generally the result of various fortunate accidents4 ,let us consider1 in the first place1 the chances for the preservation of remains of the present fauna and flora of a country4 ,the surface of the land may be densely clothed with forest and abundantly peopled with animal life4 ,but the trees die and moulder into soil4 ,the animals1 too1 disappear1 generation after generation1 and leave few or no perceptible traces of their existence4 ,if we were not aware from authentic records that central and northern ,europe were covered with vast forests at the beginning of our era1 how could we know this fact8 ,what has become of the herds of wild oxen1 the bears1 wolves and other denizens of primeval ,europe8 ,how could we prove from the examination of the surface soil of any country that those creatures had once abounded there8 ,the conditions for the preservation of any relics of the plant and animal life of a terrestrial surface must obviously be always exceptional4 ,they are supplied only where the organic remains can be protected from the air and superficial decay4 ,hence they may be observed in 7#a7 the deposits on the floors of lakes2 7#b7 in peat-mosses2 7#c7 in deltas at river-mouths2 and 7#d7 under the stalagmite of caverns in limestone districts4 ,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 preserved4 ,in the second place1 although in the sea the conditions for the preservation of organic remains are in many respects more favourable than on land1 they are apt to be frustrated by many adverse circumstances4 ,while the level of the land remains stationary1 there can be but little effective entombment of marine organisms in littoral deposits2 for only a limited accumulation of sediment will be formed until subsidence of the sea-floor takes place4 ,in the trifling beds of sand or gravel thrown up on a stationary shore1 only the harder and more durable forms of life1 such as gastropods and lamellibranchs1 which can withstand the triturating effects of the beach waves1 are likely to remain uneffaced4 ,below tide-marks1 along the margin of the land where sediment is gradually deposited1 the conditions are more favourable for the preservation of marine organisms4 ,in the sheets of sand and mud there laid down the harder parts of many forms of life may be entombed and protected from decay4 ,but only a small proportion of the total marine fauna may be expected to appear in such deposits4 ,at the best1 merely littoral and shallow-water forms will occur1 and1 even under the most favourable conditions1 they will represent but a fraction of the whole assemblage of life in these juxta-terrestrial parts of the ocean4 ,as we recede from the land the rate of deposition of sediment on the sea-floor must become feebler1 until1 in the remote central abysses1 it reaches a hardly appreciable minimum4 ,except1 therefore1 where some kind of ooze or other deposit is accumulating in these more pelagic regions1 the conditions must be on the whole unfavourable for the preservation of any adequate representation of the deep-sea fauna4 ,hard durable objects1 such as teeth and bones1 may slowly accumulate1 and be protected by a coating of peroxide of manganese1 or of some of the silicates now forming here and there over the deep-sea bottom2 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 disappear1 owing to decay1 before they can be covered up and preserved4 ,any such deep-sea formation1 if raised into land1 would supply but a meagre picture of the whole life of the sea4 ,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 land1 to which rivers and currents bring continual supplies of sediment4 ,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 subsidence4 ,if the rate of depression and that of deposit were equal1 or nearly so1 the movement might proceed for a vast period without producing any great apparent change in marine geography1 and even without seriously affecting the distribution of life over the sea-floor within the area of subsidence4 ,hundreds or thousands of feet of sedimentary strata might in this way be heaped up round the continents1 containing a fragmentary series of organic remains belonging to those forms of comparatively shallow-water life which had hard parts capable of preservation4 ,there can be little doubt that such has1 in fact1 been the history of the main mass of stratified formations in the earth's crust4 ,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 sediment4 ,the enormous thickness to which they attain seems only explicable by prolonged and repeated movements of subsidence1 interrupted1 however1 as we know1 by other movements of a contrary kind4 ,since the conditions for the preservation of organic remains exist more favourably under the sea than on land1 marine organisms must be far more abundantly conserved than those of the land4 ,this is true to-day1 and has1 as far as known1 been true in all past geological time4 ,hence for the purposes of the geologist the fossil remains of marine forms of life far surpass all others in value4 ,among them there will necessarily be a gradation of importance1 regulated chiefly by their relative abundance4 ,now1 of all the marine tribes which live within the juxta-terrestrial belt of sedimentation1 unquestionably the ,mollusca stand in the place of pre-eminence as regards their aptitude for becoming fossils4 ,they almost all possess a hard1 durable shell1 capable of resisting considerable abrasion and readily passing into a mineralized condition4 ,they are extremely abundant both as to individuals and genera4 ,they occur on the shore within tide mark1 and range thence down into the abysses4 ,moreover1 they appear to have possessed these qualifications from early geological times4 ,in the marine ,mollusca1 therefore1 we have a common ground of comparison between the stratified formations of different periods4 ,they have been styled the alphabet of palaeontological inquiry4 ,there are two main purposes to which fossils may be put in geological research3 7#a7 to throw light upon former conditions of physical geography1 such as the presence of land1 rivers1 lakes and seas1 in places where they do not now exist1 changes of climate1 and the former distribution of plants and animals2 and 7#b7 to furnish a guide in geological chronology whereby rocks may be classified according to relative date1 and the facts of geological history may be arranged and interpreted as a connected record of the earth's progress4 #a4 ,as examples of the first of these two directions of inquiry reference may be made to 7a7 former land-surfaces revealed by the occurrence of layers of soil with tree-stumps and roots still in the position of growth 7see ,,purbeckian72 7b7 ancient lakes proved by beds of marl or limestone full of lacustrine shells2 7c7 old sea-bottoms marked by the occurrence of marine organisms2 7d7 variations in the quality of the water1 such as freshness or saltness1 indicated by changes in the size and shape of the fossils2 7e7 proximity to former land1 suggested by the occurrence of abundant drift-wood in the strata2 7f7 former conditions of climate1 different from the present1 as evidenced by such organisms as tropical types of plants and animals intercalated among the strata of temperate or northern countries4 #b4 ,in applying fossils to the determination of geological chronology it is first necessary to ascertain the order of superposition of the rocks4 ,obviously1 in a continuous series of undisturbed sedimentary deposits the lowest must necessarily be the oldest1 and the plants or animals which they contain must have lived and died before any of the organisms that occur in the overlying strata4 ,this order of superposition having been settled in a series of formations1 it is found that the fossils at the bottom are not quite the same as those at the top of the series4 ,tracing the beds upward1 we discover that species after species of the lowest platforms disappears1 until perhaps not one of them is found4 ,with the cessation of these older species others make their entrance4 ,these1 in turn1 are found to die out1 and to be replaced by newer forms4 ,after patient examination of the rocks1 it has been ascertained that every well-marked formation1 or group of strata1 is characterized by its own species or genera1 or by a general assemblage1 or facies1 of organic forms4 ,such a generalization can only1 of course1 be determined by actual practical experience over an area of some size4 ,when the typical fossils of a formation are known1 they serve to identify that formation in its progress across a country4 ,thus1 in tracts where the true order of superposition cannot be determined1 owing to the want of sections or to the disturbed condition of the rocks1 fossils serve as a means of identification and furnish a guide to the succession of the rocks4 ,they even demonstrate that in some mountainous ground the beds have been turned completely upside down1 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 them4 ,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 order4 ,each formation is distinguished by its own peculiar assemblage of organic remains1 by means of which it can be followed and recognized1 even amid the crumplings and dislocations of a disturbed region4 ,the same general succession of organic types can be observed over a large part of the world1 though1 of course1 with important modifications in different countries4 ,this similarity of succession has been termed homotaxis1 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 regions4 ,it is evident that1 in this way1 a reliable method of comparison is furnished1 whereby the stratified formations of different parts of the earth's crust can be brought into relation with each other4 ,had the geologist continued to remain1 as in the days of ,werner1 hampered by the limitations imposed by a reliance on mere lithological characters1 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 crust4 ,just as1 at the present time1 sheets of gravel in one place are contemporaneous with sheets of mud at another1 so in the past all kinds of sedimentation have been in progress simultaneously1 and those of one period may not be distinguishable in themselves from those of another4 ,little or no reliance can be placed upon lithological resemblances or differences in comparing the sedimentary formations of different countries4 ,in making use of fossil evidence for the purpose of subdividing the stratified rocks of the earth's crust1 it is found to be applicable to the smaller details of stratigraphy as well as to the definition of large groups of strata4 ,thus a particular stratum may be marked by the occurrence in it of various fossils1 one or more of which may be distinctive1 either from occurring in no other bed above and below or from special abundance in that stratum4 ,one or more of these species is therefore used as a guide to the occurrence of the bed in question1 which is called by the name of the most abundant species4 ,in this way what is called a geological horizon1 or zone1 is marked off1 and its exact position in the series of formations is fixed4 ,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 stratigraphy1 which1 in itself1 was only a further development of ,william ,smith's famous idea1 ,strata identified by ,organized ,fossils4 ,it was first carried out in detail by various palaeontologists in reference to the ,jurassic formations1 notably by ,f4,a4 von ,quenstedt and ,c4,a4 ,oppel in ,germany and ,a4,d4 d',orbigny in ,france4 ,the publication of ,oppel's classic work ,die ,juraformation ,englands1 ,frankreichs und des sudwestlichen ,deutschlands 7#ahef-aheh7 marked an epoch in the development of stratigraphical geology4 ,combining what had been done by various observers with his own laborious researches in ,france1 ,england1 ,wurttemberg and ,bavaria1 he drew up a classification of the ,jurassic system1 grouping its several formations into zones1 each characterized by some distinctly predominant fossil after which it was named 7see ,,lias74 ,the same method of classification was afterwards extended to the ,cretaceous series by ,a4,d4 d',orbigny1 ,e4 ,hebert and others1 until the whole ,mesozoic rocks from the ,trias to the top of the ,chalk has now been partitioned into zones1 each named after some characteristic species or genus of fossils4 ,more recently the principle has been extended to the ,palaeozoic formations1 though as yet less fully than to the younger parts of the geological record4 ,it has been successfully applied by ,professor ,c4 ,lapworth to the investigation of the ,silurian series 7see ,,silurian2 ,,ordovician ,,system74 ,he found that the species of graptolites have each a comparatively narrow vertical range1 and they may consequently be used for stratigraphical purposes4 ,applying the method1 in the first instance1 to the highly plicated ,silurian rocks of the south of ,scotland1 he found that by means of graptolites he was able to work out the structure of the ground4 ,each great group of strata was seen to possess its own graptolitic zones1 and by their means could be identified not only in the original complex ,scottish area1 but in ,england and ,wales and in ,ireland4 ,it was eventually ascertained that the succession of zones in ,great ,britain could be recognized on the ,continent1 in ,north ,america and even in ,australia4 ,the brachiopods and trilobites have likewise been made use of for zonal purposes among the oldest sedimentary formations4 ,the most ancient of the ,palaeozoic systems has as its fitting base the ,olenellus zone4 ,within undefined and no doubt variable geographical limits palaeontological zones have been found to be remarkably persistent4 ,they follow each other in the same general order1 but not always with equal definiteness4 ,the type fossil may appear in some districts on a higher or a lower platform than it does in others4 ,only to a limited degree is there any coincidence between lithological variations in the strata and the sequence of the zones4 ,in the ,jurassic formations1 indeed1 where frequent alternations of different sedimentary materials are to be met with1 it is in some cases possible to trace a definite upward or downward limit for a zone by some abrupt change in the sedimentation1 such as from limestone to shale4 ,but such a precise demarcation is impossible where no distinct bands of different sediments are to be seen4 ,the zones can then only be vaguely determined by finding their characteristic fossils1 and noting where these begin to appear in the strata and where they cease4 ,it would seem1 therefore1 that the sequence of palaeontological zones1 or life-horizons1 has not depended merely upon changes in the nature of the conditions under which the organisms lived4 ,we should naturally expect that these changes would have had a marked influence2 that1 for instance1 a difference should be perceptible between the character of the fossils in a limestone and that of those in a shale or a sandstone4 ,the environment1 when a limestone was in course of deposition1 would generally be one of clear water1 favourable for a more vigorous and more varied fauna than where a shale series was accumulating1 when the water would be discoloured1 and only such animals would continue to live in it1 or on the bottom1 as could maintain themselves in the midst of mud4 ,but no such lithological reason1 betokening geographical changes that would affect living creatures1 can be adduced as a universally applicable explanation of the occurrence and limitation of palaeontological zones4 ,one of these zones may be only a few inches1 or feet or yards in vertical extent1 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 below4 ,there is often little or no evidence of any serious change in the conditions of sedimentation1 still less of any widespread physical disturbance1 such as the catastrophes by which the older geologists explained the extinction of successive types of life4 ,it has been suggested that1 where the life-zones are well defined1 sedimentation has been extremely slow1 and that though these zones follow each other with no break in the sedimentation1 they were really separated by prolonged intervals of time during which organic evolution could come effectively into play4 ,but it is not easy to explain how1 for example in the ,lower ,lias1 there could have been a succession of prodigious intervals1 when practically no sediment was laid down1 and yet that the strata should show no sign of contemporaneous disturbance or denudation1 but succeed each other as if they had been accumulated by one continuous process of deposit4 ,it must be admitted that the problem of life-zones in stratigraphical geology has not yet been solved4 ,as ,darwin first cogently showed1 the history of life has been very imperfectly registered in the stratified parts of the earth's crust4 ,apart from the fact that1 even under the most favourable conditions1 only a small proportion of the total flora and fauna of any period would be preserved in the fossil state1 enormous gaps occur where no record has survived at all4 ,it is as if whole chapters and books were missing from a historical work4 ,some of these lacunae are sufficiently obvious4 ,thus1 in some cases1 powerful dislocations have thrown considerable portions of the rocks out of sight4 ,sometimes extensive metamorphism has so affected them that their original characters1 including their organic contents1 have been destroyed4 ,oftenest of all1 denudation has come into play1 and vast masses of fossiliferous rock have been entirely worn away1 as is demonstrated by the abundant unconformabilities in the structure of the earth's crust4 ,while the mere fact that one series of rocks lies unconformably on another proves the lapse of a considerable interval between their respective dates1 the relative length of this interval may sometimes be proved by means of fossil evidence1 and by this alone4 ,let us suppose1 for example1 that a certain group of formations has been disturbed1 upraised1 denuded and covered unconformably by a second group4 ,in lithological characters the two may closely resemble each other1 and there may be nothing to show that the gap represented by their unconformability is of an important character4 ,in many cases1 indeed1 it would be quite impossible to pronounce any well-grounded judgment as to the amount of interval1 even measured by the vague relative standards of geological chronology4 ,but if each group contains a well-preserved suite of organic remains1 it may not only be possible1 but easy1 to say exactly how much of the geological record has been left out between the two sets of formations4 ,by comparing the fossils with those obtained from regions where the geological record is more complete1 it may be ascertained1 perhaps1 that the lower rocks belong to a certain platform or stage in geological history which for our present purpose we may call ,d1 and that the upper rocks can in like manner be paralleled with stage ,h4 ,it would be then apparent that at this locality the chronicles of three great geological periods ,e1 ,f1 and ,g were wanting1 which are elsewhere found to be intercalated between ,d and ,h4 ,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 continuous4 ,fossil evidence may be made to prove the existence of gaps which are not otherwise apparent4 ,as has been already remarked1 changes in organic forms must1 on the whole1 have been extremely slow in the geological past4 ,the whole species of a sea-floor could not pass entirely away1 and be replaced by other forms1 without the lapse of long periods of time4 ,if then among the conformable stratified formations of former ages we encounter sudden and abrupt changes in the facies of the fossils1 we may be certain that these must mark omissions in the record1 which we may hope to fill in from a more perfect series elsewhere4 ,the complete biological contrasts between the fossil contents of unconformable strata are sufficiently explicable4 ,it is not so easy to give a satisfactory account of those which occur where the beds are strictly conformable1 and where no evidence can be observed of any considerable change of physical conditions at the time of deposit4 ,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 line4 ,a few species may pass from the one into the other1 or perhaps every species may be different4 ,in cases of this kind1 when proved to be not merely local but persistent over wide areas1 we must admit1 notwithstanding the apparently undisturbed and continuous character of the original deposition of the strata1 that the abrupt transition from the one facies of fossils to the other represents a long interval of time which has not been recorded by the deposit of strata4 ,a4,c4 ,ramsay1 who called attention to these gaps1 termed them breaks in the succession of organic remains4 ,he showed that they occur abundantly among the ,palaeozoic and ,secondary rocks of ,england4 ,it is obvious1 of course1 that such breaks1 even though traceable over wide regions1 were not general over the whole globe4 ,there have never been any universal interruptions in the continuity of the chain of being1 so far as geological evidence can show4 ,but the physical changes which caused the breaks may have been general over a zoological district or minor region4 ,they no doubt often caused the complete extinction of genera and species which had a small geographical range4 ,from all these facts it is clear that the geological record1 as it now exists1 is at the best but an imperfect chronicle of geological history4 ,in no country is it complete4 ,the lacunae of one region must be supplied from another4 ,yet in proportion to the geographical distance between the localities where the gaps occur and those whence the missing intervals are supplied1 the element of uncertainty in our reading of the record is increased4 ,the most desirable method of research is to exhaust the evidence for each area or province1 and to compare the general order of its succession as a whole with that which can be established for other provinces4 ,,part ,,vii4--,,stratigraphical ,,geology ,this branch of the science arranges the rocks of the earth's crust in the order of their appearance1 and interprets the sequence of events of which they form the records4 ,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 planet1 and of each continent and country1 from the earliest times of which the rocks have preserved any memorial4 ,thus from mineralogy and petrography it contains information regarding the origin and subsequent mutations of minerals and rocks4 ,from dynamical geology it learns by what agencies the materials of the earth's crust have been formed1 altered1 broken1 upheaved and melted4 ,from geotectonic geology it understands the various processes whereby these materials were put together so as to build up the complicated crust of the earth4 ,from palaeontological geology it receives in well-determined fossil remains a clue by which to discriminate the different stratified formations1 and to trace the grand onward march of organized existence upon this planet4 ,stratigraphical geology thus gathers up the sum of all that is made known by the other departments of the science1 and makes it subservient to the interpretation of the geological history of the earth4 ,the leading principles of stratigraphy may be summed up as follows3 #a4 ,in every stratigraphical research the fundamental requisite is to establish the order of superposition of the strata4 ,until this is accomplished it is impossible to arrange the dates1 and make out the sequence of geological history4 #b4 ,the stratified portion of the earth's crust1 or what has been called the geological record1 can be subdivided into natural groups1 or series of strata1 characterized by distinctive organic remains and recognizable by these remains1 in spite of great changes in lithological character from place to place4 ,a bed1 or a number of beds1 linked together by containing one or more distinctive species or genera of fossils is termed a zone or horizon1 and usually bears the name of one of its more characteristic fossils1 as the ,planorbis-zone of the ,lower ,lias1 which is so called from the prevalence in it of the ammonite ,psiloceras planorbis4 ,two or more such zones related to each other by the possession of a number of the same characteristic species or genera have been designated beds or an assise4 ,two or more sets of beds or assises similarly related form a group or stage2 a number of groups or stages make a series1 formation or section1 and a succession of formations may be united into a system4 #c4 ,some living species of plants and animals can be traced downwards through the more recent geological formations2 but the number which can be so followed grows smaller as the examination is pursued into more ancient deposits4 ,with their disappearance other species or genera present themselves which are no longer living4 ,these in turn may be traced backward into earlier formations1 till they too cease and their places are taken by yet older forms4 ,it is thus shown that the stratified rocks contain the records of a gradual progression of organic forms4 ,a species which has once died out does not seem ever to have reappeared4 #d4 ,when the order of succession of organic remains among the stratified rocks has been determined1 they become an invaluable guide in the investigation of the relative age of rocks and the structure of the land4 ,each zone and formation1 being characterized by its own species or genera1 may be recognized by their means1 and the true succession of strata may thus be confidently established even in a country wherein the rocks have been shattered by dislocation1 folded1 inverted or metamorphosed4 #e4 ,though local differences exist in regard to the precise zone in which a given species of organism may make its first appearance1 the general order of succession of the organic forms found in the rocks is never inverted4 ,the record is nowhere complete in any region1 but the portions represented1 even though extremely imperfect1 always follow each other in their proper chronological order1 unless where disturbance of the crust has intervened to destroy the original sequence4 #f4 ,the relative chronological value of the divisions of the geological record is not to be measured by mere depth of strata4 ,while it may be reasonably assumed that1 in general1 a great thickness of stratified rock must mark the passage of a long period of time1 it cannot safely be affirmed that a much less thickness elsewhere must represent a correspondingly diminished period4 ,the need for this caution may sometimes be made evident by an unconformability between two sets of rocks1 as has already been explained4 ,the total depth of both groups together may be1 say #ajjj ft4 ,elsewhere we may find a single unbroken formation reaching a depth of #aj1jjj ft42 but it would be unwarrantable to assume that the latter represents ten times the length of time indicated by the former two4 ,so far from this being the case1 it might not be difficult to show that the minor thickness of rock really denotes by far the longer geological interval4 ,if1 for instance1 it could be proved that the upper part of both the sections lies on one and the same geological platform1 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 other1 then it would be clear that the gap marked by the unconformability really indicates a longer period than the massive succession of deposits4 #g4 ,fossil evidence furnishes the chief means of comparing the relative value of formations and groups of rock4 ,a break in the succession of organic remains1 as already explained1 marks an interval of time often unrepresented by strata at the place where the break is found4 ,the relative importance of these breaks1 and therefore1 probably1 the comparative intervals of time which they mark1 may be estimated by the difference of the facies or general character of the fossils on each side4 ,if1 for example1 in one case we find every species to be dissimilar above and below a certain horizon1 while in another locality only half of the species on each side are peculiar1 we naturally infer1 if the total number of species seems large enough to warrant the inference1 that the interval marked by the former break was much longer than that marked by the second4 ,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 them4 ,three formations of fossiliferous strata1 ,a1 ,c1 and ,h1 may occur conformably above each other4 ,by a comparison of the fossil contents of all parts of ,a1 it may be ascertained that1 while some species are peculiar to its lower1 others to its higher portions1 yet the majority extend throughout the formation4 ,if now it is found that of the total number of species in the upper portion of ,a only one-third passes up into ,c1 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 ,a4 ,it might even be possible to discover elsewhere a thick intermediate formation ,b filling up the gap between ,a and ,c4 ,in like manner were it to be discovered that1 while the whole of the formation ,c is characterized by a common suite of fossils1 not one of the species and only one half of the genera pass up into ,h1 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 ,c4 ,and thus we reach the remarkable conclusion that1 thick though the stratified formations of a country may be1 in some cases they may not represent so long a total period of time as do the gaps in their succession1--in other words1 that non-deposition was more frequent and prolonged than deposition1 or that the intervals of time which have been recorded by strata have not been so long as those which have not been so recorded4 ,in all speculations of this nature1 however1 it is necessary to reason from as wide a basis of observation as possible1 seeing that so much of the evidence is negative4 ,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 that1 onward from the time marked by that line1 these species1 owing to some change in the conditions of life1 were compelled to migrate or became locally extinct or1 from some alteration in the conditions of fossilization1 were no longer imbedded and preserved as fossils4 ,they may have continued to flourish abundantly in neighbouring districts for a long period afterward4 ,many examples of this obvious truth might be cited4 ,thus in a great succession of mingled marine1 brackish-water and terrestrial strata1 like that of the ,carboniferous ,limestone series of ,scotland1 corals1 crinoids and brachiopods abound in the limestones and accompanying shales1 but disappear as the sandstones1 ironstones1 clays1 coals and bituminous shales supervene4 ,an observer meeting for the first time with an instance of this disappearance1 and remembering what he had read about breaks in succession1 might be tempted to speculate about the extinction of these organisms1 and their replacement by other and later forms of life1 such as the ferns1 lycopods1 estuarine or fresh-water shells1 ganoid fishes and other fossils so abundant in the overlying strata4 ,but further research would show him that high above the plant-bearing sandstones and coals other limestones and shales might be observed1 once more charged with the same marine fossils as before1 and still farther overlying groups of sandstones1 coals and carbonaceous beds followed by yet higher marine limestones4 ,he would thus learn that the same organisms1 after being locally exterminated1 returned again and again to the same area4 ,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 life4 ,some breaks in the succession may thus be extremely local1 one set of organisms having been driven to a different part of the same region1 while another set occupied their place until the first was enabled to return4 #h4 ,the geological record is at the best but an imperfect chronicle of the geological history of the earth4 ,it abounds in gaps1 some of which have been caused by the destruction of strata owing to metamorphism1 denudation or otherwise1 others by original non-deposition1 as above explained4 ,nevertheless from this record alone can the progress of the earth be traced4 ,it contains the registers of the appearance and disappearance of tribes of plants and animals which have from time to time flourished on the earth4 ,only a small proportion of the total number of species which have lived in past time have been thus chronicled1 yet by collecting the broken fragments of the record an outline at least of the history of life upon the earth can be deciphered4 ,it cannot be too frequently stated1 nor too prominently kept in view1 that1 although gaps occur in the succession of organic remains as recorded in the rocks1 they do not warrant the conclusion that any such blank intervals ever interrupted the progress of plant and animal life upon the globe4 ,there is every reason to believe that the march of life has been unbroken1 onward and upward4 ,geological history1 therefore1 if its records in the stratified formations were perfect1 ought to show a blending and gradation of epoch with epoch4 ,but the progress has been constantly interrupted1 now by upheaval1 now by volcanic outbursts1 now by depression4 ,these interruptions serve as natural divisions in the chronicle1 and enable the geologist to arrange his history into periods4 ,as the order of succession among stratified rocks was first made out in ,europe1 and as many of the gaps in that succession were found to be widespread over the ,european area1 the divisions which experience established for that portion of the globe came to be regarded as typical1 and the names adopted for them were applied to the rocks of other and far distant regions4 ,this application has brought out the fact that some of the most marked breaks in the ,european series do not exist elsewhere1 and1 on the other hand1 that some portions of that series are much more complete than the corresponding sections in other regions4 ,hence1 while the general similarity of succession may remain1 different subdivisions and nomenclature are required as we pass from continent to continent4 ,the nomenclature adopted for the subdivisions of the geological record bears witness to the rapid growth of geology4 ,it is a patch-work in which no system nor language has been adhered to1 but where the influences by which the progress of the science has been moulded may be distinctly traced4 ,some of the earliest names are lithological1 and remind us of the fact that mineralogy and petrography preceded geology in the order of birth--,chalk1 ,oolite1 ,greensand1 ,millstone ,grit4 ,others are topographical1 and often recall the labours of the early geologists of ,england--,london ,clay1 ,oxford ,clay1 ,purbeck1 ,portland1 ,kimmeridge beds4 ,others are taken from local ,english provincial names1 and remind us of the debt we owe to ,william ,smith1 by whom so many of them were first used--,lias1 ,gault1 ,crag1 ,cornbrash4 ,others of later date recognize an order of superposition as already established among formations--,old ,red ,sandstone1 ,new ,red ,sandstone4 ,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 use4 ,cambrian1 ,silurian1 ,devonian1 ,permian1 ,jurassic are of this class1 and have been adopted all over the globe4 ,but whatever be the name chosen to designate a particular group of strata1 it soon comes to be used as a chronological or homotaxial term1 apart altogether from the stratigraphical character of the strata to which it is applied4 ,thus we speak of the ,chalk or ,cretaceous system1 and embrace under that term formations which may contain no chalk2 and we may describe as ,silurian a series of strata utterly unlike in lithological characters to the formations in the typical ,silurian country4 ,in using these terms we unconsciously allow the idea of relative date to arise prominently before us4 ,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 represent4 ,we speak of the ,cretaceous1 ,jurassic1 and ,cambrian periods1 and of the ,cretaceous fauna1 the ,jurassic flora1 the ,cambrian trilobites1 as if these adjectives denoted simply epochs of geological time4 ,the stratified formations of the earth's crust1 or geological record1 are classified into five main divisions1 which in their order of antiquity are as follows3 7#a7 ,archean or ,pre-,cambrian1 called also sometimes ,azoic 7lifeless7 or ,eozoic 7dawn of life72 7#b7 ,palaeozoic 7ancient life7 or ,primary2 7#c7 ,mesozoic 7middle life7 or ,secondary2 7#d7 ,cainozoic 7recent life7 or ,tertiary2 7#e7 ,quaternary or ,post-,tertiary4 ,these divisions are further ranged into systems1 formations1 groups or stages1 assises and zones4 ,accounts of the various subdivisions named are given in separate articles under their own headings4 ,in order1 however1 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 page4 ,,part ,,viii4--,,physiographical ,,geology ,this department of geological inquiry investigates the origin and history of the present topographical features of the land4 ,as these features must obviously be related to those of earlier time which are recorded in the rocks of the earth's crust1 they cannot be satisfactorily studied until at least the main outlines of the history of these rocks have been traced4 ,hence physiographical research comes appropriately after the other branches of the science have been considered4 ,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 formations2 and therefore that the present land is not an aboriginal portion of the earth's surface1 but has been overspread by the sea in which its rocks were mainly accumulated4 ,we further discover that this submergence of the land did not happen once only1 but again and again in past ages and in all parts of the world4 ,yet although the terrestrial areas varied much from age to age in their extent and in their distribution1 being at one time more continental1 at another more insular1 there is reason to believe that these successive diminutions and expansions have on the whole been effected within1 or not far outside1 the limits of the existing continents4 ,there is no evidence that any portion of the present land ever lay under the deeper parts of the ocean4 ,the abysmal deposits of the ocean-floor have no true representatives among the sedimentary formations anywhere visible on the land4 ,nor1 on the other hand1 can it be shown that any part of the existing ocean abysses ever rose above sea-level into dry land4 ,hence geologists have drawn the inference that the ocean basins have probably been always where they now are2 and that although the continental areas have often been narrowed by submergence and by denudation1 there has probably seldom or never been a complete disappearance of land4 ,the fact that the sedimentary formations of each successive geological period consist to so large an extent of mechanically formed terrigenous detritus1 affords good evidence of the coexistence of tracts of land as well as of extensive denudation4 ,the ,geological ,record or ,order of ,succession of the ,stratified ,formations of the ,earth's ,crust4 ---------------------------------------- ---------------------------------------- --- ,europe4 ,north ,america4 ---------------------------------------- ---------------------------------------- --- ,q ,historic1 up to the present time4 ,similar to the ,european u ,prehistoric1 comprising deposits of development1 but with scantier a the ,iron1 ,bronze1 and later traces of the presence of man4 t ,stone ,ages4 e ,neolithic--alluvium1 peat1 lake- r ,recent1 dwellings1 loess1 c4 n ,post- ,palaeolithic--river-gravels1 cave- a glacial deposits1 c4 r or y ,human4 o r ---------------------------------------- ---------------------------------------- ,pleist- ,older ,loess and valley-gravels2 ,as in ,europe1 it is hardly ,p ocene cave-deposits4 possible to assign a definite o or ,strand-lines or raised beaches2 chronological place to each of s ,glacial4 youngest moraines4 the various deposits of this t ,upper ,boulder-clays2 eskers2 marine period1 terrestrial and marine4 sands and clays4 ,they generally resemble the ,t ,interglacial deposits4 ,european series4 ,the e ,lower boulder-clay or ,till1 with characteristic marine1 r striated rock-surfaces below4 fluviatile and lacustrine t terraces1 which overlie the i _/ older drifts1 have been a _/ classed as the ,champlain ,group4 r _/ y _/ 4 _/ ---------------------------------------- ---------------------------------------- --- ,p ,newer3--,english ,forest-,bed ,group2 ,red and ,on the ,atlantic border l ,norwich ,crag2 ,amstelian and ,scaldesian represented by the marine i groups of ,belgium and ,holland2 ,sicilian ,floridian series2 in the o and ,astian of ,france and ,italy4 interior by a subaerial and c ,older3--,english ,coralline ,crag2 ,diestian lacustrine series2 and on the e of ,belgium2 ,plaisancian of southern ,pacific border by the thick n ,france and ,italy4 marine series of ,san ,francisco4 e 4 ---------------------------------------- ---------------------------------------- ,m ,wanting in ,britain2 well developed in ,represented in the ,eastern ,states i ,france1 ,s4 ,e4 ,europe and ,italy2 by a marine series 7,yorktown or ,c o divisible into the following groups in ,chesapeake1 ,chipola and a c descending order3 7#a7 ,pontian2 7#b7 ,chattahoochee groups71 and in i e ,sarmatian2 7#c7 ,tortonian2 7#d7 ,helvetian2 the interior by the lacustrine n n 7#e7 ,langhian 7,burdigalian74 ,loup ,fork 7,nebraska71 ,deep o e ,river1 and ,john ,day groups4 z 4 o ---------------------------------------- ---------------------------------------- i ,in ,britain the fluvio-marine series of ,on the ,atlantic border no c ,o the ,isle of ,wight2 also the volcanic equivalents have been l plateaux of ,antrim and ,inner ,hebrides satisfactorily recognised1 but o i and those of the ,faeroe ,isles and on the ,pacific side there are r g ,iceland4 ,in continental ,europe the marine deposits in ,n4 ,w4 o following subdivisions have been ,oregon1 which may represent ,t c established in descending order3 7#a7 this division4 ,in the interior e e ,aquitanian1 7#b7 ,stampian 7,rupelian71 the equivalent is believed to r n 7#c7 ,tongrain 7,sannoisian74 be the fresh-water ,white ,river t e series1 including 7#a7 i 4 ,protoceras beds1 7#b7 a ,oreodon beds1 and 7#c7 r ,titanothervum beds4 y ---------------------------------------- ---------------------------------------- 4 ,barton sands and clays2 ,ludian series of ,woodstock and ,aquia ,creek groups ,france4 of ,potomac ,river2 ,vicksburg1 ,bracklesham ,beds2 ,lutetian 7,calcaire ,jackson1 ,claiborne1 ,buhrstone1 ,e grossier and ,caillasses7 of ,paris and ,lignitic groups of o basin4 ,mississippi4 c ,london clay1 ,woolwich and ,reading ,beds2 ,in the interior a thick series of e ,thanet sands2 ,ypresian or ,londinian of fresh-water formations1 n ,n4 ,france and ,belgium2 ,sparnacian and comprising1 in descending e ,thanetian groups4 order1 the ,uinta1 ,bridger1 4 ,wind ,river1 ,wasatch1 ,torrejon1 and ,puerco groups4 ,on the ,pacific side the marine ,tejon series of ,oregon and ,california4 ---------------------------------------- ---------------------------------------- --- ,upper ,on the ,atlantic border both marine strata and others ,danian--wanting in ,britain2 uppermost containing a terrestrial flora limestone of ,denmark4 represent the ,cretaceous series ,senonian--,upper ,chalk with ,flints of of formations4 ,england2 ,aturian and ,emscherian stages ,in the interior there is also a on the ,european continent4 commingling of marine with ,turonian--,middle ,chalk with few flints1 lacustrine deposits4 ,at the top and comprising the ,angoumian and stages4 lies the ,laramie or ,lignitic ,c ,cenomanian--,lower ,chalk and ,chalk ,marl4 series with an abundant r terrestrial flora1 passing down e ,lower into the lacustrine and t brackish-water ,montana series4 a ,albian--,upper ,greensand and ,gault4 ,of older date1 the ,colorado c ,aptian--,lower ,greensand2 ,marls and series contains an abundant e limestones of ,provence1 c4 marine fauna1 yet includes also o ,urgonian 7,barremian7--,atherfield clay2 some ,niobrara marls and u massive ,hippurite limestones of limestones are likewise of s southern ,france4 marine origin1 but the lower 4 ,neocomian--,weald clay and ,hastings sand2 members of the series 7,benton ,hauterivian and ,valanginian sub-stages and ,dakota7 show another great of ,switzerland and ,france4 representation of fresh-water ,m sedimentation with lignites and e coals4 s ,in ,california a vast succession o of marine deposits 7,shasta- z ,chico7 represents the o ,cretaceous system2 and in i western ,british ,n4 ,america c coal-seams also occur4 ---------------------------------------- ---------------------------------------- o ,purbeckian--,purbeck beds2 ,munder ,mergel2 ,representatives of the ,middle and r largely present in ,westphalia4 lower ,jurassic formations have ,portlandian--,portland group of ,england1 been found in ,california and ,s represented in ,s4 ,france by the thick ,oregon1 and farther north among e ,tithonian limestones4 the ,arctic islands4 c ,kimmeridgian--,kimmeridge ,clay of ,england2 ,strata containing ,lower ,jurassic o ,virgulian and ,pterocerian groups of ,n4 marine fossils appear in n ,j ,france2 represented by thick limestones ,wyoming and ,dakota2 and above d u in the ,mediterranean basin4 them come the ,atlantosaurus a r ,corallian--,coral ,rag1 ,coralline ,oolite2 and ,baptanodon beds1 which r a ,sequanian stages of the ,continent1 have yielded so large a y s comprising the sub-stages of ,astartian variety of deinosaurs and other 4 s and ,rauracian4 vertebrates1 and especially the i ,oxfordian--,oxford ,clay2 ,axgovian and remains of a number of genera c ,neuvizyan stages4 of small mammals4 4 ,callovian--,kellaways ,rock1 ,divesian sub-stage of ,n4 ,france4 ,bathonian--series of ,english strata from ,cornbrash down to ,fuller's ,earth4 ,bajocian--,inferior ,oolite of ,england4 ,lassic--divisible into 7#a7 ,upper ,lias or ,toarcian1 7#b7 ,middle ,lias1 ,marlstone or ,charmouthian1 7#c7 ,lower ,lias of ,sinemurian and ,hettangian4 ---------------------------------------- ---------------------------------------- ,in ,germany and western ,europe this ,in ,new ,york1 ,connecticut1 ,new ,t division represents the deposits of ,brunswick1 and ,nova ,scotia r inland seas or lagoons1 and is divisible a series of red sandstone i into the following stages in descending 7,newark series7 contains land- a order3 7#a7 ,rhaetic1 7#b7 ,keuper1 7#c7 plants and labyrinthodonts s ,muschelkalk1 7#d7 ,bunter4 ,in the like the lagoon type of central s eastern ,alps and the ,mediterranean and western ,europe4 ,on the i basin the contemporaneous sedimentary ,pacific slope1 however1 marine c formations are those of open clear equivalents occur1 representing 4 sea1 in which a thickness of many the pelagic type of south- thousand feet of strata was accumulated4 eastern ,europe4 ---------------------------------------- ---------------------------------------- --- ,p ,thuringian--,zechstein1 ,magnesian ,to this division of the geologi- e ,limestone2 named from its development cal record the ,upper ,barren r in ,thuringia2 well represented ,measures of the coal-fields of m also in ,saxony1 ,bavaria and ,bohemia4 ,pennsylvania1 ,prince ,edward i ,saxonian--,rothliegendes ,group2 ,red ,island1 ,nova ,scotia and a ,sandstones1 c4 ,new ,brunswick have been n ,autunian--where the strata present the assigned4 4 lagoon facies1 well displayed at ,autun ,farther south in ,kansas1 ,texas1 in ,france2 where the marine type is and ,nebraska the representa- predominant1 as in ,russia1 the group tives of the division have an has been termed ,artinskian4 abundant marine fauna4 ---------------------------------------- ---------------------------------------- ,c ,stephanian or ,uralian--represented in ,upper productive ,coal-measures4 a ,russia by marine formations1 and in ,lower ,barren measures4 r central and western ,europe by numerous ,lower productive ,coal-measures4 b small basins containing a peculiar ,pottsville conglomerate4 o flora and in some places a great variety ,mauch ,chunk shales2 limestones n of insects4 of ,chester1 ,st ,louis1 c4 i ,westphalian or ,moscovian--,coal-measures1 ,pocono series2 ,kinderhook f ,millstone ,grit4 limestone4 e ,culm or ,dinantian--,carboniferous ,limestone r and ,calciferous ,sandstone series4 o u s 4 ---------------------------------------- ---------------------------------------- ,devonian and ,old ,red ,sandstone4 ,p ---------------------------------------- ---------------------------------------- a ,devonian type4 ,old ,red ,sandstone l type4 a ---------------------------------------- ------ _/ ,catskill red sandstone2 ,old e _/ ,famennian4 ,yellow and red ,red ,sandstone type3 the o ,upper sandstone with strata below show the z ,frasnian4 ,holoptychius1 ,devonian type4 o ,bothriolepis1c4 ,chemung ,group4 i ,genesee c _/ ,givetian4 ,caithness ,flagstones o ,middle with ,osteolepus1 _/ ,hamilton ,group4 r ,eifelian4 ,dipterus1 ,marcellus ,homosteus1 c4 ,p r ,red and purple _/ ,corniferous ,limestone4 _/ ,upper i _/,coblentizian4 sandstones and ,held- m ,lower conglomerates with ,onondaga ,limestone4 erberg a ,gedinnian4 ,cephalaspis1 ,group4 r ,pteraspis1 c4 ,oriskany ,sandstone4 y ---------------------------------------- --------------------------------------- 4 _/ ,lower ,helderberg ,group4 ,s _/ ,ludlow ,group4 ,water-,lime4 i ,upper ,wenlock ,niagara ,shale and ,limestone4 l ,llandovery ,clinton ,group4 u ,medina r i _/ ,cincinnati ,group4 a ,lower _/ ,caradoc or ,bala ,group4 ,utica n 7,ordovician7 ,llandeilo ,trenton 4 ,arenig ,chazy ,calciferous ---------------------------------------- ---------------------------------------- ,c ,upper or ,olenus series--,tremadoc ,upper or ,potsdam series with a slates and ,lingula ,flags4 ,olenus and ,dicelocephalus m ,middle or ,pardoxides series--,menevian fauna4 b ,group4 ,middle or ,acadian series with r ,lower or ,olenellus series--,llanberis ,paradoxides fauna4 i and ,harlech ,group1 and ,olenellus- ,lower or ,georgian series with a zone4 ,olenellus fauna4 n 4 ---------------------------------------- ---------------------------------------- --- ,archean1 ,pre-,cambrian1 ,eozoic4 ---------------------------------------- ---------------------------------------- --- ,in ,scotland1 underneath the ,cambrian ,in ,canada and the ,lake ,superior ,olenellus group1 lies unconformably region of the ,united ,states a mass of red sandstone and con- a vast succession of rocks of glomerate 7,torridonian7 #hjjj or #aj1jjj ,pre-,cambrian age has been ft4 thick1 which rests with a strong grouped into the following gneisses and schists 7,lewisian74 ,a subdivisions in descending thick series of slates and phyllites order3 7#a7 ,keweenwan1 lying lies below the oldest ,palaeozoic rocks unconformably on 7#b7 ,animikie1 in central ,europe1 with coarse separated by a strong gneisses below4 unconformability from 7#c7 ,upper ,huronian1 7#d7 ,lower ,huronian with an unconformable base1 7#e7 ,goutchiching1 7#f7 ,laurentian4 ,in the eastern part of ,canada1 ,newfoundland1 c41 and also in ,montana1 sedimentary formations of great thickness below the lowest ,cambrian zone have been found to contain some obscure organisms4 ---------------------------------------- ---------------------------------------- --- ,from these general considerations we proceed to inquire how the existing topographical features of the land arose4 ,obviously the co-operation of the two great geological agencies of hypogene and epigene energy1 which have been at work from the beginning of our globe's decipherable history1 must have been the cause to which these features are to be assigned2 and the task of the geologist is to ascertain1 if possible1 the part that has been taken by each4 ,there is a natural tendency to see in a stupendous piece of scenery1 such as a deep ravine1 a range of hills1 a line of precipice or a chain of mountains1 evidence only of subterranean convulsion2 and before the subject was taken up as a matter of strict scientific induction1 an appeal to former cataclysms was considered a sufficient solution of the problems presented by such features of landscape4 ,the rise of the modern ,huttonian school1 however1 led to a more careful examination of these problems4 ,the important share taken by erosion in the determination of the present features of landscape was then recognized1 while a fuller appreciation of the relative parts played by the hypogene and epigene causes has gradually been reached4 #a4 ,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-day1 it would yet suffice in a comparatively brief geological period to reduce the dry land to below the sea-level4 ,but not only would the area of the land be diminished by denudation1 it could hardly fail to be more or less involved in those widespread movements of subsidence1 during which the thick sedimentary formations of the crust appear to have been accumulated4 ,it is thus manifest that there must have been from time to time during the history of our globe upward movements of the crust1 whereby the balance between land and sea was redressed4 ,proofs of such movements have been abundantly preserved among the stratified formations4 ,we there learn that the uplifts have usually followed each other at long intervals between which subsidence prevailed1 and thus that there has been a prolonged oscillation of the crust over the great continental areas of the earth's surface4 ,an examination of that surface leads to the recognition of two great types of upheaval4 ,in the one1 the sea-floor1 with all its thick accumulations of sediment1 has been carried upwards1 sometimes for several thousand feet1 so equably that the strata retain their original flatness with hardly any sensible disturbance for hundreds of square miles4 ,in the other type the solid crust has been plicated1 corrugated and dislocated1 especially along particular lines1 and has attained its most stupendous disruption in lofty chains of mountains4 ,between these two phases of uplift many intermediate stages have been developed1 according to the direction and intensity of the subterranean force and the varying nature and disposition of the rocks ,of the crust4 7a7 ,where the uplift has extended over wide spaces1 without appreciable deformation of the crust1 the flat strata have given rise to low plains1 or if the amount of uprise has been great enough1 to high plains1 plateaux or tablelands4 ,the plains of ,russia1 for example1 lie for the most part on such tracts of equably uplifted strata4 ,the great plains of the western interior of the ,united ,states form a great plateau or tableland1 #ejjj or #fjjj ft4 above the sea1 and many thousands of square miles in extent1 on which the ,rocky ,mountains have been ridged up4 7b7 ,it is in a great mountain-chain that the complicated structures developed during disturbances of the earth's crust can best be studied 7see ,parts ,,iv4 and ,v4 of this article71 and where the influence of these structures on the topography of the surface is most effectively displayed4 ,such a chain may be the result of one colossal disturbance2 but those of high geological antiquity usually furnish proofs of successive uplifts with more or less intervening denudation4 ,formed along lines of continental displacement in the crust1 they have again and again given relief from the strain of compression by fresh crumpling1 fracture and uprise4 ,the chief guide in tracing these successive stages of growth is supplied by unconformability4 ,if1 for example1 a mountain-range consists of upraised ,silurian rocks1 upon the upturned and denuded edges of which the ,carboniferous ,limestone lies transgressively1 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 formations4 ,if1 as the range is followed along its course1 the ,carboniferous ,limestone is found to be also highly inclined and covered unconformably by the ,upper ,coal-measures1 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-measures4 ,by this simple and obvious kind of evidence the relative ages of different mountain-chains may be compared4 ,in most great chains1 however1 the rocks have been so intensely crumpled1 and even inverted1 that much labour may be required before their true relations can be determined4 ,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 development4 ,the first beginnings of the chain may have been upraised before the oldest ,palaeozoic formations were laid down4 ,there are at least traces of land and shore-lines in the ,carboniferous period4 ,subsequent submergences and uplifts appear to have occurred during the ,mesozoic periods4 ,there is evidence that thereafter the whole region sank deep under the sea1 in which the older ,tertiary sediments were accumulated1 and which seems to have spread right across the heart of the ,old ,world4 ,but after the deposition of the ,eocene formations came the gigantic disruptions whereby all the rocks of the ,alpine region were folded over each other1 crushed1 corrugated1 fractured and displaced1 some of their older portions1 including the fundamental gneisses and schists1 being squeezed up1 torn off1 and pushed horizontally for many miles over the younger rocks4 ,but this upheaval1 though the most momentous1 was not the last which the chain has undergone1 for at a later epoch in ,tertiary time renewed disturbance gave rise to a further series of ruptures and plications4 ,the chain thus successively upheaved has been continuously exposed to denudation and has consequently lost much of its original height4 ,that it has been left in a state of instability is indicated by the frequent earthquakes of the ,alpine region1 which doubtless arise from the sudden snapping of rocks under intense strain4 ,a distinct type of mountain due to direct hypogene action is to be seen in a volcano4 ,it has been already pointed out 7,part ,,iv4 sect4 #a7 that at the vents which maintain a communication between the molten magma of the earth's interior and the surface1 eruptions take place whereby quantities of lava and fragmentary materials are heaped round each orifice of discharge4 ,a typical volcanic mountain takes the form of a perfect cone1 but as it grows in size and its main vent is choked1 while the sides of the cone are unable to withstand the force of the explosions or the pressure of the ascending column of lava1 eruptions take place laterally1 and numerous parasitic cones arise on the flanks of the parent mountain4 ,where lava flows out from long fissures1 it may pile up vast sheets of rock1 and bury the surrounding country under several thousand feet of solid stone1 covering many hundreds of square miles4 ,in this way volcanic tablelands have been formed which1 attacked by the denuding forces1 are gradually trenched by valleys and ravines1 until the original level surface of the lava-field may be almost or wholly lost4 ,as striking examples of this physiographical type reference may be made to the plateau of ,abyssinia1 the ,ghats of ,india1 the plateaux of ,antrim1 the ,inner ,hebrides and ,iceland1 and the great lava-plains of the western territories of the ,united ,states4 #b4 ,but while the subterranean movements have upraised portions of the surface of the lithosphere above the level of the ocean1 and have thus been instrumental in producing the existing tracts of land1 the detailed topographical features of a landscape are not solely1 nor in general even chiefly1 attributable to these movements4 ,from the time that any portion of the sea-floor appears above sea-level1 it undergoes erosion by the various epigene agents4 ,each climate and geological region has its own development of these agents1 which include air1 aridity1 rapid and frequent alternations of wetness and dryness or of heat and cold1 rain1 springs1 frosts1 rivers1 glaciers1 the sea1 plant and animal life4 ,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 climate4 ,but it must be remembered that1 however much they may vary in activity and in the results which they effect1 the epigene forces work without intermission1 while the hypogene forces bring about the upheaval of land only after long intervals4 ,hence1 trifling as the results during a human life may appear1 if we realize the multiplying influence of time we are led to perceive that the apparently feeble superficial agents can1 in the course of ages1 achieve stupendous transformations in the aspect of the land4 ,if this efficacy may be deduced from what can be seen to be in progress now1 it may not less convincingly be shown1 from the nature of the sedimentary rocks of the earth's crust1 to have been in progress from the early beginnings of geological history4 ,side by side with the various upheavals and subsidences1 there has been a continuous removal of materials from the land1 and an equally persistent deposit of these materials under water1 with the consequent growth of new rocks4 ,denudation has been aptly compared to a process of sculpturing wherein1 while each of the implements employed by nature1 like a special kind of graving tool1 produces its own characteristic impress on the land1 they all combine harmoniously towards the achievement of their one common task4 ,hence the present contours of the land depend partly on the original configuration of the ground1 and the influence it may have had in guiding the operations of the erosive agents1 partly on the vigour with which these agents perform their work1 and partly on the varying structure and powers of resistance possessed by the rocks on which the erosion is carried on4 ,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 chain1 the influence of the hypogene forces on the topography attains its highest development4 ,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 perceived4 ,no more striking illustration of this feature can be found than that supplied by the ,alps1 nor one where the geotectonic structures have been so fully studied in detail4 ,on the outer flanks of these mountains the longitudinal ridges and valleys of the ,jura correspond with lines of anticline and syncline4 ,yet though the dominant topographical elements of the region have obviously been produced by the plication of the stratified formations1 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 view1 while on every slope proofs may be seen of extensive denudation4 ,if from these long wave-like undulations of the ground1 where the relations between the disposition of the rocks below and the forms of the surface are so clearly traceable1 the observer proceeds inwards to the main chain1 he finds that the plications and displacements of the various formations assume an increasingly complicated character2 and that although proofs of great denudation continue to abound1 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 removed4 ,along the central heights the mountains lift themselves towards the sky like the storm-swept crests of vast earth-billows4 ,the whole aspect of the ground suggests intense commotion1 and the impression thus given is often much intensified by the twisted and crumpled strata1 visible from a long distance1 on the crags and crests4 ,on this broken-up surface the various agents of denudation have been ceaselessly engaged since it emerged from the sea4 ,they have excavated valleys1 sometimes along depressions provided for them by the subterranean disturbances1 sometimes down the slopes of the disrupted blocks of ground4 ,so powerful has been this erosion that valleys cut out along lines of anticline1 which were natural ridges1 have sometimes become more important than those in lines of syncline1 which were structurally depressions4 ,the same subaerial forces have eroded lake-basins1 dug out corries or cirques1 notched the ridges1 splintered the crests and furrowed the slopes1 leaving no part of the original surface of the uplifted chain unmodified4 ,it has often been noted with surprise that features of underground structure which1 it might have been confidently anticipated1 should have exercised a marked influence on the topography of the surface have not been able to resist the levelling action of the denuding agents1 and do not now affect the surface at all4 ,this result is conspicuously seen in coal-fields where the strata are abundantly traversed by faults4 ,these dislocations1 having sometimes a displacement of several hundred feet1 might have been expected to break up the surface into a network of cliffs and plains2 yet in general they do not modify the level character of the ground above4 ,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-measures4 ,it can be traced across ,belgium into the ,boulonnais1 and may not improbably run beneath the ,secondary and ,tertiary rocks of the south of ,england4 ,it is crossed by the valleys of the ,meuse and other northerly-flowing streams4 ,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 ground1 and even an experienced geologist1 until he had learned the structure of the district1 would scarcely detect any fault at all4 ,where faults have influenced the superficial topography1 it is usually by giving rise to a hollow along which the subaerial agents and especially running water can act effectively4 ,such a hollow may be eventually widened and deepened into a valley4 ,on bare crags and crests1 lines of fault are apt to be marked by notches or clefts1 and they thus help to produce the pinnacles and serrated outlines of these exposed uplands4 ,it was cogently enforced by ,hutton and ,playfair1 and independently by ,lamarck1 that no co-operation of underground agency is needed to produce such topography as may be seen in a great part of the world1 but that if a tract of sea-floor were upraised into a wide plain1 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 now4 ,no such plain would be a dead-level4 ,it would have inequalities on its surface which would serve as channels to guide the drainage from the first showers of rain4 ,and these channels would be slowly widened and deepened until they would become ravines and valleys1 while the ground between them would be left projecting as ridges and hills4 ,nor would the erosion of such a system of water-courses require a long series of geological periods for its accomplishment4 ,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 #hjj ft4 deep might be carved out in less than a million years4 ,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 strata4 ,systems of stream-courses and valleys1 river gorges unexampled elsewhere in the world for depth and length1 vast winding lines of escarpment1 like ranges of sea-cliffs1 terraced slopes rising from plateau to plateau1 huge buttresses and solitary stacks standing like islands out of the plains1 great mountain-masses towering into picturesque peaks and pinnacles cleft by innumerable gullies1 yet everywhere marked by the parallel bars of the horizontal strata out of which they have been carved--these are the orderly symmetrical characteristics of a country where the scenery is due entirely to the action of subaerial agents on the one hand and the varying resistance of perfectly regular stratified rocks on the other4 ,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 employed4 ,the joints by which all rocks are traversed have been especially serviceable as dominant lines down which the rain has filtered1 up which the springs have risen and into which the frost wedges have been driven4 ,on the high bare scarps of a lofty mountain the inner structure of the mass is laid open1 and there the system of joints even more than faults is seen to have determined the lines of crest1 the vertical walls of cliff and precipice1 the forms of buttress and recess1 the position of cleft and chasm1 the outline of spire and pinnacle4 ,on the lower slopes1 even under the tapestry of verdure which nature delights to hang where she can over her naked rocks1 we may detect the same pervading influence of the joints upon the forms assumed by ravines and crags4 ,each kind of stone1 too1 gives rise to its own characteristic form of scenery4 ,massive crystalline rocks1 such as granite1 break up along their joints and often decay into sand or earth along their exposed surfaces1 giving rise to rugged crags with long talus slopes at their base4 ,the stratified rocks besides splitting at their joints are especially distinguished by parallel ledges1 cornices and recesses1 produced by the irregular decay of their component strata1 so that they often assume curiously architectural types of scenery4 ,but besides this family feature they display many minor varieties of aspect according to their lithological composition4 ,a range of sandstone hills1 for example1 presents a marked contrast to one of limestone1 and a line of chalk downs to the escarpments formed by alternating bands of harder and softer clays and shales4 ,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 geology4 ,a true mountain-chain1 viewed from the geological side1 is a mass of high ground which owes its prominence to a ridging-up of the earth's crust1 and the intense plication and rupture of the rocks of which it is composed4 ,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 ground1 such as a high plateau or tableland4 ,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 circumdenudation4 ,tablelands1 as already pointed out1 may be produced either by the upheaval of tracts of horizontal strata from the sea-floor into land2 or by the uprise of plains of denudation1 where rocks of various composition1 structure and age have been levelled down to near or below the level of the sea by the co-operation of the various erosive agents4 ,most of the great tablelands of the globe are platforms of little-disturbed strata which have been upraised bodily to a considerable elevation4 ,no sooner1 however1 are they placed in that position than they are attacked by running water1 and begin to be hollowed out into systems of valleys4 ,as the valleys sink1 the platforms between them grow into narrower and more definite ridges1 until eventually the level tableland is converted into a complicated network of hills and valleys1 wherein1 nevertheless1 the key to the whole arrangement is furnished by a knowledge of the disposition and effects of the flow of water4 ,the examples of this process brought to light in ,colorado1 ,wyoming1 ,nevada and the other western regions by ,newberry1 ,king1 ,hayden1 ,powell and other explorers1 are among the most striking monuments of geological operations in the world4 ,examples of ancient and much decayed tablelands formed by the denudation of much disturbed rocks are furnished by the ,highlands of ,scotland and of ,norway4 ,each of these tracts of high ground consists of some of the oldest and most dislocated formations of ,europe1 which at a remote period were worn down into a plain1 and in that condition may have lain long submerged under the sea and may possibly have been overspread there with younger formations4 ,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 denudation1 whereby each of them has been so deeply channeled into glens and fjords that it presents to-day a surface of rugged hills1 either isolated or connected along the flanks1 while only fragments of the general surface of the tableland can here and there be recognized amidst the general destruction4 ,valleys have in general been hollowed out by the greater erosive action of running water along the channels of drainage4 ,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 land4 ,sometimes these irregularities have been produced by folds of the terrestrial crust1 sometimes by faults1 sometimes by the irregularities on the surface of an uplifted platform of deposition or of denudation4 ,two dominant trends may be observed among them4 ,some are longitudinal and run along the line of flexures in the upraised tract of land1 others are transverse where the drainage has flowed down the slopes of the ridges into the longitudinal valleys or into the sea4 ,the forms of valleys have been governed partly by the structure and composition of the rocks1 and partly by the relative potency of the different denuding agents4 ,where the influence of rain and frost has been slight1 and the streams1 supplied from distant sources1 have had sufficient declivity1 deep1 narrow1 precipitous ravines or gorges have been excavated4 ,the canyons of the arid region of the ,colorado are a magnificent example of this result4 ,where1 on the other hand1 ordinary atmospheric action has been more rapid1 the sides of the river channels have been attacked1 and open sloping glens and valleys have been hollowed out4 ,a gorge or defile is usually due to the action of a waterfall1 which1 beginning with some abrupt declivity or precipice in the course of the river when it first commenced to flow1 or caused by some hard rock crossing the channel1 has eaten its way backward4 ,lakes have been already referred to1 and their modes of origin have been mentioned4 ,as they are continually being filled up with the detritus washed into them from the surrounding regions they cannot be of any great geological antiquity1 unless where by some unknown process their basins are from time to time widened and deepened4 ,in the general subaerial denudation of a country1 innumerable minor features are worked out as the structure of the rocks controls the operations of the eroding agents4 ,thus1 among comparatively undisturbed strata1 a hard bed resting upon others of a softer kind is apt to form along its outcrop a line of cliff or escarpment4 ,though a long range of such cliffs resembles a coast that has been worn by the sea1 it may be entirely due to mere atmospheric waste4 ,again1 the more resisting portions of a rock may be seen projecting as crags or knolls4 ,an igneous mass will stand out as a bold hill from amidst the more decomposable strata through which it has risen4 ,these features1 often so marked on the lower grounds1 attain their most conspicuous development among the higher and barer parts of the mountains1 where subaerial disintegration is most rapid4 ,the torrents tear out deep gullies from the sides of the declivities4 ,corries or cirques are scooped out on the one hand and naked precipices are left on the other4 ,the harder bands of rock project as massive ribs down the slopes1 shoot up into prominent aiguilles1 or help to give to the summits the notched saw-like outlines they so often present4 ,the materials worn from the surface of the higher are spread out over the lower grounds4 ,the streams as they descend begin to drop their freight of sediment when1 by the lessening of their declivity1 their carrying power is diminished4 ,the great plains of the earth's surface are due to this deposit of gravel1 sand and loam4 ,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 fell4 ,every pebble and particle of their soil1 once part of the distant mountains1 has travelled slowly and fitfully to lower levels4 ,again and again have these materials been shifted1 ever moving downward and sea-ward4 ,for centuries1 perhaps1 they have taken their share in the fertility of the plains and have ministered to the nurture of flower and tree1 of the bird of the air1 the beast of the field and of man himself4 ,but their destiny is still the great ocean4 ,in that bourne alone can they find undisturbed repose1 and there1 slowly accumulating in massive beds1 they will remain until1 in the course of ages1 renewed upheaval shall raise them into future land1 there once more to pass through the same cycle of change4 7,a4 ,ge47 ,,literature4--,historical3 ,the standard work is ,karl ,a4 von ,zittel's ,geschichte der ,geologie und ,palaontologie 7#ahii71 of which there is an abbreviated1 but still valuable1 ,english translation2 ,d',archiac1 ,histoire des progres de la geologie1 deals especially with the period #ahcd-ahej2 ,keferstein1 ,geschichte und ,literatur der ,geognosie1 gives a summary up to #ahdj2 while ,sir ,a4 ,geikie's ,founders of ,geology 7#ahig2 #bnd ed41 #aijf7 deals more particularly with the period #agej-ahbj4 ,general treatises3 ,sir ,charles ,lyell's ,principles of ,geology is a classic4 ,of modern ,english works1 ,sir ,a4 ,geikie's ,text ,book of ,geology 7#dth ed41 #aijc7 occupies the first place2 the work of ,t4,c4 ,chamberlin and ,r4,d4 ,salisbury1 ,geology2 ,earth ,history 7#c vols41 #aije-aijf71 is especially valuable for ,american geology4 ,a4 de ,lapparent's ,traite de geologie 7#eth ed41 #aijf71 is the standard ,french work4 ,h4 ,credner's ,elemente der ,geologie has gone through several editions in ,germany4 ,dynamical and physiographical geology are elaborately treated by ,e4 ,suess1 ,das ,antlitz der ,erde1 translated into ,english1 with the title ,the ,face of the ,earth4 ,the practical study of the science is treated of by ,f4 von ,richthofen1 ,fuhrer fur ,forschungsreisende 7#ahhf72 ,g4,a4 ,cole1 ,aids in ,practical ,geology 7#eth ed41 #aijf72 ,a4 ,geikie1 ,outlines of ,field ,geology 7#eth ed41 #aijj74 ,the practical applications of ,geology are discussed by ,j4,v4 ,elsden1 ,applied ,geology 7#ahih-ahii74 ,the relations of ,geology to scenery are dealt with by ,sir ,a4 ,geikie1 ,scenery of ,scotland 7#crd ed41 #aija72 ,j4,e4 ,marr1 ,the ,scientific ,study of ,scenery 7#aijj72 ,lord ,avebury1 ,the ,scenery of ,switzerland 7#ahif72 ,the ,scenery of ,england 7#aijb72 and ,j4 ,geikie1 ,earth ,sculpture 7#ahih74 ,a detailed bibliography is given in ,sir ,a4 ,geikie's ,text ,book of ,geology4 ,see also the separate articles on geological subjects for special references to authorities4 ,,footnotes3 #a ,in ,de ,luc's ,lettres physiques et morales sur les montagnes 7#aggh71 the word cosmology is used for our science1 the author stating that geology is more appropriate1 but it was not a word in use4 ,in a completed edition1 published in #aggi1 the same statement is made1 but geology occurs in the text2 in the same year ,de ,saussure used the word without any explanation1 as if it were well known4 #b ,the subject of the age of the earth has also been discussed by ,professor ,j4 ,joly and ,professor ,w4,j4 ,sollas4 ,the former geologist1 approaching the question from a novel point of view1 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 land4 ,dividing the one sum by the other1 he arrives at the result that the probable age of the earth is between #ij and #ajj millions of years 7,trans4 ,roy4 ,dublin ,soc4 ser4 ii4 vol4 vii41 #ahii1 p4 #bc3 ,geol4 ,mag41 #aijj1 p4 #bbj74 ,professor ,sollas believes that this limit exceeds what is required for the evolution of geological history1 that the lower limit assigned by ,lord ,kelvin falls short of what the facts demand1 and that geological time will probably be found to have been comprised within some indeterminate period between these limits4 7,address to ,section ,c1 ,brit4 ,assoc4 ,report1 #aijj2 ,age of the ,earth1 ,london1 #aije47 ,,geometrical ,,continuity4 ,in a report of the ,institute prefixed to ,jean ,victor ,poncelet's ,traite des proprietes projectives des figures 7,paris1 #ahbb71 it is said that he employed ce qu'il appelle le principe de continuite4 ,the law or principle thus named by him had1 he tells us1 been tacitly assumed as axiomatic by les plus savans geometres4 ,it had in fact been enunciated as lex continuationis1 and la loi de la continuite1 by ,gottfried ,wilhelm ,leibnitz 7,oxf4 ,n4,e4,d471 and previously under another name by ,johann ,kepler in cap4 iv4 #d of his ,ad ,vitellionem paralipomena quibus astronomiae pars optica traditur 7,francofurti1 #afjd74 ,of sections of the cone1 he says1 there are five species from the recta linea or line-pair to the circle4 ,from the line-pair we pass through an infinity of hyperbolas to the parabola1 and thence through an infinity of ellipses to the circle4 ,related to the sections are certain remarkable points which have no name4 ,kepler calls them foci4 ,the circle has one focus at the centre1 an ellipse or hyperbola two foci equidistant from the centre4 ,the parabola has one focus within it1 and another1 the caecus focus1 which may be imagined to be at infinity on the axis within or without the curve4 ,the line from it to any point of the section is parallel to the axis4 ,to carry out the analogy we must speak paradoxically1 and say that the line-pair likewise has foci1 which in this case coalesce as in the circle and fall upon the lines themselves2 for our geometrical terms should be subject to analogy4 ,kepler dearly loves analogies1 his most trusty teachers1 acquainted with all the secrets of nature1 omnium naturae arcanorum conscios4 ,and they are to be especially regarded in geometry as1 by the use of however absurd expressions1 classing extreme limiting forms with an infinity of intermediate cases1 and placing the whole essence of a thing clearly before the eyes4 ,here1 then1 we find formulated by ,kepler the doctrine of the concurrence of parallels at a single point at infinity and the principle of continuity 7under the name analogy7 in relation to the infinitely great4 ,such conceptions so strikingly propounded in a famous work could not escape the notice of contemporary mathematicians4 ,henry ,briggs1 in a letter to ,kepler from ,merton ,college1 ,oxford1 dated #aj ,cal4 ,martiis #afbe1 suggests improvements in the ,ad ,vitellionem paralipomena1 and gives the following construction3 ,draw a line ,,cbadc1 and let an ellipse1 a parabola1 and a hyperbola have ,b and ,a for focus and vertex4 ,let ,,cc be the other foci of the ellipse and the hyperbola4 ,make ,,ad equal to ,,ab1 and with centres ,,cc and radius in each case equal to ,,cd describe circles4 ,then any point of the ellipse is equidistant from the focus ,b and one circle1 and any point of the hyperbola from the focus ,b and the other circle4 ,any point ,p of the parabola1 in which the second focus is missing or infinitely distant1 is equidistant from the focus ,b and the line through ,d which we call the directrix1 this taking the place of either circle when its centre ,c is at infinity1 and every line ,,cp being then parallel to the axis4 ,thus ,briggs1 and we know not how many savans geometres who have left no record1 had already taken up the new doctrine in geometry in its author's lifetime4 ,six years after ,kepler's death in #afcj ,girard ,desargues1 the ,monge of his age1 brought out the first of his remarkable works founded on the same principles1 a short tract entitled ,methode universelle de mettre en perspective les objets donnes reellement ou en devis 7,paris1 #afcf72 but ,le privilege etoit de #afcj4 7,poudra1 ,,oeuvres de ,des41 i4 #ee74 ,kepler as a modern geometer is best known by his ,new ,stereometry of ,wine ,casks 7,lincii1 #afae71 in which he replaces the circuitous ,archimedean method of exhaustion by a direct royal road of infinitesimals1 treating a vanishing arc as a straight line and regarding a curve as made up of a succession of short chords4 ,some #bjjj years previously one ,antipho1 probably the well-known opponent of ,socrates1 has regarded a circle in like manner as the limiting form of a many-sided inscribed rectilinear figure4 ,antipho's notion was rejected by the men of his day as unsound1 and when reproduced by ,kepler it was again stoutly opposed as incapable of any sort of geometrical demonstration--not altogether without reason1 for it rested on an assumed law of continuity rather than on palpable proof4 ,to complete the theory of continuity1 the one thing needful was the idea of imaginary points implied in the algebraical geometry of ,rene ,descartes1 in which equations between variables representing co-ordinates were found often to have imaginary roots4 ,newton1 in his two sections on ,inventio orbium 7,principia i4 #d1 #e71 shows in his brief way that he is familiar with the principles of modern geometry4 ,in two propositions he uses an auxiliary line which is supposed to cut the conic in ,x and ,y1 but1 as he remarks at the end of the second 7prop4 #bd71 it may not cut it at all4 ,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 trajectory4 ,in the scholium appended to prop4 #bg1 after saying that an asymptote is a tangent at infinity1 he gives an unexplained general construction for the axes of a conic1 which seems to imply that it has asymptotes4 ,in all such cases1 having equations to his loci in the background1 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 reality4 ,roger ,joseph ,boscovich1 a careful student of ,newton's works1 has a full and thorough discussion of geometrical continuity in the third and last volume of his ,elementa universae matheseos 7ed4 prim4 ,venet1 #ageg71 which contains ,sectionum conicarum elementa nova quadam methodo concinnata et dissertationem de transformatione locorum geometricorum1 ubi de continuitatis lege1 et de quibusdam infiniti mysteriis4 ,his first principle is that all varieties of a defined locus have the same properties1 so that what is demonstrable of one should be demonstrable in like manner of all1 although some artifice may be required to bring out the underlying analogy between them4 ,the opposite extremities of an infinite straight line1 he says1 are to be regarded as joined1 as if the line were a circle having its centre at the infinity on either side of it4 ,this leads up to the idea of a veluti plus quam infinita extensio1 a line-circle containing1 as we say1 the line infinity4 ,change from the real to the imaginary state is contingent upon the passage of some element of a figure through zero or infinity and never takes place per saltum4 ,lines being some positive and some negative1 there must be negative rectangles and negative squares1 such as those of the exterior diameters of a hyperbola4 ,boscovich's first principle was that of ,kepler1 by whose quantumvis absurdis locutionibus the boldest applications of it are covered1 as when we say with ,poncelet that all concentric circles in a plane touch one another in two imaginary fixed points at infinity4 ,in ,g4,k4 ,ch4 von ,staudt's ,geometrie der ,lage and ,beitrage zur ,g4 der ,l4 7,nurnberg1 #ahdg1 #ahef-ahfj7 the geometry of position1 including the extension of the field of pure geometry to the infinite and the imaginary1 is presented as an independent science1 welche des ,messens nicht bedarf4 7,see ,,geometry3 ,projective47 ,ocular illusions due to distance1 such as ,roger ,bacon notices in the ,opus majus 7i4 #abf1 ii4 #ajh1 #dig2 ,oxford1 #ahig71 lead up to or illustrate the mathematical uses of the infinite and its reciprocal the infinitesimal4 ,specious objections can1 of course1 be made to the anomalies of the law of continuity1 but they are inherent in the higher geometry1 which has taught us so much of the secrets of nature4 ,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 vol4 xviii4 of the ,transactions of the ,cambridge ,philosophical ,society 7#aijj74 ,it had been generally overlooked1 until attention was called to it by the present writer in a note read in #ahhj 7,proc4 ,c4,p4,s4 iv4 #ad-ag71 and shortly afterwards in ,the ,ancient and ,modern ,geometry of ,conics1 with ,historical ,notes and ,prolegomena 7,cambridge #ahha74 7,c4 ,t4997 ,,geometry1 the general term for the branch of mathematics which has for its province the study of the properties of space4 ,from experience1 or possibly intuitively1 we characterize existent space by certain fundamental qualities1 termed axioms1 which are insusceptible of proof2 and these axioms1 in conjunction with the mathematical entities of the point1 straight line1 curve1 surface and solid1 appropriately defined1 are the premises from which the geometer draws conclusions4 ,the geometrical axioms are merely conventions2 on the one hand1 the system may be based upon inductions from experience1 in which case the deduced geometry may be regarded as a branch of physical science2 or1 on the other hand1 the system may be formed by purely logical methods1 in which case the geometry is a phase of pure mathematics4 ,obviously the geometry with which we are most familiar is that of existent space--the three-dimensional space of experience2 this geometry may be termed ,euclidean1 after its most famous expositor4 ,but other geometries exist1 for it is possible to frame systems of axioms which definitely characterize some other kind of space1 and from these axioms to deduce a series of non-contradictory propositions2 such geometries are called non-,euclidean4 ,it is convenient to discuss the subject-matter of geometry under the following headings3 ,i4 ,euclidean ,geometry3 a discussion of the axioms of existent space and of the geometrical entities1 followed by a synoptical account of ,euclid's ,elements4 ,,ii4 ,projective ,geometry3 primarily ,euclidean1 but differing from ,i4 in employing the notion of geometrical continuity 7q4v47--points and lines at infinity4 ,,iii4 ,descriptive ,geometry3 the methods for representing upon planes figures placed in space of three dimensions4 ,,iv4 ,analytical ,geometry3 the representation of geometrical figures and their relations by algebraic equations4 ,v4 ,line ,geometry3 an analytical treatment of the line regarded as the space element4 ,,vi4 ,non-,euclidean ,geometry3 a discussion of geometries other than that of the space of experience4 ,,vii4 ,axioms of ,geometry3 a critical analysis of the foundations of geometry4 ,special subjects are treated under their own headings3 e4g4 ,,projection1 ,,perspective2 ,,curve1 ,,surface2 ,,circle1 ,,conic ,,section2 ,,triangle1 ,,polygon1 ,,polyhedron2 there are also articles on special curves and figures1 e4g4 ,,ellipse1 ,,parabola1 ,,hyperbola2 ,,tetrahedron1 ,,cube1 ,,octahedron1 ,,dodecahedron1 ,,icosahedron2 ,,cardioid1 ,,catenary1 ,,cissoid1 ,,conchoid1 ,,cycloid1 ,,epicycloid1 ,,limacon1 ,,oval1 ,,quadratrix1 ,,spiral1 c4 ,history4--,the origin of geometry 7,gr4 ,greek3 ge1 earth1 ,greek3 metron1 a measure7 is1 according to ,herodotus1 to be found in the etymology of the word4 ,its birthplace was ,egypt1 and it arose from the need of surveying the lands inundated by the ,nile floods4 ,in its infancy it therefore consisted of a few rules1 very rough and approximate1 for computing the areas of triangles and quadrilaterals2 and1 with the ,egyptians1 it proceeded no further1 the geometrical entities--the point1 line1 surface and solid--being only discussed in so far as they were involved in practical affairs4 ,the point was realized as a mark or position1 a straight line as a stretched string or the tracing of a pole1 a surface as an area2 but these units were not abstracted2 and for the ,egyptians geometry was only an art--an auxiliary to surveying4#a ,the first step towards its elevation to the rank of a science was made by ,thales 7q4v47 of ,miletus1 who transplanted the elementary ,egyptian mensuration to ,greece4 ,thales clearly abstracted the notions of points and lines1 founding the geometry of the latter unit1 and discovering per saltum many propositions concerning areas1 the circle1 c4 ,the empirical rules of the ,egyptians were corrected and developed by the ,ionic ,school which he founded1 especially by ,anaximander and ,anaxagoras1 and in the #fth century ,b4,c4 passed into the care of the ,pythagoreans4 ,from this time geometry exercised a powerful influence on ,greek thought4 ,pythagoras 7q4v471 seeking the key of the universe in arithmetic and geometry1 investigated logically the principles underlying the known propositions2 and this resulted in the formulation of definitions1 axioms and postulates which1 in addition to founding a science of geometry1 permitted a crystallization1 fractional1 it is true1 of the amorphous collection of material at hand4 ,pythagorean geometry was essentially a geometry of areas and solids2 its goal was the regular solids--the tetrahedron1 cube1 octahedron1 dodecahedron and icosahedron--which symbolized the five elements of ,greek cosmology4 ,the geometry of the circle1 previously studied in ,egypt and much more seriously by ,thales1 was somewhat neglected1 although this curve was regarded as the most perfect of all plane figures and the sphere the most perfect of all solids4 ,the circle1 however1 was taken up by the ,sophists1 who made most of their discoveries in attempts to solve the classical problems of squaring the circle1 doubling the cube and trisecting an angle4 ,these problems1 besides stimulating pure geometry1 i4e4 the geometry of constructions made by the ruler and compasses1 exercised considerable influence in other directions4 ,the first problem led to the discovery of the method of exhaustion for determining areas4 ,antiphon inscribed a square in a circle1 and on each side an isosceles triangle having its vertex on the circle2 on the sides of the octagon so obtained1 isosceles triangles were again constructed1 the process leading to inscribed polygons of #h1 #af and #cb sides2 and the areas of these polygons1 which are easily determined1 are successive approximations to the area of the circle4 ,bryson of ,heraclea took an important step when he circumscribed1 in addition to inscribing1 polygons to a circle1 but he committed an error in treating the circle as the mean of the two polygons4 ,the method of ,antiphon1 in assuming that by continued division a polygon can be constructed coincident with the circle1 demanded that magnitudes are not infinitely divisible4 ,much controversy ranged about this point2 ,aristotle supported the doctrine of infinite divisibility2 ,zeno attempted to show its absurdity4 ,the mechanical tracing of loci1 a principle initiated by ,archytas of ,tarentum to solve the last two problems1 was a frequent subject for study1 and several mechanical curves were thus discovered at subsequent dates 7cissoid1 conchoid1 quadratrix74 ,mention may be made of ,hippocrates1 who1 besides developing the known methods1 made a study of similar figures1 and1 as a consequence1 of proportion4 ,this step is important as bringing into line discontinuous number and continuous magnitude4 ,a fresh stimulus was given by the succeeding ,platonists1 who1 accepting in part the ,pythagorean cosmology1 made the study of geometry preliminary to that of philosophy4 ,the many discoveries made by this school were facilitated in no small measure by the clarification of the axioms and definitions1 the logical sequence of propositions which was adopted1 and1 more especially1 by the formulation of the analytic method1 i4e4 of assuming the truth of a proposition and then reasoning to a known truth4 ,the main strength of the ,platonist geometers lies in stereometry or the geometry of solids4 ,the ,pythagoreans had dealt with the sphere and regular solids1 but the pyramid1 prism1 cone and cylinder were but little known until the ,platonists took them in hand4 ,eudoxus established their mensuration1 proving the pyramid and cone to have one-third the content of a prism and cylinder on the same base and of the same height1 and was probably the discoverer of a proof that the volumes of spheres are as the cubes of their radii4 ,the discussion of sections of the cone and cylinder led to the discovery of the three curves named the parabola1 ellipse and hyperbola 7see ,,conic ,,section72 it is difficult to over-estimate the importance of this discovery2 its investigation marks the crowning achievement of ,greek geometry1 and led in later years to the fundamental theorems and methods of modern geometry4 ,the presentation of the subject-matter of geometry as a connected and logical series of propositions1 prefaced by ,greek3 ,horoi or foundations1 had been attempted by many2 but it is to ,euclid that we owe a complete exposition4 ,little indeed in the ,elements is probably original except the arrangement2 but in this ,euclid surpassed such predecessors as ,hippocrates1 ,leon1 pupil of ,neocleides1 and ,theudius of ,magnesia1 devising an apt logical model1 although when scrutinized in the light of modern mathematical conceptions the proofs are riddled with fallacies4 ,according to the commentator ,proclus1 the ,elements were written with a twofold object1 first1 to introduce the novice to geometry1 and secondly1 to lead him to the regular solids2 conic sections found no place therein4 ,what ,euclid did for the line and circle1 ,apollonius did for the conic sections1 but there we have a discoverer as well as editor4 ,these two works1 which contain the greatest contributions to ancient geometry1 are treated in detail in ,section ,i4 ,euclidean ,geometry and the articles ,,euclid2 ,,conic ,,section2 ,,appolonius4 ,between ,euclid and ,apollonius there flourished the illustrious ,archimedes1 whose geometrical discoveries are mainly concerned with the mensuration of the circle and conic sections1 and of the sphere1 cone and cylinder1 and whose greatest contribution to geometrical method is the elevation of the method of exhaustion to the dignity of an instrument of research4 ,apollonius was followed by ,nicomedes1 the inventor of the conchoid2 ,diocles1 the inventor of the cissoid2 ,zenodorus1 the founder of the study of isoperimetrical figures2 ,hipparchus1 the founder of trigonometry2 and ,heron the elder1 who wrote after the manner of the ,egyptians1 and primarily directed attention to problems of practical surveying4 ,of the many isolated discoveries made by the later ,alexandrian mathematicians1 those of ,menelaus are of importance4 ,he showed how to treat spherical triangles1 establishing their properties and determining their congruence2 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 transversals4 ,these propositions1 and also those of ,hipparchus1 were utilized and developed by ,ptolemy 7q4v471 the expositor of trigonometry and discoverer of many isolated propositions4 ,mention may be made of the commentator ,pappus1 whose ,mathematical ,collections is valuable for its wealth of historical matter2 of ,theon1 an editor of ,euclid's ,elements and commentator of ,ptolemy's ,almagest2 of ,proclus1 a commentator of ,euclid2 and of ,eutocius1 a commentator of ,apollonius and ,archimedes4 ,the ,romans1 essentially practical and having no inclination to study science qua science1 only had a geometry which sufficed for surveying2 and even here there were abundant inaccuracies1 the empirical rules employed being akin to those of the ,egyptians and ,heron4 ,the ,hindus1 likewise1 gave more attention to computation1 and their geometry was either of ,greek origin or in the form presented in trigonometry1 more particularly connected with arithmetic4 ,it had no logical foundations2 each proposition stood alone2 and the results were empirical4 ,the ,arabs more closely followed the ,greeks1 a plan adopted as a sequel to the translation of the works of ,euclid1 ,apollonius1 ,archimedes and many others into ,arabic4 ,their chief contribution to geometry is exhibited in their solution of algebraic equations by intersecting conics1 a step already taken by the ,greeks in isolated cases1 but only elevated into a method by ,omar al ,hayyami1 who flourished in the #aath century4 ,during the middle ages little was added to ,greek and ,arabic geometry4 ,leonardo of ,pisa wrote a ,practica geometriae 7#abbj71 wherein ,euclidean methods are employed2 but it was not until the #adth century that geometry1 generally ,euclid's ,elements1 became an essential item in university curricula4 ,there was1 however1 no sign of original development1 other branches of mathematics1 mainly algebra and trigonometry1 exercising a greater fascination until the #afth century1 when the subject again came into favour4 ,the extraordinary mathematical talent which came into being in the #afth and #agth centuries reacted on geometry and gave rise to all those characters which distinguish modern from ancient geometry4 ,the first innovation of moment was the formulation of the principle of geometrical continuity by ,kepler4 ,the notion of infinity which it involved permitted generalizations and systematizations hitherto unthought of 7see ,,geometrical ,,continuity72 and the method of indefinite division applied to rectification1 and quadrature and cubature problems avoided the cumbrous method of exhaustion and provided more accurate results4 ,further progress was made by ,bonaventura ,cavalieri1 who1 in his ,geometria indivisibilibus continuorum 7#afbj71 devised a method intermediate between that of exhaustion and the infinitesimal calculus of ,leibnitz and ,newton4 ,the logical basis of his system was corrected by ,roberval and ,pascal2 and their discoveries1 taken in conjunction with those of ,leibnitz1 ,newton1 and many others in the fluxional calculus1 culminated in the branch of our subject known as differential geometry 7see ,,infinitesimal ,,calculus2 ,,curve2 ,,surface74 ,a second important advance followed the recognition that conics could be regarded as projections of a circle1 a conception which led at the hands of ,desargues and ,pascal to modern projective geometry and perspective4 ,a third1 and perhaps the most important1 advance attended the application of algebra to geometry by ,descartes1 who thereby founded analytical geometry4 ,the new fields thus opened up were diligently explored1 but the calculus exercised the greatest attraction and relatively little progress was made in geometry until the beginning of the #aith century1 when a new era opened4 ,gaspard ,monge was the first important contributor1 stimulating analytical and differential geometry and founding descriptive geometry in a series of papers and especially in his lectures at the ,ecole polytechnique4 ,projective geometry1 founded by ,desargues1 ,pascal1 ,monge and ,l4,n4,m4 ,carnot1 was crystallized by ,j4,v4 ,poncelet1 the creator of the modern methods4 ,in his ,traite des proprietes des figures 7#ahbb7 the line and circular points at infinity1 imaginaries1 polar reciprocation1 homology1 cross-ratio and projection are systematically employed4 ,in ,germany1 ,a4,f4 ,mobius1 ,j4 ,plucker and ,j4 ,steiner were making far-reaching contributions4 ,mobius1 in his ,barycentrische ,calcul 7#ahbg71 introduced homogeneous co-ordinates1 and also the powerful notion of geometrical transformation1 including the special cases of collineation and duality2 ,plucker1 in his ,analytisch-geometrische ,entwickelungen 7#ahbh-ahca71 and his ,system der analytischen ,geometrie 7#ahce71 introduced the abridged notation1 line and plane co-ordinates1 and the conception of generalized space elements2 while ,steiner1 besides enriching geometry in numerous directions1 was the first to systematically generate figures by projective pencils4 ,we may also notice ,m4 ,chasles1 whose ,apercu historique 7#ahcg7 is a classic4 ,synthetic geometry1 characterized by its fruitfulness and beauty1 attracted most attention1 and it so happened that its originally weak logical foundations became replaced by a more substantial set of axioms4 ,these were found in the anharmonic ratio1 a device leading to the liberation of synthetic geometry from metrical relations1 and in involution1 which yielded rigorous definitions of imaginaries4 ,these innovations were made by ,k4,j4,c4 von ,staudt4 ,analytical geometry was stimulated by the algebra of invariants1 a subject much developed by ,a4 ,cayley1 ,g4 ,salmon1 ,s4,h4 ,aronhold1 ,l4,o4 ,hesse1 and more particularly by ,r4,f4,a4 ,clebsch4 ,the introduction of the line as a space element1 initiated by ,h4 ,grassmann 7#ahdd7 and ,cayley 7#ahei71 yielded at the hands of ,plucker a new geometry1 termed line geometry1 a subject developed more notably by ,f4 ,klein1 ,clebsch1 ,c4,t4 ,reye and ,f4,o4,r4 ,sturm 7see ,section ,v41 ,line ,geometry74 ,non-euclidean geometries1 having primarily their origin in the discussion of ,euclidean parallels1 and treated by ,wallis1 ,saccheri and ,lambert1 have been especially developed during the #aith century4 ,four lines of investigation may be distinguished3--the naive-synthetic1 associated with ,lobatschewski1 ,bolyai1 ,gauss2 the metric differential1 studied by ,riemann1 ,helmholtz1 ,beltrami2 the projective1 developed by ,cayley1 ,klein1 ,clifford2 and the critical-synthetic1 promoted chiefly by the ,italian mathematicians ,peano1 ,veronese1 ,burali-,forte1 ,levi ,civitta1 and the ,germans ,pasch and ,hilbert4 7,c4 ,e4997 ,i4 ,,euclidean ,,geometry ,this branch of the science of geometry is so named since its methods and arrangement are those laid down in ,euclid's ,elements4 ,s #a4 ,axioms4--,the object of geometry is to investigate the properties of space4 ,the first step must consist in establishing those fundamental properties from which all others follow by processes of deductive reasoning4 ,they are laid down in the ,axioms1 and these ought to form such a system that nothing need be added to them in order fully to characterize space1 and that nothing may be omitted without making the system incomplete4 ,they must1 in fact1 completely define space4 ,s #b4 ,definitions4--,the axioms of ,euclidean ,geometry are obtained from inspection of existent space and of solids in existent space1--hence from experience4 ,the same source gives us the notions of the geometrical entities to which the axioms relate1 viz4 solids1 surfaces1 lines or curves1 and points4 ,a solid is directly given by experience2 we have only to abstract all material from it in order to gain the notion of a geometrical solid4 ,this has shape1 size1 position1 and may be moved4 ,its boundary or boundaries are called surfaces4 ,they separate one part of space from another1 and are said to have no thickness4 ,their boundaries are curves or lines1 and these have length only4 ,their boundaries1 again1 are points1 which have no magnitude but only position4 ,we thus come in three steps from solids to points which have no magnitude2 in each step we lose one extension4 ,hence we say a solid has three dimensions1 a surface two1 a line one1 and a point none4 ,space itself1 of which a solid forms only a part1 is also said to be of three dimensions4 ,the same thing is intended to be expressed by saying that a solid has length1 breadth and thickness1 a surface length and breadth1 a line length only1 and a point no extension whatsoever4 ,euclid gives the essence of these statements as definitions3-- ,def4 #a1 ,i4 ,a point is that which has no parts1 or which has no magnitude4 ,def4 #b1 ,i4 ,a line is length without breadth4 ,def4 #e1 ,i4 ,a superficies is that which has only length and breadth4 ,def4 #a1 ,,xi4 ,a solid is that which has length1 breadth and thickness4 ,it is to be noted that the synthetic method is adopted by ,euclid2 the analytical derivation of the successive ideas of surface1 line1 and point from the experimental realization of a solid does not find a place in his system1 although possessing more advantages4 ,if we allow motion in geometry1 we may generate these entities by moving a point1 a line1 or a surface1 thus3-- ,the path of a moving point is a line4 ,the path of a moving line is1 in general1 a surface4 ,the path of a moving surface is1 in general1 a solid4 ,and we may then assume that the lines1 surfaces and solids1 as defined before1 can all be generated in this manner4 ,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 points1 and so on2 and thus we come back to the considerations with which we started4 ,euclid points this out in his definitions1--,def4 #c1 ,i41 ,def4 #f1 ,i41 and ,def4 #b1 ,,xi4 ,he does not1 however1 show the connexion which these definitions have with those mentioned before4 ,when points and lines have been defined1 a statement like ,def4 #c1 ,i41 ,the extremities of a line are points1 is a proposition which either has to be proved1 and then it is a theorem1 or which has to be taken for granted1 in which case it is an axiom4 ,and so with ,def4 #f1 ,i41 and ,def4 #b1 ,,xi4 ,s #c4 ,euclid's definitions mentioned above are attempts to describe1 in a few words1 notions which we have obtained by inspection of and abstraction from solids4 ,a few more notions have to be added to these1 principally those of the simplest line--the straight line1 and of the simplest surface--the flat surface or plane4 ,these notions we possess1 but to define them accurately is difficult4 ,euclid's ,definition #d1 ,i41 ,a straight line is that which lies evenly between its extreme points1 must be meaningless to any one who has not the notion of straightness in his mind4 ,neither does it state a property of the straight line which can be used in any further investigation4 ,such a property is given in ,axiom #aj1 ,i4 ,it is really this axiom1 together with ,postulates #b and #c1 which characterizes the straight line4 ,whilst for the straight line the verbal definition and axiom are kept apart1 ,euclid mixes them up in the case of the plane4 ,here the ,definition #g1 ,i41 includes an axiom4 ,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 surface4 ,but if we take a straight line and a point in such a surface1 and draw all straight lines which join the latter to all points in the first line1 the surface will be fully determined4 ,this construction is therefore sufficient as a definition4 ,that every other straight line which joins any two points in this surface lies altogether in it is a further property1 and to assume it gives another axiom4 ,thus a number of ,euclid's axioms are hidden among his first definitions4 ,a still greater confusion exists in the present editions of ,euclid between the postulates and axioms so called1 but this is due to later editors and not to ,euclid himself4 ,the latter had the last three axioms put together with the postulates ,greek3 7aitemata71 so that these were meant to include all assumptions relating to space4 ,the remaining assumptions1 which relate to magnitudes in general1 viz4 the first eight axioms in modern editions1 were called common notions ,greek3 7koivai ennoiai74 ,of the latter a few may be said to be definitions4 ,thus the eighth might be taken as a definition of equal1 and the seventh of halves4 ,if we wish to collect the axioms used in ,euclid's ,elements1 we have therefore to take the three postulates1 the last three axioms as generally given1 a few axioms hidden in the definitions1 and an axiom used by ,euclid in the proof of ,prop4 #d1 ,i1 and on a few other occasions1 viz4 that figures may be moved in space without change of shape or size4 ,s #d4 ,postulates4--,the assumptions actually made by ,euclid may be stated as follows3-- 7#a7 ,straight lines exist which have the property that any one of them may be produced both ways without limit1 that through any two points in space such a line may be drawn1 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 other4 7,this gives the contents of ,def4 #d1 part of ,def4 #ce1 the first two ,postulates1 and ,axiom #aj47 7#b7 ,plane surfaces or planes exist having the property laid down in ,def4 #g1 that every straight line joining any two points in such a surface lies altogether in it4 7#c7 ,right angles1 as defined in ,def4 #aj1 are possible1 and all right angles are equal2 that is to say1 wherever in space we take a plane1 and wherever in that plane we construct a right angle1 all angles thus constructed will be equal1 so that any one of them may be made to coincide with any other4 7,axiom #aa47 7#d7 ,the #abth ,axiom of ,euclid4 ,this we shall not state now1 but only introduce it when we cannot proceed any further without it4 7#e7 ,figures maybe freely moved in space without change of shape or size4 ,this is assumed by ,euclid1 but not stated as an axiom4 7#f7 ,in any plane a circle may be described1 having any point in that plane as centre1 and its distance from any other point in that plane as radius4 7,postulate #c47 ,the definitions which have not been mentioned are all nominal definitions1 that is to say1 they fix a name for a thing described4 ,many of them overdetermine a figure4 ,s #e4 ,euclid's ,elements 7see ,,euclid7 are contained in thirteen books4 ,of these the first four and the sixth are devoted to plane geometry1 as the investigation of figures in a plane is generally called4 ,the #eth book contains the theory of proportion which is used in ,book ,,vi4 ,the #gth1 #hth and #ith books are purely arithmetical1 whilst the #ajth contains a most ingenious treatment of geometrical irrational quantities4 ,these four books will be excluded from our survey4 ,the remaining three books relate to figures in space1 or1 as it is generally called1 to solid geometry4 ,the #gth1 #hth1 #ith1 #ajth1 #acth and part of the #aath and #abth books are now generally omitted from the school editions of the ,elements4 ,in the first four and in the #fth book it is to be understood that all figures are drawn in a plane4 ,,book ,i4 ,,of ,,euclid',s ,,elements4 ,s #f4 ,according to the third postulate it is possible to draw in any plane a circle which has its centre at any given point1 and its radius equal to the distance of this point from any other point given in the plane4 ,this makes it possible 7,prop4 #a7 to construct on a given line ,,ab an equilateral triangle1 by drawing first a circle with ,a as centre and ,,ab as radius1 and then a circle with ,b as centre and ,,ba as radius4 ,the point where these circles intersect--that they intersect ,euclid quietly assumes--is the vertex of the required triangle4 ,euclid does not suppose1 however1 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 centre4 ,this implies also that we are not supposed to be able to make any straight line equal to any other straight line1 or to carry a distance about in space4 ,euclid therefore next solves the problem3 ,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 plane4 ,this is done in two steps4 ,it is shown in ,prop4 #b how a straight line may be drawn from a given point equal in length to another given straight line not drawn from that point4 ,and then the problem itself is solved in ,prop4 #c1 by drawing first through the given point some straight line of the required length1 and then about the same point as centre a circle having this length as radius4 ,this circle will cut off from the given straight line a length equal to the required one4 ,nowadays1 instead of going through this long process1 we take a pair of compasses and set off the given length by its aid4 ,this assumes that we may move a length about without changing it4 ,but ,euclid has not assumed it1 and this proceeding would be fully justified by his desire not to take for granted more than was necessary1 if he were not obliged at his very next step actually to make this assumption1 though without stating it4 ,s #g4 ,we now come 7in ,prop4 #d7 to the first theorem4 ,it is the fundamental theorem of ,euclid's whole system1 there being only a very few propositions 7like ,props4 #ac1 #ad1 #ae1 ,i471 except those in the #eth book and the first half of the #aath1 which do not depend upon it4 ,it is stated very accurately1 though somewhat clumsily1 as follows3-- ,if two triangles have two sides of the one equal to two sides of the other1 each to each1 and have also the angles contained by those sides equal to one another1 they shall also have their bases or third sides equal2 and the two triangles shall be equal2 and their other angles shall be equal1 each to each1 namely1 those to which the equal sides are opposite4 ,that is to say1 the triangles are identically equal1 and one may be considered as a copy of the other4 ,the proof is very simple4 ,the first triangle is taken up and placed on the second1 so that the parts of the triangles which are known to be equal fall upon each other4 ,it is then easily seen that also the remaining parts of one coincide with those of the other1 and that they are therefore equal4 ,this process of applying one figure to another ,euclid scarcely uses again1 though many proofs would be simplified by doing so4 ,the process introduces motion into geometry1 and includes1 as already stated1 the axiom that figures may be moved without change of shape or size4 ,if the last proposition be applied to an isosceles triangle1 which has two sides equal1 we obtain the theorem 7,prop4 #e71 if two sides of a triangle are equal1 then the angles opposite these sides are equal4 ,euclid's proof is somewhat complicated1 and a stumbling-block to many schoolboys4 ,the proof becomes much simpler if we consider the isosceles triangle ,,abc 7,,ab ,,ac7 twice over1 once as a triangle ,,bac1 and once as a triangle ,,cab2 and now remember that ,,ab1 ,,ac in the first are equal respectively to ,,ac1 ,,ab in the second1 and the angles included by these sides are equal4 ,hence the triangles are equal1 and the angles in the one are equal to those in the other1 viz4 those which are opposite equal sides1 i4e4 angle ,,abc in the first equals angle ,,acb in the second1 as they are opposite the equal sides ,,ac and ,,ab in the two triangles4 ,there follows the converse theorem 7,prop4 #f74 ,if two angles in a triangle are equal1 then the sides opposite them are equal1--i4e4 the triangle is isosceles4 ,the proof given consists in what is called a reductio ad absurdum1 a kind of proof often used by ,euclid1 and principally in proving the converse of a previous theorem4 ,it assumes that the theorem to be proved is wrong1 and then shows that this assumption leads to an absurdity1 i4e4 to a conclusion which is in contradiction to a proposition proved before--that therefore the assumption made cannot be true1 and hence that the theorem is true4 ,it is often stated that ,euclid invented this kind of proof1 but the method is most likely much older4 ,s #h4 ,it is next proved that two triangles which have the three sides of the one equal respectively to those of the other are identically equal1 hence that the angles of the one are equal respectively to those of the other1 those being equal which are opposite equal sides4 ,this is ,prop4 #h1 ,prop4 #g containing only a first step towards its proof4 ,these theorems allow now of the solution of a number of problems1 viz43-- ,to bisect a given angle 7,prop4 #i74 ,to bisect a given finite straight line 7,prop4 #aj74 ,to draw a straight line perpendicularly to a given straight line through a given point in it 7,prop4 #aa71 and also through a given point not in it 7,prop4 #ab74 ,the solutions all depend upon properties of isosceles triangles4 ,s #i4 ,the next three theorems relate to angles only1 and might have been proved before ,prop4 #d1 or even at the very beginning4 ,the first 7,prop4 #ac7 says1 ,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 angles4 ,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 line1 and had besides defined the sum of two angles4 ,its converse 7,prop4 #ad7 is of great use1 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 other4 ,so also is ,prop4 #ae4 ,if two straight lines cut one another1 the vertical or opposite angles shall be equal4 ,s #aj4 ,euclid returns now to properties of triangles4 ,of great importance for the next steps 7though afterwards superseded by a more complete theorem7 is ,prop4 #af4 ,if one side of a triangle be produced1 the exterior angle shall be greater than either of the interior opposite angles4 ,prop4 #ag4 ,any two angles of a triangle are together less than two right angles1 is an immediate consequence of it4 ,by the aid of these two1 the following fundamental properties of triangles are easily proved3-- ,prop4 #ah4 ,the greater side of every triangle has the greater angle opposite to it2 ,its converse1 ,prop4 #ai4 ,the greater angle of every triangle is subtended by the greater side1 or has the greater side opposite to it2 ,prop4 #bj4 ,any two sides of a triangle are together greater than the third side2 ,and also ,prop4 #ba4 ,if from the ends of the side of a triangle there be drawn two straight lines to a point within the triangle1 these shall be less than the other two sides of the triangle1 but shall contain a greater angle4 ,s #aa4 ,having solved two problems 7,props4 #bb1 #bc71 he returns to two triangles which have two sides of the one equal respectively to two sides of the other4 ,it is known 7,prop4 #d7 that if the included angles are equal then the third sides are equal2 and conversely 7,prop4 #h71 if the third sides are equal1 then the angles included by the first sides are equal4 ,from this it follows that if the included angles are not equal1 the third sides are not equal2 and conversely1 that if the third sides are not equal1 the included angles are not equal4 ,euclid now completes this knowledge by proving1 that if the included angles are not equal1 then the third side in that triangle is the greater which contains the greater angle2 and conversely1 that if the third sides are unequal1 that triangle contains the greater angle which contains the greater side4 ,these are ,prop4 #bd and ,prop4 #be4 ,s #ab4 ,the next theorem 7,prop4 #bf7 says that if two triangles have one side and two angles of the one equal respectively to one side and two angles of the other1 viz4 in both triangles either the angles adjacent to the equal side1 or one angle adjacent and one angle opposite it1 then the two triangles are identically equal4 ,this theorem belongs to a group with ,prop4 #d and ,prop4 #h4 ,its first case might have been given immediately after ,prop4 #d1 but the second case requires ,prop4 #af for its proof4 ,s #ac4 ,we come now to the investigation of parallel straight lines1 i4e4 of straight lines which lie in the same plane1 and cannot be made to meet however far they be produced either way4 ,the investigation which starts from ,prop4 #af1 will become clearer if a few names be explained which are not all used by ,euclid4 ,if two straight lines be cut by a third1 the latter is now generally called a transversal of the figure4 ,it forms at the two points where it cuts the given lines four angles with each4 ,those of the angles which lie between the given lines are called interior angles1 and of these1 again1 any two which lie on opposite sides of the transversal but one at each of the two points are called alternate angles4 ,we may now state ,prop4 #af thus3--,if two straight lines which meet are cut by a transversal1 their alternate angles are unequal4 ,for the lines will form a triangle1 and one of the alternate angles will be an exterior angle to the triangle1 the other interior and opposite to it4 ,from this follows at once the theorem contained in ,prop4 #bg4 ,if two straight lines which are cut by a transversal make alternate angles equal1 the lines cannot meet1 however far they be produced1 hence they are parallel4 ,this proves the existence of parallel lines4 ,prop4 #bh states the same fact in different forms4 ,if a straight line1 falling on two other straight lines1 make the exterior angle equal to the interior and opposite angle on the same side of the line1 or make the interior angles on the same side together equal to two right angles1 the two straight lines shall be parallel to one another4 ,hence we know that1 if two straight lines which are cut by a transversal meet1 their alternate angles are not equal2 and hence that1 if alternate angles are equal1 then the lines are parallel4 ,the question now arises1 ,are the propositions converse to these true or not8 ,that is to say1 ,if alternate angles are unequal1 do the lines meet8 ,and if the lines are parallel1 are alternate angles necessarily equal8 ,the answer to either of these two questions implies the answer to the other4 ,but it has been found impossible to prove that the negation or the affirmation of either is true4 ,the difficulty which thus arises is overcome by ,euclid assuming that the first question has to be answered in the affirmative4 ,this gives his last axiom 7#ab71 which we quote in his own words4 ,axiom #ab4--;,if a straight line meet two straight lines1 so as to make the two interior angles on the same side of it taken together less than two right angles1 these straight lines1 being continually produced1 shall at length meet on that side on which are the angles which are less than two right angles4 ,the answer to the second of the above questions follows from this1 and gives the theorem ,prop4 #bi3--;,if a straight line fall on two parallel straight lines1 it makes the alternate angles equal to one another1 and the exterior angle equal to the interior and opposite angle on the same side1 and also the two interior angles on the same side together equal to two right angles4 ,s #ad4 ,with this a new part of elementary geometry begins4 ,the earlier propositions are independent of this axiom1 and would be true even if a wrong assumption had been made in it4 ,they all relate to figures in a plane4 ,but a plane is only one among an infinite number of conceivable surfaces4 ,we may draw figures on any one of them and study their properties4 ,we may1 for instance1 take a sphere instead of the plane1 and obtain spherical in the place of plane geometry4 ,if on one of these surfaces lines and figures could be drawn1 answering to all the definitions of our plane figures1 and if the axioms with the exception of the last all hold1 then all propositions up to the #bhth will be true for these figures4 ,this is the case in spherical geometry if we substitute shortest line or great circle for straight line1 small circle for circle1 and if1 besides1 we limit all figures to a part of the sphere which is less than a hemisphere1 so that two points on it cannot be opposite ends of a diameter1 and therefore determine always one and only one great circle4 ,for spherical triangles1 therefore1 all the important propositions #d1 #h1 #bf2 #e and #f2 and #ah1 #ai and #bj will hold good4 ,this remark will be sufficient to show the impossibility of proving ,euclid's last axiom1 which would mean proving that this axiom is a consequence of the others1 and hence that the theory of parallels would hold on a spherical surface1 where the other axioms do hold1 whilst parallels do not even exist4 ,it follows that the axiom in question states an inherent difference between the plane and other surfaces1 and that the plane is only fully characterized when this axiom is added to the other assumptions4 ,s #ae4 ,the introduction of the new axiom and of parallel lines leads to a new class of propositions4 ,after proving 7,prop4 #cj7 that two lines which are each parallel to a third are parallel to each other1 we obtain the new properties of triangles contained in ,prop4 #cb4 ,of these the second part is the most important1 viz4 the theorem1 ,the three interior angles of every triangle are together equal to two right angles4 ,as easy deductions not given by ,euclid but added by ,simson follow the propositions about the angles in polygons1 they are given in ,english editions as corollaries to ,prop4 #cb4 ,these theorems do not hold for spherical figures4 ,the sum of the interior angles of a spherical triangle is always greater than two right angles1 and increases with the area4 ,s #af4 ,the theory of parallels as such may be said to be finished with ,props4 #cc and #cd1 which state properties of the parallelogram1 i4e4 of a quadrilateral formed by two pairs of parallels4 ,they are-- ,prop4 #cc4 ,the straight lines which join the extremities of two equal and parallel straight lines towards the same parts are themselves equal and parallel2 and ,prop4 #cd4 ,the opposite sides and angles of a parallelogram are equal to one another1 and the diameter 7diagonal7 bisects the parallelogram1 that is1 divides it into two equal parts4 ,s #ag4 ,the rest of the first book relates to areas of figures4 ,the theory is made to depend upon the theorems-- ,prop4 #ce4 ,parallelograms on the same base and between the same parallels are equal to one another2 and ,prop4 #cf4 ,parallelograms on equal bases and between the same parallels are equal to one another4 ,as each parallelogram is bisected by a diagonal1 the last theorems hold also if the word parallelogram be replaced by triangle1 as is done in ,props4 #cg and #ch4 ,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 line4 ,the theorems converse to the last form the contents of the next three propositions1 viz43 ,props1 #dj and #da4--;,equal triangles1 on the same or on equal bases1 in the same straight line1 and on the same side of it1 are between the same parallels4 ,that the two cases here stated are given by ,euclid in two separate propositions proved separately is characteristic of his method4 ,s #ah4 ,to compare areas of other figures1 ,euclid shows first1 in ,prop4 #db1 how to draw a parallelogram which is equal in area to a given triangle1 and has one of its angles equal to a given angle4 ,if the given angle is right1 then the problem is solved to draw a rectangle equal in area to a given triangle4 ,next this parallelogram is transformed into another parallelogram1 which has one of its sides equal to a given straight line1 whilst its angles remain unaltered4 ,this may be done by aid of the theorem in ,prop4 #dc4 ,the complements of the parallelograms which are about the diameter of any parallelogram are equal to one another4 ,thus the problem 7,prop4 #dd7 is solved to construct a parallelogram on a given line1 which is equal in area to a given triangle1 and which has one angle equal to a given angle 7generally a right angle74 ,as every polygon can be divided into a number of triangles1 we can now construct a parallelogram having a given angle1 say a right angle1 and being equal in area to a given polygon4 ,for each of the triangles into which the polygon has been divided1 a parallelogram may be constructed1 having one side equal to a given straight line and one angle equal to a given angle4 ,if these parallelograms be placed side by side1 they may be added together to form a single parallelogram1 having still one side of the given length4 ,this is done in ,prop4 #de4 ,herewith a means is found to compare areas of different polygons4 ,we need only construct two rectangles equal in area to the given polygons1 and having each one side of given length4 ,by comparing the unequal sides we are enabled to judge whether the areas are equal1 or which is the greater4 ,euclid does not state this consequence1 but the problem is taken up again at the end of the second book1 where it is shown how to construct a square equal in area to a given polygon4 ,prop4 #df is3 ,to describe a square on a given straight line4 ,s #ai4 ,the first book concludes with one of the most important theorems in the whole of geometry1 and one which has been celebrated since the earliest times4 ,it is stated1 but on doubtful authority1 that ,pythagoras discovered it1 and it has been called by his name4 ,if we call that side in a right-angled triangle which is opposite the right angle the hypotenuse1 we may state it as follows3-- ,theorem of ,pythagoras 7,prop4 #dg74--;,in every right-angled triangle the square on the hypotenuse is equal to the sum of the squares of the other sides4 ,and conversely-- ,prop4 #dh4 ,if the square described on one of the sides of a triangle be equal to the squares described on the other sides1 then the angle contained by these two sides is a right angle4 ,on this theorem 7,prop4 #dg7 almost all geometrical measurement depends1 which cannot be directly obtained4 ,,book ,,ii4 ,s #bj4 ,the propositions in the second book are very different in character from those in the first2 they all relate to areas of rectangles and squares4 ,their true significance is best seen by stating them in an algebraic form4 ,this is often done by expressing the lengths of lines by aid of numbers1 which tell how many times a chosen unit is contained in the lines4 ,if there is a unit to be found which is contained an exact number of times in each side of a rectangle1 it is easily seen1 and generally shown in the teaching of arithmetic1 that the rectangle contains a number of unit squares equal to the product of the numbers which measure the sides1 a unit square being the square on the unit line4 ,if1 however1 no such unit can be found1 this process requires that connexion between lines and numbers which is only established by aid of ratios of lines1 and which is therefore at this stage altogether inadmissible4 ,but there exists another way of connecting these propositions with algebra1 based on modern notions which seem destined greatly to change and to simplify mathematics4 ,we shall introduce here as much of it as is required for our present purpose4 ,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 angles2 in the text this phraseology is extended by speaking of rectangles contained by any two straight lines1 meaning the rectangle which has two adjacent sides equal to the two straight lines4 ,we shall denote a finite straight line by a single small letter1 a1 b1 c1 444 x1 and the area of the rectangle contained by two lines a and b by ab1 and this we shall call the product of the two lines a and b4 ,it will be understood that this definition has nothing to do with the definition of a product of numbers4 ,we define as follows3-- ,the sum of two straight lines a and b means a straight line c which may be divided in two parts equal respectively to a and b4 ,this sum is denoted by a b4 ,the difference of two lines a and b 7in symbols1 a-b7 means a line c which when added to b gives a2 that is1 a - b c if b c a4 ,the product of two lines a and b 7in symbols1 ab7 means the area of the rectangle contained by the lines a and b4 ,for aa1 which means the square on the line a1 we write a#b4 ,s #ba4 ,the first ten of the fourteen propositions of the second book may then be written in the form of formulae as follows3-- ,prop4 #a4 a7b c d 444 7 ab ac ad 444 #b4 ab ac a#b if b c a4 #c4 a7a b7 a#b ab4 #d4 7a b7#b a#b #b;ab b#b4 #e4 7a b77a - b7 b#b a#b4 #f4 7a b77a - b7 b#b a#b4 #g4 a#b 7a - b7#b #b;a7a - b7 b#b4 #h4 #d7;a b7a b#b 7#b;a b7#b4 #i4 7a b7#b 7a - b7#b #b;a#b #b;b#b4 #aj4 7a b7#b 7a - b7#b #b;a#b #b;b#b4 ,it will be seen that #e and #f1 and also #i and #aj1 are identical4 ,in ,euclid's statement they do not look the same1 the figures being arranged differently4 ,if the letters a1 b1 c1 444 denoted numbers1 it follows from algebra that each of these formulae is true4 ,but this does not prove them in our case1 where the letters denote lines1 and their products areas without any reference to numbers4 ,to prove them we have to discover the laws which rule the operations introduced1 viz4 addition and multiplication of segments4 ,this we shall do now2 and we shall find that these laws are the same with those which hold in algebraical addition and multiplication4 ,s #bb4 ,in a sum of numbers we may change the order in which the numbers are added1 and we may also add the numbers together in groups and then add these groups4 ,but this also holds for the sum of segments and for the sum of rectangles1 as a little consideration shows4 ,that the sum of rectangles has always a meaning follows from the ,props4 #dc-de in the first book4 ,these laws about addition are reducible to the two-- a b b a 7#a71 a 7b c7 a b c 7#b72 or1 when expressed for rectangles1 ab ed ed ab 7#c71 ab 7cd ef7 ab cd ef 7#d74 ,the brackets mean that the terms in the bracket have been added together before they are added to another term4 ,the more general cases for more terms may be deduced from the above4 ,for the product of two numbers we have the law that it remains unaltered if the factors be interchanged4 ,this also holds for our geometrical product4 ,for if ab denotes the area of the rectangle which has a as base and b as altitude1 then ba will denote the area of the rectangle which has b as base and a as altitude4 ,but in a rectangle we may take either of the two lines which contain it as base1 and then the other will be the altitude4 ,this gives ab ba 7#e74 ,in order further to multiply a sum by a number1 we have in algebra the rule3--,multiply each term of the sum1 and add the products thus obtained4 ,that this holds for our geometrical products is shown by ,euclid in his first proposition of the second book1 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 bases4 ,in symbols this gives1 in the simplest case1 a7b c7 ab ac 7#f74 and 7b c7a ba ca _/ ,to these laws1 which have been investigated by ,sir ,william ,hamilton and by ,hermann ,grassmann1 the former has given special names4 ,he calls the laws expressed in 7#a7 and 7#c7 the commutative law for addition2 7#e7 multiplication2 7#b7 and 7#d7 the associative laws for addition2 7#f7 the distributive law4 ,s #bc4 ,having proved that these six laws hold1 we can at once prove every one of the above propositions in their algebraical form4 ,the first is proved geometrically1 it being one of the fundamental laws4 ,the next two propositions are only special cases of the first4 ,of the others we shall prove one1 viz4 the fourth3-- 7a b7#b 7a b77a b7 7a b7a 7a b7b by 7#f74 ,but 7a b7a aa ba by 7#f71 aa ab by 7#e72 and 7a b7b ab bb by 7#f74 ,therefore 7a b7#b aa ab 7ab bb7 aa 7ab ab7 bb by 7#d74 aa #b;ab bb _/ ,this gives the theorem in question4 ,in the same manner every one of the first ten propositions is proved4 ,it will be seen that the operations performed are exactly the same as if the letters denoted numbers4 ,props4 #e and #f may also be written thus-- 7a b77a - b7 a#b - b#b4 ,prop4 #g1 which is an easy consequence of ,prop4 #d1 may be transformed4 ,if we denote by c the line a b1 so that c a b1 a c - b1 we get c#b 7c - b7#b #b;c7c - b7 b#b #b;c#b - #b;bc b#b4 ,subtracting c#b from both sides1 and writing a for c1 we get 7a - b7#b a#b - #b;ab b#b4 ,in ,euclid's ,elements this form of the theorem does not appear1 all propositions being so stated that the notion of subtraction does not enter into them4 ,s #bd4 ,the remaining two theorems 7,props4 #ab and #ac7 connect the square on one side of a triangle with the sum of the squares on the other sides1 in case that the angle between the latter is acute or obtuse4 ,they are important theorems in trigonometry1 where it is possible to include them in a single theorem4 ,s #be4 ,there are in the second book two problems1 ,props4 #aa and #ad4 ,if written in the above symbolic language1 the former requires to find a line x such that a7a - x7 x#b4 ,prop4 #aa contains1 therefore1 the solution of a quadratic equation1 which we may write x#b ax a#b4 ,the solution is required later on in the construction of a regular decagon4 ,more important is the problem in the last proposition 7,prop4 #ad74 ,it requires the construction of a square equal in area to a given rectangle1 hence a solution of the equation x#b ab4 ,in ,book ,i41 #db-de1 it has been shown how a rectangle may be constructed equal in area to a given figure bounded by straight lines4 ,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 figure1 or we can square any such figure4 ,as of two squares that is the greater which has the greater side1 it follows that now the comparison of two areas has been reduced to the comparison of two lines4 ,the problem of reducing other areas to squares is frequently met with among ,greek mathematicians4 ,we need only mention the problem of squaring the circle 7see ,,circle74 ,in the present day the comparison of areas is performed in a simpler way by reducing all areas to rectangles having a common base4 ,their altitudes give then a measure of their areas4 ,the construction of a rectangle having the base u1 and being equal in area to a given rectangle1 depends upon ,prop4 #dc1 ,i4 ,this therefore gives a solution of the equation ab ux1 where x denotes the unknown altitude4 ,,book ,,iii4 ,s #bf4 ,the third book of the ,elements relates exclusively to properties of the circle4 ,a circle and its circumference have been defined in ,book ,i41 ,def4 #ae4 ,we restate it here in slightly different words3-- ,definition4--,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 plane4 ,this point is called the centre of the circle4 ,of the new definitions1 of which eleven are given at the beginning of the third book1 a few only require special mention4 ,the first1 which says that circles with equal radii are equal1 is in part a theorem1 but easily proved by applying the one circle to the other4 ,or it may be considered proved by aid of ,prop4 #bd1 equal circles not being used till after this theorem4 ,in the second definition is explained what is meant by a line which touches a circle4 ,such a line is now generally called a tangent to the circle4 ,the introduction of this name allows us to state many of ,euclid's propositions in a much shorter form4 ,for the same reason we shall call a straight line joining two points on the circumference of a circle a chord4 ,definitions #d and #e may be replaced with a slight generalization by the following3-- ,definition4--,by the distance of a point from a line is meant the length of the perpendicular drawn from the point to the line4 ,s #bg4 ,from the definition of a circle it follows that every circle has a centre4 ,prop4 #a requires to find it when the circle is given1 i4e4 when its circumference is drawn4 ,to solve this problem a chord is drawn 7that is1 any two points in the circumference are joined71 and through the point where this is bisected a perpendicular to it is erected4 ,euclid then proves1 first1 that no point off this perpendicular can be the centre1 hence that the centre must lie in this line2 and1 secondly1 that of the points on the perpendicular one only can be the centre1 viz4 the one which bisects the parts of the perpendicular bounded by the circle4 ,in the second part ,euclid silently assumes that the perpendicular there used does cut the circumference in two1 and only in two points4 ,the proof therefore is incomplete4 ,the proof of the first part1 however1 is exact4 ,by drawing two non-parallel chords1 and the perpendiculars which bisect them1 the centre will be found as the point where these perpendiculars intersect4 ,s #bh4 ,in ,prop4 #b it is proved that a chord of a circle lies altogether within the circle4 ,what we have called the first part of ,euclid's solution of ,prop4 #a may be stated as a theorem3-- ,every straight line which bisects a chord1 and is at right angles to it1 passes through the centre of the circle4 ,the converse to this gives ,prop4 #c1 which may be stated thus3-- ,if a straight line through the centre of a circle bisect a chord1 then it is perpendicular to the chord1 and if it be perpendicular to the chord it bisects it4 ,an easy consequence of this is the following theorem1 which is essentially the same as ,prop4 #d3-- ,two chords of a circle1 of which neither passes through the centre1 cannot bisect each other4 ,these last three theorems are fundamental for the theory of the circle4 ,it is to be remarked that ,euclid never proves that a straight line cannot have more than two points in common with a circumference4 ,s #bi4 ,the next two propositions 7#e and #f7 might be replaced by a single and a simpler theorem1 viz3-- ,two circles which have a common centre1 and whose circumferences have one point in common1 coincide4 ,or1 more in agreement with ,euclid's form3-- ,two different circles1 whose circumferences have a point in common1 cannot have the same centre4 ,that ,euclid treats of two cases is characteristic of ,greek mathematics4 ,the next two propositions 7#g and #h7 again belong together4 ,they may be combined thus3-- ,if from a point in a plane of a circle1 which is not the centre1 straight lines be drawn to the different points of the circumference1 then of all these lines one is the shortest1 and one the longest1 and these lie both in that straight line which joins the given point to the centre4 ,of all the remaining lines each is equal to one and only one other1 and these equal lines lie on opposite sides of the shortest or longest1 and make equal angles with them4 ,euclid distinguishes the two cases where the given point lies within or without the circle1 omitting the case where it lies in the circumference4 ,from the last proposition it follows that if from a point more than two equal straight lines can be drawn to the circumference1 this point must be the centre4 ,this is ,prop4 #i4 ,as a consequence of this we get ,if the circumferences of the two circles have three points in common they coincide4 ,for in this case the two circles have a common centre1 because from the centre of the one three equal lines can be drawn to points on the circumference of the other4 ,but two circles which have a common centre1 and whose circumferences have a point in common1 coincide4 7,compare above statement of ,props4 #e and #f47 ,this theorem may also be stated thus3-- ,through three points only one circumference may be drawn2 or1 ,three points determine a circle4 ,euclid does not give the theorem in this form4 ,he proves1 however1 that the two circles cannot cut another in more than two points 7,prop4 #aj71 and that two circles cannot touch one another in more points than one 7,prop4 #ac74 ,s #cj4 ,propositions #aa and #ab assert that if two circles touch1 then the point of contact lies on the line joining their centres4 ,this gives two propositions1 because the circles may touch either internally or externally4 ,s #ca4 ,propositions #ad and #ae relate to the length of chords4 ,the first says that equal chords are equidistant from the centre1 and that chords which are equidistant from the centre are equal2 ,whilst ,prop4 #ae compares unequal chords1 viz4 ,of all chords the diameter is the greatest1 and of other chords that is the greater which is nearer to the centre2 and conversely1 the greater chord is nearer to the centre4 ,s #cb4 ,in ,prop4 #af the tangent to a circle is for the first time introduced4 ,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 tangent4 ,the proposition itself does not state this4 ,it runs thus3-- ,prop4 #af4 ,the straight line drawn at right angles to the diameter of a circle1 from the extremity of it1 falls without the circle2 and no straight line can be drawn from the extremity1 between that straight line and the circumference1 so as not to cut the circle4 ,corollary4--,the straight line at right angles to a diameter drawn through the end point of it touches the circle4 ,the statement of the proposition and its whole treatment show the difficulties which the tangents presented to ,euclid4 ,prop4 #ag solves the problem through a given point1 either in the circumference or without it1 to draw a tangent to a given circle4 ,closely connected with ,prop4 #af are ,props4 #ah and #ai1 which state 7,prop4 #ah71 that the line joining the centre of a circle to the point of contact of a tangent is perpendicular to the tangent2 and conversely 7,prop4 #ai71 that the straight line through the point of contact of1 and perpendicular to1 a tangent to a circle passes through the centre of the circle4 ,s #cc4 ,the rest of the book relates to angles connected with a circle1 viz4 angles which have the vertex either at the centre or on the circumference1 and which are called respectively angles at the centre and angles at the circumference4 ,between these two kinds of angles exists the important relation expressed as follows3-- ,prop4 #bj4 ,the angle at the centre of a circle is double of the angle at the circumference on the same base1 that is1 on the same arc4 ,this is of great importance for its consequences1 of which the two following are the principal3-- ,prop4 #ba4 ,the angles in the same segment of a circle are equal to one another2 ,prop4 #bb4 ,the opposite angles of any quadrilateral figure inscribed in a circle are together equal to two right angles4 ,further consequences are3-- ,prop4 #bc4 ,on the same straight line1 and on the same side of it1 there cannot be two similar segments of circles1 not coinciding with one another2 ,prop4 #bd4 ,similar segments of circles on equal straight lines are equal to one another4 ,the problem ,prop4 #be4 ,a segment of a circle being given to describe the circle of which it is a segment1 may be solved much more easily by aid of the construction described in relation to ,prop4 #a1 ,,iii41 in ,s #bg4 ,s #cd4 ,there follow four theorems connecting the angles at the centre1 the arcs into which they divide the circumference1 and the chords subtending these arcs4 ,they are expressed for angles1 arcs and chords in equal circles1 but they hold also for angles1 arcs and chords in the same circle4 ,the theorems are3-- ,prop4 #bf4 ,in equal circles equal angles stand on equal arcs1 whether they be at the centres or circumferences2 ,prop4 #bg4 7converse to ,prop4 #bf74 ,in equal circles the angles which stand on equal arcs are equal to one another1 whether they be at the centres or the circumferences2 ,prop4 #bh4 ,in equal circles equal straight lines 7equal chords7 cut off equal arcs1 the greater equal to the greater1 and the less equal to the less2 ,prop4 #bi 7converse to ,prop4 #bh74 ,in equal circles equal arcs are subtended by equal straight lines4 ,s #ce4 ,other important consequences of ,props4 #bj-bb are3-- ,prop4 #ca4 ,in a circle the angle in a semicircle is a right angle2 but the angle in a segment greater than a semicircle is less than a right angle2 and the angle in a segment less than a semicircle is greater than a right angle2 ,prop4 #cb4 ,if a straight line touch a circle1 and from the point of contact a straight line be drawn cutting the circle1 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 circle4 ,s #cf4 ,propositions #cj1 #cc1 #cd1 contain problems which are solved by aid of the propositions preceding them3-- ,prop4 #cj4 ,to bisect a given arc1 that is1 to divide it into two equal parts2 ,prop4 #cc4 ,on a given straight line to describe a segment of a circle containing an angle equal to a given rectilineal angle2 ,prop4 #cd4 ,from a given circle to cut off a segment containing an angle equal to a given rectilineal angle4 ,s #cg4 ,if we draw chords through a point ,a within a circle1 they will each be divided by ,a into two segments4 ,between these segments the law holds that the rectangle contained by them has the same area on whatever chord through ,a the segments are taken4 ,the value of this rectangle changes1 of course1 with the position of ,a4 ,a similar theorem holds if the point ,a be taken without the circle4 ,on every straight line through ,a1 which cuts the circle in two points ,b and ,c1 we have two segments ,,ab and ,,ac1 and the rectangles contained by them are again equal to one another1 and equal to the square on a tangent drawn from ,a to the circle4 ,the first of these theorems gives ,prop4 #ce1 and the second ,prop4 #cf1 with its corollary1 whilst ,prop4 #cg1 the last of ,book ,,iii41 gives the converse to ,prop4 #cf4 ,the first two theorems may be combined in one3-- ,if through a point ,a in the plane of a circle a straight line be drawn cutting the circle in ,b and ,c1 then the rectangle ,,ab4,,ac has a constant value so long as the point ,a be fixed2 and if from ,a a tangent ,,ad can be drawn to the circle1 touching at ,d1 then the above rectangle equals the square on ,,ad4 ,prop4 #cg may be stated thus3-- ,if from a point ,a without a circle a line be drawn cutting the circle in ,b and ,c1 and another line to a point ,d on the circle1 and ,,ab4,,ac ,,ad#b1 then the line ,,ad touches the circle at ,d4 ,it is not difficult to prove also the converse to the general proposition as above stated4 ,this proposition and its converse may be expressed as follows3-- ,if four points ,,abcd be taken on the circumference of a circle1 and if the lines ,,ab1 ,,cd1 produced if necessary1 meet at ,e1 then ,,ea4,,eb ,,ec4,,ed2 and conversely1 if this relation holds then the four points lie on a circle1 that is1 the circle drawn through three of them passes through the fourth4 ,that a circle may always be drawn through three points1 provided that they do not lie in a straight line1 is proved only later on in ,book ,,iv4 ,,book ,,iv4 ,s #ch4 ,the fourth book contains only problems1 all relating to the construction of triangles and polygons inscribed in and circumscribed about circles1 and of circles inscribed in or circumscribed about triangles and polygons4 ,they are nearly all given for their own sake1 and not for future use in the construction of figures1 as are most of those in the former books4 ,in seven definitions at the beginning of the book it is explained what is understood by figures inscribed in or described about other figures1 with special reference to the case where one figure is a circle4 ,instead1 however1 of saying that one figure is described about another1 it is now generally said that the one figure is circumscribed about the other4 ,we may then state the definitions #c or #d thus3-- ,definition4--,a polygon is said to be inscribed in a circle1 and the circle is said to be circumscribed about the polygon1 if the vertices of the polygon lie in the circumference of the circle4 ,and definitions #e and #f thus3-- ,definition4--,a polygon is said to be circumscribed about a circle1 and a circle is said to be inscribed in a polygon1 if the sides of the polygon are tangents to the circle4 ,s #ci4 ,the first problem is merely constructive4 ,it requires to draw in a given circle a chord equal to a given straight line1 which is not greater than the diameter of the circle4 ,the problem is not a determinate one1 inasmuch as the chord may be drawn from any point in the circumference4 ,this may be said of almost all problems in this book1 especially of the next two4 ,they are3-- ,prop4 #b4 ,in a given circle to inscribe a triangle equiangular to a given triangle2 ,prop4 #c4 ,about a given circle to circumscribe a triangle equiangular to a given triangle4 ,s #dj4 ,of somewhat greater interest are the next problems1 where the triangles are given and the circles to be found4 ,prop4 #d4 ,to inscribe a circle in a given triangle4 ,the result is that the problem has always a solution1 viz4 the centre of the circle is the point where the bisectors of two of the interior angles of the triangle meet4 ,the solution shows1 though ,euclid does not state this1 that the problem has but one solution2 and also1 ,the three bisectors of the interior angles of any triangle meet in a point1 and this is the centre of the circle inscribed in the triangle4 ,the solutions of most of the other problems contain also theorems4 ,of these we shall state those which are of special interest2 ,euclid does not state any one of them4 ,s #da4 ,prop4 #e4 ,to circumscribe a circle about a given triangle4 ,the one solution which always exists contains the following3-- ,the three straight lines which bisect the sides of a triangle at right angles meet in a point1 and this point is the centre of the circle circumscribed about the triangle4 ,euclid adds in a corollary the following property3-- ,the centre of the circle circumscribed about a triangle lies within1 on a side of1 or without the triangle1 according as the triangle is acute-angled1 right-angled or obtuse-angled4 ,s #db4 ,whilst it is always possible to draw a circle which is inscribed in or circumscribed about a given triangle1 this is not the case with quadrilaterals or polygons of more sides4 ,of those for which this is possible the regular polygons1 i4e4 polygons which have all their sides and angles equal1 are the most interesting4 ,in each of them a circle may be inscribed1 and another may be circumscribed about it4 ,euclid does not use the word regular1 but he describes the polygons in question as equiangular and equilateral4 ,we shall use the name regular polygon4 ,the regular triangle is equilateral1 the regular quadrilateral is the square4 ,euclid considers the regular polygons of #d1 #e1 #f and #ae sides4 ,for each of the first three he solves the problems--7#a7 to inscribe such a polygon in a given circle2 7#b7 to circumscribe it about a given circle2 7#c7 to inscribe a circle in1 and 7#d7 to circumscribe a circle about1 such a polygon4 ,for the regular triangle the problems are not repeated1 because more general problems have been solved4 ,props4 #f1 #g1 #h and #i solve these problems for the square4 ,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 parts1 or what comes to the same thing1 of drawing from the centre of the circle n radii such that the angles between consecutive radii are equal1 that is1 to divide the space about the centre into n equal angles4 ,thus1 if it is required to inscribe a square in a circle1 we have to draw four lines from the centre1 making the four angles equal4 ,this is done by drawing two diameters at right angles to one another4 ,the ends of these diameters are the vertices of the required square4 ,if1 on the other hand1 tangents be drawn at these ends1 we obtain a square circumscribed about the circle4 ,s #dc4 ,to construct a regular pentagon1 we find it convenient first to construct a regular decagon4 ,this requires to divide the space about the centre into ten equal angles4 ,each will be #a_/ajth of a right angle1 or #a_/eth of two right angles4 ,if we suppose the decagon constructed1 and if we join the centre to the end of one side1 we get an isosceles triangle1 where the angle at the centre equals #a_/eth of two right angles2 hence each of the angles at the base will be #b_/eths of two right angles1 as all three angles together equal two right angles4 ,thus we have to construct an isosceles triangle1 having the angle at the vertex equal to half an angle at the base4 ,this is solved in ,prop4 #aj1 by aid of the problem in ,prop4 #aa of the second book4 ,if we make the sides of this triangle equal to the radius of the given circle1 then the base will be the side of the regular decagon inscribed in the circle4 ,this side being known the decagon can be constructed1 and if the vertices are joined alternately1 leaving out half their number1 we obtain the regular pentagon4 7,prop4 #aa47 ,euclid does not proceed thus4 ,he wants the pentagon before the decagon4 ,this1 however1 does not change the real nature of his solution1 nor does his solution become simpler by not mentioning the decagon4 ,once the regular pentagon is inscribed1 it is easy to circumscribe another by drawing tangents at the vertices of the inscribed pentagon4 ,this is shown in ,prop4 #ab4 ,props4 #ac and #ad teach how a circle may be inscribed in or circumscribed about any given regular pentagon4 ,s #dd4 ,the regular hexagon is more easily constructed1 as shown in ,prop4 #ae4 ,the result is that the side of the regular hexagon inscribed in a circle is equal to the radius of the circle4 ,for this polygon the other three problems mentioned are not solved4 ,s #de4 ,the book closes with ,prop4 #af4 ,to inscribe a regular quindecagon in a given circle4 ,if we inscribe a regular pentagon and a regular hexagon in the circle1 having one vertex in common1 then the arc from the common vertex to the next vertex of the pentagon is #a_/eth of the circumference1 and to the next vertex of the hexagon is #a_/fth of the circumference4 ,the difference between these arcs is1 therefore1 #a_/e - #a_/f #a_/cjth of the circumference4 ,the latter may1 therefore1 be divided into thirty1 and hence also in fifteen equal parts1 and the regular quindecagon be described4 ,s #df4 ,we conclude with a few theorems about regular polygons which are not given by ,euclid4 ,the straight lines perpendicular to and bisecting the sides of any regular polygon meet in a point4 ,the straight lines bisecting the angles in the regular polygon meet in the same point4 ,this point is the centre of the circles circumscribed about and inscribed in the regular polygon4 ,we can bisect any given arc 7,prop4 #cj1 ,,iii474 ,hence we can divide a circumference into #bn equal parts as soon as it has been divided into n equal parts1 or as soon as a regular polygon of n sides has been constructed4 ,hence-- ,if a regular polygon of n sides has been constructed1 then a regular polygon of #bn sides1 of #dn1 of #hn sides1 c41 may also be constructed4 ,euclid shows how to construct regular polygons of #c1 #d1 #e and #ae sides4 ,it follows that we can construct regular polygons of #c1 #f1 #ab1 #bd sides #d1 #h1 #af1 #cb #e1 #aj1 #bj1 #dj #ae1 #cj1 #fj1 #abj ,the construction of any new regular polygon not included in one of these series will give rise to a new series4 ,till the beginning of the #aith century nothing was added to the knowledge of regular polygons as given by ,euclid4 ,then ,gauss1 in his celebrated ,arithmetic1 proved that every regular polygon of #bn #a sides may be constructed if this number #bn #a be prime1 and that no others except those with #bm7#bn #a7 sides can be constructed by elementary methods4 ,this shows that regular polygons of #g1 #i1 #ac sides cannot thus be constructed1 but that a regular polygon of #ag sides is possible2 for #ag #bd #a4 ,the next polygon is one of #beg sides4 ,the construction becomes already rather complicated for #ag sides4 ,,book ,v4 ,s #dg4 ,the fifth book of the ,elements is not exclusively geometrical4 ,it contains the theory of ratios and proportion of quantities in general4 ,the treatment1 as here given1 is admirable1 and in every respect superior to the algebraical method by which ,euclid's theory is now generally replaced4 ,we shall treat the subject in order to show why the usual algebraical treatment of proportion is not really sound4 ,we begin by quoting those definitions at the beginning of ,book ,v4 which are most important4 ,these definitions have given rise to much discussion4 ,the only definitions which are essential for the fifth book are ,defs4 #a1 #b1 #d1 #e1 #f and #g4 ,of the remainder #c1 #h and #i are more than useless1 and probably not ,euclid's1 but additions of later editors1 of whom ,theon of ,alexandria was the most prominent4 ,defs4 #aj and #aa belong rather to the sixth book1 whilst all the others are merely nominal4 ,the really important ones are #d1 #e1 #f and #g4 ,s #dh4 ,to define a magnitude is not attempted by ,euclid4 ,the first two definitions state what is meant by a part1 that is1 a submultiple or measure1 and by a multiple of a given magnitude4 ,the meaning of ,def4 #d is that two given quantities can have a ratio to one another only in case that they are comparable as to their magnitude1 that is1 if they are of the same kind4 ,def4 #c1 which is probably due to ,theon1 professes to define a ratio1 but is as meaningless as it is uncalled for1 for all that is wanted is given in ,defs4 #e and #g4 ,in ,def4 #e it is explained what is meant by saying that two magnitudes have the same ratio to one another as two other magnitudes1 and in ,def4 #g what we have to understand by a greater or a less ratio4 ,the #fth definition is only nominal1 explaining the meaning of the word proportional4 ,euclid represents magnitudes by lines1 and often denotes them either by single letters or1 like lines1 by two letters4 ,we shall use only single letters for the purpose4 ,if a and b denote two magnitudes of the same kind1 their ratio will be denoted by a 3 b2 if c and d are two other magnitudes of the same kind1 but possibly of a different kind from a and b1 then if c and d have the same ratio to one another as a and b1 this will be expressed by writing-- a 3 b 33 c 3 d4 ,further1 if m is a 7whole7 number1 ma shall denote the multiple of a which is obtained by taking it m times4 ,s #di4 ,the whole theory of ratios is based on ,def4 #e4 ,def4 #e4 ,the first of four magnitudes is said to have the same ratio to the second that the third has to the fourth when1 any equimultiples whatever of the first and the third being taken1 and any equimultiples whatever of the second and the fourth1 if the multiple of the first be less than that of the second1 the multiple of the third is also less than that of the fourth2 and if the multiple of the first is equal to that of the second1 the multiple of the third is also equal to that of the fourth2 and if the multiple of the first is greater than that of the second1 the multiple of the third is also greater than that of the fourth4 ,it will be well to show at once in an example how this definition can be used1 by proving the first part of the first proposition in the sixth book4 ,triangles of the same altitude are to one another as their bases1 or if a and b are the bases1 and alpha and beta the areas1 of two triangles which have the same altitude1 then a 3 b 33 alpha 3 beta4 ,to prove this1 we have1 according to ,definition #e1 to show-- if ma nb1 then malpha nbeta1 if ma nb1 then malpha nbeta1 if ma nb1 then malpha nbeta4 ,that this is true is in our case easily seen4 ,we may suppose that the triangles have a common vertex1 and their bases in the same line4 ,we set off the base a along the line containing the bases m times2 we then join the different parts of division to the vertex1 and get m triangles all equal to alpha4 ,the triangle on ma as base equals1 therefore1 malpha4 ,if we proceed in the same manner with the base b1 setting it off n times1 we find that the area of the triangle on the base nb equals nbeta1 the vertex of all triangles being the same4 ,but if two triangles have the same altitude1 then their areas are equal if the bases are equal2 hence malpha nbeta if ma nb1 and if their bases are unequal1 then that has the greater area which is on the greater base2 in other words1 malpha is greater than1 equal to1 or less than nbeta1 according as ma is greater than1 equal to1 or less than nb1 which was to be proved4 ,s #ej4 ,it will be seen that even in this example it does not become evident what a ratio really is4 ,it is still an open question whether ratios are magnitudes which we can compare4 ,we do not know whether the ratio of two lines is a magnitude of the same kind as the ratio of two areas4 ,though we might say that ,def4 #e defines equal ratios1 still we do not know whether they are equal in the sense of the axiom1 that two things which are equal to a third are equal to one another4 ,that this is the case requires a proof1 and until this proof is given we shall use the 33 instead of the sign 1 which1 however1 we shall afterwards introduce4 ,as soon as it has been established that all ratios are like magnitudes1 it becomes easy to show that1 in some cases at least1 they are numbers4 ,this step was never made by ,greek mathematicians4 ,they distinguished always most carefully between continuous magnitudes and the discrete series of numbers4 ,in modern times it has become the custom to ignore this difference4 ,if1 in determining the ratio of two lines1 a common measure can be found1 which is contained m times in the first1 and n times in the second1 then the ratio of the two lines equals the ratio of the two numbers m 3 n4 ,this is shown by ,euclid in ,prop4 #e1 ,x4 ,but the ratio of two numbers is1 as a rule1 a fraction1 and the ,greeks did not1 as we do1 consider fractions as numbers4 ,far less had they any notion of introducing irrational numbers1 which are neither whole nor fractional1 as we are obliged to do if we wish to say that all ratios are numbers4 ,the incommensurable numbers which are thus introduced as ratios of incommensurable quantities are nowadays as familiar to us as fractions2 but a proof is generally omitted that we may apply to them the rules which have been established for rational numbers only4 ,euclid's treatment of ratios avoids this difficulty4 ,his definitions hold for commensurable as well as for incommensurable quantities4 ,even the notion of incommensurable quantities is avoided in ,book ,v4 ,but he proves that the more elementary rules of algebra hold for ratios4 ,we shall state all his propositions in that algebraical form to which we are now accustomed4 ,this may1 of course1 be done without changing the character of ,euclid's method4 ,s4 #ea4 ,using the notation explained above we express the first propositions as follows3-- ,prop4 #a4 ,if a ma'1 b mb'1 c mc'1 then a b c m7a' b' c'74 ,prop4 #b4 ,if a mb1 and c md1 e nb1 and f nd1 then a e is the same multiple of b as c f is of d1 viz43-- a e 7m n7b1 and c f 7m n7d4 ,prop4 #c4 ,if a mb1 c md1 then is na the same multiple of b that nc is of d1 viz4 na nmb1 nc nmd4 ,prop4 #d4 ,if a 3 b 33 c 3 d1 then ma 3 nb 33 mc 3 nd4 ,prop4 #e4 ,if a mb1 and c md1 then a - c m7b - d74 ,prop4 #f4 ,if a mb1 c md1 then are a - nb and c - nd either equal to1 or equimultiples of1 b and d1 viz4 a - nb 7m - n7b and c - nd 7m - n7d1 where m - n may be unity4 ,all these propositions relate to equimultiples4 ,now follow propositions about ratios which are compared as to their magnitude4 ,s #eb4 ,prop4 #g4 ,if a b1 then a 3 c 33 b 3 c and c 3 a 33 c 3 b4 ,the proof is simply this4 ,as a b we know that ma mb2 therefore if ma nc1 then mb nc1 if ma nc1 then mb nc1 if ma nc1 then mb nc1 therefore the first proportion holds by ,definition #e4 ,prop4 #h4 ,if a b1 then a 3 c b 3 c1 and c 3 a c 3 b4 ,the proof depends on ,definition #g4 ,prop4 #i 7converse to ,prop4 #g74 ,if a 3 c 33 b 3 c1 or if c 3 a 33 c 3 b1 then a b4 ,prop4 #aj 7converse to ,prop4 #h74 ,if a 3 c b 3 c1 then a b1 and if c 3 a c 3 b1 then a b4 ,prop4 #aa4 ,if a 3 b 33 c 3 d1 and a 3 b 33 e 3 f1 then c 3 d 33 e 3 f4 ,in words1 if too ratios are equal to a third1 they are equal to one another4 ,after these propositions have been proved1 we have a right to consider a ratio as a magnitude1 for only now can we consider a ratio as something for which the axiom about magnitudes holds3 things which are equal to a third are equal to one another4 ,we shall indicate this by writing in future the sign instead of 334 ,the remaining propositions1 which explain themselves1 may then be stated as follows3 ,s #ec4 ,prop4 #ab4 ,if a 3 b c 3 d e 3 f1 then a c e 3 b d f a 3 b4 ,prop4 #ac4 ,if a 3 b c 3 d and c 3 d e 3 f1 then a 3 b e 3 f4 ,prop4 #ad4 ,if a 3 b c 3 d1 and a c1 then b d4 ,prop4 #ae4 ,magnitudes have the same ratio to one another that their equimultiples have-- ma 3 mb a 3 b4 ,prop4 #af4 ,if a1 b1 c1 d are magnitudes of the same kind1 and if a 3 b c 3 d1 then a 3 c b 3 d4 ,prop4 #ag4 ,if a b 3 b c d 3 d1 then a 3 b c 3 d4 ,prop4 #ah 7converse to #ag74 ,if a 3 b c 3 d then a b 3 b c d 3 d4 ,prop4 #ai4 ,if a1 b1 c1 d are quantities of the same kind1 and if a 3 b c 3 d1 then a - c 3 b - d a 3 b4 ,s #ed4 ,prop4 #bj4 ,if there be three magnitudes1 and another three1 which have the same ratio1 taken two and two1 then if the first be greater than the third1 the fourth shall be greater than the sixth3 and if equal1 equal2 and if less1 less4 ,if we understand by a 3 b 3 c 3 d 3 e 3 444 a' 3 b' 3 c' 3 d' 3 e' 3 444 that the ratio of any two consecutive magnitudes on the first side equals that of the corresponding magnitudes on the second side1 we may write this theorem in symbols1 thus3-- ,if a1 b1 c be quantities of one1 and d1 e1 f magnitudes of the same or any other kind1 such that a 3 b 3 c d 3 e 3 f1 and if a c1 then d f1 but if a c1 then d f1 and if a c1 then d f4 ,prop4 #ba4 ,if a 3 b e 3 f and b 3 c d 3 e1 or if a 3 b 3 c #a_/;f 3 #a_/;e 3 #a_/;d1 and if a c1 then d f1 but if a c1 then d f1 and if a c1 then d f4 ,by aid of these two propositions the following two are proved4 ,s #ee4 ,prop4 #bb4 ,if there be any number of magnitudes1 and as many others1 which have the same ratio1 taken two and two in order1 the first shall have to the last of the first magnitudes the same ratio which the first of the others has to the last4 ,we may state it more generally1 thus3 ,if a 3 b 3 c 3 d 3 e3 444 a' 3 b' 3 c' 3 d' 3 e' 3 444 1 then not only have two consecutive1 but any two magnitudes on the first side1 the same ratio as the corresponding magnitudes on the other4 ,for instance-- a 3 c a' 3 c'2 b 3 e b' 3 e'1 c4 ,prop4 #bc we state only in symbols1 viz43-- ,if a 3 b 3 c 3 d 3 e 3 444 #a_/;a' 3 #a_/;b' 3 #a_/;c' 3 #a_/;d' 3 #a_/;e' 4441 then a 3 c c' 3 a'1 b 3 e e' 3 b'1 and so on4 ,prop4 #bd comes to this3 ,if a 3 b c 3 d and e 3 b f 3 d1 then a e 3 b c f 3 d4 ,some of the proportions which are considered in the above propositions have special names4 ,these we have omitted1 as being of no use1 since algebra has enabled us to bring the different operations contained in the propositions under a common point of view4 ,s #ef4 ,the last proposition in the fifth book is of a different character4 ,prop4 #be4 ,if four magnitudes of the same kind be proportional1 the greatest and least of them together shall be greater than the other two together4 ,in symbols-- ,if a1 b1 c1 d be magnitudes of the same kind1 and if a 3 b c 3 d1 and if a is the greatest1 hence d the least1 then a d b c4 ,s #eg4 ,we return once again to the question4 ,what is a ratio8 ,we have seen that we may treat ratios as magnitudes1 and that all ratios are magnitudes of the same kind1 for we may compare any two as to their magnitude4 ,it will presently be shown that ratios of lines may be considered as quotients of lines1 so that a ratio appears as answer to the question1 ,how often is one line contained in another8 ,but the answer to this question is given by a number1 at least in some cases1 and in all cases if we admit incommensurable numbers4 ,considered from this point of view1 we may say the fifth book of the ,elements shows that some of the simpler algebraical operations hold for incommensurable numbers4 ,in the ordinary algebraical treatment of numbers this proof is altogether omitted1 or given by a process of limits which does not seem to be natural to the subject4 ,,book ,,vi4 ,s #eh4 ,the sixth book contains the theory of similar figures4 ,after a few definitions explaining terms1 the first proposition gives the first application of the theory of proportion4 ,prop4 #a4 ,triangles and parallelograms of the same altitude are to one another as their bases4 ,the proof has already been considered in ,s #di4 ,from this follows easily the important theorem ,prop4 #b4 ,if a straight line be drawn parallel to one of the sides of a triangle it shall cut the other sides1 or those sides produced1 proportionally2 and if the sides or the sides produced be cut proportionally1 the straight line which joins the points of section shall be parallel to the remaining side of the triangle4 ,s #ei4 ,the next proposition1 together with one added by ,simson as ,prop4 ,a1 may be expressed more conveniently if we introduce a modern phraseology1 viz4 if in a line ,,ab we assume a point ,c between ,a and ,b1 we shall say that ,c divides ,,ab internally in the ratio ,,ac 3 ,,cb2 but if ,c be taken in the line ,,ab produced1 we shall say that ,,ab is divided externally in the ratio ,,ac 3 ,,cb4 ,the two propositions then come to this3 ,prop4 #c4 ,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 angle2 and conversely1 if a line through the vertex of a triangle divide the base internally in the ratio of the two other sides1 then that line bisects the angle at the vertex4 ,simson's ,prop4 ,a4 ,the line which bisects an exterior angle of a triangle divides the opposite side externally in the ratio of the other sides2 and conversely1 if a line through the vertex of a triangle divide the base externally in the ratio of the sides1 then it bisects an exterior angle at the vertex of the triangle4 ,if we combine both we have-- ,the two lines which bisect the interior and exterior angles at one vertex of a triangle divide the opposite side internally and externally in the same ratio1 viz4 in the ratio of the other two sides4 ,s #fj4 ,the next four propositions contain the theory of similar triangles1 of which four cases are considered4 ,they may be stated together4 ,two triangles are similar1-- #a4 7,prop4 #d74 ,if the triangles are equiangular3 #b4 7,prop4 #e74 ,if the sides of the one are proportional to those of the other2 #c4 7,prop4 #f74 ,if two sides in one are proportional to two sides in the other1 and if the angles contained by these sides are equal2 #d4 7,prop4 #g74 ,if two sides in one are proportional to two sides in the other1 if the angles opposite homologous sides are equal1 and if the angles opposite the other homologous sides are both acute1 both right or both obtuse2 homologous sides being in each case those which are opposite equal angles4 ,an important application of these theorems is at once made to a right-angled triangle1 viz43-- ,prop4 #h4 ,in a right-angled triangle1 if a perpendicular be drawn from the right angle to the base1 the triangles on each side of it are similar to the whole triangle1 and to one another4 ,corollary4--,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 base1 and also that each of the sides is a mean proportional between the base and the segment of the base adjacent to that side4 ,s #fa4 ,there follow four propositions containing problems1 in language slightly different from ,euclid's1 viz43-- ,prop4 #i4 ,to divide a straight line into a given number of equal parts4 ,prop4 #aj4 ,to divide a straight line in a given ratio4 ,prop4 #aa4 ,to find a third proportional to two given straight lines4 ,prop4 #ab4 ,to find a fourth proportional to three given straight lines4 ,prop4 #ac4 ,to find a mean proportional between two given straight lines4 ,the last three may be written as equations with one unknown quantity--viz4 if we call the given straight lines a1 b1 c1 and the required line x1 we have to find a line x so that ,prop4 #aa4 a 3 b b 3 x2 ,prop4 #ab4 a 3 b c 3 x2 ,prop4 #ac4 a 3 x x 3 b4 ,we shall see presently how these may be written without the signs of ratios4 ,s #fb4 ,euclid considers next proportions connected with parallelograms and triangles which are equal in area4 ,prop4 #ad4 ,equal parallelograms which have one angle of the one equal to one angle of the other have their sides about the equal angles reciprocally proportional2 and parallelograms which have one angle of the one equal to one angle of the other1 and their sides about the equal angles reciprocally proportional1 are equal to one another4 ,prop4 #ae4 ,equal triangles which have one angle of the one equal to one angle of the other1 have their sides about the equal angles reciprocally proportional2 and triangles which have one angle of the one equal to one angle of the other1 and their sides about the equal angles reciprocally proportional1 are equal to one another4 ,illustration ,the latter proposition is really the same as the former1 for if1 as in the accompanying diagram1 in the figure belonging to the former the two equal parallelograms ,,ab and ,,bc be bisected by the lines ,,df and ,,eg1 and if ,,ef be drawn1 we get the figure belonging to the latter4 ,it is worth noticing that the lines ,,fe and ,,dg are parallel4 ,we may state therefore the theorem-- ,if two triangles are equal in area1 and have one angle in the one vertically opposite to one angle in the other1 then the two straight lines which join the remaining two vertices of the one to those of the other triangle are parallel4 ,s #fc4 ,a most important theorem is ,prop4 #af4 ,if four straight lines be proportionals1 the rectangle contained by the extremes is equal to the rectangle contained by the means2 and if the rectangle contained by the extremes be equal to the rectangle contained by the means1 the four straight lines are proportionals4 ,in symbols1 if a1 b1 c1 d are the four lines1 and if a 3 b c 3 d1 then ad bc2 and conversely1 if ad bc1 then a 3 b c 3 d1 where ad and bc denote 7as in ,s #bj71 the areas of the rectangles contained by a and d and by b and c respectively4 ,this allows us to transform every proportion between four lines into an equation between two products4 ,it shows further that the operation of forming a product of two lines1 and the operation of forming their ratio are each the inverse of the other4 ,if we now define a quotient a_/b of two lines as the number which multiplied into b gives a1 so that a -- b a1 b we see that from the equality of two quotients a c -- -- b d follows1 if we multiply both sides by bd1 a c -- b4d -- d4b1 b d ad cb4 ,but from this it follows1 according to the last theorem1 that a 3 b c 3 d4 ,hence we conclude that the quotient a_/b and the ratio a 3 b are different forms of the same magnitude1 only with this important difference that the quotient a_/b would have a meaning only if a and b have a common measure1 until we introduce incommensurable numbers1 while the ratio a 3 b has always a meaning1 and thus gives rise to the introduction of incommensurable numbers4 ,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 ,,ii4 ,it will also be seen that if we write the ratios in ,book ,v4 as quotients1 or rather as fractions1 then most of the theorems state properties of quotients or of fractions4 ,s #fd4 ,prop4 #ag4 ,if three straight lines are proportional the rectangle contained by the extremes is equal to the square on the mean2 and conversely1 is only a special case of #af4 ,after the problem1 ,prop4 #ah1 ,on a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure1 there follows another fundamental theorem3 ,prop4 #ai4 ,similar triangles are to one another in the duplicate ratio of their homologous sides4 ,in other words1 the areas of similar triangles are to one another as the squares on homologous sides4 ,this is generalized in3 ,prop4 #bj4 ,similar polygons may be divided into the same number of similar triangles1 having the same ratio to one another that the polygons have2 and the polygons are to one another in the duplicate ratio of their homologous sides4 ,s #fe4 ,prop4 #ba4 ,rectilineal figures which are similar to the same rectilineal figure are also similar to each other1 is an immediate consequence of the definition of similar figures4 ,as similar figures may be said to be equal in shape but not in size1 we may state it also thus3 ,figures which are equal in shape to a third are equal in shape to each other4 ,prop4 #bb4 ,if four straight lines be proportionals1 the similar rectilineal figures similarly described on them shall also be proportionals2 and if the similar rectilineal figures similarly described on four straight lines be proportionals1 those straight lines shall be proportionals4 ,this is essentially the same as the following3-- ,if a 3 b c 3 d1 then a#b 3 b#b c#b 3 d#b4 ,s #ff4 ,now follows a proposition which has been much discussed with regard to ,euclid's exact meaning in saying that a ratio is compounded of two other ratios1 viz43 ,prop4 #bc4 ,parallelograms which are equiangular to one another1 have to one another the ratio which is compounded of the ratios of their sides4 ,the proof of the proposition makes its meaning clear4 ,in symbols the ratio a 3 c is compounded of the two ratios a 3 b and b 3 c1 and if a 3 b a' 3 b'1 b 3 c b 3 c1 then a 3 c is compounded of a' 3 b' and b 3 c4 ,if we consider the ratios as numbers1 we may say that the one ratio is the product of those of which it is compounded1 or in symbols1 a a b a' b a a' b b -- -- 4 -- -- 4 --1 if -- -- and -- --4 c b c b' c b b' c c ,the theorem in ,prop4 #bc is the foundation of all mensuration of areas4 ,from it we see at once that two rectangles have the ratio of their areas compounded of the ratios of their sides4 ,if ,a is the area of a rectangle contained by a and b1 and ,b that of a rectangle contained by c and d1 so that ,a ab1 ,b cd1 then ,a 3 ,b ab 3 cd1 and this is1 the theorem says1 compounded of the ratios a 3 c and b 3 d4 ,in forms of quotients1 a b ab -- 4 -- --4 c d cd ,this shows how to multiply quotients in our geometrical calculus4 ,further1 ,two triangles have the ratios of their areas compounded of the ratios of their bases and their altitude4 ,for a triangle is equal in area to half a parallelogram which has the same base and the same altitude4 ,s #fg4 ,to bring these theorems to the form in which they are usually given1 we assume a straight line u as our unit of length 7generally an inch1 a foot1 a mile1 c471 and determine the number alpha which expresses how often u is contained in a line a1 so that alpha denotes the ratio a 3 u whether commensurable or not1 and that a alphau4 ,we call this number alpha the numerical value of a4 ,if in the same manner beta be the numerical value of a line b we have a 3 b alpha 3 beta2 in words3 ,the ratio of two lines 7and of two like quantities in general7 is equal to that of their numerical values4 ,this is easily proved by observing that a alphau1 b betau1 therefore a 3 b alphau 3 betau1 and this may without difficulty be shown to equal alpha 3 beta4 ,if now a1 b be base and altitude of one1 a'1 b' those of another parallelogram1 alpha1 beta and alpha'1 beta' their numerical values respectively1 and ,a1 ,a' their areas1 then ,a a b alpha beta alphabeta -- -- 4 -- -------- 4 ------ ---------------4 ,a' a' b' alpha' beta' alpha'beta' ,in words3 ,the areas of two parallelograms are to each other as the products of the numerical values of their bases and altitudes4 ,if especially the second parallelogram is the unit square1 i4e4 a square on the unit of length1 then alpha' beta' #a1 ,a' u#b1 and we have ,a -- alphabeta or ,a alphabeta 4 u#b4 ,a' ,this gives the theorem3 ,the number of unit squares contained in a parallelogram equals the product of the numerical values of base and altitude1 and similarly the number of unit squares contained in a triangle equals half the product of the numerical values of base and altitude4 ,this is often stated by saying that the area of a parallelogram is equal to the product of the base and the altitude1 meaning by this product the product of the numerical values1 and not the product as defined above in ,s #bj4 ,s #fh4 ,propositions #bd and #bf relate to parallelograms about diagonals1 such as are considered in ,book ,i41 #dc4 ,they are-- ,prop4 #bd4 ,parallelograms about the diameter of any parallelogram are similar to the whole parallelogram and to one another2 and its converse 7,prop4 #bf71 ,if two similar parallelograms have a common angle1 and be similarly situated1 they are about the same diameter4 ,between these is inserted a problem4 ,prop4 #be4 ,to describe a rectilineal figure which shall be similar to one given rectilinear figure1 and equal to another given rectilineal figure4 ,s #fi4 ,prop4 #bg contains a theorem relating to the theory of maxima and minima4 ,we may state it thus3 ,prop4 #bg4 ,if a parallelogram be divided into two by a straight line cutting the base1 and if on half the base another parallelogram be constructed similar to one of those parts1 then this third parallelogram is greater than the other part4 ,of far greater interest than this general theorem is a special case of it1 where the parallelograms are changed into rectangles1 and where one of the parts into which the parallelogram is divided is made a square2 for then the theorem changes into one which is easily recognized to be identical with the following3-- ,of all rectangles which have the same perimeter the square has the greatest area4 ,this may also be stated thus3-- ,of all rectangles which have the same area the square has the least perimeter4 ,s #gj4 ,the next three propositions contain problems which may be said to be solutions of quadratic equations4 ,the first two are1 like the last1 involved in somewhat obscure language4 ,we transcribe them as follows3 ,problem4--,to describe on a given base a parallelogram1 and to divide it either internally 7,prop4 #bh7 or externally 7,prop4 #bi7 from a point on the base into two parallelograms1 of which the one has a given size 7is equal in area to a given figure71 whilst the other has a given shape 7is similar to a given parallelogram74 ,if we express this again in symbols1 calling the given base a1 the one part x1 and the altitude y1 we have to determine x and y in the first case from the equations 7a - x7y k#b1 x p -- --1 y q k#b being the given size of the first1 and p and q the base and altitude of the parallelogram which determine the shape of the second of the required parallelograms4 ,if we substitute the value of y1 we get pk#b 7a - x7x ----1 q or1 ax - x#b b#b1 where a and b are known quantities1 taking b#b pk#b_/q4 ,the second case 7,prop4 #bi7 gives rise1 in the same manner1 to the quadratic ax x#b b#b4 ,the next problem-- ,prop4 #cj4 ,to cut a given straight line in extreme and mean ratio1 leads to the equation ax x#b a#b4 ,this is1 therefore1 only a special case of the last1 and is1 besides1 an old acquaintance1 being essentially the same problem as that proposed in ,,ii4 #aa4 ,prop4 #cj may therefore be solved in two ways1 either by aid of ,prop4 #bi or by aid of ,,ii4 #aa4 ,euclid gives both solutions4 ,s #ga4 ,prop4 #ca 7,theorem74 ,in any right-angled triangle1 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 angle1--is a pretty generalization of the theorem of ,pythagoras 7,i4 #dg74 ,leaving out the next proposition1 which is of little interest1 we come to the last in this book4 ,prop4 #cc4 ,in equal circles angles1 whether at the centres or the circumferences1 have the same ratio which the arcs on which they stand have to one another2 so also have the sectors4 ,of this1 the part relating to angles at the centre is of special importance2 it enables us to measure angles by arcs4 ,with this closes that part of the ,elements which is devoted to the study of figures in a plane4 ,,book ,,xi4 ,s #gb4 ,in this book figures are considered which are not confined to a plane1 viz4 first relations between lines and planes in space1 and afterwards properties of solids4 ,of new definitions we mention those which relate to the perpendicularity and the inclination of lines and planes4 ,def4 #c4 ,a straight line is perpendicular1 or at right angles1 to a plane when it makes right angles with every straight line meeting it in that plane4 ,the definition of perpendicular planes 7,def4 #d7 offers no difficulty4 ,euclid defines the inclination of lines to planes and of planes to planes 7,defs4 #e and #f7 by aid of plane angles1 included by straight lines1 with which we have been made familiar in the first books4 ,the other important definitions are those of parallel planes1 which never meet 7,def4 #h71 and of solid angles formed by three or more planes meeting in a point 7,def4 #i74 ,to these we add the definition of a line parallel to a plane as a line which does not meet the plane4 ,s #gc4 ,before we investigate the contents of ,book ,,xi41 it will be well to recapitulate shortly what we know of planes and lines from the definitions and axioms of the first book4 ,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 it4 ,this is equivalent to saying that a straight line which has two points in a plane has all points in the plane4 ,hence1 a straight line which does not lie in the plane cannot have more than one point in common with the plane4 ,this is virtually the same as ,euclid's ,prop4 #a1 viz43-- ,prop4 #a4 ,one part of a straight line cannot be in a plane and another part without it4 ,it also follows1 as was pointed out in ,s #c1 in discussing the definitions of ,book ,i41 that a plane is determined already by one straight line and a point without it1 viz4 if all lines be drawn through the point1 and cutting the line1 they will form a plane4 ,this may be stated thus3-- ,a plane is determined-- #ast1 ,by a straight line and a point which does not lie on it2 #bnd1 ,by three points which do not lie in a straight line2 for if two of these points be joined by a straight line we have case #a2 #crd1 ,by two intersecting straight lines2 for the point of intersection and two other points1 one in each line1 give case #b2 #dth1 ,by two parallel lines 7,def4 #ce1 ,i474 ,the third case of this theorem is ,euclid's ,prop4 #b4 ,two straight lines which cut one another are in one plane1 and three straight lines which meet one another are in one plane4 ,and the fourth is ,euclid's ,prop4 #g4 ,if two straight lines be parallel1 the straight line drawn from any point in one to any point in the other is in the same plane with the parallels4 ,from the definition of a plane further follows ,prop4 #c4 ,if two planes cut one another1 their common section is a straight line4 ,s #gd4 ,whilst these propositions are virtually contained in the definition of a plane1 the next gives us a new and fundamental property of space1 showing at the same time that it is possible to have a straight line perpendicular to a plane1 according to ,def4 #c4 ,it states-- ,prop4 #d4 ,if a straight line is perpendicular to two straight lines in a plane which it meets1 then it is perpendicular to all lines in the plane which it meets1 and hence it is perpendicular to the plane4 ,def4 #c may be stated thus3 ,if a straight line is perpendicular to a plane1 then it is perpendicular to every line in the plane which it meets4 ,the converse to this would be ,all straight lines which meet a given straight line in the same point1 and are perpendicular to it1 lie in a plane which is perpendicular to that line4 ,this ,euclid states thus3 ,prop4 #e4 ,if three straight lines meet all at one point1 and a straight line stands at right angles to each of them at that point1 the three straight lines shall be in one and the same plane4 ,s #ge4 ,there follow theorems relating to the theory of parallel lines in space1 viz43-- ,prop4 #f4 ,any two lines which are perpendicular to the same plane are parallel to each other2 and conversely ,prop4 #h4 ,if of two parallel straight lines one is perpendicular to a plane1 the other is so also4 ,prop4 #g4 ,if two straight lines are parallel1 the straight line which joins any point in one to any point in the other is in the same plane as the parallels4 7,see above1 ,s #gc47 ,prop4 #i4 ,two straight lines which are each of them parallel to the same straight line1 and not in the same plane with it1 are parallel to one another2 where the words1 and not in the same plane with it1 may be omitted1 for they exclude the case of three parallels in a plane1 which has been proved before2 and ,prop4 #aj4 ,if two angles in different planes have the two limits of the one parallel to those of the other1 then the angles are equal4 ,that their planes are parallel is shown later on in ,prop4 #ae4 ,this theorem is not necessarily true1 for the angles in question may be supplementary2 but then the one angle will be equal to that which is adjacent and supplementary to the other1 and this latter angle will also have its limits parallel to those of the first4 ,from this theorem it follows that if we take any two straight lines in space which do not meet1 and if we draw through any point ,p in space two lines parallel to them1 then the angle included by these lines will always be the same1 whatever the position of the point ,p may be4 ,this angle has in modern times been called the angle between the given lines3-- ,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 lines4 ,s #gf4 ,it is now possible to solve the following two problems3-- ,to draw a straight line perpendicular to a given plane from a given point which lies #a4 ,not in the plane 7,prop4 #aa74 #b4 ,in the plane 7,prop4 #ab74 ,the second case is easily reduced to the first--viz4 if by aid of the first we have drawn any perpendicular to the plane from some point without it1 we need only draw through the given point in the plane a line parallel to it1 in order to have the required perpendicular given4 ,the solution of the first part is of interest in itself4 ,it depends upon a construction which may be expressed as a theorem4 ,if from a point ,a without a plane a perpendicular ,,ab be drawn to the plane1 and if from the foot ,b of this perpendicular another perpendicular ,,bc be drawn to any straight line in the plane1 then the straight line joining ,a to the foot ,c of this second perpendicular will also be perpendicular to the line in the plane4 ,the theory of perpendiculars to a plane is concluded by the theorem-- ,prop4 #ac4 ,through any point in space1 whether in or without a plane1 only one straight line can be drawn perpendicular to the plane4 ,s #gg4 ,the next four propositions treat of parallel planes4 ,it is shown that planes which have a common perpendicular are parallel 7,prop4 #ad72 that two planes are parallel if two intersecting straight lines in the one are parallel respectively to two straight lines in the other plane 7,prop4 #ae72 that parallel planes are cut by any plane in parallel straight lines 7,prop4 #af72 and lastly1 that any two straight lines are cut proportionally by a series of parallel planes 7,prop4 #ag74 ,this theory is made more complete by adding the following theorems1 which are easy deductions from the last3 ,two parallel planes have common perpendiculars 7converse to #ad72 and ,two planes which are parallel to a third plane are parallel to each other4 ,it will be noted that ,prop4 #ae at once allows of the solution of the problem3 ,through a given point to draw a plane parallel to a given plane4 ,and it is also easily proved that this problem allows always of one1 and only of one1 solution4 ,s #gh4 ,we come now to planes which are perpendicular to one another4 ,two theorems relate to them4 ,prop4 #ah4 ,if a straight line be at right angles to a plane1 every plane which passes through it shall be at right angles to that plane4 ,prop4 #ai4 ,if two planes which cut one another be each of them perpendicular to a third plane1 their common section shall be perpendicular to the same plane4 ,s #gi4 ,if three planes pass through a common point1 and if they bound each other1 a solid angle of three faces1 or a trihedral angle1 is formed1 and similarly by more planes a solid angle of more faces1 or a polyhedral angle4 ,these have many properties which are quite analogous to those of triangles and polygons in a plane4 ,euclid states some1 viz43-- ,prop4 #bj4 ,if a solid angle be contained by three plane angles1 any two of them are together greater than the third4 ,but the next-- ,prop4 #ba4 ,every solid angle is contained by plane angles1 which are together less than four right angles--has no analogous theorem in the plane4 ,we may mention1 however1 that the theorems about triangles contained in the propositions of ,book ,i41 which do not depend upon the theory of parallels 7that is all up to ,prop4 #bg71 have their corresponding theorems about trihedral angles4 ,the latter are formed1 if for side of a triangle we write plane angle or face of trihedral angle1 and for angle of triangle we substitute angle between two faces where the planes containing the solid angle are called its faces4 ,we get1 for instance1 from ,i4 #d1 the theorem1 ,if two trihedral angles have the angles of two faces in the one equal to the angles of two faces in the other1 and have likewise the angles included by these faces equal1 then the angles in the remaining faces are equal1 and the angles between the other faces are equal each to each1 viz4 those which are opposite equal faces4 ,the solid angles themselves are not necessarily equal1 for they may be only symmetrical like the right hand and the left4 ,the connexion indicated between triangles and trihedral angles will also be recognized in ,prop4 #bb4 ,if every two of three plane angles be greater than the third1 and if the straight lines which contain them be all equal1 a triangle may be made of the straight lines that join the extremities of those equal straight lines4 ,and ,prop4 #bc solves the problem1 ,to construct a trihedral angle having the angles of its faces equal to three given plane angles1 any two of them being greater than the third4 ,it is1 of course1 analogous to the problem of constructing a triangle having its sides of given length4 ,two other theorems of this kind are added by ,simson in his edition of ,euclid's ,elements4 ,s #hj4 ,these are the principal properties of lines and planes in space1 but before we go on to their applications it will be well to define the word distance4 ,in geometry distance means always shortest distance2 viz4 the distance of a point from a straight line1 or from a plane1 is the length of the perpendicular from the point to the line or plane4 ,the distance between two non-intersecting lines is the length of their common perpendicular1 there being but one4 ,the distance between two parallel lines or between two parallel planes is the length of the common perpendicular between the lines or the planes4 ,s #ha4 ,parallelepipeds4--,the rest of the book is devoted to the study of the parallelepiped4 ,in ,prop4 #bd the possibility of such a solid is proved1 viz43-- ,prop4 #bd4 ,if a solid be contained by six planes two and two of which are parallel1 the opposite planes are similar and equal parallelograms4 ,euclid calls this solid henceforth a parallelepiped1 though he never defines the word4 ,either face of it may be taken as base1 and its distance from the opposite face as altitude4 ,prop4 #be4 ,if a solid parallelepiped be cut by a plane parallel to two of its opposite planes1 it divides the whole into two solids1 the base of one of which shall be to the base of the other as the one solid is to the other4 ,this theorem corresponds to the theorem 7,,vi4 #a7 that parallelograms between the same parallels are to one another as their bases4 ,a similar analogy is to be observed among a number of the remaining propositions4 ,s #hb4 ,after solving a few problems we come to ,prop4 #bh4 ,if a solid parallelepiped be cut by a plane passing through the diagonals of two of the opposite planes1 it shall be cut in two equal parts4 ,in the proof of this1 as of several other propositions1 ,euclid neglects the difference between solids which are symmetrical like the right hand and the left4 ,prop4 #ca4 ,solid parallelepipeds1 which are upon equal bases1 and of the same altitude1 are equal to one another4 ,props4 #bi and #cj contain special cases of this theorem leading up to the proof of the general theorem4 ,as consequences of this fundamental theorem we get ,prop4 #cb4 ,solid parallelepipeds1 which have the same altitude1 are to one another as their bases2 and ,prop4 #cc4 ,similar solid parallelepipeds are to one another in the triplicate ratio of their homologous sides4 ,if we consider1 as in ,s #fg1 the ratios of lines as numbers1 we may also say-- ,the ratio of the volumes of similar parallelepipeds is equal to the ratio of the third powers of homologous sides4 ,parallelepipeds which are not similar but equal are compared by aid of the theorem ,prop4 #cd4 ,the bases and altitudes of equal solid parallelepipeds are reciprocally proportional2 and if the bases and altitudes be reciprocally proportional1 the solid parallelepipeds are equal4 ,s #hc4 ,of the following propositions the #cgth and #djth are of special interest4 ,prop4 #cg4 ,if four straight lines be proportionals1 the similar solid parallelepipeds1 similarly described from them1 shall also be proportionals2 and if the similar parallelepipeds similarly described from four straight lines be proportionals1 the straight lines shall be proportionals4 ,in symbols it says-- ,if a 3 b c 3 d1 then a#c 3 b#c c#c 3 d#c4 ,prop4 #dj teaches how to compare the volumes of triangular prisms with those of parallelepipeds1 by proving that a triangular prism is equal in volume to a parallelepiped1 which has its altitude and its base equal to the altitude and the base of the triangular prism4 ,s #hd4 ,from these propositions follow all results relating to the mensuration of volumes4 ,we shall state these as we did in the case of areas4 ,the starting-point is the rectangular parallelepiped1 which has every edge perpendicular to the planes it meets1 and which takes the place of the rectangle in the plane4 ,if this has all its edges equal we obtain the cube4 ,if we take a certain line u as unit length1 then the square on u is the unit of area1 and the cube on u the unit of volume1 that is to say1 if we wish to measure a volume we have to determine how many unit cubes it contains4 ,a rectangular parallelepiped has1 as a rule1 the three edges unequal1 which meet at a point4 ,every other edge is equal to one of them4 ,if a1 b1 c be the three edges meeting at a point1 then we may take the rectangle contained by two of them1 say by b and c1 as base and the third as altitude4 ,let ,v be its volume1 ,v' that of another rectangular parallelepiped which has the edges a'1 b1 c1 hence the same base as the first4 ,it follows then easily1 from ,prop4 #be or #cb1 that ,v 3 ,v' a 3 a'2 or in words1 ,rectangular parallelepipeds on equal bases are proportional to their altitudes4 ,if we have two rectangular parallelepipeds1 of which the first has the volume ,v and the edges a1 b1 c1 and the second1 the volume ,v' and the edges a'1 b'1 c'1 we may compare them by aid of two new ones which have respectively the edges a'1 b1 c and a'1 b'1 c1 and the volumes ,v#a and ,v#b4 ,we then have ,v 3 ,v#a a 3 a'2 ,v#a 3 ,v#b b 3 b'1 ,v#b 3 ,v' c 3 c'4 ,compounding these1 we have ,v 3 ,v' 7a 3 a'77b 3 b'77c 3 c'71 or ,v a b c -- -- 4 -- 4 --4 ,v' a' b' c' ,hence1 as a special case1 making ,v' equal to the unit cube ,u on u we get ,v a b c -- -- 4 -- 4 -- alpha4beta4gamma1 ,u u u u where alpha1 beta1 gamma are the numerical values of a1 b1 c2 that is1 ,the number of unit cubes in a rectangular parallelepiped is equal to the product of the numerical values of its three edges4 ,this is generally expressed by saying the volume of a rectangular parallelepiped is measured by the product of its sides1 or by the product of its base into its altitude1 which in this case is the same4 ,prop4 #ca allows us to extend this to any parallelepipeds1 and ,props4 #bh or #dj1 to triangular prisms4 ,the volume of any parallelepiped1 or of any triangular prism1 is measured by the product of base and altitude4 ,the consideration that any polygonal prism may be divided into a number of triangular prisms1 which have the same altitude and the sum of their bases equal to the base of the polygonal prism1 shows further that the same holds for any prism whatever4 ,,book ,,xii4 ,s #he4 ,in the last part of ,book ,,xi4 we have learnt how to compare the volumes of parallelepipeds and of prisms4 ,in order to determine the volume of any solid bounded by plane faces we must determine the volume of pyramids1 for every such solid may be decomposed into a number of pyramids4 ,as every pyramid may again be decomposed into triangular pyramids1 it becomes only necessary to determine their volume4 ,this is done by the ,theorem4--,every triangular pyramid is equal in volume to one third of a triangular prism having the same base and the same altitude as the pyramid4 ,this is an immediate consequence of ,euclid's ,prop4 #g4 ,every prism having a triangular base may be divided into three pyramids that have triangular bases1 and are equal to one another4 ,the proof of this theorem is difficult1 because the three triangular pyramids into which the prism is divided are by no means equal in shape1 and cannot be made to coincide4 ,it has first to be proved that two triangular pyramids have equal volumes1 if they have equal bases and equal altitudes4 ,this ,euclid does in the following manner4 ,he first shows 7,prop4 #c7 that a triangular pyramid may be divided into four parts1 of which two are equal triangular pyramids similar to the whole pyramid1 whilst the other two are equal triangular prisms1 and further1 that these two prisms together are greater than the two pyramids1 hence more than half the given pyramid4 ,he next shows 7,prop4 #d7 that if two triangular pyramids are given1 having equal bases and equal altitudes1 and if each be divided as above1 then the two triangular prisms in the one are equal to those in the other1 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 other4 ,hence to these pyramids the same process is again applicable4 ,we are thus enabled to cut out of the two given pyramids equal parts1 each greater than half the original pyramid4 ,of the remainder we can again cut out equal parts greater than half these remainders1 and so on as far as we like4 ,this process may be continued till the last remainder is smaller than any assignable quantity1 however small4 ,it follows1 so we should conclude at present1 that the two volumes must be equal1 for they cannot differ by any assignable quantity4 ,to ,greek mathematicians this conclusion offers far greater difficulties4 ,they prove elaborately1 by a reductio ad absurdum1 that the volumes cannot be unequal4 ,this proof must be read in the ,elements4 ,we must1 however1 state that we have in the above not proved ,euclid's ,prop4 #e1 but only a special case of it4 ,euclid does not suppose that the bases of the two pyramids to be compared are equal1 and hence he proves that the volumes are as the bases4 ,the reasoning of the proof becomes clearer in the special case1 from which the general one may be easily deduced4 ,s #hf4 ,prop4 #f extends the result to pyramids with polygonal bases4 ,from these results follow again the rules at present given for the mensuration of solids1 viz4 a pyramid is the third part of a triangular prism having the same base and the same altitude4 ,but a triangular prism is equal in volume to a parallelepiped which has the same base and altitude4 ,hence if ,b is the base and h the altitude1 we have ,volume of prism ,bh1 ,volume of pyramid #a_/c;,bh1 statements which have to be taken in the sense that ,b means the number of square units in the base1 h the number of units of length in the altitude1 or that ,b and h denote the numerical values of base and altitude4 ,s #hg4 ,a method similar to that used in proving ,prop4 #e leads to the following results relating to solids bounded by simple curved surfaces3-- ,prop4 #aj4 ,every cone is the third part of a cylinder which has the same base1 and is of an equal altitude with it4 ,prop4 #aa4 ,cones or cylinders of the same altitude are to one another as their bases4 ,prop4 #ab4 ,similar cones or cylinders have to one another the triplicate ratio of that which the diameters of their bases have4 ,prop4 #ac4 ,if a cylinder be cut by a plane parallel to its opposite planes or bases1 it divides the cylinder into two cylinders1 one of which is to the other as the axis of the first to the axis of the other2 which may also be stated thus3-- ,cylinders on the same base are proportional to their altitudes4 ,prop4 #ad4 ,cones or cylinders upon equal bases are to one another as their altitudes4 ,prop4 #ae4 ,the bases and altitudes of equal cones or cylinders are reciprocally proportional1 and if the bases and altitudes be reciprocally proportional1 the cones or cylinders are equal to one another4 ,these theorems again lead to formulae in mensuration1 if we compare a cylinder with a prism having its base and altitude equal to the base and altitude of the cylinder4 ,this may be done by the method of exhaustion4 ,we get1 then1 the result that their bases are equal1 and have1 if ,b denotes the numerical value of the base1 and h that of the altitude1 ,volume of cylinder ,bh1 ,volume of cone #a_/c;,bh4 ,s #hh4 ,the remaining propositions relate to circles and spheres4 ,of the sphere only one property is proved1 viz43-- ,prop4 #ah4 ,spheres have to one another the triplicate ratio of that which their diameters have4 ,the mensuration of the sphere1 like that of the circle1 the cylinder and the cone1 had not been settled in the time of ,euclid4 ,it was done by ,archimedes4 ,,book ,,xiii4 ,s #hi4 ,the #acth and last book of ,euclid's ,elements is devoted to the regular solids 7see ,,polyhedron74 ,it is shown that there are five of them1 viz43-- #a4 ,the regular tetrahedron1 with #d triangular faces and #d vertices2 #b4 ,the cube1 with #h vertices and #f square faces2 #c4 ,the octahedron1 with #f vertices and #h triangular faces2 #d4 ,the dodecahedron1 with #ab pentagonal faces1 #c at each of the #bj vertices2 #e4 ,the icosahedron1 with #bj triangular faces1 #e at each of the #ab vertices4 ,it is shown how to inscribe these solids in a given sphere1 and how to determine the lengths of their edges4 ,s #ij4 ,the #acth book1 and therefore the ,elements1 conclude with the scholium1 that no other regular solid exists besides the five ones enumerated4 ,the proof is very simple4 ,each face is a regular polygon1 hence the angles of the faces at any vertex must be angles in equal regular polygons1 must be together less than four right angles 7,,xi4 #ba71 and must be three or more in number4 ,each angle in a regular triangle equals two-thirds of one right angle4 ,hence it is possible to form a solid angle with three1 four or five regular triangles or faces4 ,these give the solid angles of the tetrahedron1 the octahedron and the icosahedron4 ,the angle in a square 7the regular quadrilateral7 equals one right angle4 ,hence three will form a solid angle1 that of the cube1 and four will not4 ,the angle in the regular pentagon equals #f_/e of a right angle4 ,hence three of them equal #ah_/e 7i4e4 less than #d7 right angles1 and form the solid angle of the dodecahedron4 ,three regular polygons of six or more sides cannot form a solid angle4 ,therefore no other regular solids are possible4 7,o4 ,h47 ,,ii4 ,,projective ,,geometry ,it is difficult1 at the outset1 to characterize projective geometry as compared with ,euclidean4 ,but a few examples will at least indicate the practical differences between the two4 ,in ,euclid's ,elements almost all propositions refer to the magnitude of lines1 angles1 areas or volumes1 and therefore to measurement4 ,the statement that an angle is right1 or that two straight lines are parallel1 refers to measurement4 ,on the other hand1 the fact that a straight line does or does not cut a circle is independent of measurement1 it being dependent only upon the mutual position of the line and the circle4 ,this difference becomes clearer if we project any figure from one plane to another 7see ,,projection74 ,by this the length of lines1 the magnitude of angles and areas1 is altered1 so that the projection1 or shadow1 of a square on a plane will not be a square2 it will1 however1 be some quadrilateral4 ,again1 the projection of a circle will not be a circle1 but some other curve more or less resembling a circle4 ,but one property may be stated at once--no straight line can cut the projection of a circle in more than two points1 because no straight line can cut a circle in more than two points4 ,there are1 then1 some properties of figures which do not alter by projection1 whilst others do4 ,to the latter belong nearly all properties relating to measurement1 at least in the form in which they are generally given4 ,the others are said to be projective properties1 and their investigation forms the subject of projective geometry4 ,different as are the kinds of properties investigated in the old and the new sciences1 the methods followed differ in a still greater degree4 ,in ,euclid each proposition stands by itself2 its connexion with others is never indicated2 the leading ideas contained in its proof are not stated2 general principles do not exist4 ,in the modern methods1 on the other hand1 the greatest importance is attached to the leading thoughts which pervade the whole2 and general principles1 which bring whole groups of theorems under one aspect1 are given rather than separate propositions4 ,the whole tendency is towards generalization4 ,a straight line is considered as given in its entirety1 extending both ways to infinity1 while ,euclid never admits anything but finite quantities4 ,the treatment of the infinite is in fact another fundamental difference between the two methods3 ,euclid avoids it2 in modern geometry it is systematically introduced4 ,of the different modern methods of geometry1 we shall treat principally of the methods of projection and correspondence which have proved to be the most powerful4 ,these have become independent of ,euclidean ,geometry1 especially through the ,geometrie der ,lage of ,v4 ,staudt and the ,ausdehnungslehre of ,grassmann4 ,for the sake of brevity we shall presuppose a knowledge of ,euclid's ,elements1 although we shall use only a few of his propositions4 ,s #a4 ,geometrical ,elements4 ,we consider space as filled with points1 lines and planes1 and these we call the elements out of which our figures are to be formed1 calling any combination of these elements a figure4 ,by a line we mean a straight line in its entirety1 extending both ways to infinity2 and by a plane1 a plane surface1 extending in all directions to infinity4 ,we accept the three-dimensional space of experience--the space assumed by ,euclid--which has for its properties 7among others73-- ,through any two points in space one and only one line may be drawn2 ,through any three points which are not in a line1 one and only one plane may be placed2 ,the intersection of two planes is a line2 ,a line which has two points in common with a plane lies in the plane1 hence the intersection of a line and a plane is a single point2 and ,three planes which do not meet in a line have one single point in common4 ,these results may be stated differently in the following form3-- ,i4 ,a plane is determined-- ,a point is determined-- #a4 ,by three points which do #a4 ,by three planes which do not lie in a line2 not pass through a line2 #b4 ,by two intersecting lines2 #b4 ,by two intersecting lines2 #c4 ,by a line and a point #c4 ,by a plane and a line which does not lie in it4 which does not lie in it4 ,,ii4 ,a line is determined-- #a4 ,by two points2 #b4 ,by two planes4 ,it will be observed that not only are planes determined by points1 but also points by planes2 that therefore the planes may be considered as elements1 like points2 and also that in any one of the above statements we may interchange the words point and plane1 and we obtain again a correct statement1 provided that these statements themselves are true4 ,as they stand1 we ought1 in several cases1 to add if they are not parallel1 or some such words1 parallel lines and planes being evidently left altogether out of consideration4 ,to correct this we have to reconsider the theory of parallels4 ,illustration3 ,,fig4 #a4 ,s #b4 ,parallels4 ,point at ,infinity4--,let us take in a plane a line p 7fig4 #a71 a point ,s not in this line1 and a line q drawn through ,s4 ,then this line q will meet the line p in a point ,a4 ,if we turn the line q about ,s towards q'1 its point of intersection with p will move along p towards ,b1 passing1 on continued turning1 to a greater and greater distance1 until it is moved out of our reach4 ,if we turn q still farther1 its continuation will meet p1 but now at the other side of ,a4 ,the point of intersection has disappeared to the right and reappeared to the left4 ,there is one intermediate position where q is parallel to p--that is where it does not cut p4 ,in every other position it cuts p in some finite point4 ,if1 on the other hand1 we move the point ,a to an infinite distance in p1 then the line q which passes through ,a will be a line which does not cut p at any finite point4 ,thus we are led to say3 ,every line through ,s which joins it to any point at an infinite distance in p is parallel to p4 ,but by ,euclid's #abth axiom there is but one line parallel to p through ,s4 ,the difficulty in which we are thus involved is due to the fact that we try to reason about infinity as if we1 with our finite capabilities1 could comprehend the infinite4 ,to overcome this difficulty1 we may say that all points at infinity in a line appear to us as one1 and may be replaced by a single ideal point4 ,we may therefore now give the following definitions and axiom3-- ,definition4--,lines which meet at infinity are called parallel4 ,axiom4--,all points at an infinite distance in a line may be considered as one single point4 ,definition4--,this ideal point is called the point at infinity in the line4 ,the axiom is equivalent to ,euclid's ,axiom #ab1 for it follows from either that through any point only one line may be drawn parallel to a given line4 ,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 infinity4 ,a line thus appears closed by this point1 and we speak as if we could move a point along the line from one position ,a to another ,b in two ways1 either through the point at infinity or through finite points only4 ,it must never be forgotten that this point at infinity is ideal2 in fact1 the whole notion of infinity is only a mathematical conception1 and owes its introduction 7as a method of research7 to the working generalizations which it permits4 ,s #c4 ,line and ,plane at ,infinity4--,having arrived at the notion of replacing all points at infinity in a line by one ideal point1 there is no difficulty in replacing all points at infinity in a plane by one ideal line4 ,to make this clear1 let us suppose that a line p1 which cuts two fixed lines a and b in the points ,a and ,b1 moves parallel to itself to a greater and greater distance4 ,it will at last cut both a and b at their points at infinity1 so that a line which joins the two points at infinity in two intersecting lines lies altogether at infinity4 ,every other line in the plane will meet it therefore at infinity1 and thus it contains all points at infinity in the plane4 ,all points at infinity in a plane lie in a line1 which is called the line at infinity in the plane4 ,it follows that parallel planes must be considered as planes having a common line at infinity1 for any other plane cuts them in parallel lines which have a point at infinity in common4 ,if we next take two intersecting planes1 then the point at infinity in their line of intersection lies in both planes1 so that their lines at infinity meet4 ,hence every line at infinity meets every other line at infinity1 and they are therefore all in one plane4 ,all points at infinity in space may be considered as lying in one ideal plane1 which is called the plane at infinity4 ,s #d4 ,parallelism4--,we have now the following definitions3-- ,parallel lines are lines which meet at infinity2 ,parallel planes are planes which meet at infinity2 ,a line is parallel to a plane if it meets it at infinity4 ,theorems like this--,lines 7or planes7 which are parallel to a third are parallel to each other--follow at once4 ,this view of parallels leads therefore to no contradiction of ,euclid's ,elements4 ,as immediate consequences we get the propositions3-- ,every line meets a plane in one point1 or it lies in it2 ,every plane meets every other plane in a line2 ,any two lines in the same plane meet4 ,s #e4 ,aggregates of ,geometrical ,elements4--,we have called points1 lines and planes the elements of geometrical figures4 ,we also say that an element of one kind contains one of the other if it lies in it or passes through it4 ,all the elements of one kind which are contained in one or two elements of a different kind form aggregates which have to be enumerated4 ,they are the following3-- ,i4 ,of one dimension4 #a4 ,the row1 or range1 of points formed by all points in a line1 which is called its base4 #b4 ,the flat pencil formed by all the lines through a point in a plane4 ,its base is the point in the plane4 #c4 ,the axial pencil formed by all planes through a line which is called its base or axis4 ,,ii4 ,of two dimensions4 #a4 ,the field of points and lines--that is1 a plane with all its points and all its lines4 #b4 ,the pencil of lines and planes--that is1 a point in space with all lines and all planes through it4 ,,iii4 ,of three dimensions4 ,the space of points--that is1 all points in space4 ,the space of planes--that is1 all planes in space4 ,,iv4 ,of four dimensions4 ,the space of lines1 or all lines in space4 ,s #f4 ,meaning of ,dimensions4--,the word dimension in the above needs explanation4 ,if in a plane we take a row p and a pencil with centre ,q1 then through every point in p one line in the pencil will pass1 and every ray in ,q will cut p in one point1 so that we are entitled to say a row contains as many points as a flat pencil lines1 and1 we may add1 as an axial pencil planes1 because an axial pencil is cut by a plane in a flat pencil4 ,the number of elements in the row1 in the flat pencil1 and in the axial pencil is1 of course1 infinite and indefinite too1 but the same in all4 ,this number may be denoted by infinity4 ,then a plane contains infinity#b points and as many lines4 ,to see this1 take a flat pencil in a plane4 ,it contains infinity lines1 and each line contains infinity points1 whilst each point in the plane lies on one of these lines4 ,similarly1 in a plane each line cuts a fixed line in a point4 ,but this line is cut at each point by infinity lines and contains infinity points2 hence there are infinity#b lines in a plane4 ,a pencil in space contains as many lines as a plane contains points and as many planes as a plane contains lines1 for any plane cuts the pencil in a field of points and lines4 ,hence a pencil contains infinity#b lines and infinity#b planes4 ,the field and the pencil are of two dimensions4 ,to count the number of points in space we observe that each point lies on some line in a pencil4 ,but the pencil contains infinity#b lines1 and each line infinity points2 hence space contains infinity#c points4 ,each plane cuts any fixed plane in a line4 ,but a plane contains infinity#b lines1 and through each pass infinity planes2 therefore space contains infinity#c planes4 ,hence space contains as many planes as points1 but it contains an infinite number of times more lines than points or planes4 ,to count them1 notice that every line cuts a fixed plane in one point4 ,but infinity#b lines pass through each point1 and there are infinity#b points in the plane4 ,hence there are infinity#d lines in space4 ,the space of points and planes is of three dimensions1 but the space of lines is of four dimensions4 ,a field of points or lines contains an infinite number of rows and flat pencils2 a pencil contains an infinite number of flat pencils and of axial pencils2 space contains a triple infinite number of pencils and of fields1 infinity#d rows and axial pencils and infinity#e flat pencils--or1 in other words1 each point is a centre of infinity#b flat pencils4 ,s #g4 ,the above enumeration allows a classification of figures4 ,figures in a row consist of groups of points only1 and figures in the flat or axial pencil consist of groups of lines or planes4 ,in the plane we may draw polygons2 and in the pencil or in the point1 solid angles1 and so on4 ,we may also distinguish the different measurements ,we have-- ,in the row1 length of segment2 ,in the flat pencil1 angles2 ,in the axial pencil1 dihedral angles between two planes2 ,in the plane1 areas2 ,in the pencil1 solid angles2 ,in the space of points or planes1 volumes4 ,,segments ,,of ,a ,,line ,s #h4 ,any two points ,a and ,b in space determine on the line through them a finite part1 which may be considered as being described by a point moving from ,a to ,b4 ,this we shall denote by ,,ab1 and distinguish it from ,,ba1 which is supposed as being described by a point moving from ,b to ,a1 and hence in a direction or in a sense opposite to ,,ab4 ,such a finite line1 which has a definite sense1 we shall call a segment1 so that ,,ab and ,,ba denote different segments1 which are said to be equal in length but of opposite sense4 ,the one sense is often called positive and the other negative4 ,in introducing the word sense for direction in a line1 we have the word direction reserved for direction of the line itself1 so that different lines have different directions1 unless they be parallel1 whilst in each line we have a positive and negative sense4 ,we may also say1 with ,clifford1 that ,,ab denotes the step of going from ,a to ,b4 ,illustration3 ,,fig4 #b4 ,s #i4 ,if we have three points ,a1 ,b1 ,c in a line 7fig4 #b71 the step ,,ab will bring us from ,a to ,b1 and the step ,,bc from ,b to ,c4 ,hence both steps are equivalent to the one step ,,ac4 ,this is expressed by saying that ,,ac is the sum of ,,ab and ,,bc2 in symbols-- ,,ab ,,bc ,,ac1 where account is to be taken of the sense4 ,this equation is true whatever be the position of the three points on the line4 ,as a special case we have ,,ab ,,ba #j1 7#a7 and similarly ,,ab ,,bc ,,ca #j1 7#b7 which again is true for any three points in a line4 ,we further write ,,ab -,,ba4 where - denotes negative sense4 ,we can then1 just as in algebra1 change subtraction of segments into addition by changing the sense1 so that ,,ab - ,,cb is the same as ,,ab 7-,,cb7 or ,,ab ,,bc4 ,a figure will at once show the truth of this4 ,the sense is1 in fact1 in every respect equivalent to the sign of a number in algebra4 ,s #aj4 ,of the many formulae which exist between points in a line we shall have to use only one more1 which connects the segments between any four points ,a1 ,b1 ,c1 ,d in a line4 ,we have ,,bc ,,bd ,,dc1 ,,ca ,,cd ,,da1 ,,ab ,,ad ,,db2 or multiplying these by ,,ad1 ,,bd1 ,,cd respectively1 we get ,,bc4,,ad ,,bd4,,ad ,,dc4,,ad ,,bd4,,ad - ,,cd4,,ad ,,ca4,,bd ,,cd4,,bd ,,da4,,bd ,,cd4,,bd - ,,ad4,,bd ,,ab4,,cd ,,ad4,,cd ,,db4,,cd ,,ad4,,cd - ,,bd4,,cd4 ,it will be seen that the sum of the right-hand sides vanishes1 hence that ,,bc4,,ad ,,ca4,,bd ,,ab4,,cd #j 7#c7 for any four points on a line4 ,illustration3 ,,fig4 #c4 ,s #aa4 ,if ,c is any point in the line ,,ab1 then we say that ,c divides the segment ,,ab in the ratio ,,ac_/,,cb1 account being taken of the sense of the two segments ,,ac and ,,cb4 ,if ,c lies between ,a and ,b the ratio is positive1 as ,,ac and ,,cb have the same sense4 ,but if ,c lies without the segment ,,ab1 i4e4 if ,c divides ,,ab externally1 then the ratio is negative4 ,to see how the value of this ratio changes with ,c1 we will move ,c along the whole line 7fig4 #c71 whilst ,a and ,b remain fixed4 ,if ,c lies at the point ,a1 then ,,ac #j1 hence the ratio ,,ac 3 ,,cb vanishes4 ,as ,c moves towards ,b1 ,,ac increases and ,,cb decreases1 so that our ratio increases4 ,at the middle point ,m of ,,ab it assumes the value #a1 and then increases till it reaches an infinitely large value1 when ,c arrives at ,b4 ,on passing beyond ,b the ratio becomes negative4 ,if ,c is at ,p we have ,,ac ,,ap ,,ab ,,bp1 hence ,,ac ,,ab ,,bp ,,ab -- -- -- - -- - #a4 ,,cb ,,pb ,,pb ,,bp ,in the last expression the ratio ,,ab 3 ,,bp is positive1 has its greatest value infinity when ,c coincides with ,b1 and vanishes when ,,bc becomes infinite4 ,hence1 as ,c moves from ,b to the right to the point at infinity1 the ratio ,,ac 3 ,,cb varies from -infinity to -#a4 ,if1 on the other hand1 ,c is to the left of ,a1 say at ,q1 we have ,,ac ,,aq ,,ab ,,bq ,,ab - ,,qb1 hence ,,ac_/,,cb ,,ab_/,,qb - #a4 ,here ,,ab ,,qb1 hence the ratio ,,ab 3 ,,qb is positive and always less than one1 so that the whole is negative and #a4 ,if ,c is at the point at infinity it is -#a1 and then increases as ,c moves to the right1 till for ,c at ,a we get the ratio #j4 ,hence-- ,as ,c moves along the line from an infinite distance to the left to an infinite distance at the right1 the ratio always increases2 it starts with the value -#a1 reaches #j at ,a1 #a at ,m1 infinity at ,b1 now changes sign to -infinity1 and increases till at an infinite distance it reaches again the value -#a4 ,it assumes therefore all possible values from -infinity to infinity1 and each value only once1 so that not only does every position of ,c determine a definite value of the ratio ,,ac 3 ,,cb1 but also1 conversely1 to every positive or negative value of this ratio belongs one single point in the line ,,ab4 ,relations between segments of lines are interesting as showing an application of algebra to geometry4 ,the genesis of such relations from algebraic identities is very simple4 ,for example1 if a1 b1 c1 x be any four quantities1 then a b --------------------- --------------------- 7a - b77a - c77x - a7 7b - c77b - a77x - b7 c x --------------------- ---------------------2 7c - a77c - b77x - c7 7x - a77x - b77x - c7 this may be proved1 cumbrously1 by multiplying up1 or1 simply1 by decomposing the right-hand member of the identity into partial fractions4 ,now take a line ,,abcdx1 and let ,,ab a1 ,,ac b1 ,,ad c1 ,,ax x4 ,then obviously 7a - b7 ,,ab - ,,ac -,,bc1 paying regard to signs2 7a - c7 ,,ab - ,,ad ,,db1 and so on4 ,substituting these values in the identity we obtain the following relation connecting the segments formed by five points on a line3-- ,,ab ,,ac ,,ad ,,ax -------- -------- -------- --------4 ,,bc4,,bd4,,bx ,,cd4,,cb4,,cx ,,db4,,dc4,,dx ,,bx4,,cx4,,dx ,conversely1 if a metrical relation be given1 its validity may be tested by reducing to an algebraic equation1 which is an identity if the relation be true4 ,for example1 if ,,abcdx be five collinear points1 prove ,,ad4,,ax ,,bd4,,bx ,,cd4,,cx ----- ----- ----- #a4 ,,ab4,,ac ,,bc4,,ba ,,ca4,,cb ,clearing of fractions by multiplying throughout by ,,ab4,,bc4,,ca1 we have to prove -,,ad4,,ax4,,bc - ,,bd4,,bx4,,ca - ,,cd4,,cx4,,ab ,,ab4,,bc4,,ca4 ,take ,a as origin and let ,,ab a1 ,,ac b1 ,,ad c1 ,,ax x4 ,substituting for the segments in terms of a1 b1 c1 x1 we obtain on simplification a#b;b - ab#b -ab#b a#b;b1 an obvious identity4 ,an alternative method of testing a relation is illustrated in the following example3-- ,if ,a1 ,b1 ,c1 ,d1 ,e1 ,f be six collinear points1 then ,,ae4,,af ,,be4,,bf ,,ce4,,cf ,,de4,,df -------- -------- -------- -------- #j4 ,,ab4,,ac4,,ad ,,bc4,,bd4,,ba ,,cd4,,ca4,,cb ,,da4,,db4,,dc ,clearing of fractions by multiplying throughout by ,,ab4,,bc4,,cd4,,da1 and reducing to a common origin ,o 7calling ,,oa a1 ,,ob b1 c471 an equation containing the second and lower powers of ,,oa 7 a71 c41 is obtained4 ,calling ,,oa x1 it is found that x b1 x c1 x d are solutions4 ,hence the quadratic has three roots2 consequently it is an identity4 ,the relations connecting five points which we have instanced above may be readily deduced from the six-point relation2 the first by taking ,d at infinity1 and the second by taking ,f at infinity1 and then making the obvious permutations of the points4 ,,projection ,,and ,,cross-,,ratios ,s #ab4 ,if we join a point ,a to a point ,s1 then the point where the line ,,sa cuts a fixed plane pi is called the projection of ,a on the plane pi from ,s as centre of projection4 ,if we have two planes pi and pi' and a point ,s1 we may project every point ,a in pi to the other plane4 ,if ,a' is the projection of ,a1 then ,a is also the projection of ,a'1 so that the relations are reciprocal4 ,to every figure in pi we get as its projection a corresponding figure in pi'4 ,we shall determine such properties of figures as remain true for the projection1 and which are called projective properties4 ,for this purpose it will be sufficient to consider at first only constructions in one plane4 ,illustration3 ,,fig4 #d4 ,illustration3 ,,fig4 #e4 ,let us suppose we have given in a plane two lines p and p' and a centre ,s 7fig4 #d72 we may then project the points in p from ,s to p'4 ,let ,a'1 ,b' 444 be the projections of ,a1 ,b 4441 the point at infinity in p which we shall denote by ,i will be projected into a finite point ,i' in p'1 viz4 into the point where the parallel to p through ,s cuts p'4 ,similarly one point ,j in p will be projected into the point ,j' at infinity in p'4 ,this point ,j is of course the point where the parallel to p' through ,s cuts p4 ,we thus see that every point in p is projected into a single point in p'4 ,fig4 #e shows that a segment ,,ab will be projected into a segment ,a',b' which is not equal to it1 at least not as a rule2 and also that the ratio ,,ac 3 ,,cb is not equal to the ratio ,a',c' 3 ,c',b' formed by the projections4 ,these ratios will become equal only if p and p' are parallel1 for in this case the triangle ,,sab is similar to the triangle ,,sa',b'4 ,between three points in a line and their projections there exists therefore in general no relation4 ,but between four points a relation does exist4 ,s #ac4 ,let ,a1 ,b1 ,c1 ,d be four points in p1 ,a'1 ,b'1 ,c1 ,d' their projections in p'1 then the ratio of the two ratios ,,ac 3 ,,cb and ,,ad 3 ,,db into which ,c and ,d divide the segment ,,ab is equal to the corresponding expression between ,a'1 ,b'1 ,c'1 ,d'4 ,in symbols we have ,,ac ,,ad ,a',c' ,a',d' -- 3 -- ---- 3 ----4 ,,cb ,,db ,c',b' ,d',b' ,this is easily proved by aid of similar triangles4 ,illustration3 ,,fig4 #f4 ,through the points ,a and ,b on p draw parallels to p'1 which cut the projecting rays in ,c#b1 ,d#b1 ,b#b and ,a#a1 ,c#a1 ,d#a1 as indicated in fig4 #f4 ,the two triangles ,,acc#b and ,,bcc#a will be similar1 as will also be the triangles ,,add#b and ,,bdd#a4 ,the proof is left to the reader4 ,this result is of fundamental importance4 ,the expression ,,ac_/,,cb 3 ,,ad_/,,db has been called by ,chasles the anharmonic ratio of the four points ,a1 ,b1 ,c1 ,d4 ,professor ,clifford proposed the shorter name of cross-ratio4 ,we shall adopt the latter4 ,we have then the ,,fundamental ,,theorem4--,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 it4 ,s #ad4 ,before we draw conclusions from this result1 we must investigate the meaning of a cross-ratio somewhat more fully4 ,if four points ,a1 ,b1 ,c1 ,d are given1 and we wish to form their cross-ratio1 we have first to divide them into two groups of two1 the points in each group being taken in a definite order4 ,thus1 let ,a1 ,b be the first1 ,c1 ,d the second pair1 ,a and ,c being the first points in each pair4 ,the cross-ratio is then the ratio ,,ac 3 ,,cb divided by ,,ad 3 ,,db4 ,this will be denoted by 7,,ab1 ,,cd71 so that ,,ac ,,ad 7,,ab1 ,,cd7 -- 3 --4 ,,cb ,,db ,this is easily remembered4 ,in order to write it out1 make first the two lines for the fractions1 and put above and below these the letters ,a and ,b in their places1 thus1 ,a99_/,b 3 ,a99_/,b2 and then fill up1 crosswise1 the first by ,c and the other by ,d4 ,s #ae4 ,if we take the points in a different order1 the value of the cross-ratio will change4 ,we can do this in twenty-four different ways by forming all permutations of the letters4 ,but of these twenty-four cross-ratios groups of four are equal1 so that there are really only six different ones1 and these six are reciprocals in pairs4 ,we have the following rules3-- ,i4 ,if in a cross-ratio the two groups be interchanged1 its value remains unaltered1 i4e4 7,,ab1 ,,cd7 7,,cd1 ,,ab7 7,,ba1 ,,dc7 7,,dc1 ,,ba74 ,,ii4 ,if in a cross-ratio the two points belonging to one of the two groups be interchanged1 the cross-ratio changes into its reciprocal1 i4e4 7,,ab1 ,,cd7 #a_/7;,,ab1 ,,dc7 #a_/7;,,ba1 ,,cd7 #a_/7;,,cd1 ,,ba7 #a_/7;,,dc1 ,,ab74 ,from ,i4 and ,,ii4 we see that eight cross-ratios are associated with 7,,ab1 ,,cd74 ,,iii4 ,if in a cross-ratio the two middle letters be interchanged1 the cross-ratio alpha changes into its complement #a - alpha1 i4e4 7,,ab1 ,,cd7 #a - 7,,ac1 ,,bd74 ,s #af4 ,if lambda 7,,ab1 ,,cd71 mu 7,,ac1 ,,db71 nu 7,,ad1 ,,bc71 then lambda1 mu1 nu and their reciprocals #a_/lambda1 #a_/mu1 #a_/nu are the values of the total number of twenty-four cross-ratios4 ,moreover1 lambda1 mu1 nu are connected by the relations lambda #a_/mu mu #a_/nu nu #a_/lambda -lambdamunu #a2 this proposition may be proved by substituting for lambda1 mu1 nu and reducing to a common origin4 ,there are therefore four equations between three unknowns2 hence if one cross-ratio be given1 the remaining twenty-three are determinate4 ,moreover1 two of the quantities lambda1 mu1 nu are positive1 and the remaining one negative4 ,the following scheme shows the twenty-four cross-ratios expressed in terms of lambda1 mu1 nu4 ---------------------------------------- ---------------------- 7,,ab1 ,,cd7 7,,ba1 ,,dc7 lambda #a - mu #a_/7a - nu7 7,,cd1 ,,ab7 7,,dc1 ,,ba7 ---------------------------------------- ---------------------- 7,,ac1 ,,db7 7,,bd1 ,,ca7 #a_/7a - lambda7 #a_/mu 7nu - #a7_/nu 7,,ca1 ,,bd7 7,,db1 ,,ac7 ---------------------------------------- ---------------------- 7,,ab1 ,,dc7 7,,ba1 ,,cd7 #a_/lambda #a_/7a - mu7 #a - nu 7,,cd1 ,,ba7 7,,dc1 ,,ab7 ---------------------------------------- ---------------------- 7,,ad1 ,,bc7 7,,bc1 ,,ad7 7lambda - #a7_/lambdamu_/7mu - #a7 nu 7,,cb1 ,,da7 7,,da1 ,,cb7 ---------------------------------------- ---------------------- 7,,ac1 ,,bd7 7,,bd1 ,,ac7 #a - lambda mu nu_/7nu - #a7 7,,ca1 ,,db7 7,,db1 ,,ca7 ---------------------------------------- ---------------------- 7,,ad1 ,,cb7 7,,bc1 ,,da7 lambda_/7lambda - #a77mu - #a7_/mu #a_/nu 7,,cb1 ,,ad7 7,,da1 ,,bc7 ---------------------------------------- ---------------------- ,s #ag4 ,if one of the points of which a cross-ratio is formed is the point at infinity in the line1 the cross-ratio changes into a simple ratio4 ,it is convenient to let the point at infinity occupy the last place in the symbolic expression for the cross-ratio4 ,thus if ,i is a point at infinity1 we have 7,,ab1 ,,ci7 -,,ac_/,,cb1 because ,,ai 3 ,,ib -#a4 ,every common ratio of three points in a line may thus be expressed as a cross-ratio1 by adding the point at infinity to the group of points4 ,,harmonic ,,ranges ,s #ah4 ,if the points have special positions1 the cross-ratios may have such a value that1 of the six different ones1 two and two become equal4 ,if the first two shall be equal1 we get lambda #a_/lambda1 or lambda#b #a1 lambda -#a4 ,if we take lambda #a1 we have 7,,ab1 ,,cd7 #a1 or ,,ac_/,,cb ,,ad_/,,db2 that is1 the points ,c and ,d coincide1 provided that ,a and ,b are different4 ,if we take lambda -#a1 so that 7,,ab1 ,,cd7 -#a1 we have ,,ac_/,,cb -,,ad_/,,db4 ,hence ,c and ,d divide ,,ab internally and externally in the same ratio4 ,the four points are in this case said to be harmonic points1 and ,c and ,d are said to be harmonic conjugates with regard to ,a and ,b4 ,but we have also 7,,cd1 ,,ab7 -#a1 so that ,a and ,b are harmonic conjugates with regard to ,c and ,d4 ,the principal property of harmonic points is that their cross-ratio remains unaltered if we interchange the two points belonging to one pair1 viz4 7,,ab1 ,,cd7 7,,ab1 ,,dc7 7,,ba1 ,,cd74 ,for four harmonic points the six cross-ratios become equal two and two3 lambda lambda -#a1 #a - lambda #b1 ------------ #a_/b1 lambda - #a #a #a lambda - #a -------- -#a1 ------------ #a_/b1 ------------ #b4 lambda #a - lambda lambda ,hence if we get four points whose cross-ratio is #b or #a_/b1 then they are harmonic1 but not arranged so that conjugates are paired4 ,if this is the case the cross-ratio -#a4 ,s #ai4 ,if we equate any two of the above six values of the cross-ratios1 we get either lambda #a1 #j1 infinity1 or lambda -#a1 #b1 #a_/b1 or else lambda becomes a root of the equation lambda#b - lambda #a #j1 that is1 an imaginary cube root of -#a4 ,in this case the six values become three and three equal1 so that only two different values remain4 ,this case1 though important in the theory of cubic curves1 is for our purposes of no interest1 whilst harmonic points are all-important4 ,s #bj4 ,from the definition of harmonic points1 and by aid of ,s #aa1 the following properties are easily deduced4 ,if ,c and ,d are harmonic conjugates with regard to ,a and ,b1 then one of them lies in1 the other without ,,ab2 it is impossible to move from ,a to ,b without passing either through ,c or through ,d2 the one blocks the finite way1 the other the way through infinity4 ,this is expressed by saying ,a and ,b are separated by ,c and ,d4 ,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 ,a1 ,b4 ,if ,a and ,b are different points1 and if ,c coincides with ,a or ,b1 ,d does4 ,but if ,a and ,b coincide1 one of the points ,c or ,d1 lying between them1 coincides with them1 and the other may be anywhere in the line4 ,it follows that1 if of four harmonic conjugates two coincide1 then a third coincides with them1 and the fourth may be any point in the line4 ,if ,c is the middle point between ,a and ,b1 then ,d is the point at infinity2 for ,,ac 3 ,,cb #a1 hence ,,ad 3 ,,db must be equal to -#a4 ,the harmonic conjugate of the point at infinity in a line with regard to two points ,a1 ,b is the middle point of ,,ab4 ,this important property gives a first example how metric properties are connected with projective ones4 ,s #ba4 ,harmonic properties of the complete quadrilateral and quadrangle4 ,illustration3 ,,fig4 #g4 ,illustration3 ,,fig4 #h4 ,a figure formed by four lines in a plane is called a complete quadrilateral1 or1 shorter1 a four-side4 ,the four sides meet in six points1 named the vertices1 which may be joined by three lines 7other than the sides71 named the diagonals or harmonic lines4 ,the diagonals enclose the harmonic triangle of the quadrilateral4 ,in fig4 #g1 ,a',b',c'1 ,b',,ac1 ,c',,ab1 ,,cba' are the sides1 ,a1 ,a'1 ,b1 ,b'1 ,c1 ,c' the vertices1 ,,aa'1 ,,bb'1 ,,cc' the harmonic lines1 and alphabetagamma the harmonic triangle of the quadrilateral4 ,a figure formed by four coplanar points is named a complete quadrangle1 or1 shorter1 a four-point4 ,the four points may be joined by six lines1 named the sides1 which intersect in three other points1 termed the diagonal or harmonic points4 ,the harmonic points are the vertices of the harmonic triangle of the complete quadrangle4 ,in fig4 #h1 ,,aa'1 ,,bb' are the points1 ,,aa'1 ,,bb'1 ,a',b'1 ,b',a1 ,,ab1 ,,ba' are the sides1 ,l1 ,m1 ,n are the diagonal points1 and ,,lmn is the harmonic triangle of the quadrangle4 ,the harmonic property of the complete quadrilateral is3 ,any diagonal or harmonic line is harmonically divided by the other two2 and of a complete quadrangle3 ,the angle at any harmonic point is divided harmonically by the joins to the other harmonic points4 ,to prove the first theorem1 we have to prove 7,,aa'1 betagamma71 7,,bb'1 gammaalpha71 7,,cc'1 betaalpha7 are harmonic4 ,consider the cross-ratio 7,,cc'1 alphabeta74 ,then projecting from ,a on ,,bb' we have ,a7,,cc'1 alphabeta7 ,a7,b',b1 alphagamma74 ,projecting from ,a' on ,,bb'1 ,a'7,,cc'1 alphabeta7 ,a'7,,bb'1 alphagamma74 ,hence 7,b',b1 alphagamma7 7,,bb'1 alphagamma71 i4e4 the cross-ratio 7,,bb'1 alphagamma7 equals that of its reciprocal2 hence the range is harmonic4 ,the second theorem states that the pencils ,l7,,ba1 ,,nm71 ,m7,b',a1 ,,ln71 ,n7,,ba1 ,,lm7 are harmonic4 ,deferring the subject of harmonic pencils to the next section1 it will suffice to state here that any transversal intersects an harmonic pencil in an harmonic range4 ,consider the pencil ,l7,,ba1 ,,nm71 then it is sufficient to prove 7,,ba'1 ,,nm'7 is harmonic4 ,this follows from the previous theorem by considering ,a',b as a diagonal of the quadrilateral ,,alb',m4 ,this property of the complete quadrilateral allows the solution of the problem3 ,to construct the harmonic conjugate ,d to a point ,c with regard to two given points ,a and ,b4 ,through ,a draw any two lines1 and through ,c one cutting the former two in ,g and ,h4 ,join these points to ,b1 cutting the former two lines in ,e and ,f4 ,the point ,d where ,,ef cuts ,,ab will be the harmonic conjugate required4 ,this remarkable construction requires nothing but the drawing of lines1 and is therefore independent of measurement4 ,in a similar manner the harmonic conjugate of the line ,,va for two lines ,,vc1 ,,vd is constructed with the aid of the property of the complete quadrangle4 ,s #bb4 ,harmonic ,pencils4--,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 pencil4 ,we have seen 7,s #ac7 that if the lines which join four points ,a1 ,b1 ,c1 ,d to any point ,s be cut by any other line in ,a'1 ,b'1 ,c'1 ,d'1 then 7,,ab1 ,,cd7 7,a',b'1 ,c',d'74 ,in other words1 four lines in a flat pencil are cut by every other line in four points whose cross-ratio is constant4 ,definition4--,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 line4 ,if a1 b1 c1 d be the lines1 then this cross-ratio is denoted by 7ab1 cd74 ,definition4--,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 planes1 or1 what is the same thing1 the cross-ratio of the four rays in which any plane cuts the four planes4 ,in order that this definition may have a meaning1 it has to be proved that all lines cut the pencil in points which have the same cross-ratio4 ,this is seen at once for two intersecting lines1 as their plane cuts the axial pencil in a flat pencil1 which is itself cut by the two lines4 ,the cross-ratio of the four points on one line is therefore equal to that on the other1 and equal to that of the four rays in the flat pencil4 ,if two non-intersecting lines p and q cut the four planes in ,a1 ,b1 ,c1 ,d and ,a'1 ,b'1 ,c'1 ,d'1 draw a line r to meet both p and q1 and let this line cut the planes in ,a1 ,b1 ,c1 ,d4 ,then 7,,ab1 ,,cd7 7,a',b'1 ,c',d'71 for each is equal to 7,a,b1 ,c,d74 ,s #bc4 ,we may now also extend the notion of harmonic elements1 viz4 ,definition4--,four rays in a flat pencil and four planes in an axial pencil are said to be harmonic if their cross-ratio equals -#a1 that is1 if they are cut by a line in four harmonic points4 ,if we understand by a median line of a triangle a line which joins a vertex to the middle point of the opposite side1 and by a median line of a parallelogram a line joining middle points of opposite sides1 we get as special cases of the last theorem3 ,the diagonals and median lines of a parallelogram form an harmonic pencil2 and ,at a vertex of any triangle1 the two sides1 the median line1 and the line parallel to the base form an harmonic pencil4 ,taking the parallelogram a rectangle1 or the triangle isosceles1 we get3 ,any two lines and the bisections of their angles form an harmonic pencil4 ,or3 ,in an harmonic pencil1 if two conjugate rays are perpendicular1 then the other two are equally inclined to them2 and1 conversely1 if one ray bisects the angle between conjugate rays1 it is perpendicular to its conjugate4 ,this connects perpendicularity and bisection of angles with projective properties4 ,s #bd4 ,we add a few theorems and problems which are easily proved or solved by aid of harmonics4 ,an harmonic pencil is cut by a line parallel to one of its rays in three equidistant points4 ,through a given point to draw a line such that the segment determined on it by a given angle is bisected at that point4 ,having given two parallel lines1 to bisect on either any given segment without using a pair of compasses4 ,having given in a line a segment and its middle point1 to draw through any given point in the plane a line parallel to the given line4 ,to draw a line which joins a given point to the intersection of two given lines which meet off the drawing paper 7by aid of ,s #ba74 ,,correspondence4 ,,homographic ,,and ,,perspective ,,ranges ,s #be4 ,two rows1 p and p'1 which are one the projection of the other 7as in fig4 #e71 stand in a definite relation to each other1 characterized by the following properties4 #a4 ,to each point in either corresponds one point in the other2 that is1 those points are said to correspond which are projections of one another4 #b4 ,the cross-ratio of any four points in one equals that of the corresponding points in the other4 #c4 ,the lines joining corresponding points all pass through the same point4 ,if we suppose corresponding points marked1 and the rows brought into any other position1 then the lines joining corresponding points will no longer meet in a common point1 and hence the third of the above properties will not hold any longer2 but we have still a correspondence between the points in the two rows possessing the first two properties4 ,such a correspondence has been called a one-one correspondence1 whilst the two rows between which such correspondence has been established are said to be projective or homographic4 ,two rows which are each the projection of the other are therefore projective4 ,we shall presently see1 also1 that any two projective rows may always be placed in such a position that one appears as the projection of the other4 ,if they are in such a position the rows are said to be in perspective position1 or simply to be in perspective4 ,s #bf4 ,the notion of a one-one correspondence between rows may be extended to flat and axial pencils1 viz4 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 second1 and if the cross-ratio of four rays in one equals that of the corresponding rays in the second4 ,similarly an axial pencil may be projective to an axial pencil4 ,but a flat pencil may also be projective to an axial pencil1 or either pencil may be projective to a row4 ,the definition is the same in each case3 there is a one-one correspondence between the elements1 and four elements have the same cross-ratio as the corresponding ones4 ,s #bg4 ,there is also in each case a special position which is called perspective1 viz4 #a4 ,two projective rows are perspective if they lie in the same plane1 and if the one row is a projection of the other4 #b4 ,two projective flat pencils are perspective--7#a7 if they lie in the same plane1 and have a row as a common section2 7#b7 if they lie in the same pencil 7in space71 and are both sections of the same axial pencil2 7#c7 if they are in space and have a row as common section1 or are both sections of the same axial pencil1 one of the conditions involving the other4 #c4 ,two projective axial pencils1 if their axes meet1 and if they have a flat pencil as a common section4 #d4 ,a row and a projective flat pencil1 if the row is a section of the pencil1 each point lying in its corresponding line4 #e4 ,a row and a projective axial pencil1 if the row is a section of the pencil1 each point lying in its corresponding line4 #f4 ,a flat and a projective axial pencil1 if the former is a section of the other1 each ray lying in its corresponding plane4 ,that in each case the correspondence established by the position indicated is such as has been called projective follows at once from the definition4 ,it is not so evident that the perspective position may always be obtained4 ,we shall show in ,s #cj this for the first three cases4 ,first1 however1 we shall give a few theorems which relate to the general correspondence1 not to the perspective position4 ,s #bh4 ,two rows or pencils1 flat or axial1 which are projective to a third are projective to each other2 this follows at once from the definitions4 ,s #bi4 ,if two rows1 or two pencils1 either flat or axial1 or a row and a pencil1 be projective1 we may assume to any three elements in the one the three corresponding elements in the other1 and then the correspondence is uniquely determined4 ,for if in two projective rows we assume that the points ,a1 ,b1 ,c in the first correspond to the given points ,a'1 ,b'1 ,c' in the second1 then to any fourth point ,d in the first will correspond a point ,d' in the second1 so that 7,,ab1 ,,cd7 7,a',b'1 ,c',d'74 ,but there is only one point1 ,d'1 which makes the cross-ratio 7,a',b'1 ,c',d'7 equal to the given number 7,,ab1 ,,cd74 ,the same reasoning holds in the other cases4 ,s #cj4 ,if two rows are perspective1 then the lines joining corresponding points all meet in a point1 the centre of projection2 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 second4 ,this follows from the definition4 ,the converse also holds1 viz4 ,if two projective rows have such a position that one point in the one coincides with its corresponding point in the other1 then they are perspective1 that is1 the lines joining corresponding points all pass through a common point1 and form a flat pencil4 ,for let ,a1 ,b1 ,c1 ,d 444 be points in the one1 and ,a'1 ,b'1 ,c'1 ,d' 444 the corresponding points in the other row1 and let ,a be made to coincide with its corresponding point ,a'4 ,let ,s be the point where the lines ,,bb' and ,,cc' meet1 and let us join ,s to the point ,d in the first row4 ,this line will cut the second row in a point ,d1 so that ,a1 ,b1 ,c1 ,d are projected from ,s into the points ,a1 ,b'1 ,c'1 ,d4 ,the cross-ratio 7,,ab1 ,,cd7 is therefore equal to 7,,ab'1 ,c',d71 and by hypothesis it is equal to 7,a',b'1 ,c',d'74 ,hence 7,a',b'1 ,c',d7 7,a',b'1 ,c',d'71 that is1 ,d is the same point as ,d'4 ,s #ca4 ,if two projected flat pencils in the same plane are in perspective1 then the intersections of corresponding lines form a row1 and the line joining the two centres as a line in the first pencil corresponds to the same line as a line in the second4 ,and conversely1 ,if two projective pencils in the same plane1 but with different centres1 have one line in the one coincident with its corresponding line in the other1 then the two pencils are perspective1 that is1 the intersection of corresponding lines lie in a line4 ,the proof is the same as in ,s #cj4 ,s #cb4 ,if two projective flat pencils in the same point 7pencil in space71 but not in the same plane1 are perspective1 then the planes joining corresponding rays all pass through a line 7they form an axial pencil71 and the line common to the two pencils 7in which their planes intersect7 corresponds to itself4 ,and conversely3-- ,if two flat pencils which have a common centre1 but do not lie in a common plane1 are placed so that one ray in the one coincides with its corresponding ray in the other1 then they are perspective1 that is1 the planes joining corresponding lines all pass through a line4 ,s #cc4 ,if two projective axial pencils are perspective1 then the intersection of corresponding planes lie in a plane1 and the plane common to the two pencils 7in which the two axes lie7 corresponds to itself4 ,and conversely3-- ,if two projective axial pencils are placed in such a position that a plane in the one coincides with its corresponding plane1 then the two pencils are perspective1 that is1 corresponding planes meet in lines which lie in a plane4 ,the proof again is the same as in ,s #cj4 ,s #cd4 ,these theorems relating to perspective position become illusory if the projective rows of pencils have a common base4 ,we then have3-- ,in two projective rows on the same line--and also in two projective and concentric flat pencils in the same plane1 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 other4 ,proof 7in case of two rows74--,between four elements ,a1 ,b1 ,c1 ,d and their corresponding elements ,a'1 ,b'1 ,c'1 ,d' exists the relation 7,,abcd7 7,a',b',c',d'74 ,if now ,a'1 ,b'1 ,c' coincide respectively with ,a1 ,b1 ,c1 we get 7,,ab1 ,,cd7 7,,ab1 ,,cd'71 hence ,d and ,d' coincide4 ,the last theorem may also be stated thus3-- ,in two projective rows or pencils1 which have a common base but are not identical1 not more than two elements in the one can coincide with their corresponding elements in the other4 ,thus two projective rows on the same line cannot have more than two pairs of coincident points unless every point coincides with its corresponding point4 ,it is easy to construct two projective rows on the same line1 which have two pairs of corresponding points coincident4 ,let the points ,a1 ,b1 ,c as points belonging to the one row correspond to ,a1 ,b1 and ,c' as points in the second4 ,then ,a and ,b coincide with their corresponding points1 but ,c does not4 ,it is1 however1 not necessary that two such rows have twice a point coincident with its corresponding point2 it is possible that this happens only once or not at all4 ,of this we shall see examples later4 ,illustration3 ,,fig4 #i4 ,s #ce4 ,if two projective rows or pencils are in perspective position1 we know at once which element in one corresponds to any given element in the other4 ,if p and q 7fig4 #i7 are two projective rows1 so that ,k corresponds to itself1 and if we know that to ,a and ,b in p correspond ,a' and ,b' in q1 then the point ,s1 where ,,aa' meets ,,bb'1 is the centre of projection1 and hence1 in order to find the point ,c' corresponding to ,c1 we have only to join ,c to ,s2 the point ,c'1 where this line cuts q1 is the point required4 ,illustration3 ,,fig4 #aj4 ,if two flat pencils1 ,s#a and ,s#b1 in a plane are perspective 7fig4 #aj71 we need only to know two pairs1 a1 a' and b1 b'1 of corresponding rays in order to find the axis s of projection4 ,this being known1 a ray c' in ,s#b1 corresponding to a given ray c in ,s#a1 is found by joining ,s#b to the point where c cuts the axis s4 ,a similar construction holds in the other cases of perspective figures4 ,on this depends the solution of the following general problem4 ,s #cf4 ,three pairs of corresponding elements in two projective rows or pencils being given1 to determine for any element in one the corresponding element in the other4 ,we solve this in the two cases of two projective rows and of two projective flat pencils in a plane4 ,problem ,i4--,let ,a1 ,b1 ,c be ,problem ,,ii4--,let a1 b1 c be three points in a row s1 ,a'1 ,b'1 three rays in a pencil ,s1 a'1 ,c' the corresponding points in a b'1 c' the corresponding rays in projective row s'1 both being in a projective pencil ,s'1 both a plane2 it is required to find being in the same plane2 it is for any point ,d in s the required to find for any ray d corresponding point ,d' in s'4 in ,s the corresponding ray d' in ,s'4 ,the solution is made to depend on the construction of an auxiliary row or pencil which is perspective to both the given ones4 ,this is found as follows3-- ,illustration3 ,,fig4 #aa4 ,solution of ,problem ,i4--,on the line joining two corresponding points1 say ,,aa' 7fig4 #aa71 take any two points1 ,s and ,s'1 as centres of auxiliary pencils4 ,join the intersection ,b#a of ,,sb and ,s',b' to the intersection ,c#a of ,,sc and ,s',c' by the line s#a4 ,then a row on s#a will be perspective to s with ,s as centre of projection1 and to s' with ,s' as centre4 ,to find now the point ,d' on s' corresponding to a point ,d on s we have only to determine the point ,d#a1 where the line ,,sd cuts s#a1 and to draw ,s',d#a2 the point where this line cuts s' will be the required point ,d'4 ,proof4--,the rows s and s' are both perspective to the row s#a1 hence they are projective to one another4 ,to ,a1 ,b1 ,c1 ,d on s correspond ,a#a1 ,b#a1 ,c#a1 ,d#a on s#a1 and to these correspond ,a'1 ,b'1 ,c'1 ,d' on s'2 so that ,d and ,d' are corresponding points as required4 ,illustration3 ,,fig4 #ab4 ,solution of ,problem ,,ii4--,through the intersection ,a of two corresponding rays a and a' 7fig4 #ab71 take two lines1 s and s'1 as bases of auxiliary rows4 ,let ,s#a be the point where the line b#a1 which joins ,b and ,b'1 cuts the line c#a1 which joins ,c and ,c'4 ,then a pencil ,s#a will be perspective to ,s with s as axis of projection4 ,to find the ray d' in ,s' corresponding to a given ray d in ,s1 cut d by s at ,d2 project this point from ,s#a to ,d' on s' and join ,d' to ,s'4 ,this will be the required ray4 ,proof4--,that the pencil ,s#a is perspective to ,s and also to ,s' follows from construction4 ,to the lines a#a1 b#a1 c#a1 d#a in ,s#a correspond the lines a1 b1 c1 d in ,s and the lines a'1 b'1 c'1 d' in ,s'1 so that d and d' are corresponding rays4 ,in the first solution the two centres1 ,s1 ,s'1 are any two points on a line joining any two corresponding points1 so that the solution of the problem allows of a great many different constructions4 ,but whatever construction be used1 the point ,d'1 corresponding to ,d1 must be always the same1 according to the theorem in ,s #bi4 ,this gives rise to a number of theorems1 into which1 however1 we shall not enter4 ,the same remarks hold for the second problem4 ,s #cg4 ,homological ,triangles4--,as a further application of the theorems about perspective rows and pencils we shall prove the following important theorem4 ,theorem4--,if ,,abc and ,a',b',c' 7fig4 #ac7 be two triangles1 such that the lines ,,aa'1 ,,bb'1 ,,cc' meet in a point ,s1 then the intersections of ,,bc and ,b',c'1 of ,,ca and ,c',a'1 and of ,,ab and ,a',b' will lie in a line4 ,such triangles are said to be homological1 or in perspective4 ,the triangles are co-axial in virtue of the property that the meets of corresponding sides are collinear and copolar1 since the lines joining corresponding vertices are concurrent4 ,proof4--,let a1 b1 c denote the lines ,,aa'1 ,,bb'1 ,,cc'1 which meet at ,s4 ,then these may be taken as bases of projective rows1 so that ,a1 ,a'1 ,s on a correspond to ,b1 ,b'1 ,s on b1 and to ,c1 ,c'1 ,s on c4 ,as the point ,s is common to all1 any two of these rows will be perspective4 ,if ,s#a be the centre of projection of rows b and c1 ,s#b c and a1 ,s#c a and b1 and if the line ,s#a,s#b cuts a in ,a#a1 and b in ,b#a1 and c in ,c#a1 then ,a#a1 ,b#a will be corresponding points in a and b1 both corresponding to ,c#a in c4 ,but a and b are perspective1 therefore the line ,a#a;,b#a1 that is ,s#a,s#b1 joining corresponding points must pass through the centre of projection ,s#c of a and b4 ,in other words1 ,s#a1 ,s#b1 ,s#c lie in a line4 ,this is ,desargues' celebrated theorem if we state it thus3-- ,illustration3 ,,fig4 #ac4 ,theorem of ,desargues4--,if each of two triangles has one vertex on each of three concurrent lines1 then the intersections of corresponding sides lie in a line1 those sides being called corresponding which are opposite to vertices on the same line4 ,the converse theorem holds also1 viz4 ,theorem4--,if the sides of one triangle meet those of another in three points which lie in a line1 then the vertices lie on three lines which meet in a point4 ,the proof is almost the same as before4 ,s #ch4 ,metrical ,relations between ,projective ,rows4--,every row contains one point which is distinguished from all others1 viz4 the point at infinity4 ,in two projective rows1 to the point ,i at infinity in one corresponds a point ,i' in the other1 and to the point ,j' at infinity in the second corresponds a point ,j in the first4 ,the points ,i' and ,j are in general finite4 ,if now ,a and ,b are any two points in the one1 ,a'1 ,b' the corresponding points in the other row1 then 7,,ab1 ,,ji7 7,a',b'1 ,j',i'71 or ,,aj_/,,jb 3 ,,ai_/,,ib ,a',j'_/,j',b' 3 ,a',i'_/,i',b'4 ,but1 by ,s #ag1 ,,ai_/,,ib ,a',j'_/,j',b' -#a2 therefore the last equation changes into ,,aj4,a',i' ,,bj4,b',i'1 that is to say-- ,theorem4--,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 constant1 viz4 ,,aj4,a',i' k4 ,steiner has called this number k the ,power of the correspondence4 ,the relation ,,aj 4 ,a',i' k shows that if ,j1 ,i' be given then the point ,a' corresponding to a specified point ,a is readily found2 hence ,a1 ,a' generate homographic ranges of which ,i and ,j' correspond to the points at infinity on the ranges4 ,if we take any two origins ,o1 ,o'1 on the ranges and reduce the expression ,,aj 4 ,a',i' k to its algebraic equivalent1 we derive an equation of the form alphaxx' betax gammax' delta #j4 ,conversely1 if a relation of this nature holds1 then points corresponding to solutions in x1 x' form homographic ranges4 ,s #ci4 ,similar ,rows4--,if the points at infinity in two projective rows correspond so that ,i' and ,j are at infinity1 this result loses its meaning4 ,but if ,a1 ,b1 ,c be any three points in one1 ,a'1 ,b'1 ,c' the corresponding ones on the other row1 we have 7,,ab1 ,,ci7 7,a',b'1 ,c',i'71 which reduces to ,,ac_/,,cb ,a',c'_/,c',b' or ,,ac_/,a',c' ,,bc_/,b',c'1 that is1 corresponding segments are proportional4 ,conversely1 if corresponding segments are proportional1 then to the point at infinity in one corresponds the point at infinity in the other4 ,if we call such rows similar1 we may state the result thus-- ,theorem4--,two projective rows are similar if to the point at infinity in one corresponds the point at infinity in the other1 and conversely1 if two rows are similar then they are projective1 and the points at infinity are corresponding points4 ,from this the well-known propositions follow3-- ,two lines are cut proportionally 7in similar rows7 by a series of parallels4 ,the rows are perspective1 with centre of projection at infinity4 ,if two similar rows are placed parallel1 then the lines joining homologous points pass through a common point4 ,s #dj4 ,if two flat pencils be projective1 then there exists in either1 one single pair of lines at right angles to one another1 such that the corresponding lines in the other pencil are again at right angles4 ,illustration3 ,,fig4 #ad4 ,to prove this1 we place the pencils in perspective position 7fig4 #ad7 by making one ray coincident with its corresponding ray4 ,corresponding rays meet then on a line p4 ,and now we draw the circle which has its centre ,o on p1 and which passes through the centres ,s and ,s' of the two pencils4 ,this circle cuts p in two points ,h and ,k4 ,the two pairs of rays1 h1 k1 and h'1 k'1 joining these points to ,s and ,s' will be pairs of corresponding rays at right angles4 ,the construction gives in general but one circle1 but if the line p is the perpendicular bisector of ,,ss'1 there exists an infinite number1 and to every right angle in the one pencil corresponds a right angle in the other4 ,,principle ,,of ,,duality ,s #da4 ,it has been stated in ,s #a that not only points1 but also planes and lines1 are taken as elements out of which figures are built up4 ,we shall now see that the construction of one figure which possesses certain properties gives rise in many cases to the construction of another figure1 by replacing1 according to definite rules1 elements of one kind by those of another4 ,the new figure thus obtained will then possess properties which may be stated as soon as those of the original figure are known4 ,we obtain thus a principle1 known as the principle of duality or of reciprocity1 which enables us to construct to any figure not containing any measurement in its construction a reciprocal figure1 as it is called1 and to deduce from any theorem a reciprocal theorem1 for which no further proof is needed4 ,it is convenient to print reciprocal propositions on opposite sides of a page broken into two columns1 and this plan will occasionally be adopted4 ,we begin by repeating in this form a few of our former statements3-- ,two points determine a line4 ,two planes determine a line4 ,three points which are not in a ,three planes which do not pass line determine a plane4 through a line determine a point4 ,a line and a point without it ,a line and a plane not through determine a plane4 it determine a point4 ,two lines in a plane determine ,two lines through a point a point4 determine a plane4 ,these propositions show that it will be possible1 when any figure is given1 to construct a second figure by taking planes instead of points1 and points instead of planes1 but lines where we had lines4 ,for instance1 if in the first figure we take a plane and three points in it1 we have to take in the second figure a point and three planes through it4 ,the three points in the first1 together with the three lines joining them two and two1 form a triangle2 the three planes in the second and their three lines of intersection form a trihedral angle4 ,a triangle and a trihedral angle are therefore reciprocal figures4 ,similarly1 to any figure in a plane consisting of points and lines will correspond a figure consisting of planes and lines passing through a point ,s1 and hence belonging to the pencil which has ,s as centre4 ,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 point4 ,in this case each figure forms a tetrahedron4 ,s #db4 ,as other examples we have the following3-- ,to a row is reciprocal an axial pencil1 a flat pencil a flat pencil1 a field of points and lines a pencil of planes and lines1 the space of points the space of planes4 ,for the row consists of a line and all the points in it1 reciprocal to it therefore will be a line with all planes through it1 that is1 an axial pencil2 and so for the other cases4 ,this correspondence of reciprocity breaks down1 however1 if we take figures which contain measurement in their construction4 ,for instance1 there is no figure reciprocal to two planes at right angles1 because there is no segment in a row which has a magnitude as definite as a right angle4 ,we add a few examples of reciprocal propositions which are easily proved4 ,theorem4--,if ,a1 ,b1 ,c1 ,d are ,theorem4--,if alpha1 beta1 any four points in space1 and if gamma1 delta are four planes the lines ,,ab and ,,cd meet1 then in space1 and if the lines all four points lie in a plane1 alphabeta and gammadelta hence also ,,ac and ,,bd1 as well meet1 then all four planes lie as ,,ad and ,,bc1 meet4 in a point 7pencil71 hence also alphagamma and betadelta1 well as alphadelta and as betagamma1 meet4 ,theorem4--,if of any number of lines every one meets every other1 whilst all do not lie in a point1 then all lie in lie in a plane1 then all lie in a plane4 a point 7pencil74 ,s #dc4 ,reciprocal figures as explained lie both in space of three dimensions4 ,if the one is confined to a plane 7is formed of elements which lie in a plane71 then the reciprocal figure is confined to a pencil 7is formed of elements which pass through a point74 ,but there is also a more special principle of duality1 according to which figures are reciprocal which lie both in a plane or both in a pencil4 ,in the plane we take points and lines as reciprocal elements1 for they have this fundamental property in common1 that two elements of one kind determine one of the other4 ,in the pencil1 on the other hand1 lines and planes have to be taken as reciprocal1 and here it holds again that two lines or planes determine one plane or line4 ,thus1 to one plane figure we can construct one reciprocal figure in the plane1 and to each one reciprocal figure in a pencil4 ,we mention a few of these4 ,at first we explain a few names3-- ,a figure consisting of n points ,a figure consisting of n lines in a plane will be called an in a plane will be called an n-point4 n-side4 ,a figure consisting of n planes ,a figure consisting of n lines in a pencil will be called an in a pencil will be called an n-flat4 n-edge4 ,it will be understood that an n-side is different from a polygon of n sides4 ,the latter has sides of finite length and n vertices1 the former has sides all of infinite extension1 and every point where two of the sides meet will be a vertex4 ,a similar difference exists between a solid angle and an n-edge or an n-flat4 ,we notice particularly-- ,a four-point has six sides1 of ,a four-side has six vertices1 of which two and two are opposite1 which two and two are opposite1 and three diagonal points1 which and three diagonals1 which join are intersections of opposite opposite vertices4 sides4 ,a four-flat has six edges1 of ,a four-edge has six faces1 of which two and two are opposite1 which two and two are opposite1 and three diagonal planes1 which and three diagonal edges1 which pass through opposite edges4 are intersections of opposite faces4 ,a four-side is usually called a complete quadrilateral1 and a four-point a complete quadrangle4 ,the above notation1 however1 seems better adapted for the statement of reciprocal propositions4 ,s #dd4 ,if a point moves in a plane it ,if a line moves in a plane it describes a plane curve4 envelopes a plane curve 7fig4 #ae74 ,if a plane moves in a pencil it ,if a line moves in a pencil it envelopes a cone4 describes a cone4 ,a curve thus appears as generated either by points1 and then we call it a locus1 or by lines1 and then we call it an envelope4 ,in the same manner a cone1 which means here a surface1 appears either as the locus of lines passing through a fixed point1 the vertex of the cone1 or as the envelope of planes passing through the same point4 ,illustration3 ,,fig4 #ae4 ,to a surface as locus of points corresponds1 in the same manner1 a surface as envelope of planes2 and to a curve in space as locus of points corresponds a developable surface as envelope of planes4 ,it will be seen from the above that we may1 by aid of the principle of duality1 construct for every figure a reciprocal figure1 and that to any property of the one a reciprocal property of the other will exist1 as long as we consider only properties which depend upon nothing but the positions and intersections of the different elements and not upon measurement4 ,for such propositions it will therefore be unnecessary to prove more than one of two reciprocal theorems4 ,,generation ,,of ,,curves ,,and ,,cones ,,of ,,second ,,order ,,or ,,second ,,class ,s #de4 ,conics4--,if we have two projective pencils in a plane1 corresponding rays will meet1 and their point of intersection will constitute some locus which we have to investigate4 ,reciprocally1 if two projective rows in a plane are given1 then the lines which join corresponding points will envelope some curve4 ,we prove first3-- ,theorem4--,if two projective ,theorem4--,if two projective flat pencils lie in a plane1 but rows lie in a plane1 but are are neither in perspective nor neither in perspective nor on a concentric1 then the locus of common base1 then the envelope intersections of corresponding of lines joining corresponding rays is a curve of the second points is a curve of the second order1 that is1 no line contains class1 that is1 through no point more than two points of the pass more than two of the locus4 enveloping lines4 ,proof4--,we draw any line t4 ,proof4--,we take any point ,t ,this cuts each of the pencils in and join it to all points in each a row1 so that we have on t two row4 ,this gives two concentric rows1 and these are projective pencils1 which are projective because the pencils are because the rows are projective4 projective4 ,if corresponding rays ,if a line joining corresponding of the two pencils meet on the points in the two rows passes line t1 their intersection will through ,t1 it will be a line in be a point in the one row which the one pencil which coincides coincides with its corresponding with its corresponding line in point in the other4 ,but two the other4 ,but two projective projective rows on the same base concentric flat pencils in the cannot have more than two same plane cannot have more than points of one coincident with two lines of one coincident with their corresponding points in their corresponding line in the the other 7,s #cd74 other 7,s #cd74 ,it will be seen that the proofs are reciprocal1 so that the one may be copied from the other by simply interchanging the words point and line1 locus and envelope1 row and pencil1 and so on4 ,we shall therefore in future prove seldom more than one of two reciprocal theorems1 and often state one theorem only1 the reader being recommended to go through the reciprocal proof by himself1 and to supply the reciprocal theorems when not given4 ,s #df4 ,we state the theorems in the pencil reciprocal to the last1 without proving them3-- ,theorem4--,if two projective ,theorem4--,if two projective flat pencils are concentric1 but axial pencils lie in the same are neither perspective nor pencil 7their axes meet in a coplanar1 then the envelope of point71 but are neither perspective the planes joining corresponding nor co-axial1 then the locus rays is a cone of the second of lines joining corresponding class2 that is1 no line through planes is a cone of the second the common centre contains more order2 that is1 no plane in the than two of the enveloping pencil contains more than two planes4 ,s #dg4 ,of theorems about cones of second order and cones of second class we shall state only very few4 ,we point out1 however1 the following connexion between the curves and cones under consideration3 ,the lines which join any point ,every plane section of a cone in space to the points on a curve of the second order is a curve of of the second order form a cone the second order4 of the second order4 ,the planes which join any ,every plane section of a cone point in space to the lines of the second class is a curve of enveloping a curve of the the second class4 second class envelope themselves a cone of the second class4 ,by its aid1 or by the principle of duality1 it will be easy to obtain theorems about them from the theorems about the curves4 ,we prove the first4 ,a curve of the second order is generated by two projective pencils4 ,these pencils1 when joined to the point in space1 give rise to two projective axial pencils1 which generate the cone in question as the locus of the lines where corresponding planes meet4 ,s#dh4 ,theorem4--,the curve of second ,theorem4--,the envelope of order which is generated by two second class which is generated projective flat pencils passes by two projective rows contains through the centres of the two the bases of these rows as pencils4 enveloping lines or tangents4 ,proof4--,if ,s and ,s' are the ,proof4--,if s and s' are the two pencils1 then to the ray ,,ss' two rows1 then to the point ss' or p' in the pencil ,s' or ,p' as a point in s' corresponds in the pencil ,s a corresponds in s a point ,p1 ray p1 which is different from which is not coincident with ,p'1 p'1 for the pencils are not for the rows are not perspective4 ,but p and p' meet perspective4 ,but ,p and ,p' are at ,s1 so that ,s is a point on joined by s1 so that s is one of the curve1 and similarly ,s'4 the enveloping lines1 and similarly s'4 ,it follows that every line in one of the two pencils cuts the curve in two points1 viz4 once at the centre ,s of the pencil1 and once where it cuts its corresponding ray in the other pencil4 ,these two points1 however1 coincide1 if the line is cut by its corresponding line at ,s itself4 ,the line p in ,s1 which corresponds to the line ,,ss' in ,s'1 is therefore the only line through ,s which has but one point in common with the curve1 or which cuts the curve in two coincident points4 ,such a line is called a tangent to the curve1 touching the latter at the point ,s1 which is called the point of contact4 ,in the same manner we get in the reciprocal investigation the result that through every point in one of the rows1 say in s1 two tangents may be drawn to the curve1 the one being s1 the other the line joining the point to its corresponding point in s'4 ,there is1 however1 one point ,p in s for which these two lines coincide4 ,such a point in one of the tangents is called the point of contact of the tangent4 ,we thus get-- ,theorem4--,to the line joining ,theorem4--,to the point of the centres of the projective intersection of the bases of two pencils as a line in one pencil projective rows as a point in corresponds in the other the one row corresponds in the other tangent at its centre4 the point of contact of its base4 ,s #di4 ,two projective pencils are determined if three pairs of corresponding lines are given4 ,hence if a#a1 b#a1 c#a are three lines in a pencil ,s#a1 and a#b1 b#b1 c#b the corresponding lines in a projective pencil ,s#b1 the correspondence and therefore the curve of the second order generated by the points of intersection of corresponding rays is determined4 ,of this curve we know the two centres ,s#a and ,s#b1 and the three points a#a;a#b1 b#a;b#b1 c#a;c#b1 hence five points in all4 ,this and the reciprocal considerations enable us to solve the following two problems3 ,problem4--,to construct a curve ,problem4--,to construct a curve of the second order1 of which of the second class1 of which five points ,s#a1 ,s#b1 ,a1 ,b1 ,c are five tangents u#a1 u#b1 a1 b1 c given4 are given4 ,in order to solve the left-hand problem1 we take two of the given points1 say ,s#a and ,s#b1 as centres of pencils4 ,these we make projective by taking the rays a#a1 b#a1 c#a1 which join ,s#a to ,a1 ,b1 ,c respectively1 as corresponding to the rays a#b1 b#b1 c#b1 which join ,s#b to ,a1 ,b1 ,c respectively1 so that three rays meet their corresponding rays at the given points ,a1 ,b1 ,c4 ,this determines the correspondence of the pencils which will generate a curve of the second order passing through ,a1 ,b1 ,c and through the centres ,s#a and ,s#b1 hence through the five given points4 ,to find more points on the curve we have to construct for any ray in ,s#a the corresponding ray in ,s#b4 ,this has been done in ,s #cf4 ,but we repeat the construction in order to deduce further properties from it4 ,we also solve the right-hand problem4 ,here we select two1 viz4 u#a1 u#b of the five given lines1 u#a1 u#b1 a1 b1 c1 as bases of two rows1 and the points ,a#a1 ,b#a1 ,c#a where a1 b1 c cut u#a as corresponding to the points ,a#b1 ,b#b1 ,c#b where a1 b1 c cut u#b4 ,we get then the following solutions of the two problems3 ,solution4--,through the point ,a ,solution4--,in the line a take draw any two lines1 u#a and u#b any two points ,s#a and ,s#b as 7fig4 #af71 the first u#a to cut centres of pencils 7fig4 #ag71 the pencil ,s#a in a row ,,ab#a;,c#a1 the first ,s#a 7,a#a;,b#a;,c#a7 to project the other u#b to cut the pencil the row u#a1 the other ,s#b ,s#b in a row ,,ab#b;,c#b4 ,these two 7,a#b;,b#b;,c#b7 to project the row u#b4 rows will be perspective1 as the ,these two pencils will be point ,a corresponds to itself1 perspective1 the line ,s#a;,a#a being and the centre of projection the same as the corresponding will be the point ,s1 where the line ,s#b;,a#b1 and the axis of lines ,b#a;,b#b and ,c#a;,c#b meet4 ,to projection will be the line u1 find now for any ray d#a in ,s#a which joins the intersection ,b its corresponding ray d#b in ,s#b1 of ,s#a;,b#a and ,s#b;,b#b to the we determine the point ,d#a where intersection ,c of ,s#a;,c#a and ,s#b;,c#b4 d#a cuts u#a1 project this point ,to find now for any point ,d#a in from ,s to ,d#b on u#b and join ,s#b u#a the corresponding point ,d#b in to ,d#b4 ,this will be the required u#b1 we draw ,s#a;,d#a and project the ray d#b which cuts d#a at some point ,d where this line cuts u point ,d on the curve4 from ,s#b to u#b4 ,this will give the required point ,d#b1 and the line d joining ,d#a to ,d#b will be a new tangent to the curve4 ,s #ej4 ,these constructions prove1 when rightly interpreted1 very important properties of the curves in question4 ,illustration3 ,,fig4 #af4 ,if in fig4 #af we draw in the pencil ,s#a the ray k#a which passes through the auxiliary centre ,s1 it will be found that the corresponding ray k#b cuts it on u#b4 ,hence-- ,theorem4--,in the above ,theorem4--,in the above construction the bases of the construction 7fig4 #ag7 the auxiliary rows u#a and u#b cut the tangents to the curve from the curve where they cut the rays centres of the auxiliary pencils ,s#b,s and ,s#a,s respectively4 ,s#a and ,s#b are the lines which pass through u#bu and u#au respectively4 ,as ,a is any given point on the curve1 and u#a any line through it1 we have solved the problems3 ,problem4--,to find the second ,problem4--,to find the second point in which any line through tangent which can be drawn from a known point on the curve cuts any point in a given tangent to the curve4 the curve4 ,if we determine in ,s#a 7fig4 #af7 the ray corresponding to the ray ,s#b,s#a in ,s#b1 we get the tangent at ,s#a4 ,similarly1 we can determine the point of contact of the tangents u#a or u#b in fig4 #ag4 ,illustration3 ,,fig4 #ag4 ,s #ea4 ,if five points are given1 of which not three are in a line1 then we can1 as has just been shown1 always draw a curve of the second order through them2 we select two of the points as centres of projective pencils1 and then one such curve is determined4 ,it will be presently shown that we get always the same curve if two other points are taken as centres of pencils1 that therefore five points determine one curve of the second order1 and reciprocally1 that five tangents determine one curve of the second class4 ,six points taken at random will therefore not lie on a curve of the second order4 ,in order that this may be the case a certain condition has to be satisfied1 and this condition is easily obtained from the construction in ,s #di1 fig4 #af4 ,if we consider the conic determined by the five points ,a1 ,s#a1 ,s#b1 ,k1 ,l1 then the point ,d will be on the curve if1 and only if1 the points on ,d#a1 ,s1 ,d#b be in a line4 ,illustration3 ,,fig4 #ah4 ,this may be stated differently if we take ,,aks#a;,,ds#b,l 7figs4 #af and #ah7 as a hexagon inscribed in the conic1 then ,,ak and ,,ds#b will be opposite sides1 so will be ,,ks#a and ,s#b,l1 as well as ,s#a;,d and ,,la4 ,the first two meet in ,d#b1 the others in ,s and ,d#a respectively4 ,we may therefore state the required condition1 together with the reciprocal one1 as follows3-- ,pascal's ,theorem4--,if a hexagon ,brianchon's ,theorem4--,if a be inscribed in a curve of the hexagon be circumscribed about second order1 then the a curve of the second class1 then intersectionsof opposite sides the lines joining opposite vertices are three points in a line4 are three lines meeting in a point4 ,these celebrated theorems1 which are known by the names of their discoverers1 are perhaps the most fruitful in the whole theory of conics4 ,before we go over to their applications we have to show that we obtain the same curve if we take1 instead of ,s#a1 ,s#b1 any two other points on the curve as centres of projective pencils4 ,s #eb4 ,we know that the curve depends only upon the correspondence between the pencils ,s#a and ,s#b1 and not upon the special construction used for finding new points on the curve4 ,the point ,a 7fig4 #af or #ah71 through which the two auxiliary rows u#a1 u#b were drawn1 may therefore be changed to any other point on the curve4 ,let us now suppose the curve drawn1 and keep the points ,s#a1 ,s#b1 ,k1 ,l and ,d1 and hence also the point ,s fixed1 whilst we move ,a along the curve4 ,then the line ,,al will describe a pencil about ,l as centre1 and the point ,d#a a row on ,s#a;,d perspective to the pencil ,l4 ,at the same time ,,ak describes a pencil about ,k and ,d#b a row perspective to it on ,s#b;,d4 ,but by ,pascal's theorem ,d#a and ,d#b will always lie in a line with ,s1 so that the rows described by ,d#a and ,d#b are perspective4 ,it follows that the pencils ,k and ,l will themselves be projective1 corresponding rays meeting on the curve4 ,this proves that we get the same curve whatever pair of the five given points we take as centres of projective pencils4 ,hence-- ,only one curve of the second ,only one curve of the second order can be drawn which passes class can be drawn which touches through five given points4 five given lines4 ,we have seen that if on a curve of the second order two points coincide at ,a1 the line joining them becomes the tangent at ,a4 ,if1 therefore1 a point on the curve and its tangent are given1 this will be equivalent to having given two points on the curve4 ,similarly1 if on the curve of second class a tangent and its point of contact are given1 this will be equivalent to two given tangents4 ,we may therefore extend the last theorem3 ,only one curve of the second ,only one curve of the second order can be drawn1 of which class can be drawn1 of which four four points and the tangent at tangents and the point of contact oneof them1 or three points at one of them1 or three tangents and the tangents at two of and the points of contact at two them1 are given4 of them1 are given4 ,s #ec4 ,at the same time it has been proved3 ,if all points on a curve of the ,all tangents to a curve of second second order be joined to any class are cut by any two of two of them1 then the two them in projective rows1 those pencils thus formed are being corresponding points which projective1 those rays being lie on the same tangent4 ,hence-- corresponding which meet on the curve4 ,hence-- ,the cross-ratio of four rays ,the cross-ratio of the four joining a point ,s on a curve of points in which any tangent u is second order to four fixed cut by four fixed tangents a1 b1 c1 points ,a1 ,b1 ,c1 ,d in the curve d is independent of the position of is independent of the position u1 and is called the cross-ratio of of ,s1 and is called the cross- the four tangents a1 b1 c1 d4 ratio of the four points ,a1 ,b1 ,c1 ,d4 ,if this cross-ratio equals -#a ,if this cross-ratio equals -#a the four points are said to be the four tangents are said to be four harmonic points4 four harmonic tangents4 ,we have seen that a curve of second order1 as generated by projective pencils1 has at the centre of each pencil one tangent2 and further1 that any point on the curve may be taken as centre of such pencil4 ,hence-- ,a curve of second order has ,a curve of second class has on at every point one tangent4 every tangent a point of contact4 ,s #ed4 ,we return to ,pascal's and ,brianchon's theorems and their applications1 and shall1 as before1 state the results both for curves of the second order and curves of the second class1 but prove them only for the former4 ,pascal's theorem may be used when five points are given to find more points on the curve1 viz4 it enables us to find the point where any line through one of the given points cuts the curve again4 ,it is convenient1 in making use of ,pascal's theorem1 to number the points1 to indicate the order in which they are to be taken in forming a hexagon1 which1 by the way1 may be done in #fj different ways4 ,it will be seen that #a #b 7leaving out #c7 #d #e are opposite sides1 so are #b #c and 7leaving out #d7 #e #f1 and also #c #d and 7leaving out #e7 #f #a4 ,if the points #a #b #c #d #e are given1 and we want a #fth point on a line drawn through #a1 we know all the sides of the hexagon with the exception of #e #f1 and this is found by ,pascal's theorem4 ,if this line should happen to pass through #a1 then #f and #a coincide1 or the line #f #a is the tangent at #a4 ,and always if two consecutive vertices of the hexagon approach nearer and nearer1 then the side joining them will ultimately become a tangent4 ,we may therefore consider a pentagon inscribed in a curve of second order and the tangent at one of its vertices as a hexagon1 and thus get the theorem3 ,every pentagon inscribed in a ,every pentagon circumscribed curve of second order has the about a curve of the second class property that the intersections has the property that the lines of two pairs of non-consecutive which join two pairs of non- sides lie in a line with the consecutive vertices meet on that point where the fifth side cuts line which joins the fifth vertex the tangent at the opposite to the point of contact of the vertex4 opposite side4 ,this enables us also to solve the following problems4 ,given five points on a curve of ,given five tangents to a curve second order to construct the of second class to construct the tangent at any one of them4 point of contact of any one of them4 ,illustration3 ,,fig4 #ai4 ,if two pairs of adjacent vertices coincide1 the hexagon becomes a quadrilateral1 with tangents at two vertices4 ,these we take to be opposite1 and get the following theorems3 ,if a quadrilateral be inscribed ,if a quadrilateral be circumscribed in a curve of second order1 the about a curve of second intersections of opposite sides1 class1 the lines joining opposite and also the intersections of vertices1 and also the lines joining the tangents at opposite points of contact of opposite vertices1 lie in a line 7fig4 sides1 meet in a point4 #ai74 ,illustration3 ,,fig4 #bj4 ,if we consider the hexagon made up of a triangle and the tangents at its vertices1 we get-- ,if a triangle is inscribed in a ,if a triangle be circumscribed curve of the second order1 the about a curve of second class1 points in which the sides are the lines which join the vertices cut by the tangents at the to the points of contact of the opposite vertices meet in a opposite sides meet in a point point4 7fig4 #bj74 ,s #ee4 ,of these theorems1 those about the quadrilateral give rise to a number of others4 ,four points ,a1 ,b1 ,c1 ,d may in three different ways be formed into a quadrilateral1 for we may take them in the order ,,abcd1 or ,,acbd1 or ,,acdb1 so that either of the points ,b1 ,c1 ,d may be taken as the vertex opposite to ,a4 ,accordingly we may apply the theorem in three different ways4 ,let ,a1 ,b1 ,c1 ,d be four points on a curve of second order 7fig4 #ba71 and let us take them as forming a quadrilateral by taking the points in the order ,,abcd1 so that ,a1 ,c and also ,b1 ,d are pairs of opposite vertices4 ,then ,p1 ,q will be the points where opposite sides meet1 and ,e1 ,f the intersections of tangents at opposite vertices4 ,the four points ,p1 ,q1 ,e1 ,f lie therefore in a line4 ,the quadrilateral ,,acbd gives us in the same way the four points ,q1 ,r1 ,g1 ,h in a line1 and the quadrilateral ,,abdc a line containing the four points ,r1 ,p1 ,i1 ,k4 ,these three lines form a triangle ,,pqr4 ,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 conic1 and the tangents at these points as a four-side circumscribed about it1--viz4 it will be seen that ,p1 ,q1 ,r are the diagonal points of the four-point ,,abcd1 whilst the sides of the triangle ,,pqr are the diagonals of the circumscribing four-side4 ,hence the theorem-- ,any four-point on a curve of the second order and the four-side formed by the tangents at these points stand in this relation that the diagonal points of the four-point lie in the diagonals of the four-side4 ,and conversely1 ,if a four-point and a circumscribed four-side stand in the above relation1 then a curve of the second order may be described which passes through the four points and touches there the four sides of these figures4 ,that the last part of the theorem is true follows from the fact that the four points ,a1 ,b1 ,c1 ,d and the line a1 as tangent at ,a1 determine a curve of the second order1 and the tangents to this curve at the other points ,b1 ,c1 ,d are given by the construction which leads to fig4 #ba4 ,illustration3 ,,fig4 #ba4 ,the theorem reciprocal to the last is-- ,any four-side circumscribed about a curve of second class and the four-point formed by the points of contact stand in this relation that the diagonals of the four-side pass through the diagonal points of the four-point4 ,and conversely1 ,if a four-side and an inscribed four-point stand in the above relation1 then a curve of the second class may be described which touches the sides of the four-side at the points of the four-point4 ,s #ef4 ,the four-point and the four-side in the two reciprocal theorems are alike4 ,hence if we have a four-point ,,abcd and a four-side abcd related in the manner described1 then not only may a curve of the second order be drawn1 but also a curve of the second class1 which both touch the lines a1 b1 c1 d at the points ,a1 ,b1 ,c1 ,d4 ,the curve of second order is already more than determined by the points ,a1 ,b1 ,c and the tangents a1 b1 c at ,a1 ,b and ,c4 ,the point ,d may therefore be any point on this curve1 and d any tangent to the curve4 ,on the other hand the curve of the second class is more than determined by the three tangents a1 b1 c and their points of contact ,a1 ,b1 ,c1 so that d is any tangent to this curve4 ,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 contact4 ,in other words1 the curve of second order is a curve of second class1 and vice versa4 ,hence the important theorems-- ,every curve of second order is ,every curve of second class is a a curve of second class4 curve of second order4 ,the curves of second order and of second class1 having thus been proved to be identical1 shall henceforth be called by the common name of ,conics4 ,for these curves hold1 therefore1 all properties which have been proved for curves of second order or of second class4 ,we may therefore now state ,pascal's and ,brianchon's theorem thus-- ,pascal's ,theorem4--,if a hexagon be inscribed in a conic1 then the intersections of opposite sides lie in a line4 ,brianchon's ,theorem4--,if a hexagon be circumscribed about a conic1 then the diagonals forming opposite centres meet in a point4 ,s #eg4 ,if we suppose in fig4 #ba that the point ,d together with the tangent d moves along the curve1 whilst ,a1 ,b1 ,c and their tangents a1 b1 c remain fixed1 then the ray ,,da will describe a pencil about ,a1 the point ,q a projective row on the fixed line ,,bc1 the point ,f the row b1 and the ray ,,ef a pencil about ,e4 ,but ,,ef passes always through ,q4 ,hence the pencil described by ,,ad is projective to the pencil described by ,,ef1 and therefore to the row described by ,f on b4 ,at the same time the line ,,bd describes a pencil about ,b projective to that described by ,,ad 7,s #ec74 ,therefore the pencil ,,bd and the row ,f on b are projective4 ,hence-- ,if on a conic a point ,a be taken and the tangent a at this point1 then the cross-ratio of the four rays which join ,a to any four points on the curve is equal to the cross-ratio of the points in which the tangents at these points cut the tangent at ,a4 ,s #eh4 ,there are theorems about cones of second order and second class in a pencil which are reciprocal to the above1 according to ,s #dc4 ,we mention only a few of the more important ones4 ,the locus of intersections of corresponding planes in two projective axial pencils whose axes meet is a cone of the second order4 ,the envelope of planes which join corresponding lines in two projective flat pencils1 not in the same plane1 is a cone of the second class4 ,cones of second order and cones of second class are identical4 ,every plane cuts a cone of the second order in a conic4 ,a cone of second order is uniquely determined by five of its edges or by five of its tangent planes1 or by four edges and the tangent plane at one of them1 c4 c4 ,pascal's ,theorem4--,if a solid angle of six faces be inscribed in a cone of the second order1 then the intersections of opposite faces are three lines in a plane4 ,brianchon's ,theorem4--,if a solid angle of six edges be circumscribed about a cone of the second order1 then the planes through opposite edges meet in a line4 ,each of the other theorems about conics may be stated for cones of the second order4 ,s #ei4 ,projective ,definitions of the ,conics4--,we now consider the shape of the conics4 ,we know that any line in the plane of the conic1 and hence that the line at infinity1 either has no point in common with the curve1 or one 7counting for two coincident points7 or two distinct points4 ,if the line at infinity has no point on the curve the latter is altogether finite1 and is called an ,ellipse 7fig4 #ba74 ,if the line at infinity has only one point in common with the conic1 the latter extends to infinity1 and has the line at infinity a tangent4 ,it is called a ,parabola 7fig4 #bb74 ,if1 lastly1 the line at infinity cuts the curve in two points1 it consists of two separate parts which each extend in two branches to the points at infinity where they meet4 ,the curve is in this case called an ,hyperbola 7see fig4 #bj74 ,the tangents at the two points at infinity are finite because the line at infinity is not a tangent4 ,they are called ,asymptotes4 ,the branches of the hyperbola approach these lines indefinitely as a point on the curves moves to infinity4 ,illustration3 ,,fig4 #bb4 ,s #fj4 ,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 equal4 ,if two points ,s#a1 ,s#b on a circle be joined to any other two points ,a and ,b on the circle1 then the angle included by the rays ,s#a;,a and ,s#a;,b is equal to that between the rays ,s#b;,a and ,s#b;,b1 so that as ,a moves along the circumference the rays ,s#a;,a and ,s#b;,a describe equal and therefore projective pencils4 ,the circle can thus be generated by two projective pencils1 and is a curve of the second order4 ,if we join a point in space to all points on a circle1 we get a 7circular7 cone of the second order 7,s #dc74 ,every plane section of this cone is a conic4 ,this conic will be an ellipse1 a parabola1 or an hyperbola1 according as the line at infinity in the plane has no1 one or two points in common with the conic in which the plane at infinity cuts the cone4 ,it follows that our curves of second order may be obtained as sections of a circular cone1 and that they are identical with the ,conic ,sections of the ,greek mathematicians4 ,s #fa4 ,any two tangents to a parabola are cut by all others in projective rows2 but the line at infinity being one of the tangents1 the points at infinity on the rows are corresponding points1 and the rows therefore similar4 ,hence the theorem-- ,the tangents to a parabola cut each other proportionally4 ,,pole ,,and ,,polar ,s #fb4 ,we return once again to fig4 #ba1 which we obtained in ,s #ee4 ,if a four-side be circumscribed about and a four-point inscribed in a conic1 so that the vertices of the second are the points of contact of the sides of the first1 then the triangle formed by the diagonals of the first is the same as that formed by the diagonal points of the other4 ,such a triangle will be called a polar-triangle of the conic1 so that ,,pqr in fig4 #ba is a polar-triangle4 ,it has the property that on the side p opposite ,p meet the tangents at ,a and ,b1 and also those at ,c and ,d4 ,from the harmonic properties of four-points and four-sides it follows further that the points ,l1 ,m1 where it cuts the lines ,,ab and ,,cd1 are harmonic conjugates with regard to ,,ab and ,,cd respectively4 ,if the point ,p is given1 and we draw a line through it1 cutting the conic in ,a and ,b1 then the point ,q harmonic conjugate to ,p with regard to ,,ab1 and the point ,h where the tangents at ,a and ,b meet1 are determined4 ,but they lie both on p1 and therefore this line is determined4 ,if we now draw a second line through ,p1 cutting the conic in ,c and ,d1 then the point ,m harmonic conjugate to ,p with regard to ,,cd1 and the point ,g where the tangents at ,c and ,d meet1 must also lie on p4 ,as the first line through ,p already determines p1 the second may be any line through ,p4 ,now every two lines through ,p determine a four-point ,,abcd on the conic1 and therefore a polar-triangle which has one vertex at ,p and its opposite side at p4 ,this result1 together with its reciprocal1 gives the theorems-- ,all polar-triangles which have one vertex in common have also the opposite side in common4 ,all polar-triangles which have one side in common have also the opposite vertex in common4 ,s #fc4 ,to any point ,p in the plane of1 but not on1 a conic corresponds thus one line p as the side opposite to ,p in all polar-triangles which have one vertex at ,p1 and reciprocally to every line p corresponds one point ,p as the vertex opposite to p in all triangles which have p as one side4 ,we call the line p the polar of ,p1 and the point ,p the pole of the line p with regard to the conic4 ,if a point lies on the conic1 we call the tangent at that point its polar2 and reciprocally we call the point of contact the pole of tangent4 ,s #fd4 ,from these definitions and former results follow-- ,the polar of any point ,p not ,the pole of any line p not a on the conic is a line p1 which tangent to the conic is a point has the following properties3-- ,p1 which has the following properties3-- #a4 ,on every line through ,p #a4 ,of all lines through a point which cuts the conic1 the polar on p from which two tangents of ,p contains the harmonic may be drawn to the conic1 the conjugate of ,p with regard to pole ,p contains the line which is those points on the conic4 harmonic conjugate to p1 with regard to the two tangents4 #b4 ,if tangents can be drawn #b4 ,if p cuts the conic1 the from ,p1 their points of contact tangents at the intersections lie on p4 meet at ,p4 #c4 ,tangents drawn at the #c4 ,the point of contact of points where any line through ,p tangents drawn from any point cuts the conic meet on p2 and on p to the conic lie in a line conversely1 with ,p2 and conversely1 #d4 ,if from any point on p1 #d4 ,tangents drawn at points tangents be drawn1 their points where any line through ,p cuts the of contact will lie in a line conic meet on p4 with ,p4 #e4 ,any four-point on the conic #e4 ,any four-side circumscribed which has one diagonal point at about a conic which has one ,p has the other two lying on p4 diagonal on p has the other two meeting at ,p4 ,the truth of #b follows from #a4 ,if ,t be a point where p cuts the conic1 then one of the points where ,,pt cuts the conic1 and which are harmonic conjugates with regard to ,,pt1 coincides with ,t2 hence the other does--that is1 ,,pt touches the curve at ,t4 ,that #d is true follows thus3 ,if we draw from a point ,h on the polar one tangent a to the conic1 join its point of contact ,a to the pole ,p1 determine the second point of intersection ,b of this line with the conic1 and draw the tangent at ,b1 it will pass through ,h1 and will therefore be the second tangent which may be drawn from ,h to the curve4 ,s #fe4 ,the second property of the polar or pole gives rise to the theorem-- ,from a point in the plane of a ,a line in the plane of a conic conic1 two1 one or no tangents has two1 one or no points in may be drawn to the conic1 common with the conic1 according as its polar has two1 as two1 one or no tangents one1 or no points in common can be drawn from its pole to the with the curve4 conic4 ,of any point in the plane of a conic we say that it was without1 on or within the curve according as two1 one or no tangents to the curve pass through it4 ,the points on the conic separate those within the conic from those without4 ,that this is true for a circle is known from elementary geometry4 ,that it also holds for other conics follows from the fact that every conic may be considered as the projection of a circle1 which will be proved later on4 ,the fifth property of pole and polar stated in ,s #fd shows how to find the polar of any point and the pole of any line by aid of the straight-edge only4 ,practically it is often convenient to draw three secants through the pole1 and to determine only one of the diagonal points for two of the four-points formed by pairs of these lines and the conic 7fig4 #bb74 ,these constructions also solve the problem-- ,from a point without a conic1 to draw the two tangents to the conic by aid of the straight-edge only4 ,for we need only draw the polar of the point in order to find the points of contact4 ,s #ff4 ,the property of a polar-triangle may now be stated thus-- ,in a polar-triangle each side is the polar of the opposite vertex1 and each vertex is the pole of the opposite side4 ,illustration3 ,,fig4 #bc4 ,if ,p is one vertex of a polar-triangle1 then the other vertices1 ,q and ,r1 lie on the polar p of ,p4 ,one of these vertices we may choose arbitrarily4 ,for if from any point ,q on the polar a secant be drawn cutting the conic in ,a and ,d 7fig4 #bc71 and if the lines joining these points to ,p cut the conic again at ,b and ,c1 then the line ,,bc will pass through ,q4 ,hence ,p and ,q are two of the vertices on the polar-triangle which is determined by the four-point ,,abcd4 ,the third vertex ,r lies also on the line p4 ,it follows1 therefore1 also-- ,if ,q is a point on the polar of ,p1 then ,p is a point on the polar of ,q2 and reciprocally1 ,if q is a line through the pole of p1 then p is a line through the pole of q4 ,this is a very important theorem4 ,it may also be stated thus-- ,if a point moves along a line describing a row1 its polar turns about the pole of the line describing a pencil4 ,this pencil is projective to the row1 so that the cross-ratio of four poles in a row equals the cross-ratio of its four polars1 which pass through the pole of the row4 ,to prove the last part1 let us suppose that ,p1 ,a and ,b in fig4 #bc remain fixed1 whilst ,q moves along the polar p of ,p4 ,this will make ,,cd turn about ,p and move ,r along p1 whilst ,,qd and ,,rd describe projective pencils about ,a and ,b4 ,hence ,q and ,r describe projective rows1 and hence ,,pr1 which is the polar of ,q1 describes a pencil projective to either4 ,s #fg4 ,two points1 of which one1 and therefore each1 lies on the polar of the other1 are said to be conjugate with regard to the conic2 and two lines1 of which one1 and therefore each1 passes through the pole of the other1 are said to be conjugate with regard to the conic4 ,hence all points conjugate to a point ,p lie on the polar of ,p2 all lines conjugate to a line p pass through the pole of p4 ,if the line joining two conjugate poles cuts the conic1 then the poles are harmonic conjugates with regard to the points of intersection2 hence one lies within the other without the conic1 and all points conjugate to a point within a conic lie without it4 ,of a polar-triangle any two vertices are conjugate poles1 any two sides conjugate lines4 ,if1 therefore1 one side cuts a conic1 then one of the two vertices which lie on this side is within and the other without the conic4 ,the vertex opposite this side lies also without1 for it is the pole of a line which cuts the curve4 ,in this case therefore one vertex lies within1 the other two without4 ,if1 on the other hand1 we begin with a side which does not cut the conic1 then its pole lies within and the other vertices without4 ,hence-- ,every polar-triangle has one and only one vertex within the conic4 ,we add1 without a proof1 the theorem-- ,the four points in which a conic is cut by two conjugate polars are four harmonic points in the conic4 ,s #fh4 ,if two conics intersect in four points 7they cannot have more points in common1 ,s #eb71 there exists one and only one four-point which is inscribed in both1 and therefore one polar-triangle common to both4 ,theorem4--,two conics which intersect in four points have always one and only one common polar-triangle2 and reciprocally1 ,two conics which have four common tangents have always one and only one common polar-triangle4 ,,diameters ,,and ,,axes ,,of ,,conics ,s #fi4 ,diameters4--,the theorems about the harmonic properties of poles and polars contain1 as special cases1 a number of important metrical properties of conics4 ,these are obtained if either the pole or the polar is moved to infinity1--it being remembered that the harmonic conjugate to a point at infinity1 with regard to two points ,a1 ,b1 is the middle point of the segment ,,ab4 ,the most important properties are stated in the following theorems3-- ,the middle points of parallel chords of a conic lie in a line--viz4 on the polar to the point at infinity on the parallel chords4 ,this line is called a diameter4 ,the polar of every point at infinity is a diameter4 ,the tangents at the end points of a diameter are parallel1 and are parallel to the chords bisected by the diameter4 ,all diameters pass through a common point1 the pole of the line at infinity4 ,all diameters of a parabola are parallel1 the pole to the line at infinity being the point where the curve touches the line at infinity4 ,in case of the ellipse and hyperbola1 the pole to the line at infinity is a finite point called the centre of the curve4 ,a centre of a conic bisects every chord through it4 ,the centre of an ellipse is within the curve1 for the line at infinity does not cut the ellipse4 ,the centre of an hyperbola is without the curve1 because the line at infinity cuts the curve4 ,hence also-- ,from the centre of an hyperbola two tangents can be drawn to the curve which have their point of contact at infinity4 ,these are called ,asymptotes 7,s #ei74 ,to construct a diameter of a conic1 draw two parallel chords and join their middle points4 ,to find the centre of a conic1 draw two diameters2 their intersection will be the centre4 ,s #gj4 ,conjugate ,diameters4--,a polar-triangle with one vertex at the centre will have the opposite side at infinity4 ,the other two sides pass through the centre1 and are called conjugate diameters1 each being the polar of the point at infinity on the other4 ,of two conjugate diameters each bisects the chords parallel to the other1 and if one cuts the curve1 the tangents at its ends are parallel to the other diameter4 ,further-- ,every parallelogram inscribed in a conic has its sides parallel to two conjugate diameters2 and ,every parallelogram circumscribed about a conic has as diagonals two conjugate diameters4 ,this will be seen by considering the parallelogram in the first case as an inscribed four-point1 in the other as a circumscribed four-side1 and determining in each case the corresponding polar-triangle4 ,the first may also be enunciated thus-- ,the lines which join any point on an ellipse or an hyperbola to the ends of a diameter are parallel to two conjugate diameters4 ,s #ga4 ,if every diameter is perpendicular to its conjugate the conic is a circle4 ,for the lines which join the ends of a diameter to any point on the curve include a right angle4 ,a conic which has more than one pair of conjugate diameters at right angles to each other is a circle4 ,illustration3 ,,fig4 #bd4 ,let ,,aa' and ,,bb' 7fig4 #bd7 be one pair of conjugate diameters at right angles to each other1 ,,cc and ,,dd' a second pair4 ,if we draw through the end point ,a of one diameter a chord ,,ap parallel to ,,dd'1 and join ,p to ,a'1 then ,,pa and ,,pa' are1 according to ,s #gj1 parallel to two conjugate diameters4 ,but ,,pa is parallel to ,,dd'1 hence ,,pa' is parallel to ,,cc1 and therefore ,,pa and ,,pa' are perpendicular4 ,if we further draw the tangents to the conic at ,a and ,a'1 these will be perpendicular to ,,aa'1 they being parallel to the conjugate diameter ,,bb'4 ,we know thus five points on the conic1 viz4 the points ,a and ,a' with their tangents1 and the point ,p4 ,through these a circle may be drawn having ,,aa' as diameter2 and as through five points one conic only can be drawn1 this circle must coincide with the given conic4 ,s #gb4 ,axes4--,conjugate diameters perpendicular to each other are called axes1 and the points where they cut the curve vertices of the conic4 ,in a circle every diameter is an axis1 every point on it is a vertex2 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 intersection4 ,illustration3 ,,fig4 #be4 ,if we describe on a diameter ,,ab of an ellipse or hyperbola a circle concentric to the conic1 it will cut the latter in ,a and ,b 7fig4 #be74 ,each of the semicircles in which it is divided by ,,ab will be partly within1 partly without the curve1 and must cut the latter therefore again in a point4 ,the circle and the conic have thus four points ,a1 ,b1 ,c1 ,d1 and therefore one polar-triangle1 in common 7,s #fh74 ,of this the centre is one vertex1 for the line at infinity is the polar to this point1 both with regard to the circle and the other conic4 ,the other two sides are conjugate diameters of both1 hence perpendicular to each other4 ,this gives-- ,an ellipse as well as an hyperbola has one pair of axes4 ,this reasoning shows at the same time how to construct the axis of an ellipse or of an hyperbola4 ,a parabola has one axis1 if we define an axis as a diameter perpendicular to the chords which it bisects4 ,it is easily constructed4 ,the line which bisects any two parallel chords is a diameter4 ,chords perpendicular to it will be bisected by a parallel diameter1 and this is the axis4 ,s #gc4 ,the first part of the right-hand theorem in ,s #fd may be stated thus3 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 conic4 ,if we take instead of ,p the centre ,c of an hyperbola1 then the conjugate lines become conjugate diameters1 and the tangents asymptotes4 ,hence-- ,any two conjugate diameters of an hyperbola are harmonic conjugates with regard to the asymptotes4 ,as the axes are conjugate diameters at right angles to one another1 it follows 7,s #bc7-- ,the axes of an hyperbola bisect the angles between the asymptotes4 ,illustration3 ,,fig4 #bf4 ,let ,o be the centre of the hyperbola 7fig4 #bf71 t any secant which cuts the hyperbola in ,c1 ,d and the asymptotes in ,e1 ,f1 then the line ,,om which bisects the chord ,,cd is a diameter conjugate to the diameter ,,ok which is parallel to the secant t1 so that ,,ok and ,,om are harmonic with regard to the asymptotes4 ,the point ,m therefore bisects ,,ef4 ,but by construction ,m bisects ,,cd4 ,it follows that ,,df ,,ec1 and ,,ed ,,cf2 or ,on any secant of an hyperbola the segments between the curve and the asymptotes are equal4 ,if the chord is changed into a tangent1 this gives-- ,the segment between the asymptotes on any tangent to an hyperbola is bisected by the point of contact4 ,the first part allows a simple solution of the problem to find any number of points on an hyperbola1 of which the asymptotes and one point are given4 ,this is equivalent to three points and the tangents at two of them4 ,this construction requires measurement4 ,s #gd4 ,for the parabola1 too1 follow some metrical properties4 ,a diameter ,,pm 7fig4 #bg7 bisects every chord conjugate to it1 and the pole ,p of such a chord ,,bc lies on the diameter4 ,but a diameter cuts the parabola once at infinity4 ,hence-- ,the segment ,,pm which joins the middle point ,m of a chord of a parabola to the pole ,p of the chord is bisected by the parabola at ,a4 ,s #ge4 ,two asymptotes and any two tangents to an hyperbola may be considered as a quadrilateral circumscribed about the hyperbola4 ,but in such a quadrilateral the intersections of the diagonals and the points of contact of opposite sides lie in a line 7,s #ed74 ,if therefore ,,defg 7fig4 #bh7 is such a quadrilateral1 then the diagonals ,,df and ,,ge will meet on the line which joins the points of contact of the asymptotes1 that is1 on the line at infinity2 hence they are parallel4 ,from this the following theorem is a simple deduction3 ,all triangles formed by a tangent and the asymptotes of an hyperbola are equal in area4 ,illustration3 ,,fig4 #bg4 ,illustration3 ,,fig4 #bh4 ,if we draw at a point ,p 7fig4 #bh7 on an hyperbola a tangent1 the part ,,hk between the asymptotes is bisected at ,p4 ,the parallelogram ,,pqoq' formed by the asymptotes and lines parallel to them through ,p will be half the triangle ,,ohk1 and will therefore be constant4 ,if we now take the asymptotes ,,ox and ,,oy as oblique axes of co-ordinates1 the lines ,,oq and ,,qp will be the co-ordinates of ,p1 and will satisfy the equation xy const4 a#b4 ,for the asymptotes as axes of co-ordinates the equation of the hyperbola is xy const4 ,,involution ,illustration3 ,,fig4 #bi4 ,s #gf4 ,if we have two projective rows1 ,,abc on u and ,a',b',c' on u'1 and place their bases on the same line1 then each point in this line counts twice1 once as a point in the row u and once as a point in the row u'4 ,in fig4 #bi we denote the points as points in the one row by letters above the line ,a1 ,b1 ,c 4441 and as points in the second row by ,a'1 ,b'1 ,c' 444 below the line4 ,let now ,a and ,b' be the same point1 then to ,a will correspond a point ,a' in the second1 and to ,b' a point ,b in the first row4 ,in general these points ,a' and ,b will be different4 ,it may1 however1 happen that they coincide4 ,then the correspondence is a peculiar one1 as the following theorem shows3 ,if two projective rows lie on the same base1 and if it happens that to one point in the base the same point corresponds1 whether we consider the point as belonging to the first or to the second row1 then the same will happen for every point in the base--that is to say1 to every point in the line corresponds the same point in the first as in the second row4 ,illustration3 ,,fig4 #cj4 ,in order to determine the correspondence1 we may assume three pairs of corresponding points in two projective rows4 ,let then ,a'1 ,b'1 ,c'1 in fig4 #cj1 correspond to ,a1 ,b1 ,c1 so that ,a and ,b'1 and also ,b and ,a'1 denote the same point4 ,let us further denote the point ,c' when considered as a point in the first row by ,d2 then it is to be proved that the point ,d'1 which corresponds to ,d1 is the same point as ,c4 ,we know that the cross-ratio of four points is equal to that of the corresponding row4 ,hence 7,,ab1 ,,cd7 7,a',b'1 ,c',d'7 but replacing the dashed letters by those undashed ones which denote the same points1 the second cross-ratio equals 7,,ba1 ,,dd'71 which1 according to ,s #ae1 equals 7,,ab1 ,d',d72 so that the equation becomes 7,,ab1 ,,cd7 7,,ab1 ,d',d74 ,this requires that ,c and ,d' coincide4 ,s #gg4 ,two projective rows on the same base1 which have the above property1 that to every point1 whether it be considered as a point in the one or in the other row1 corresponds the same point1 are said to be in involution1 or to form an involution of points on the line4 ,we mention1 but without proving it1 that any two projective rows may be placed so as to form an involution4 ,an involution may be said to consist of a row of pairs of points1 to every point ,a corresponding a point ,a'1 and to ,a' again the point ,a4 ,these points are said to be conjugate1 or1 better1 one point is termed the mate of the other4 ,from the definition1 according to which an involution may be considered as made up of two projective rows1 follow at once the following important properties3 #a4 ,the cross-ratio of four points equals that of the four conjugate points4 #b4 ,if we call a point which coincides with its mate a focus or double point of the involution1 we may say3 ,an involution has either two foci1 or one1 or none1 and is called respectively a hyperbolic1 parabolic or elliptic involution 7,s #cd74 #c4 ,in an hyperbolic involution any two conjugate points are harmonic conjugates with regard to the two foci4 ,for if ,a1 ,a' be two conjugate points1 ,f#a1 ,f#b the two foci1 then to the points ,f#a1 ,f#b1 ,a1 ,a' in the one row correspond the points ,f#a1 ,f#b1 ,a'1 ,a in the other1 each focus corresponding to itself4 ,hence 7,f#a;,f#b1 ,,aa'7 7,f#a;,f#b1 ,a',a7--that is1 we may interchange the two points ,,aa' without altering the value of the cross-ratio1 which is the characteristic property of harmonic conjugates 7,s #ah74 #d4 ,the point conjugate to the point at infinity is called the centre of the involution4 ,every involution has a centre1 unless the point at infinity be a focus1 in which case we may say that the centre is at infinity4 ,in an hyperbolic involution the centre is the middle point between the foci4 #e4 ,the product of the distances of two conjugate points ,a1 ,a' from the centre ,o is constant3 ,,oa 4 ,,oa' c4 ,for let ,a1 ,a' and ,b1 ,b' be two pairs of conjugate points1 the centre1 ,i the point at infinity1 then 7,,ab1 ,,oi7 7,a',b'1 ,,io71 or ,,oa 4 ,,oa' ,,ob 4 ,,ob'4 ,in order to determine the distances of the foci from the centre1 we write ,f for ,a and ,a' and get ,,of#b c2 ,,of -rootc4 ,hence if c is positive ,,of is real1 and has two values1 equal and opposite4 ,the involution is hyperbolic4 ,if c #j1 ,,of #j1 and the two foci both coincide with the centre4 ,if c is negative1 rootc becomes imaginary1 and there are no foci4 ,hence we may write-- ,in an hyperbolic involution1 ,,oa4,,oa' k#b1 ,in a parabolic involution1 ,,oa4,,oa' #j1 ,in an elliptic involution1 ,,oa4,,oa' -k#b4 ,from these expressions it follows that conjugate points ,a1 ,a' in an hyperbolic involution lie on the same side of the centre1 and in an elliptic involution on opposite sides of the centre1 and that in a parabolic involution one coincides with the centre4 ,in the first case1 for instance1 ,,oa4,,oa' is positive2 hence ,,oa and ,,oa' have the same sign4 ,it also follows that two segments1 ,,aa' and ,,bb'1 between pairs of conjugate points have the following positions3 in an hyperbolic involution they lie either one altogether within or altogether without each other2 in a parabolic involution they have one point in common2 and in an elliptic involution they overlap1 each being partly within and partly without the other4 ,proof4--,we have ,,oa4,,oa' ,,ob4,,ob' k#b in case of an hyperbolic involution4 ,let ,a and ,b be the points in each pair which are nearer to the centre ,o4 ,if now ,a1 ,a' and ,b1 ,b' lie on the same side of ,o1 and if ,b is nearer to ,o than ,a1 so that ,,ob ,,oa1 then ,,ob' ,,oa'2 hence ,b' lies farther away from ,o than ,a'1 or the segment ,,aa' lies within ,,bb'4 ,and so on for the other cases4 #f4 ,an involution is determined-- 7alpha7 ,by two pairs of conjugate points4 ,hence also 7beta7 ,by one pair of conjugate points and the centre2 7gamma7 ,by the two foci2 7delta7 ,by one focus and one pair of conjugate points2 7epsilon7 ,by one focus and the centre4 #g4 ,the condition that ,a1 ,b1 ,c and ,a'1 ,b'1 ,c' may form an involution may be written in one of the forms-- 7,,ab1 ,,cc'7 7,a',b'1 ,c',c71 or 7,,ab1 ,,ca'7 7,a',b'1 ,c',a71 or 7,,ab1 ,c',a'7 7,a',b'1 ,,ca71 for each expresses that in the two projective rows in which ,a1 ,b1 ,c and ,a'1 ,b'1 ,c' are conjugate points two conjugate elements may be interchanged4 #h4 ,any three pairs4 ,a1 ,a'1 ,b1 ,b'1 ,c1 ,c'1 of conjugate points are connected by the relations3 ,,ab'4,,bc'4,,ca' ,,ab'4,,bc4,c',a' ,,ab4,b',c'4,,ca' ,,ab4,b',c4,c',a' ----------- ----------- ----------- ----------- -#a4 ,a',b4,b',c4,c',a ,a',b4,b',c'4,,ca ,a',b'4,,bc4,c',a ,a',b'4,,bc'4,,ca ,these relations readily follow by working out the relations in 7#g7 7above74 ,s #gh4 ,involution of a quadrangle4--,the sides of any four-point are cut by any line in six points in involution1 opposite sides being cut in conjugate points4 ,let ,a#a;,b#a;,c#a;,d#a 7fig4 #ca7 be the four-point4 ,if its sides be cut by the line p in the points ,a1 ,a'1 ,b1 ,b'1 ,c1 ,c'1 if further1 ,c#a;,d#a cuts the line ,a#a;,b#a in ,c#b1 and if we project the row ,a#a;,b#a;,c#b;,c to p once from ,d#a and once from ,c#a1 we get 7,a',b'1 ,c',c7 7,,ba1 ,c',c74 ,interchanging in the last cross-ratio the letters in each pair we get 7,a',b'1 ,c',c7 7,,ab1 ,,cc'74 ,hence by ,s #gg 7#g7 the points are in involution4 ,the theorem may also be stated thus3 ,the three points in which any line cuts the sides of a triangle and the projections1 from any point in the plane1 of the vertices of the triangle on to the same line are six points in involution4 ,illustration3 ,,fig4 #ca4 ,or again-- ,the projections from any point on to any line of the six vertices of a four-side are six points in involution1 the projections of opposite vertices being conjugate points4 ,this property gives a simple means to construct1 by aid of the straight edge only1 in an involution of which two pairs of conjugate points are given1 to any point its conjugate4 ,s #gi4 ,pencils in ,involution4--,the theory of involution may at once be extended from the row to the flat and the axial pencil--viz4 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 points4 ,an involution in a pencil consists of pairs of conjugate rays or planes2 it has two1 one or no focal rays 7double lines7 or planes1 but nothing corresponding to a centre4 ,an involution in a flat pencil contains always one1 and in general only one1 pair of conjugate rays which are perpendicular to one another4 ,for in two projective flat pencils exist always two corresponding right angles 7,s #dj74 ,each involution in an axial pencil contains in the same manner one pair of conjugate planes at right angles to one another4 ,as a rule1 there exists but one pair of conjugate lines or planes at right angles to each other4 ,but it is possible that there are more1 and then there is an infinite number of such pairs4 ,an involution in a flat pencil1 in which every ray is perpendicular to its conjugate ray1 is said to be circular4 ,that such involution is possible is easily seen thus3 if in two concentric flat pencils each ray on one is made to correspond to that ray on the other which is perpendicular to it1 then the two pencils are projective1 for if we turn the one pencil through a right angle each ray in one coincides with its corresponding ray in the other4 ,but these two projective pencils are in involution4 ,a circular involution has no focal rays1 because no ray in a pencil coincides with the ray perpendicular to it4 ,s #hj4 ,every elliptical involution in a row may be considered as a section of a circular involution4 ,in an elliptical involution any two segments ,,aa' and ,,bb' lie partly within and partly without each other 7fig4 #cb74 ,hence two circles described on ,,aa' and ,,bb' as diameters will intersect in two points ,e and ,e'4 ,the line ,,ee' cuts the base of the involution at a point ,o1 which has the property that ,,oa4,,oa' ,,ob 4 ,,ob'1 for each is equal to ,,oe 4 ,,oe'4 ,the point ,o is therefore the centre of the involution4 ,if we wish to construct to any point ,c the conjugate point ,c'1 we may draw the circle through ,,cee'4 ,this will cut the base in the required point ,c' for ,,oc4,,oc' ,,oa4,,oa'4 ,but ,,ec and ,,ec' are at right angles4 ,hence the involution which is obtained by joining ,e or ,e' to the points in the given involution is circular4 ,this may also be expressed thus3 ,illustration3 ,,fig4 #cb4 ,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 angle4 ,at the same time the following problem has been solved3 ,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 given4 ,s #ha4 ,involution ,range on a ,conic4--,by the aid of ,s #ec1 the points on a conic may be made to correspond to those on a line1 so that the row of points on the conic is projective to a row of points on a line4 ,we may also have two projective rows on the same conic1 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 belongs4 ,an involution of points on a conic will have the property 7as follows from its definition1 and from ,s #ec7 that the lines which join conjugate points of the involution to any point on the conic are conjugate lines of an involution in a pencil1 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 tangent4 ,for such involution on a conic the following theorem holds3 ,the lines which join corresponding points in an involution on a conic all pass through a fixed point2 and reciprocally1 the points of intersection of conjugate lines in an involution among tangents to a conic lie on a line4 ,illustration3 ,,fig4 #cc ,we prove the first part only4 ,the involution is determined by two pairs of conjugate points1 say by ,a1 ,a' and ,b1 ,b' 7fig4 #cc74 ,let ,,aa' and ,,bb' meet in ,p4 ,if we join the points in involution to any point on the conic1 and the conjugate points to another point on the conic1 we obtain two projective pencils4 ,we take ,a and ,a' as centres of these pencils1 so that the pencils ,a7,a',,bb'7 and ,a'7,,ab',b7 are projective1 and in perspective position1 because ,,aa' corresponds to ,a',a4 ,hence corresponding rays meet in a line1 of which two points are found by joining ,,ab' to ,a',b and ,,ab to ,a',b'4 ,it follows that the axis of perspective is the polar of the point ,p1 where ,,aa' and ,,bb' meet4 ,if we now wish to construct to any other point ,c on the conic the corresponding point ,c'1 we join ,c to ,a' and the point where this line cuts p to ,a4 ,the latter line cuts the conic again in ,c'4 ,but we know from the theory of pole and polar that the line ,,cc' passes through ,p4 ,the point of concurrence is called the pole of the involution1 and the line of collinearity of the meets is called the axis of the involution4 ,,involution ,,determined ,,by ,a ,,conic ,,on ,a ,,line4--,,foci ,s #hb4 ,the polars1 with regard to a conic1 of points in a row p form a pencil ,p projective to the row 7,s #ff74 ,this pencil cuts the base of the row p in a projective row4 ,if ,a is a point in the given row1 ,a' the point where the polar of ,a cuts p1 then ,a and ,a' will be corresponding points4 ,if we take ,a' a point in the first row1 then the polar of ,a' will pass through ,a1 so that ,a corresponds to ,a'--in other words1 the rows are in involution4 ,the conjugate points in this involution are conjugate points with regard to the conic4 ,conjugate points coincide only if the polar of a point ,a passes through ,a--that is1 if ,a lies on the conic4 ,hence-- ,a conic determines on every line in its plane an involution1 in which those points are conjugate which are also conjugate with regard to the conic4 ,if the line cuts the conic the involution is hyperbolic1 the points of intersection being the foci4 ,if the line touches the conic the involution is parabolic1 the two foci coinciding at the point of contact4 ,if the line does not cut the conic the involution is elliptic1 having no foci4 ,if1 on the other hand1 we take a point ,p in the plane of a conic1 we get to each line a through ,p one conjugate line which joins ,p to the pole of a4 ,these pairs of conjugate lines through ,p form an involution in the pencil at ,p4 ,the focal rays of this involution are the tangents drawn from ,p to the conic4 ,this gives the theorem reciprocal to the last1 viz3-- ,a conic determines in every pencil in its plane an involution1 corresponding lines being conjugate lines with regard to the conic4 ,if the point is without the conic the involution is hyperbolic1 the tangents from the points being the focal rays4 ,if the point lies on the conic the involution is parabolic1 the tangent at the point counting for coincident focal rays4 ,if the point is within the conic the involution is elliptic1 having no focal rays4 ,it will further be seen that the involution determined by a conic on any line p is a section of the involution1 which is determined by the conic at the pole ,p of p4 ,s #hc4 ,foci4--,the centre of a pencil in which the conic determines a circular involution is called a focus of the conic4 ,in other words1 a focus is such a point that every line through it is perpendicular to its conjugate line4 ,the polar to a focus is called a directrix of the conic4 ,from the definition it follows that every focus lies on an axis1 for the line joining a focus to the centre of the conic is a diameter to which the conjugate lines are perpendicular2 and every line joining two foci is an axis1 for the perpendiculars to this line through the foci are conjugate to it4 ,these conjugate lines pass through the pole of the line1 the pole lies therefore at infinity1 and the line is a diameter1 hence by the last property an axis4 ,it follows that all foci lie on one axis1 for no line joining a point in one axis to a point in the other can be an axis4 ,as the conic determines in the pencil which has its centre at a focus a circular involution1 no tangents can be drawn from the focus to the conic4 ,hence each focus lies within a conic2 and a directrix does not cut the conic4 ,further properties are found by the following considerations3 ,s #hd4 ,through a point ,p one line p can be drawn1 which is with regard to a given conic conjugate to a given line q1 viz4 that line which joins the point ,p to the pole of the line q4 ,if the line q is made to describe a pencil about a point ,q1 then the line p will describe a pencil about ,p4 ,these two pencils will be projective1 for the line p passes through the pole of q1 and whilst q describes the pencil ,q1 its pole describes a projective row1 and this row is perspective to the pencil ,p4 ,we now take the point ,p on an axis of the conic1 draw any line p through it1 and from the pole of p draw a perpendicular q to p4 ,let q cut the axis in ,q4 ,then1 in the pencils of conjugate lines1 which have their centres at ,p and ,q1 the lines p and q are conjugate lines at right angles to one another4 ,besides1 to the axis as a ray in either pencil will correspond in the other the perpendicular to the axis 7,s #gb74 ,the conic generated by the intersection of corresponding lines in the two pencils is therefore the circle on ,,pq as diameter1 so that every line in ,p is perpendicular to its corresponding line in ,q4 ,to every point ,p on an axis of a conic corresponds thus a point ,q1 such that conjugate lines through ,p and ,q are perpendicular4 ,we shall show that these point-pairs ,p1 ,q form an involution4 ,to do this let us move ,p along the axis1 and with it the line p1 keeping the latter parallel to itself4 ,then ,p describes a row1 p a perspective pencil 7of parallels71 and the pole of p a projective row4 ,at the same time the line q describes a pencil of parallels perpendicular to p1 and perspective to the row formed by the pole of p4 ,the point ,q1 therefore1 where q cuts the axis1 describes a row projective to the row of points ,p4 ,the two points ,p and ,q describe thus two projective rows on the axis2 and not only does ,p as a point in the first row correspond to ,q1 but also ,q as a point in the first corresponds to ,p4 ,the two rows therefore form an involution4 ,the centre of this involution1 it is easily seen1 is the centre of the conic4 ,a focus of this involution has the property that any two conjugate lines through it are perpendicular2 hence1 it is a focus to the conic4 ,such involution exists on each axis4 ,but only one of these can have foci1 because all foci lie on the same axis4 ,the involution on one of the axes is elliptic1 and appears 7,s #hj7 therefore as the section of two circular involutions in two pencils whose centres lie in the other axis4 ,these centres are foci1 hence the one axis contains two foci1 the other axis none2 or every central conic has two foci which lie on one axis equidistant from the centre4 ,the axis which contains the foci is called the principal axis2 in case of an hyperbola it is the axis which cuts the curve1 because the foci lie within the conic4 ,in case of the parabola there is but one axis4 ,the involution on this axis has its centre at infinity4 ,one focus is therefore at infinity1 the one focus only is finite4 ,a parabola has only one focus4 ,illustration3 ,,fig4 #cd4 ,s #he4 ,if through any point ,p 7fig4 #cd7 on a conic the tangent ,,pt and the normal ,,pn 7i4e4 the perpendicular to the tangent through the point of contact7 be drawn1 these will be conjugate lines with regard to the conic1 and at right angles to each other4 ,they will therefore cut the principal axis in two points1 which are conjugate in the involution considered in ,s #hd2 hence they are harmonic conjugates with regard to the foci4 ,if therefore the two foci ,f#a and ,f#b be joined to ,p1 these lines will be harmonic with regard to the tangent and normal4 ,as the latter are perpendicular1 they will bisect the angles between the other pair4 ,hence-- ,the lines joining any point on a conic to the two foci are equally inclined to the tangent and normal at that point4 ,in case of the parabola this becomes-- ,the line joining any point on a parabola to the focus and the diameter through the point1 are equally inclined to the tangent and normal at that point4 ,from the definition of a focus it follows that-- ,the segment of a tangent between the directrix and the point of contact is seen from the focus belonging to the directrix under a right angle1 because the lines joining the focus to the ends of this segment are conjugate with regard to the conic1 and therefore perpendicular4 ,with equal ease the following theorem is proved3 ,the two lines which join the points of contact of two tangents each to one focus1 but not both to the same1 are seen from the intersection of the tangents under equal angles4 ,s #hf4 ,other focal properties of a conic are obtained by the following considerations3 ,illustration3 ,,fig4 #ce4 ,let ,f 7fig4 #ce7 be a focus to a conic1 f the corresponding directrix1 ,a and ,b the points of contact of two tangents meeting at ,t1 and ,p the point where the line ,,ab cuts the directrix4 ,then ,,tf will be the polar of ,p 7because polars of ,f and ,t meet at ,p74 ,hence ,,tf and ,,pf are conjugate lines through a focus1 and therefore perpendicular4 ,they are further harmonic conjugates with regard to ,,fa and ,,fb 7,,ss #fd and #ac71 so that they bisect the angles formed by these lines4 ,this by the way proves-- ,the segments between the point of intersection of two tangents to a conic and their points of contact are seen from a focus under equal angles4 ,if we next draw through ,a and ,b lines parallel to ,,tf1 then the points ,a#a1 ,b#a where these cut the directrix will be harmonic conjugates with regard to ,p and the point where ,,ft cuts the directrix4 ,the lines ,,ft and ,,fp bisect therefore also the angles between ,,fa#a and ,,fb#a4 ,from this it follows easily that the triangles ,,faa#a and ,,fbb#a are equiangular1 and therefore similar1 so that ,,fa 3 ,,aa#a ,,fb 3 ,,bb#a4 ,the triangles ,,aa#a;,a#b and ,,bb#a;,b#b formed by drawing perpendiculars from ,a and ,b to the directrix are also similar1 so that ,,aa#a 3 ,,aa#b ,,bb#a 3 ,,bb#b4 ,this1 combined with the above proportion1 gives ,,fa 3 ,,aa#b ,,fb 3 ,,bb#b4 ,hence the theorem3 ,the ratio of the distances of any point on a conic from a focus and the corresponding directrix is constant4 ,to determine this ratio we consider its value for a vertex on the principal axis4 ,in an ellipse the focus lies between the two vertices on this axis1 hence the focus is nearer to a vertex than to the corresponding directrix4 ,similarly1 in an hyperbola a vertex is nearer to the directrix than to the focus4 ,in a parabola the vertex lies halfway between directrix and focus4 ,it follows in an ellipse the ratio between the distance of a point from the focus to that from the directrix is less than unity1 in the parabola it equals unity1 and in the hyperbola it is greater than unity4 ,it is here the same which focus we take1 because the two foci lie symmetrical to the axis of the conic4 ,if now ,p is any point on the conic having the distances r#a and r#b from the foci and the distances d#a and d#b from the corresponding directrices1 then r#a_/;d#a r#b_/;d#b e1 where e is constant4 ,hence also r#a - r#b _/ d#a - d#b e4 ,in the ellipse1 which lies between the directrices1 d#a d#b is constant1 therefore also r#a r#b4 ,in the hyperbola on the other hand d#a - d#b is constant1 equal to the distance between the directrices1 therefore in this case r#a - r#b is constant4 ,if we call the distances of a point on a conic from the focus its focal distances we have the theorem3 ,in an ellipse the sum of the focal distances is constant2 and in an hyperbola the difference of the focal distances is constant4 ,this constant sum or difference equals in both cases the length of the principal axis4 ,,pencil ,,of ,,conics ,s #hg4 ,through four points ,a1 ,b1 ,c1 ,d in a plane1 of which no three lie in a line1 an infinite number of conics may be drawn1 viz4 through these four points and any fifth one single conic4 ,this system of conics is called a pencil of conics4 ,similarly1 all conics touching four fixed lines form a system such that any fifth tangent determines one and only one conic4 ,we have here the theorems3 ,the pairs of points in which ,the pairs of tangents which any line is cut by a system of can be drawn from a point to conics through four fixed points a system of conics touching four are in involution4 fixed lines are in involution4 ,illustration3 ,,fig4 #cf4 ,we prove the first theorem only4 ,let ,,abcd 7fig4 #cf7 be the four-point1 then any line t will cut two opposite sides ,,ac1 ,,bd in the points ,e1 ,e'1 the pair ,,ad1 ,,bc in points ,f1 ,f'1 and any conic of the system in ,m1 ,n1 and we have ,a7,,cd1 ,,mn7 ,b7,,cd1 ,,mn74 ,if we cut these pencils by t we get 7,,ef1 ,,mn7 7,f',e'1 ,,mn7 or 7,,ef1 ,,mn7 7,e',f'1 ,,nm74 ,but this is1 according to ,s #gg 7#g71 the condition that ,m1 ,n are corresponding points in the involution determined by the point pairs ,e1 ,e'1 ,f1 ,f' in which the line t cuts pairs of opposite sides of the four-point ,,abcd4 ,this involution is independent of the particular conic chosen4 ,s #hh4 ,there follow several important theorems3 ,through four points two1 one1 or no conics may be drawn which touch any given line1 according as the involution determined by the given four-point on the line has real1 coincident or imaginary foci4 ,two1 one1 or no conics may be drawn which touch four given lines and pass through a given point1 according as the involution determined by the given four-side at the point has real1 coincident or imaginary focal rays4 ,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 line4 ,as a special case we get1 by taking the line at infinity3 ,through four points of which none is at infinity either two or no parabolas may be drawn4 ,the problem of drawing a conic through four points and touching a given line is solved by determining the points of contact on the line1 that is1 by determining the foci of the involution in which the line cuts the sides of the four-point4 ,the corresponding remark holds for the problem of drawing the conics which touch four lines and pass through a given point4 ,,ruled ,,quadric ,,surfaces ,s #hi4 ,we have considered hitherto projective rows which lie in the same plane1 in which case lines joining corresponding points envelop a conic4 ,we shall now consider projective rows whose bases do not meet4 ,in this case1 corresponding points will be joined by lines which do not lie in a plane1 but on some surface1 which like every surface generated by lines is called a ruled surface4 ,this surface clearly contains the bases of the two rows4 ,if the points in either row be joined to the base of the other1 we obtain two axial pencils which are also projective1 those planes being corresponding which pass through corresponding points in the given rows4 ,if ,a'1 ,a be two corresponding points1 alpha1 alpha' the planes in the axial pencils passing through them1 then ,,aa' will be the line of intersection of the corresponding planes alpha1 alpha' and also the line joining corresponding points in the rows4 ,if we cut the whole figure by a plane this will cut the axial pencils in two projective flat pencils1 and the curve of the second order generated by these will be the curve in which the plane cuts the surface4 ,hence ,the locus of lines joining corresponding points in two projective rows which do not lie in the same plane is a surface which contains the bases of the rows1 and which can also be generated by the lines of intersection of corresponding planes in two projective axial pencils4 ,this surface is cut by every plane in a curve of the second order1 hence either in a conic or in a line-pair4 ,no line which does not lie altogether on the surface can have more than two points in common with the surface1 which is therefore said to be of the second order or is called a ruled quadric surface4 ,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 line1 for this will cut the surface in a conic4 ,it follows also that a line which contains more than two points of the surface lies altogether on the surface4 ,s #ij4 ,through any point in space one line can always be drawn cutting two given lines which do not themselves meet4 ,if therefore three lines in space be given of which no two meet1 then through every point in either one line may be drawn cutting the other two4 ,if a line moves so that it always cuts three given lines of which no two meet1 then it generates a ruled quadric surface4 ,let a1 b1 c be the given lines1 and p1 q1 r 444 lines cutting them in the points ,a1 ,a'1 ,a 4442 ,b1 ,b'1 ,b 4442 ,c1 ,c'1 ,c 444 respectively2 then the planes through a containing p1 q1 r1 and the planes through b containing the same lines1 may be taken as corresponding planes in two axial pencils which are projective1 because both pencils cut the line c in the same row1 ,c1 ,c'1 ,c 4442 the surface can therefore be generated by projective axial pencils4 ,of the lines p1 q1 r 444 no two can meet1 for otherwise the lines a1 b1 c which cut them would also lie in their plane4 ,there is a single infinite number of them1 for one passes through each point of a4 ,these lines are said to form a set of lines on the surface4 ,if now three of the lines p1 q1 r be taken1 then every line d cutting them will have three points in common with the surface1 and will therefore lie altogether on it4 ,this gives rise to a second set of lines on the surface4 ,from what has been said the theorem follows3 ,a ruled quadric surface contains two sets of straight lines4 ,every line of one set cuts every line of the other1 but no two lines of the same set meet4 ,any two lines of the same set may be taken as bases of two projective rows1 or of two projective pencils which generate the surface4 ,they are cut by the lines of the other set in two projective rows4 ,the plane at infinity like every other plane cuts the surface either in a conic proper or in a line-pair4 ,in the first case the surface is called an ,hyperboloid of one sheet1 in the second an ,hyperbolic ,paraboloid4 ,the latter may be generated by a line cutting three lines of which one lies at infinity1 that is1 cutting two lines and remaining parallel to a given plane4 ,,quadric ,,surfaces ,s #ia4 ,the conics1 the cones of the second order1 and the ruled quadric surfaces complete the figures which can be generated by projective rows or flat and axial pencils1 that is1 by those aggregates of elements which are of one dimension 7,,ss #e1 #f74 ,we shall now consider the simpler figures which are generated by aggregates of two dimensions4 ,the space at our disposal will not1 however1 allow us to do more than indicate a few of the results4 ,s #ib4 ,we establish a correspondence between the lines and planes in pencils in space1 or reciprocally between the points and lines in two or more planes1 but consider principally pencils4 ,in two pencils we may either make planes correspond to planes and lines to lines1 or else planes to lines and lines to planes4 ,if hereby the condition be satisfied that to a flat1 or axial1 pencil corresponds in the first case a projective flat1 or axial1 pencil1 and in the second a projective axial1 or flat1 pencil1 the pencils are said to be projective in the first case and reciprocal in the second4 ,for instance1 two pencils which join two points ,s#a and ,s#b to the different points and lines in a given plane pi are projective 7and in perspective position71 if those lines and planes be taken as corresponding which meet the plane pi in the same point or in the same line4 ,in this case every plane through both centres ,s#a and ,s#b of the two pencils will correspond to itself4 ,if these pencils are brought into any other position they will be projective 7but not perspective74 ,the correspondence between two projective pencils is uniquely determined1 if to four rays 7or planes7 in the one the corresponding rays 7or planes7 in the other are given1 provided that no three rays of either set lie in a plane4 ,let a1 b1 c1 d be four rays in the one1 a'1 b'1 c'1 d' the corresponding rays in the other pencil4 ,we shall show that we can find for every ray e in the first a single corresponding ray e' in the second4 ,to the axial pencil a 7b1 c1 d 4447 formed by the planes which join a to b1 c1 d 4441 respectively corresponds the axial pencil a' 7b'1 c'1 d' 444 71 and this correspondence is determined4 ,hence1 the plane a'e' which corresponds to the plane ae is determined4 ,similarly the plane b'e' may be found and both together determine the ray e'4 ,similarly the correspondence between two reciprocal pencils is determined if for four rays in the one the corresponding planes in the other are given4 ,s #ic4 ,we may now combine-- #a4 ,two reciprocal pencils4 ,each ray cuts its corresponding plane in a point1 the locus of these points is a quadric surface4 #b4 ,two projective pencils4 ,each plane cuts its corresponding plane in a line1 but a ray as a rule does not cut its corresponding ray4 ,the locus of points where a ray cuts its corresponding ray is a twisted cubic4 ,the lines where a plane cuts its corresponding plane are secants4 #c4 ,three projective pencils4 ,the locus of intersection of corresponding planes is a cubic surface4 ,of these we consider only the first two cases4 ,s #id4 ,if two pencils are reciprocal1 then to a plane in either corresponds a line in the other1 to a flat pencil an axial pencil1 and so on4 ,every line cuts its corresponding plane in a point4 ,if ,s#a and ,s#b be the centres of the two pencils1 and ,p be a point where a line a#a in the first cuts its corresponding plane alpha#b1 then the line b#b in the pencil ,s#b which passes through ,p will meet its corresponding plane beta#a in ,p4 ,for b#b is a line in the plane alpha#b4 ,the corresponding plane beta#a must therefore pass through the line a#a1 hence through ,p4 ,the points in which the lines in ,s#a cut the planes corresponding to them in ,s#b are therefore the same as the points in which the lines in ,s#b cut the planes corresponding to them in ,s#a4 ,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 surface4 ,the surface itself is therefore called a quadric surface1 or a surface of the second order4 ,to prove this we consider any line p in space4 ,the flat pencil in ,s#a which lies in the plane drawn through p and the corresponding axial pencil in ,s#b determine on p two projective rows1 and those points in these which coincide with their corresponding points lie on the surface4 ,but there exist only two1 or one1 or no such points1 unless every point coincides with its corresponding point4 ,in the latter case the line lies altogether on the surface4 ,this proves also that a plane cuts the surface in a curve of the second order1 as no line can have more than two points in common with it4 ,to show that this is a curve of the same kind as those considered before1 we have to show that it can be generated by projective flat pencils4 ,we prove first that this is true for any plane through the centre of one of the pencils1 and afterwards that every point on the surface may be taken as the centre of such pencil4 ,let then alpha#a be a plane through ,s#a4 ,to the flat pencil in ,s#a which it contains corresponds in ,s#b a projective axial pencil with axis a#b and this cuts alpha#a in a second flat pencil4 ,these two flat pencils in alpha#a are projective1 and1 in general1 neither concentric nor perspective4 ,they generate therefore a conic4 ,but if the line a#b passes through ,s#a the pencils will have ,s#a as common centre1 and may therefore have two1 or one1 or no lines united with their corresponding lines4 ,the section of the surface by the plane alpha#a will be accordingly a line-pair or a single line1 or else the plane alpha#a will have only the point ,s#a in common with the surface4 ,every line l#a through ,s#a cuts the surface in two points1 viz4 first in ,s#a and then at the point where it cuts its corresponding plane4 ,if now the corresponding plane passes through ,s#a1 as in the case just considered1 then the two points where l#a cuts the surface coincide at ,s#a1 and the line is called a tangent to the surface with ,s#a as point of contact4 ,hence if l#a be a tangent1 it lies in that plane tau#a which corresponds to the line ,s#b,s#a as a line in the pencil ,s#b4 ,the section of this plane has just been considered4 ,it follows that-- ,all tangents to quadric surface at the centre of one of the reciprocal pencils lie in a plane which is called the tangent plane to the surface at that point as point of contact4 ,to the line joining the centres of the two pencils as a line in one corresponds in the other the tangent plane at its centre4 ,the tangent plane to a quadric surface either cuts the surface in two lines1 or it has only a single line1 or else only a single point in common with the surface4 ,in the first case the point of contact is said to be hyperbolic1 in the second parabolic1 in the third elliptic4 ,s #ie4 ,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 surface4 ,let ,s be any point on the surface ,phi' generated by the reciprocal pencils ,s#a and ,s#b4 ,we have to establish a reciprocal correspondence between the pencils ,s and ,s#a1 so that the surface generated by them is identical with ,phi4 ,to do this we draw two planes alpha#a and beta#a through ,s#a1 cutting the surface ,phi in two conics which we also denote by alpha#a and beta#a4 ,these conics meet at ,s#a1 and at some other point ,t where the line of intersection of alpha#a and beta#a cuts the surface4 ,in the pencil ,s we draw some plane sigma which passes through ,t1 but not through ,s#a or ,s#b4 ,it will cut the two conics first at ,t1 and therefore each at some other point which we call ,a and ,b respectively4 ,these we join to ,s by lines a and b1 and now establish the required correspondence between the pencils ,s#a and ,s as follows3--,to ,s#a,t shall correspond the plane sigma1 to the plane alpha#a the line a1 and to beta#a the line b1 hence to the flat pencil in alpha#a the axial pencil a4 ,these pencils are made projective by aid of the conic in alpha#a4 ,in the same manner the flat pencil in beta#a is made projective to the axial pencil b by aid of the conic in beta#a1 corresponding elements being those which meet on the conic4 ,this determines the correspondence1 for we know for more than four rays in ,s#a the corresponding planes in ,s4 ,the two pencils ,s and ,s#a thus made reciprocal generate a quadric surface ,phi'1 which passes through the point ,s and through the two conics alpha#a and beta#a4 ,the two surfaces ,phi and ,phi' have therefore the points ,s and ,s#a and the conics alpha#a and beta#a in common4 ,to show that they are identical1 we draw a plane through ,s and ,s#b1 cutting each of the conics alpha#a and beta#a in two points1 which will always be possible4 ,this plane cuts ,phi and ,phi' in two conics which have the point ,s and the points where it cuts alpha#a and beta#a in common1 that is five points in all4 ,the conics therefore coincide4 ,this proves that all those points ,p on ,phi' lie on ,phi which have the property that the plane ,,ss#b,p cuts the conics alpha#a1 beta#a in two points each4 ,if the plane ,,ss#b,p has not this property1 then we draw a plane ,,ss#a,p4 ,this cuts each surface in a conic1 and these conics have in common the points ,s1 ,s#a1 one point on each of the conics alpha#a1 beta#a1 and one point on one of the conics through ,s and ,s#b which lie on both surfaces1 hence five points4 ,they are therefore coincident1 and our theorem is proved4 ,s #if4 ,the following propositions follow3-- ,a quadric surface has at every point a tangent plane4 ,every plane section of a quadric surface is a conic or a line-pair4 ,every line which has three points in common with a quadric surface lies on the surface4 ,every conic which has five points in common with a quadric surface lies on the surface4 ,through two conics which lie in different planes1 but have two points in common1 and through one external point always one quadric surface may be drawn4 ,s #ig4 ,every plane which cuts a quadric surface in a line-pair is a tangent plane4 ,for every line in this plane through the centre of the line-pair 7the point of intersection of the two lines7 cuts the surface in two coincident points and is therefore a tangent to the surface1 the centre of the line-pair being the point of contact4 ,if a quadric surface contains a line1 then every plane through this line cuts the surface in a line-pair 7or in two coincident lines74 ,for this plane cannot cut the surface in a conic4 ,hence3-- ,if a quadric surface contains one line p then it contains an infinite number of lines1 and through every point ,q on the surface1 one line q can be drawn which cuts p4 ,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 line4 ,no two such lines q on the surface can meet4 ,for as both meet p their plane would contain p and therefore cut the surface in a triangle4 ,every line which cuts three lines q will be on the surface2 for it has three points in common with it4 ,hence the quadric surfaces which contain lines are the same as the ruled quadric surfaces considered in ,,ss #hi-ic1 but with one important exception4 ,in the last investigation we have left out of consideration the possibility of a plane having only one line 7two coincident lines7 in common with a quadric surface4 ,s #ih4 ,to investigate this case we suppose first that there is one point ,a on the surface through which two different lines a1 b can be drawn1 which lie altogether on the surface4 ,if ,p is any other point on the surface which lies neither on a nor b1 then the plane through ,p and a will cut the surface in a second line a' which passes through ,p and which cuts a4 ,similarly there is a line b' through ,p which cuts b4 ,these two lines a' and b' may coincide1 but then they must coincide with ,,pa4 ,if this happens for one point ,p1 it happens for every other point ,q4 ,for if two different lines could be drawn through ,q1 then by the same reasoning the line ,,pq would be altogether on the surface1 hence two lines would be drawn through ,p against the assumption4 ,from this follows3-- ,if there is one point on a quadric surface through which one1 but only one1 line can be drawn on the surface1 then through every point one line can be drawn1 and all these lines meet in a point4 ,the surface is a cone of the second order4 ,if through one point on a quadric surface1 two1 and only two1 lines can be drawn on the surface1 then through every point two lines may be drawn1 and the surface is ruled quadric surface4 ,if through one point on a quadric surface no line on the surface can be drawn1 then the surface contains no lines4 ,using the definitions at the end of ,s #ie1 we may also say3-- ,on a quadric surface the points are all hyperbolic1 or all parabolic1 or all elliptic4 ,as an example of a quadric surface with elliptical points1 we mention the sphere which may be generated by two reciprocal pencils1 where to each line in one corresponds the plane perpendicular to it in the other4 ,s #ii4 ,poles and ,polar ,planes4--,the theory of poles and polars with regard to a conic is easily extended to quadric surfaces4 ,let ,p be a point in space not on the surface1 which we suppose not to be a cone4 ,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 intersection4 ,through one of these lines we draw two planes alpha and beta4 ,the locus of the points ,q in alpha is a line a1 the polar of ,p with regard to the conic in which alpha cuts the surface4 ,similarly the locus of points ,q in beta is a line b4 ,this cuts a1 because the line of intersection of alpha and beta contains but one point ,q4 ,the locus of all points ,q therefore is a plane4 ,this plane is called the polar plane of the point ,p1 with regard to the quadric surface4 ,if ,p lies on the surface we take the tangent plane of ,p as its polar4 ,the following propositions hold3-- #a4 ,every point has a polar plane1 which is constructed by drawing the polars of the point with regard to the conics in which two planes through the point cut the surface4 #b4 ,if ,q is a point in the polar of ,p1 then ,p is a point in the polar of ,q1 because this is true with regard to the conic in which a plane through ,,pq cuts the surface4 #c4 ,every plane is the polar plane of one point1 which is called the ,pole of the plane4 ,the pole to a plane is found by constructing the polar planes of three points in the plane4 ,their intersection will be the pole4 #d4 ,the points in which the polar plane of ,p cuts the surface are points of contact of tangents drawn from ,p to the surface1 as is easily seen4 ,hence3-- #e4 ,the tangents drawn from a point ,p to a quadric surface form a cone of the second order1 for the polar plane of ,p cuts it in a conic4 #f4 ,if the pole describes a line a1 its polar plane will turn about another line a'1 as follows from #b4 ,these lines a and a' are said to be conjugate with regard to the surface4 ,s #ajj4 ,the pole of the line at infinity is called the centre of the surface4 ,if it lies at the infinity1 the plane at infinity is a tangent plane1 and the surface is called a paraboloid4 ,the polar plane to any point at infinity passes through the centre1 and is called a diametrical plane4 ,a line through the centre is called a diameter4 ,it is bisected at the centre4 ,the line conjugate to it lies at infinity4 ,if a point moves along a diameter its polar plane turns about the conjugate line at infinity2 that is1 it moves parallel to itself1 its centre moving on the first line4 ,the middle points of parallel chords lie in a plane1 viz4 in the polar plane of the point at infinity through which the chords are drawn4 ,the centres of parallel sections lie in a diameter which is a line conjugate to the line at infinity in which the planes meet4 ,,twisted ,,cubics ,s #aja4 ,if two pencils with centres ,s#a and ,s#b are made projective1 then to a ray in one corresponds a ray in the other1 to a plane a plane1 to a flat or axial pencil a projective flat or axial pencil1 and so on4 ,there is a double infinite number of lines in a pencil4 ,we shall see that a single infinite number of lines in one pencil meets its corresponding ray1 and that the points of intersection form a curve in space4 ,of the double infinite number of planes in the pencils each will meet its corresponding plane4 ,this gives a system of a double infinite number of lines in space4 ,we know 7,s #e7 that there is a quadruple infinite number of lines in space4 ,from among these we may select those which satisfy one or more given conditions4 ,the systems of lines thus obtained were first systematically investigated and classified by ,plucker1 in his ,geometrie des ,raumes4 ,he uses the following names3-- ,a treble infinite number of lines1 that is1 all lines which satisfy one condition1 are said to form a complex of lines2 e4g4 all lines cutting a given line1 or all lines touching a surface4 ,a double infinite number of lines1 that is1 all lines which satisfy two conditions1 or which are common to two complexes1 are said to form a congruence of lines2 e4g4 all lines in a plane1 or all lines cutting two curves1 or all lines cutting a given curve twice4 ,a single infinite number of lines1 that is1 all lines which satisfy three conditions1 or which belong to three complexes1 form a ruled surface2 e4g4 one set of lines on a ruled quadric surface1 or developable surfaces which are formed by the tangents to a curve4 ,it follows that all lines in which corresponding planes in two projective pencils meet form a congruence4 ,we shall see this congruence consists of all lines which cut a twisted cubic twice1 or of all secants to a twisted cubic4 ,s #ajb4 ,let l#a be the line ,s#a,s#b as a line in the pencil ,s#a4 ,to it corresponds a line l#b in ,s#b4 ,at each of the centres two corresponding lines meet4 ,the two axial pencils with l#a and l#b as axes are projective1 and1 as1 their axes meet at ,s#b1 the intersections of corresponding planes form a cone of the second order 7,s #eh71 with ,s#b as centre4 ,if pi#a and pi#b be corresponding planes1 then their intersection will be a line p#b which passes through ,s#b4 ,corresponding to it in ,s#a will be a line p#a which lies in the plane pi#a1 and which therefore meets p#b at some point ,p4 ,conversely1 if p#b be any line in ,s#b which meets its corresponding line p#a at a point ,p1 then to the plane l#bp#b will correspond the plane l#ap#a1 that is1 the plane ,s#a,s#b,p4 ,these planes intersect in p#b1 so that p#b is a line on the quadric cone generated by the axial pencils l#a and l#b4 ,hence3-- ,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 pencil1 and passes through the centre of the second4 ,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 common1 and form therefore1 together with the line ,s#a,s#b or l#a1 the intersection of these cones4 ,any plane cuts each of the cones in a conic4 ,these two conics have necessarily that point in common in which it cuts the line l#a1 and therefore besides either one or three other points4 ,it follows that the curve is of the third order as a plane may cut it in three1 but not in more than three1 points4 ,hence3-- ,the locus of points in which corresponding lines on two projective pencils meet is a curve of the third order or a twisted cubic k1 which passes through the centres of the pencils1 and which appears as the intersection of two cones of the second order1 which have one line in common4 ,a line belonging to the congruence determined by the pencils is a secant of the cubic2 it has two1 or one1 or no points in common with this cubic1 and is called accordingly a secant proper1 a tangent1 or a secant improper of the cubic4 ,a secant improper may be considered1 to use the language of coordinate geometry1 as a secant with imaginary points of intersection4 ,s #ajc4 ,if a#a and a#b be any two corresponding lines in the two pencils1 then corresponding planes in the axial pencils having a#a and a#b as axes generate a ruled quadric surface4 ,if ,p be any point on the cubic k1 and if p#a1 p#b be the corresponding rays in ,s#a and ,s#b which meet at ,p1 then to the plane a#ap#a in ,s#a corresponds a#bp#b in ,s#b4 ,these therefore meet in a line through ,p4 ,this may be stated thus3-- ,those secants of the cubic which cut a ray a#a1 drawn through the centre ,s#a of one pencil1 form a ruled quadric surface which passes through both centres1 and which contains the twisted cubic k4 ,of such surfaces an infinite number exists4 ,every ray through ,s#a or ,s#b which is not a secant determines one of them4 ,if1 however1 the rays a#a and a#b are secants meeting at ,a1 then the ruled quadric surface becomes a cone of the second order1 having ,a as centre4 ,or all lines of the congruence which pass through a point on the twisted cubic k form a cone of the second order4 ,in other words1 the projection of a twisted cubic from any point in the curve on to any plane is a conic4 ,if a#a is not a secant1 but made to pass through any point ,q in space1 the ruled quadric surface determined by a#a will pass through ,q4 ,there will therefore be one line of the congruence passing through ,q1 and only one4 ,for if two such lines pass through ,q1 then the lines ,s#a,q and ,s#b,q will be corresponding lines2 hence ,q will be a point on the cubic k1 and an infinite number of secants will pass through it4 ,hence3-- ,through every point in space not on the twisted cubic one and only one secant to the cubic can be drawn4 ,s #ajd4 ,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 cubic4 ,if we take any two points ,s1 ,s' on the cubic1 and draw the secants through each of them1 we obtain two quadric cones1 which have the line ,,ss' in common1 and which intersect besides along the cubic4 ,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 cubic1 the correspondence is determined4 ,these two pencils will generate a cubic1 and the two cones of secants having ,s and ,s' as centres will be identical with the above cones1 for each has five rays in common with one of the first1 viz4 the line ,,ss' and the four lines determined for the correspondence2 therefore these two cones intersect in the original cubic4 ,this gives the theorem3-- ,on a twisted cubic any two points may be taken as centres of projective pencils which generate the cubic1 corresponding planes being those which meet on the same secant4 ,of the two projective pencils at ,s and ,s' we may keep the first fixed1 and move the centre of the other along the curve4 ,the pencils will hereby remain projective1 and a plane alpha in ,s will be cut by its corresponding plane alpha' always in the same secant a4 ,whilst ,s' moves along the curve the plane alpha' will turn about a1 describing an axial pencil4 ,,authorities4--,in this article we have given a purely geometrical theory of conics1 cones of the second order1 quadric surfaces1 c4 ,in doing so we have followed1 to a great extent1 ,reye's ,geometrie der ,lage1 and to this excellent work those readers are referred who wish for a more exhaustive treatment of the subject4 ,other works especially valuable as showing the development of the subject are3 ,monge1 ,geometrie descriptive3 ,carnot1 ,geometrie de position 7#ahjc71 containing a theory of transversals2 ,poncelet's great work ,traite des proprietes projectives des figures 7#ahbb72 ,mobins1 ,barycentrischer ,calcul 7#ahbf72 ,steiner1 ,abhangigkeit geometrischer ,gestalten 7#ahcb71 containing the first full discussion of the projective relations between rows1 pencils1 c42 ,von ,staudt1 ,geometrie der ,lage 7#ahdg7 and ,beitrage zur ,geometrie der ,lage 7#ahef-ahfj71 in which a system of geometry is built up from the beginning without any reference to number1 so that ultimately a number itself gets a geometrical definition1 and in which imaginary elements are systematically introduced into pure geometry2 ,chasles1 ,apercu historique 7#ahcg71 in which the author gives a brilliant account of the progress of modern geometrical methods1 pointing out the advantages of the different purely geometrical methods as compared with the analytical ones1 but without taking as much account of the ,german as of the ,french authors2 ,id41 ,rapport sur les progres de la geometrie 7#ahgj71 a continuation of the ,apercu2 ,id41 ,traite de geometrie superieure 7#aheb72 ,cremona1 ,introduzione ad una teoria geometrica delle curve piane 7#ahfb7 and its continuation ,preliminari di una teoria geometrica delle superficie 7,german translations by ,curtze74 ,as more elementary books1 we mention3 ,cremona1 ,elements of ,projective ,geometry1 translated from the ,italian by ,c4 ,leudesdorf 7#bnd ed41 #ahid72 ,j4,w4 ,russell1 ,pure ,geometry 7#bnd ed41 #aije74 7,o4 ,h47 ,,iii4 ,,descriptive ,,geometry ,this branch of geometry is concerned with the methods for representing solids and other figures in three dimensions by drawings in one plane4 ,the most important method is that which was invented by ,monge towards the end of the #ahth century4 ,it is based on parallel projections to a plane by rays perpendicular to the plane4 ,such a projection is called orthographic 7see ,,projection1 ,s #ah74 ,if the plane is horizontal the projection is called the plan of the figure1 and if the plane is vertical the elevation4 ,in ,monge's method a figure is represented by its plan and elevation4 ,it is therefore often called drawing in plan and elevation1 and sometimes simply orthographic projection4 ,s #a4 ,we suppose then that we have two planes1 one horizontal1 the other vertical1 and these we call the planes of plan and of elevation respectively1 or the horizontal and the vertical plane1 and denote them by the letters pi#a and pi#b4 ,their line of intersection is called the axis1 and will be denoted by xy4 ,if the surface of the drawing paper is taken as the plane of the plan1 then the vertical plane will be the plane perpendicular to it through the axis xy4 ,to bring this also into the plane of the drawing paper we turn it about the axis till it coincides with the horizontal plane4 ,this process of turning one plane down till it coincides with another is called rabatting one to the other4 ,of course there is no necessity to have one of the two planes horizontal1 but even when this is not the case it is convenient to retain the above names4 ,illustration3 ,,fig4 #cg4 ,illustration3 ,,fig4 #ch4 ,the whole arrangement will be better understood by referring to fig4 #cg4 ,a point ,a in space is there projected by the perpendicular ,,aa#a and ,,aa#b to the planes pi#a and pi#b so that ,a#a and ,a#b are the horizontal and vertical projections of ,a4 ,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 plane1 we see that the axis which is perpendicular both to ,,aa#a and to ,,aa#b is also perpendicular to ,a#a;,a#j and to ,a#b;,a#j because these four lines are all in the same plane4 ,hence1 if the plane pi#b be turned about the axis till it coincides with the plane pi#a1 then ,a#b;,a#j will be the continuation of ,a#a;,a#j4 ,this position of the planes is represented in fig4 #ch1 in which the line ,a#a;,a#b is perpendicular to the axis x4 ,conversely any two points ,a#a1 ,a#b in a line perpendicular to the axis will be the projections of some point in space when the plane pi#b is turned about the axis till it is perpendicular to the plane pi#a1 because in this position the two perpendiculars to the planes pi#a and pi#b through the points ,a#a and ,a#b will be in a plane and therefore meet at some point ,a4 ,representation of ,points4--,we have thus the following method of representing in a single plane the position of points in space3--we take in the plane a line xy as the axis1 and then any pair of points ,a#a1 ,a#b in the plane on a line perpendicular to the axis represent a point ,a in space4 ,if the line ,a#a;,a#b cuts the axis at ,a#j1 and if at ,a#a a perpendicular be erected to the plane1 then the point ,a will be in it at a height ,a#a;,a ,a#j;,a#b above the plane4 ,this gives the position of the point ,a relative to the plane pi#a4 ,in the same way1 if in a perpendicular to pi#b through ,a#b a point ,a be taken such that ,a#b;,a ,a#j;,a#a1 then this will give the point ,a relative to the plane pi#b4 ,illustration3 ,,fig4 #ci4 ,s #b4 ,the two planes pi#a1 pi#b in their original position divide space into four parts4 ,these are called the four quadrants4 ,we suppose that the plane pi#b is turned as indicated in fig4 #cg1 so that the point ,p comes to ,q and ,r to ,s1 then the quadrant in which the point ,a lies is called the first1 and we say that in the first quadrant a point lies above the horizontal and in front of the vertical plane4 ,now we go round the axis in the sense in which the plane pi#b is turned and come in succession to the second1 third and fourth quadrant4 ,in the second a point lies above the plane of the plan and behind the plane of elevation1 and so on4 ,in fig4 #ci1 which represents a side view of the planes in fig4 #cg the quadrants are marked1 and in each a point with its projection is taken4 ,fig4 #ch shows how these are represented when the plane pi#b is turned down4 ,we see that ,a point lies in the first quadrant if the plan lies below1 the elevation above the axis2 in the second if plan and elevation both lie above2 in the third if the plan lies above1 the elevation below2 in the fourth if plan and elevation both lie below the axis4 ,if a point lies in the horizontal plane1 its elevation lies in the axis and the plan coincides with the point itself4 ,if a point lies in the vertical plane1 its plan lies in the axis and the elevation coincides with the point itself4 ,if a point lies in the axis1 both its plan and elevation lie in the axis and coincide with it4 ,of each of these propositions1 which will easily be seen to be true1 the converse holds also4 ,s #c4 ,representation of a ,plane4--,as we are thus enabled to represent points in a plane1 we can represent any finite figure by representing its separate points4 ,it is1 however1 not possible to represent a plane in this way1 for the projections of its points completely cover the planes pi#a and pi#b1 and no plane would appear different from any other4 ,but any plane alpha cuts each of the planes pi#a1 pi#b in a line4 ,these are called the traces of the plane4 ,they cut each other in the axis at the point where the latter cuts the plane alpha4 ,a plane is determined by its two traces1 which are two lines that meet on the axis1 and1 conversely1 any two lines which meet on the axis determine a plane4 ,if the plane is parallel to the axis its traces are parallel to the axis4 ,of these one may be at infinity2 then the plane will cut one of the planes of projection at infinity and will be parallel to it4 ,thus a plane parallel to the horizontal plane of the plan has only one finite trace1 viz4 that with the plane of elevation4 ,illustration3 ,,fig4 #dj4 ,if the plane passes through the axis both its traces coincide with the axis4 ,this is the only case in which the representation of the plane by its two traces fails4 ,a third plane of projection is therefore introduced1 which is best taken perpendicular to the other two4 ,we call it simply the third plane and denote it by pi#c4 ,as it is perpendicular to pi#a1 it may be taken as the plane of elevation1 its line of intersection gamma with pi#a being the axis1 and be turned down to coincide with pi#a4 ,this is represented in fig4 #dj4 ,,oc is the axis xy whilst ,,oa and ,,ob are the traces of the third plane4 ,they lie in one line gamma4 ,the plane is rabatted about gamma to the horizontal plane4 ,a plane alpha through the axis xy will then show in it a trace alpha#c4 ,in fig4 #dj the lines ,,oc and ,,op will thus be the traces of a plane through the axis xy1 which makes an angle ,,poq with the horizontal plane4 ,we can also find the trace which any other plane makes with pi#c4 ,in rabatting the plane pi#c its trace ,,ob with the plane pi#b will come to the position ,,od4 ,hence a plane beta having the traces ,,ca and ,,cb will have with the third plane the trace beta#c1 or ,,ad if ,,od ,,ob4 ,it also follows immediately that-- ,if a plane alpha is perpendicular to the horizontal plane1 then every point in it has its horizontal projection in the horizontal trace of the plane1 as all the rays projecting these points lie in the plane itself4 ,any plane which is perpendicular to the horizontal plane has its vertical trace perpendicular to the axis4 ,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 trace4 ,s #d4 ,representation of a ,line4--,a line is determined either by two points in it or by two planes through it4 ,we get accordingly two representations of it either by projections or by traces4 ,first4--,a line a is represented by its projections a#a and a#b on the two planes pi#a and pi#b4 ,these may be any two lines1 for1 bringing the planes pi#a1 pi#b into their original position1 the planes through these lines perpendicular to pi#a and pi#b respectively will intersect in some line a which has a#a1 a#b as its projections4 ,secondly4--,a line a is represented by its traces--that is1 by the points in which it cuts the two planes pi#a1 pi#b4 ,any two points may be taken as the traces of a line in space1 for it is determined when the planes are in their original position as the line joining the two traces4 ,this representation becomes undetermined if the two traces coincide in the axis4 ,in this case we again use a third plane1 or else the projections of the line4 ,the fact that there are different methods of representing points and planes1 and hence two methods of representing lines1 suggests the principle of duality 7section ii41 ,projective ,geometry1 ,s #da74 ,it is worth while to keep this in mind4 ,it is also worth remembering that traces of planes or lines always lie in the planes or lines which they represent4 ,projections do not as a rule do this excepting when the point or line projected lies in one of the planes of projection4 ,having now shown how to represent points1 planes and lines1 we have to state the conditions which must hold in order that these elements may lie one in the other1 or else that the figure formed by them may possess certain metrical properties4 ,it will be found that the former are very much simpler than the latter4 ,before we do this1 however1 we shall explain the notation used2 for it is of great importance to have a systematic notation4 ,we shall denote points in space by capitals ,a1 ,b1 ,c2 planes in space by ,greek letters alpha1 beta1 gamma2 lines in space by small letters a1 b1 c2 horizontal projections by suffixes #a1 like ,a#a1 a#a2 vertical projections by suffixes #b1 like ,a#b1 a#b2 traces by single and double dashes alpha' alpha1 a'1 a4 ,hence ,p#a will be the horizontal projection of a point ,p in space2 a line a will have the projections a#a1 a#b and the traces a' and a2 a plane alpha has the traces alpha' and alpha4 ,s #e4 ,if a point lies in a line1 the projections of the point lie in the projections of the line4 ,if a line lies in a plane1 the traces of the line lie in the traces of the plane4 ,these propositions follow at once from the definitions of the projections and of the traces4 ,if a point lies in two lines its projections must lie in the projections of both4 ,hence ,if two lines1 given by their projections1 intersect1 the intersection of their planes and the intersection of their elevations must lie in a line perpendicular to the axis1 because they must be the projections of the point common to the two lines4 ,similarly--,if two lines given by their traces lie in the same plane or intersect1 then the lines joining their horizontal and vertical traces respectively must meet on the axis1 because they must be the traces of the plane through them4 ,s #f4 ,to find the projections of a line which joins two points ,a1 ,b given by their projections ,a#a1 ,a#b and ,b#a1 ,b#b1 we join ,a#a1 ,b#a and ,a#b1 ,b#b2 these will be the projections required4 ,for example1 the traces of a line are two points in the line whose projections are known or at all events easily found4 ,they are the traces themselves and the feet of the perpendiculars from them to the axis4 ,hence if a' a 7fig4 #da7 are the traces of a line a1 and if the perpendiculars from them cut the axis in ,p and ,q respectively1 then the line a',q will be the horizontal and a,p the vertical projection of the line4 ,illustration3 ,,fig4 #da4 ,conversely1 if the projections a#a1 a#b of a line are given1 and if these cut the axis in ,q and ,p respectively1 then the perpendiculars ,pa' and ,qa to the axis drawn through these points cut the projections a#a and a#b in the traces a' and a4 ,to find the line of intersection of two planes1 we observe that this line lies in both planes2 its traces must therefore lie in the traces of both4 ,hence the points where the horizontal traces of the given planes meet will be the horizontal1 and the point where the vertical traces meet the vertical trace of the line required4 ,s #g4 ,to decide whether a point ,a1 given by its projections1 lies in a plane alpha1 given by its traces1 we draw a line p by joining ,a to some point in the plane alpha and determine its traces4 ,if these lie in the traces of the plane1 then the line1 and therefore the point ,a1 lies in the plane2 otherwise not4 ,this is conveniently done by joining ,a#a to some point p' in the trace alpha'2 this gives p#a2 and the point where the perpendicular from p' to the axis cuts the latter we join to ,a#b2 this gives p#b4 ,if the vertical trace of this line lies in the vertical trace of the plane1 then1 and then only1 does the line p1 and with it the point ,a1 lie in the plane alpha4 ,s #h4 ,parallel planes have parallel traces1 because parallel planes are cut by any plane1 hence also by pi#a and by pi#b1 in parallel lines4 ,parallel lines have parallel projections1 because points at infinity are projected to infinity4 ,if a line is parallel to a plane1 then lines through the traces of the line and parallel to the traces of the plane must meet on the axis1 because these lines are the traces of a plane parallel to the given plane4 ,s #i4 ,to draw a plane through two intersecting lines or through two parallel lines1 we determine the traces of the lines2 the lines joining their horizontal and vertical traces respectively will be the horizontal and vertical traces of the plane4 ,they will meet1 at a finite point or at infinity1 on the axis if the lines do intersect4 ,to draw a plane through a line and a point without the line1 we join the given point to any point in the line and determine the plane through this and the given line4 ,to draw a plane through three points which are not in a line1 we draw two of the lines which each join two of the given points and draw the plane through them4 ,if the traces of all three lines ,,ab1 ,,bc1 ,,ca be found1 these must lie in two lines which meet on the axis4 ,s #aj4 ,we have in the last example got more points1 or can easily get more points1 than are necessary for the determination of the figure required--in this case the traces of the plane4 ,this will happen in a great many constructions and is of considerable importance4 ,it may happen that some of the points or lines obtained are not convenient in the actual construction4 ,the horizontal traces of the lines ,,ab and ,,ac may1 for instance1 fall very near together1 in which case the line joining them is not well defined4 ,or1 one or both of them may fall beyond the drawing paper1 so that they are practically non-existent for the construction4 ,in this case the traces of the line ,,bc may be used4 ,or1 if the vertical traces of ,,ab and ,,ac are both in convenient position1 so that the vertical trace of the required plane is found and one of the horizontal traces is got1 then we may join the latter to the point where the vertical trace cuts the axis4 ,the draughtsman must remember that the lines which he draws are not mathematical lines without thickness1 and therefore every drawing is affected by some errors4 ,it is therefore very desirable to be able constantly to check the latter4 ,such checks always present themselves when the same result can be obtained by different constructions1 or when1 as in the above case1 some lines must meet on the axis1 or if three points must lie in a line4 ,a careful draughtsman will always avail himself of these checks4 ,s #aa4 ,to draw a plane through a given point parallel to a given plane alpha1 we draw through the point two lines which are parallel to the plane alpha1 and determine the plane through them2 or1 as we know that the traces of the required plane are parallel to those of the given one 7,s #h71 we need only draw one line l through the point parallel to the plane and find one of its traces1 say the vertical trace l2 a line through this parallel to the vertical trace of alpha will be the vertical trace beta of the required plane beta1 and a line parallel to the horizontal trace of alpha meeting beta on the axis will be the horizontal trace beta'4 ,illustration3 ,,fig4 #db4 ,let ,a#a ,a#b 7fig4 #db7 be the given point1 alpha' alpha the given plane1 a line l#a through ,a#a1 parallel to alpha' and a horizontal line l#b through ,a#b will be the projections of a line l through ,a parallel to the plane1 because the horizontal plane through this line will cut the plane alpha in a line c which has its horizontal projection c#a parallel to alpha'4 ,s #ab4 ,we now come to the metrical properties of figures4 ,a line is perpendicular to a plane if the projections of the line are perpendicular to the traces of the plane4 ,we prove it for the horizontal projection4 ,if a line p is perpendicular to a plane alpha1 every plane through p is perpendicular to alpha2 hence also the vertical plane which projects the line p to p#a4 ,as this plane is perpendicular both to the horizontal plane and to the plane alpha1 it is also perpendicular to their intersection--that is1 to the horizontal trace of alpha4 ,it follows that every line in this projecting plane1 therefore also p#a1 the plan of p1 is perpendicular to the horizontal trace of alpha4 ,to draw a plane through a given point ,a perpendicular to a given line p1 we first draw through some point ,o in the axis lines gamma'1 gamma perpendicular respectively to the projections p#a and p#b of the given line4 ,these will be the traces of a plane gamma which is perpendicular to the given line4 ,we next draw through the given point ,a a plane parallel to the plane gamma2 this will be the plane required4 ,other metrical properties depend on the determination of the real size or shape of a figure4 ,in general the projection of a figure differs both in size and shape from the figure itself4 ,but figures in a plane parallel to a plane of projection will be identical with their projections1 and will thus be given in their true dimensions4 ,in other cases there is the problem1 constantly recurring1 either to find the true shape and size of a plane figure when plan and elevation are given1 or1 conversely1 to find the latter from the known true shape of the figure itself4 ,to do this1 the plane is turned about one of its traces till it is laid down into that plane of projection to which the trace belongs4 ,this is technically called rabatting the plane respectively into the plane of the plan or the elevation4 ,as there is no difference in the treatment of the two cases1 we shall consider only the case of rabatting a plane alpha into the plane of the plan4 ,the plan of the figure is a parallel 7orthographic7 projection of the figure itself4 ,the results of parallel projection 7see ,,projection1 ,,ss #ag and #ah7 may therefore now be used4 ,the trace alpha' will hereby take the place of what formerly was called the axis of projection4 ,hence we see that corresponding points in the plan and in the rabatted plane are joined by lines which are perpendicular to the trace alpha' and that corresponding lines meet on this trace4 ,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 plane4 ,before1 however1 we treat of this we consider some special cases4 ,s #ac4 ,to determine the distance between two points ,a1 ,b given by their projections ,a#a1 ,b#a and ,a#b1 ,b#b1 or1 in other words1 to determine the true length of a line the plan and elevation of which are given4 ,illustration3 ,,fig4 #dc4 ,solution4--,the two points ,a1 ,b in space lie vertically above their plans ,a#a1 ,b#a 7fig4 #dc7 and ,a#a;,a ,a#j;,a#b1 ,b#a;,b ,b#j;,b#b4 ,the four points ,a1 ,b1 ,a#a1 ,b#a therefore form a plane quadrilateral on the base ,a#a;,b#a and having right angles at the base4 ,this plane we rabatt about ,a#a;,b#a by drawing ,a#a;,a and ,b#a;,b perpendicular to ,a#a;,b#a and making ,a#a;,a ,a#j;,a#b1 ,b#a;,b ,b#j;,b#b4 ,then ,,ab will give the length required4 ,the construction might have been performed in the elevation by making ,a#b;,a ,a#j;,a#a and ,b#b;,b ,b#j;,b#a on lines perpendicular to ,a#b;,b#b4 ,of course ,,ab must have the same length in both cases4 ,this figure may be turned into a model4 ,cut the paper along ,a#a;,a1 ,,ab and ,,bb#a1 and fold the piece ,a#a;,,abb#a over along ,a#a;,b#a till it stands upright at right angles to the horizontal plane4 ,the points ,a1 ,b will then be in their true position in space relative to pi#a4 ,similarly if ,b#b;,,baa#b be cut out and turned along ,a#b;,b#b through a right angle we shall get ,,ab in its true position relative to the plane pi#b4 ,lastly we fold the whole plane of the paper along the axis x till the plane pi#b is at right angles to pi#a4 ,in this position the two sets of points ,,ab will coincide if the drawing has been accurate4 ,models of this kind can be made in many cases and their construction cannot be too highly recommended in order to realize orthographic projection4 ,s #ad4 ,to find the angle between two given lines a1 b of which the projections a#a1 b#a and a#b1 b#b are given4 ,illustration3 ,,fig4 #dd4 ,solution4--,let a#a1 b#a 7fig4 #dd7 meet in ,p#a1 a#b1 b#b in ,t1 then if the line ,p#a,t is not perpendicular to the axis the two lines will not meet4 ,in this case we draw a line parallel to b to meet the line a4 ,this is easiest done by drawing first the line ,p#a,p#b perpendicular to the axis to meet a#b in ,p#b1 and then drawing through ,p#b a line c#b parallel to b#b2 then b#a1 c#b will be the projections of a line c which is parallel to b and meets a in ,p4 ,the plane alpha which these two lines determine we rabatt to the plan4 ,we determine the traces a' and c' of the lines a and c2 then a'c' is the trace alpha' of their plane4 ,on rabatting the point ,p comes to a point ,s on the line ,p#a,q perpendicular to a'c'1 so that ,,qs ,,qp4 ,but ,,qp is the hypotenuse of a triangle ,,pp#a,q with a right angle ,p#a4 ,this we construct by making ,,qr ,p#j,p#b2 then ,p#a,r ,,pq4 ,the lines a',s and c',s will therefore include angles equal to those made by the given lines4 ,it is to be remembered that two lines include two angles which are supplementary4 ,which of these is to be taken in any special case depends upon the circumstances4 ,to determine the angle between a line and a plane1 we draw through any point in the line a perpendicular to the plane 7,s #ab7 and determine the angle between it and the given line4 ,the complement of this angle is the required one4 ,to determine the angle between two planes1 we draw through any point two lines perpendicular to the two planes and determine the angle between the latter as above4 ,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 this4 ,this is especially the case if one of the planes is the horizontal or vertical plane of projection4 ,thus in fig4 #de the angle ,p#a,,qr is the angle which the plane alpha makes with the horizontal plane4 ,s #ae4 ,we return to the general case of rabatting a plane alpha of which the traces alpha' alpha are given4 ,illustration3 ,,fig4 #de4 ,here it will be convenient to determine first the position which the trace alpha--which is a line in alpha--assumes when rabatted4 ,points in this line coincide with their elevations4 ,hence it is given in its true dimension1 and we can measure off along it the true distance between two points in it4 ,if therefore 7fig4 #de7 ,p is any point in alpha originally coincident with its elevation ,p#b1 and if ,o is the point where alpha cuts the axis xy1 so that ,o is also in alpha'1 then the point ,p will after rabatting the plane assume such a position that ,,op ,,op#b4 ,at the same time the plan is an orthographic projection of the plane alpha4 ,hence the line joining ,p to the plan ,p#a will after rabatting be perpendicular to alpha'4 ,but ,p#a is known2 it is the foot of the perpendicular from ,p#b to the axis xy4 ,we draw therefore1 to find ,p1 from ,p#a a perpendicular ,p#a,q to alpha' and find on it a point ,p such that ,,op ,,op#b4 ,then the line ,,op will be the position of alpha when rabatted4 ,this line corresponds therefore to the plan of alpha--that is1 to the axis xy1 corresponding points on these lines being those which lie on a perpendicular to alpha'4 ,we have thus one pair of corresponding lines and can now find for any point ,b#a in the plan the corresponding point ,b in the rabatted plane4 ,we draw a line through ,b#a1 say ,b#a,p#a1 cutting alpha' in ,c4 ,to it corresponds the line ,,cp1 and the point where this is cut by the projecting ray through ,b#a1 perpendicular to alpha'1 is the required point ,b4 ,similarly any figure in the rabatted plane can be found when the plan is known2 but this is usually found in a different manner without any reference to the general theory of parallel projection4 ,as this method and the reasoning employed for it have their peculiar advantages1 we give it also4 ,supposing the planes pi#a and pi#b to be in their positions in space perpendicular to each other1 we take a section of the whole figure by a plane perpendicular to the trace alpha' about which we are going to rabatt the plane alpha4 ,let this section pass through the point ,q in alpha'4 ,its traces will then be the lines ,,qp#a and ,p#a,p#b 7fig4 #i74 ,these will be at right angles1 and will therefore1 together with the section ,,qp#b of the plane alpha1 form a right-angled triangle ,,qp#a,p#b with the right angle at ,p#a1 and having the sides ,p#a,q and ,p#a,p#b which both are given in their true lengths4 ,this triangle we rabatt about its base ,p#a,q1 making ,p#a,r ,p#a,p#b4 ,the line ,,qr will then give the true length of the line ,,qp in space4 ,if now the plane alpha be turned about alpha' the point ,p will describe a circle about ,q as centre with radius ,,qp ,,qr1 in a plane perpendicular to the trace alpha'4 ,hence when the plane alpha has been rabatted into the horizontal plane the point ,p will lie in the perpendicular ,p#a,q to alpha'1 so that ,,qp ,,qr4 ,if ,a#a is the plan of a point ,a in the plane alpha1 and if ,a#a lies in ,,qp#a1 then the point ,a will lie vertically above ,a#a in the line ,,qp4 ,on turning down the triangle ,,qp#a,p#b1 the point ,a will come to ,a#j1 the line ,a#a;,a#j being perpendicular to ,,qp#a4 ,hence ,a will be a point in ,,qp such that ,,qa ,,qa#j4 ,if ,b#a is the plan of another point1 but such that ,a#a;,b#a is parallel to alpha'1 then the corresponding line ,,ab will also be parallel to alpha'4 ,hence1 if through ,a a line ,,ab be drawn parallel to alpha'1 and ,b#a;,b perpendicular to alpha'1 then their intersection gives the point ,b4 ,thus of any point given in plan the real position in the plane alpha1 when rabatted1 can be found by this second method4 ,this is the one most generally given in books on geometrical drawing4 ,the first method explained is1 however1 in most cases preferable as it gives the draughtsman a greater variety of constructions4 ,it requires a somewhat greater amount of theoretical knowledge4 ,if instead of our knowing the plan of a figure the latter is itself given1 then the process of finding the plan is the reverse of the above and needs little explanation4 ,we give an example4 ,s #af4 ,it is required to draw the plan and elevation of a polygon of which the real shape and position in a given plane alpha are known4 ,we first rabatt the plane alpha 7fig4 #df7 as before so that ,p#a comes to ,p1 hence ,,op#a to ,,op4 ,let the given polygon in alpha be the figure ,,abcde4 ,we project1 not the vertices1 but the sides4 ,to project the line ,,ab1 we produce it to cut alpha' in ,f and ,,op in ,g1 and draw ,,gg#a perpendicular to alpha'2 then ,g#a corresponds to ,g1 therefore ,,fg#a to ,,fg4 ,in the same manner we might project all the other sides1 at least those which cut ,,of and ,,op in convenient points4 ,it will be best1 however1 first to produce all the sides to cut ,,op and alpha' and then to draw all the projecting rays through ,a1 ,b1 ,c 444 perpendicular to alpha'1 and in the same direction the lines ,g1 ,g#a1 c4 ,by drawing ,,fg we get the points ,a#a1 ,b#a on the projecting ray through ,a and ,b4 ,we then join ,b to the point ,m where ,,bc produced meets the trace alpha'4 ,this gives ,c#a4 ,so we go on till we have found ,e#a4 ,the line ,a#a ,e#a must then meet ,,ae in alpha'1 and this gives a check4 ,if one of the sides cuts alpha' or ,,op beyond the drawing paper this method fails1 but then we may easily find the projection of some other line1 say of a diagonal1 or directly the projection of a point1 by the former methods4 ,the diagonals may also serve to check the drawing1 for two corresponding diagonals must meet in the trace alpha'4 ,illustration3 ,,fig4 #df4 ,having got the plan we easily find the elevation4 ,the elevation of ,g is above ,g#a in alpha1 and that of ,f is at ,f#b in the axis4 ,this gives the elevation ,f#b;,g#b of ,,fg and in it we get ,a#b;,b#b in the verticals through ,a#a and ,b#a4 ,as a check we have ,,og ,,og#b4 ,similarly the elevation of the other sides and vertices are found4 ,s #ag4 ,we proceed to give some applications of the above principles to the representation of solids and of the solution of problems connected with them4 ,of a pyramid are given its base1 the length of the perpendicular from the vertex to the base1 and the point where this perpendicular cuts the base2 it is required first to develop the whole surface of the pyramid into one plane1 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 it4 #a4 ,as the planes of projection are not given we can take them as we like1 and we select them in such a manner that the solution becomes as simple as possible4 ,we take the plane of the base as the horizontal plane and the vertical plane perpendicular to the plane of the section4 ,let then 7fig4 #dg7 ,,abcd be the base of the pyramid1 ,v#a the plan of the vertex1 then the elevations of ,a1 ,b1 ,c1 ,d will be in the axis at ,a#b1 ,b#b1 ,c#b1 ,d#b1 and the vertex at some point ,v#b above ,v#a at a known distance from the axis4 ,the lines ,v#a;,a1 ,v#a;,b1 c41 will be the plans and the lines ,v#b;,a#b1 ,v#b;,b#b1 c41 the elevations of the edges of the pyramid1 of which thus plan and elevation are known4 ,we develop the surface into the plane of the base by turning each lateral face about its lower edge into the horizontal plane by the method used in ,s #ad4 ,if one face has been turned down1 say ,,abv to ,,abp1 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#a,q perpendicular to ,,bc the point ,q such that ,,bq ,,bp1 for these lines represent the same edge ,,bv of the pyramid4 ,next ,r is found by making ,,cr ,,cq1 and so on till we have got the last vertex--in this case ,s4 ,the fact that ,,as must equal ,,ap gives a convenient check4 #b4 ,the plane alpha whose section we have to determine has its horizontal trace given perpendicular to the axis1 and its vertical trace makes the given angle with the axis4 ,this determines it4 ,to find the section of the pyramid by this plane there are two methods applicable3 we find the sections of the plane either with the faces or with the edges of the pyramid4 ,we use the latter4 ,as the plane alpha is perpendicular to the vertical plane1 the trace alpha contains the projection of every figure in it2 the points ,e#b1 ,f#b1 ,g#b1 ,h#b where this trace cuts the elevations of the edges will therefore be the elevations of the points where the edges cut alpha4 ,from these we find the plans ,e#a1 ,f#a1 ,g#a1 ,h#a1 and by joining them the plan of the section4 ,if from ,e#a1 ,f#a lines be drawn perpendicular to ,,ab1 these will determine the points ,e1 ,f on the developed face in which the plane alpha cuts it2 hence also the line ,,ef4 ,similarly on the other faces4 ,of course ,,bf must be the same length on ,,bp and on ,,bq4 ,if the plane alpha be rabatted to the plan1 we get the real shape of the section as shown in the figure in ,,efgh4 ,this is done easily by making ,f#j;,f ,,of#b1 c4 ,if the figure representing the development of the pyramid1 or better a copy of it1 is cut out1 and if the lateral faces be bent along the lines ,,ab1 ,,bc1 c41 we get a model of the pyramid with the section marked on its faces4 ,this may be placed on its plan ,,abcd and the plane of elevation bent about the axis x4 ,the pyramid stands then in front of its elevations4 ,if next the plane alpha with a hole cut out representing the true section be bent along the trace alpha' till its edge coincides with alpha1 the edges of the hole ought to coincide with the lines ,,ef1 ,,fg1 c41 on the faces4 ,s #ah4 ,polyhedra like the pyramid in ,s #ag are represented by the projections of their edges and vertices4 ,but solids bounded by curved surfaces1 or surfaces themselves1 cannot be thus represented4 ,for a surface we may use1 as in case of the plane1 its traces--that is1 the curves in which it cuts the planes of projection4 ,we may also project points and curves on the surface4 ,a ray cuts the surface generally in more than one point2 hence it will happen that some of the rays touch the surface1 if two of these points coincide4 ,the points of contact of these rays will form some curve on the surface1 and this will appear from the centre of projection as the boundary of the surface or of part of the surface4 ,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 surface4 ,the projections of these contours are therefore best adapted to give an idea of the shape of a surface4 ,illustration3 ,,fig4 #dg4 ,thus the tangents drawn from any finite centre to a sphere form a right circular cone1 and this will be cut by any plane in a conic4 ,it is often called the projection of a sphere1 but it is better called the contour-line of the sphere1 as it is the boundary of the projections of all points on the sphere4 ,if the centre is at infinity the tangent cone becomes a right circular cylinder touching the sphere along a great circle1 and if the projection is1 as in our case1 orthographic1 then the section of this cone by a plane of projection will be a circle equal to the great circle of the sphere4 ,we get such a circle in the plan and another in the elevation1 their centres being plan and elevation of the centre of the sphere4 ,similarly the rays touching a cone of the second order will lie in two planes which pass through the vertex of the cone1 the contour-line of the projection of the cone consists therefore of two lines meeting in the projection of the vertex4 ,these may1 however1 be invisible if no real tangent rays can be drawn from the centre of projection2 and this happens when the ray projecting the centre of the vertex lies within the cone4 ,in this case the traces of the cone are of importance4 ,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 cone1 and in the elevation the contour1 consisting of a pair of lines intersecting in the elevation of the vertex of the cone4 ,the circle in the plan and the pair of lines in the elevation do not determine the surface1 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 plane4 ,the surface becomes only completely defined if we write down to the figure that it shall represent a cone4 ,the same holds for all surfaces4 ,even a plane is fully represented by its traces only under the silent understanding that the traces are those of a plane4 ,s #ai4 ,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 surfaces4 ,the former is constantly used in nearly all problems concerning surfaces4 ,its solution depends of course on the nature of the surface4 ,to determine the curve of intersection of two surfaces1 we take a plane and determine its section with each of the two surfaces1 rabatting this plane if necessary4 ,this gives two curves which lie in the same plane and whose intersections will give us points on both surfaces4 ,it must here be remembered that two curves in space do not necessarily intersect1 hence that the points in which their projections intersect are not necessarily the projections of points common to the two curves4 ,this will1 however1 be the case if the two curves lie in a common plane4 ,by taking then a number of plane sections of the surfaces we can get as many points on their curve of intersection as we like4 ,these planes have1 of course1 to be selected in such a way that the sections are curves as simple as the case permits of1 and such that they can be easily and accurately drawn4 ,thus when possible the sections should be straight lines or circles4 ,this not only saves time in drawing but determines all points on the sections1 and therefore also the points where the two curves meet1 with equal accuracy4 ,s #bj4 ,we give a few examples how these sections have to be selected4 ,a cone is cut by every plane through the vertex in lines1 and if it is a cone of revolution by planes perpendicular to the axis in circles4 ,a cylinder is cut by every plane parallel to the axis in lines1 and if it is a cylinder of revolution by planes perpendicular to the axis in circles4 ,a sphere is cut by every plane in a circle4 ,hence in case of two cones situated anywhere in space we take sections through both vertices4 ,these will cut both cones in lines4 ,similarly in case of two cylinders we may take sections parallel to the axis of both4 ,in case of a sphere and a cone of revolution with vertical axis1 horizontal sections will cut both surfaces in circles whose plans are circles and whose elevations are lines1 whilst vertical sections through the vertex of the cone cut the latter in lines and the sphere in circles4 ,to avoid drawing the projections of these circles1 which would in general be ellipses1 we rabatt the plane and then draw the circles in their real shape4 ,and so on in other cases4 ,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 x1 or where these projections touch the contour of one of the surfaces4 7,o4 ,h47 ,,iv4 ,,analytical ,,geometry #a4 ,in the name geometry there is a lasting record that the science had its origin in the knowledge that two distances may be compared by measurement1 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 direction4 ,the distance from an observer's eye of an object seen would be specified as soon as it was ascertained that a rod1 straight to the eye and of length taken as known1 could be given the direction of the line of vision1 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 eye4 ,moreover1 if a field had for two of its boundaries lines straight to the eye1 one running from south to north and the other from west to east1 the position of a point in the field would be specified if the rod1 when directed west1 had to be shifted from the point one observed number of times westward to meet the former boundary1 and also1 when directed south1 had to be shifted another observed number of times southward to meet the latter4 ,comparison by measurement1 the beginning of geometry1 involved counting1 the basis of arithmetic2 and the science of number was marked out from the first as of geometrical importance4 ,but the arithmetic of the ancients was inadequate as a science of number4 ,though a length might be recognized as known when measurement certified that it was so many times a standard length1 it was not every length which could be thus specified in terms of the same standard length1 even by an arithmetic enriched with the notion of fractional number4 ,the idea of possible incommensurability of lengths was introduced into ,europe by ,pythagoras2 and the corresponding idea of irrationality of number was absent from a crude arithmetic1 while there were great practical difficulties in the way of its introduction4 ,hence perhaps it arose that1 till comparatively modern times1 appeal to arithmetical aid in geometrical reasoning was in all possible ways restrained4 ,geometry figured rather as the helper of the more difficult science of arithmetic4 #b4 ,it was reserved for algebra to remove the disabilities of arithmetic1 and to restore the earliest ideas of the land-measurer to the position of controlling ideas in geometrical investigation4 ,this unified science of pure number made comparatively little headway in the hands of the ancients1 but began to receive due attention shortly after the revival of learning4 ,it expresses whole classes of arithmetical facts in single statements1 gives to arithmetical laws the form of equations involving symbols which may mean any known or sought numbers1 and provides processes which enable us to analyse the information given by an equation and derive from that equation other equations1 which express laws that are in effect consequences or causes of a law started from1 but differ greatly from it in form4 ,above all1 for present purposes1 it deals not only with integral and fractional number1 but with number regarded as capable of continuous growth1 just as distance is capable of continuous growth4 ,the difficulty of the arithmetical expression of irrational number1 a difficulty considered by the modern school of analysts to have been at length surmounted 7see ,,function71 is not vital to it4 ,it can call the ratio of the diagonal of a square to a side1 for instance1 or that of the circumference of a circle to a diameter1 a number1 and let a or x denote that number1 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 assign4 ,counting only1 and not the counting of objects1 is of the essence of arithmetic1 and of algebra4 ,but it is lawful to count objects1 and in particular to count equal lengths by measure4 ,the widened idea is that even when a or x is an irrational number we may speak of a or x unit lengths by measure4 ,we may give concrete interpretation to an algebraical equation by allowing its terms all to mean numbers of times the same unit length1 or the same unit area1 or c4 and in any equation lawfully derived from the first by algebraical processes we may do the same4 ,descartes in his ,geometrie 7#afcg7 was the first to systematize the application of this principle to the inherent first notions of geometry2 and the methods which he instituted have become the most potent methods of all in geometrical research4 ,it is hardly too much to say that1 when known facts as to a geometrical figure have once been expressed in algebraical terms1 all strictly consequential facts as to the figure can be deduced by almost mechanical processes4 ,some may well be unexpected consequences2 and in obtaining those of which there has been suggestion beforehand the often bewildering labour of constant attention to the figure is obviated4 ,these are the methods of what is now called analytical1 or sometimes algebraical1 geometry4 #c4 ,the modern use of the term analytical in geometry has obscured1 but not made obsolete1 an earlier use1 one as old as ,plato4 ,there is nothing algebraical in this analysis1 as distinguished from synthesis1 of the ,greeks1 and of the expositors of pure geometry4 ,it has reference to an order of ideas in demonstration1 or1 more frequently1 in discovering means to effect the geometrical construction of a figure with an assigned special property4 ,we have to suppose hypothetically that the construction has been performed1 drawing a rough figure which exhibits it as nearly as is practicable4 ,we then analyse or critically examine the figure1 treated as correct1 and ascertain other properties which it can only possess in association with the one in question4 ,presently one of these properties will often be found which is of such a character that the construction of a figure possessing it is simple4 ,the means of effecting synthetically a construction such as was desired is thus brought to light by what ,plato called analysis4 ,or again1 being asked to prove a theorem ,a1 we ascertain that it must be true if another theorem ,b is1 that ,b must be if ,c is1 and so on1 thus eventually finding that the theorem ,a is the consequence1 through a chain of intermediaries1 of a theorem ,z of which the establishment is easy4 ,this geometrical analysis is not the subject of the present article2 but in the reasoning from form to form of an equation or system of equations1 with the object of basing the algebraical proof of a geometrical fact on other facts of a more obvious character1 the same logic is utilized1 and the name analytical geometry is thus in part explained4 #d4 ,in algebra real positive number was alone at first dealt with1 and in geometry actual signless distance4 ,but in algebra it became of importance to say that every equation of the first degree has a root1 and the notion of negative number was introduced4 ,the negative unit had to be defined as what can be added to the positive unit and produce the sum zero4 ,the corresponding notion was readily at hand in geometry1 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 ,o1 with the result of reaching ,o again4 ,thus1 to give full interpretation in geometry to the algebraically negative1 it was only necessary to associate distinctness of sign with oppositeness of direction4 ,later it was discovered that algebraical reasoning would be much facilitated1 and that conclusions as to the real would retain all their soundness1 if a pair of imaginary units -root-#a of what might be called number were allowed to be contemplated1 the pair being defined1 though not separately1 by the two properties of having the real sum #j and the real product #a4 ,only in these two real combinations do they enter in conclusions as to the real4 ,an advantage gained was that every quadratic equation1 and not some quadratics only1 could be spoken of as having two roots4 ,these admissions of new units into algebra were final1 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 case1 and that every root has a value included in the form a b root-#a1 with a1 b1 real4 ,the corresponding enrichment could be given to geometry1 with corresponding advantages and the same absence of danger1 and this was done4 ,on a line of measurement of distance we contemplate as existing1 not only an infinite continuum of points at real distances from an origin of measurement ,o1 but a doubly infinite continuum of points1 all but the singly infinite continuum of real ones imaginary1 and imaginary in conjugate pairs1 a conjugate pair being at imaginary distances from ,o1 which have a real arithmetic and a real geometric mean4 ,to geometry enriched with this conception all algebra has its application4 #e4 ,actual geometry is one1 two or three-dimensional1 i4e4 lineal1 plane or solid4 ,in one-dimensional geometry positions and measurements in a single line only are admitted4 ,now descriptive constructions for points in a line are impossible without going out of the line4 ,it has therefore been held that there is a sense in which no science of geometry strictly confined to one dimension exists4 ,but an algebra of one variable can be applied to the study of distances along a line measured from a chosen point on it1 so that the idea of construction as distinct from measurement is not essential to a one-dimensional geometry aided by algebra4 ,in geometry of two dimensions1 the flat of the land-measurer1 the passage from one point ,o to any other point1 can be effected by two successive marches1 one east or west and one north or south1 and1 as will be seen1 an algebra of two variables suffices for geometrical exploitation4 ,in geometry of three dimensions1 that of space1 any point can be reached from a chosen one by three marches1 one east or west1 one north or south1 and one up or down2 and we shall see that an algebra of three variables is all that is necessary4 ,with three dimensions actual geometry stops2 but algebra can supply any number of variables4 ,four or more variables have been used in ways analogous to those in which one1 two and three variables are used for the purposes of one1 two and three-dimensional geometry1 and the results have been expressed in quasi-geometrical language on the supposition that a higher space can be conceived of1 though not realized1 in which four independent directions exist1 such that no succession of marches along three of them can effect the same displacement of a point as a march along the fourth2 and similarly for higher numbers than four4 ,thus analytical1 though not actual1 geometries exist for four and more dimensions4 ,they are in fact algebras furnished with nomenclature of a geometrical cast1 suggested by convenient forms of expression which actual geometry has1 in return for benefits received1 conferred on algebras of one1 two and three variables4 ,we will confine ourselves to the dimensions of actual geometry1 and will devote no space to the one-dimensional1 except incidentally as existing within the two-dimensional4 ,the analytical method will now be explained for the cases of two and three dimensions in succession4 ,the form of it originated by ,descartes1 and thence known as ,cartesian1 will alone be considered in much detail4 ,i4 ,plane ,analytical ,geometry4 ,illustration3 ,,fig4 #dh4 ,illustration3 ,,fig4 #di4 #f4 ,coordinates4--,it is assumed that the points1 lines and figures considered lie in one and the same plane1 which plane therefore need not be in any way referred to4 ,in the plane a point ,o1 and two lines x',ox1 y',oy1 intersecting in ,o1 are taken once for all1 and regarded as fixed4 ,o is called the origin1 and x',ox1 y',oy the axes of x and y respectively4 ,other positions in the plane are specified in relation to this fixed origin and these fixed axes4 ,from any point ,p we suppose ,,pm drawn parallel to the axis of y to meet the axis of x in ,m1 and may also suppose ,,pn drawn parallel to the axis of x to meet the axis of y in ,n1 so that ,,ompn is a parallelogram4 ,the position of ,p is determined when we know ,,om 7 ,,np7 and ,,mp 7 ,,on74 ,if ,,om is x times the unit of a scale of measurement chosen at pleasure1 and ,,mp is y times the unit1 so that x and y have numerical values1 we call x and y the 7,cartesian7 coordinates of ,p4 ,to distinguish them we often speak of y as the ordinate1 and of x as the abscissa4 ,it is necessary to attend to signs2 x has one sign or the other according as the point ,p is on one side or the other of the axis of y1 and y one sign or the other according as ,p is on one side or the other of the axis of x4 ,using the letters ,n1 ,e1 ,s1 ,w1 as in a map1 and considering the plane as divided into four quadrants by the axes1 the signs are usually taken to be3 x y ,for quadrant ,n ,e - ,s ,e - ,n ,w - - ,s ,w ,a point is referred to as the point 7a1 b71 when its coordinates are x a1 y b4 ,a point may be fixed1 or it may be variable1 i4e4 be regarded for the time being as free to move in the plane4 ,the coordinates 7x1 y7 of a variable point are algebraic variables1 and are said to be current coordinates4 ,the axes of x and y are usually 7as in fig4 #dh7 taken at right angles to one another1 and we then speak of them as rectangular axes1 and of x and y as rectangular coordinates of a point ,p2 ,,ompn is then a rectangle4 ,sometimes1 however1 it is convenient to use axes which are oblique to one another1 so that 7as in fig4 #di7 the angle x,oy between their positive directions is some known angle omega distinct from a right angle1 and ,,ompn is always an oblique parallelogram with given angles2 and we then speak of x and y as oblique coordinates4 ,the coordinates are as a rule taken to be rectangular in what follows4 #g4 ,equations and loci4 ,if 7x1 y7 is the point ,p1 and if we are given that x #j1 we are told that1 in fig4 #dh or fig4 #di1 the point ,m lies at ,o1 whatever value y may have1 i4e4 we are told the one fact that ,p lies on the axis of y4 ,conversely1 if ,p lies anywhere on the axis of y1 we have always ,,om #j1 i4e4 x #j4 ,thus the equation x #j is one satisfied by the coordinates 7x1 y7 of every point in the axis of y1 and not by those of any other point4 ,we say that x #j is the equation of the axis of y1 and that the axis of y is the locus represented by the equation x #j4 ,similarly y #j is the equation of the axis of x4 ,an equation x a1 where a is a constant1 expresses that ,p lies on a parallel to the axis of y through a point ,m on the axis of x such that ,,om a4 ,every line parallel to the axis of y has an equation of this form4 ,similarly1 every line parallel to the axis of x has an equation of the form y b1 where b is some definite constant4 ,these are simple cases of the fact that a single equation in the current coordinates of a variable point 7x1 y7 imposes one limitation on the freedom of that point to vary4 ,the coordinates of a point taken at random in the plane will1 as a rule1 not satisfy the equation1 but infinitely many points1 and in most cases infinitely many real ones1 have coordinates which do satisfy it1 and these points are exactly those which lie upon some locus of one dimension1 a straight line or more frequently a curve1 which is said to be represented by the equation4 ,take1 for instance1 the equation y mx1 where m is a given constant4 ,it is satisfied by the coordinates of every point ,p1 which is such that1 in fig4 #dh1 the distance ,,mp1 with its proper sign1 is m times the distance ,,om1 with its proper sign1 i4e4 by the coordinates of every point in the straight line through ,o which we arrive at by making a line1 originally coincident with x',ox1 revolve about ,o in the direction opposite to that of the hands of a watch through an angle of which m is the tangent1 and by those of no other points4 ,that line is the locus which it represents4 ,take1 more generally1 the equation y phi7x71 where phi7x7 is any given non-ambiguous function of x4 ,choosing any point ,m on x',ox in fig4 #a1 and giving to x the value of the numerical measure of ,,om1 the equation determines a single corresponding y1 and so determines a single point ,p on the line through ,m parallel to y',oy4 ,this is one point whose coordinates satisfy the equation4 ,now let ,m move from the extreme left to the extreme right of the line x',ox1 regarded as extended both ways as far as we like1 i4e4 let x take all real values from -oo to oo4 ,with every value goes a point ,p1 as above1 on the parallel to y',oy through the corresponding ,m2 and we thus find that there is a path from the extreme left to the extreme right of the figure1 all points ,p along which are distinguished from other points by the exceptional property of satisfying the equation by their coordinates4 ,this path is a locus2 and the equation y phi7x7 represents it4 ,more generally still1 take an equation f7x1 y7 #j which involves both x and y under a functional form4 ,any particular value given to x in it produces from it an equation for the determination of a value or values of y1 which go with that value of x in specifying a point or points 7x1 y71 of which the coordinates satisfy the equation f7x1 y7 #j4 ,here again1 as x takes all values1 the point or points describe a path or paths1 which constitute a locus represented by the equation4 ,except when y enters to the first degree only in f7x1 y71 it is not to be expected that all the values of y1 determined as going with a chosen value of x1 will be necessarily real2 indeed it is not uncommon for all to be imaginary for some ranges of values of x4 ,the locus may largely consist of continua of imaginary points2 but the real parts of it constitute a real curve or real curves4 ,note that we have to allow x to admit of all imaginary1 as well as of all real1 values1 in order to obtain all imaginary parts of the locus4 ,a locus or curve may be algebraically specified in another way2 viz4 we may be given two equations x f7theta71 y ,f7theta71 which express the coordinates of any point of it as two functions of the same variable parameter theta to which all values are open4 ,as theta takes all values in turn1 the point 7x1 y7 traverses the curve4 ,it is a good exercise to trace a number of curves1 taken as defined by the equations which represent them4 ,this1 in simple cases1 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 x1 and connecting the various points 7x1 y7 thus obtained4 ,but methods exist for diminishing the labour of this tentative process4 ,another problem1 which will be more attended to here1 is that of determining the equations of curves of known interest1 taken as defined by geometrical properties4 ,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 character4 #h4 ,the ,straight ,line4--,this is the simplest type of locus4 ,also the simplest type of equation in x and y is ,ax ,by ,c #j1 one of the first degree4 ,here the coefficients ,a1 ,b1 ,c are constants4 ,they are1 like the current coordinates1 x1 y1 numerical4 ,but1 in giving interpretation to such an equation1 we must of course refer to numbers ,ax1 ,by1 ,c of unit magnitudes of the same kind1 of units of counting for instance1 or unit lengths or unit squares4 ,it will now be seen that every straight line has an equation of the first degree1 and that every equation of the first degree represents a straight line4 ,illustration3 ,,fig4 #ej4 ,it has been seen 7,s #g7 that lines parallel to the axes have equations of the first degree1 free from one of the variables4 ,take now a straight line ,,abc inclined to both axes4 ,let it make a given angle alpha with the positive direction of the axis of x1 i4e4 in fig4 #ej let this be the angle through which ,ax must be revolved counter-clockwise about ,a in order to be made coincident with the line4 ,let ,c1 of coordinates 7h1 k71 be a fixed point on the line1 and ,p7x1 y7 any other point upon it4 ,draw the ordinates ,,cd1 ,,pm of ,c and ,p1 and let the parallel to the axis of x through ,c meet ,,pm1 produced if necessary1 in ,r4 ,the right-angled triangle ,,crp tells us that1 with the signs appropriate to their directions attached to ,,cr and ,,rp1 ,,rp ,,cr tan alpha1 i4e4 ,,mp - ,,dc 7,,om - ,,od7 tan alpha1 and this gives that y - k tan alpha 7x - h71 an equation of the first degree satisfied by x and y4 ,no point not on the line satisfies the same equation2 for the line from ,c to any point off the line would make with ,,cr some angle beta different from alpha1 and the point in question would satisfy an equation y - k tan beta7x - h71 which is inconsistent with the above equation4 ,the equation of the line may also be written y mx b1 where m tan alpha1 and b k - h tan alpha4 ,here b is the value obtained for y from the equation when #j is put for x1 i4e4 it is the numerical measure1 with proper sign1 of ,,ob1 the intercept made by the line on the axis of y1 measured from the origin4 ,for different straight lines1 m and b may have any constant values we like4 ,now the general equation of the first degree ,ax ,by ,c #j may be written y -7,a_/,b7x - ,c_/,b1 unless ,b #j1 in which case it represents a line parallel to the axis of y2 and -,a_/,b1 -,c_/,b are values which can be given to m and b1 so that every equation of the first degree represents a straight line4 ,it is important to notice that the general equation1 which in appearance contains three constants ,a1 ,b1 ,c1 in effect depends on two only1 the ratios of two of them to the third4 ,in virtue of this last remark1 we see that two distinct conditions suffice to determine a straight line4 ,for instance1 it is easy from the above to see that x y -- -- #a a b is the equation of a straight line determined by the two conditions that it makes intercepts ,,oa1 ,,ob on the two axes1 of which a and b are the numerical measures with proper signs3 note that in fig4 #ej a is negative4 ,again1 y#b - y#a y - y#a ------- 7x - x#a71 x#b - x#a i4e4 7y#a - y#b7x - 7x#a - x#b7y x#ay#b - x#by#a #j1 represents the line determined by the data that it passes through two given points 7x#a1 y#a7 and 7x#b1 y#b74 ,to prove this find m in the equation y - y#a m7x - x#a7 of a line through 7x#a1 y#a71 from the condition that 7x#b1 y#b7 lies on the line4 ,in this paragraph the coordinates have been assumed rectangular4 ,had they been oblique1 the doctrine of similar triangles would have given the same results1 except that in the forms of equation y - k m7x - h71 y mx b1 we should not have had m tan alpha4 #i4 ,the ,circle4--,it is easy to write down the equation of a given circle4 ,let 7h1 k7 be its given centre ,c1 and rho the numerical measure of its given radius4 ,take ,p 7x1 y7 any point on its circumference1 and construct the triangle ,,crp1 in fig4 #ej as above4 ,the fact that this is right-angled tells us that ,,cr#b ,,rp#b ,,cp#b1 and this at once gives the equation 7x - h7#b 7y - k7#b rho#b4 ,a point not upon the circumference of the particular circle is at some distance from 7h1 k7 different from rho1 and satisfies an equation inconsistent with this one2 which accordingly represents the circumference1 or1 as we say1 the circle4 ,the equation is of the form x#b y#b #b;,ax #b;,by ,c #j4 ,conversely every equation of this form represents a circle3 we have only to take -,a1 -,b1 ,a#b ,b#b - ,c for h1 k1 rho#b respectively1 to obtain its centre and radius4 ,but this statement must appear too unrestricted4 ,ought we not to require ,a#b ,b#b - ,c to be positive8 ,certainly1 if by circle we are only to mean the visible round circumference of the geometrical definition4 ,yet1 analytically1 we contemplate altogether imaginary circles1 for which rho#b is negative1 and circles1 for which rho #j1 with all their reality condensed into their centres4 ,even when rho#b is positive1 so that a visible round circumference exists1 we do not regard this as constituting the whole of the circle4 ,giving to x any value whatever in 7x - h7#b 7y - k7#b rho#b1 we obtain two values of y1 real1 coincident or imaginary1 each of which goes with the abscissa x as the ordinate of a point1 real or imaginary1 on what is represented by the equation of the circle4 ,the doctrine of the imaginary on a circle1 and in geometry generally1 is of purely algebraical inception2 but it has been in its entirety accepted by modern pure geometers1 and signal success has attended the efforts of those who1 like ,k4,g4,c4 von ,staudt1 have striven to base its conclusions on principles not at all algebraical in form1 though of course cognate to those adopted in introducing the imaginary into algebra4 ,a circle with its centre at the origin has an equation x#b y#b rho#b4 ,in oblique coordinates the general equation of a circle is x#b #bxy cos omega y#b #b;,ax #b;,by ,c #j4 #aj4 ,the conic sections are the next simplest loci2 and it will be seen later that they are the loci represented by equations of the second degree4 ,circles are particular cases of conic sections2 and they have just been seen to have for their equations a particular class of equations of the second degree4 ,another particular class of such equations is that included in the form 7,ax ,by ,c77,a'x ,b'y ,c'7 #j1 which represents two straight lines1 because the product on the left vanishes if1 and only if1 one of the two factors does1 i4e4 if1 and only if1 7x1 y7 lies on one or other of two straight lines4 ,the condition that ax#b #b;hxy by#b #b;gx #b;fy c #j1 which is often written 7a1 b1 c1 f1 g1 h77x1 y1 ,i7#b #j1 takes this form is abc #b;fgh-af#b-;bg#b - ch#b #j4 ,note that the two lines may1 in particular cases1 be parallel or coincident4 ,any equation like ,f#a7x1 y7 ,f#b7x1 y7 444 ,fn7x1 y7 #j1 of which the left-hand side breaks up into factors1 represents all the loci separately represented by ,f#a7x1 y7 #j1 ,f#b7x1 y7 #j1 444 ,fn7x1 y7 #j4 ,in particular an equation of degree n which is free from x represents n straight lines parallel to the axis of x1 and one of degree n which is homogeneous in x and y1 i4e4 one which upon division by xn1 becomes an equation in the ratio y_/x1 represents n straight lines through the origin4 ,curves represented by equations of the third degree are called cubic curves4 ,the general equation of this degree will be written 79977x1 y1 ,i7#c #j4 #aa4 ,descriptive ,geometry4--,a geometrical proposition is either descriptive or metrical3 in the former case the statement of it is independent of the idea of magnitude 7length1 inclination1 c471 and in the latter it has reference to this idea4 ,the method of coordinates seems to be by its inception essentially metrical4 ,yet in dealing by this method with descriptive propositions we are eminently free from metrical considerations1 because of our power to use general equations1 and to avoid all assumption that measurements implied are any particular measurements4 ,illustration3 ,,fig4 #ea4 #ab4 ,it is worth while to illustrate this by the instance of the well-known theorem of the radical centre of three circles4 ,the theorem is that1 given any three circles ,a1 ,b1 ,c 7fig4 #ea71 the common chords alphaalpha'1 betabeta'1 gammagamma' of the three pairs of circles meet in a point4 ,the geometrical proof is metrical throughout3-- ,take ,o the point of intersection of alphaalpha'1 betabeta'1 and joining this with gamma'1 suppose that gamma',o does not pass through gamma1 but that it meets the circles ,a1 ,b in two distinct points gamma#b1 gamma#a respectively4 ,we have then the known metrical property of intersecting chords of a circle2 viz4 in circle ,c1 where alphaalpha'1 betabeta'1 are chords meeting at a point ,o1 ,oalpha4,oalpha' ,obeta4,obeta'1 where1 as well as in what immediately follows1 ,oalpha1 c4 denote1 of course1 lengths or distances4 ,similarly in circle ,a1 ,obeta4,obeta' ,ogamma#b4,ogamma'1 and in circle ,b1 ,oalpha4,oalpha' ,ogamma#a4,ogamma'4 ,consequently ,ogamma#a4,ogamma' ,ogamma#b4,ogamma'1 that is1 ,ogamma#a ,ogamma#b1 or the points gamma#a and gamma#b coincide2 that is1 they each coincide with gamma4 ,we contrast this with the analytical method3-- ,here it only requires to be known that an equation ,ax ,by ,c #j represents a line1 and an equation x#b y#b ,ax ,by ,c #j represents a circle4 ,a1 ,b1 ,c have1 in the two cases respectively1 metrical significations2 but these we are not concerned with4 ,using ,s to denote the function x#b y#b ,ax ,by ,c1 the equation of a circle is ,s o4 ,let the equation of any other circle be ,s'1 x#b y#b ,a'x ,b'y ,c' #j2 the equation ,s - ,s' #j is a linear equation 7,s - ,s' is in fact 7,a - ,a'7x 7,b - ,b'7y ,c - ,c71 and it thus represents a line2 this equation is satisfied by the coordinates of each of the points of intersection of the two circles 7for at each of these points ,s #j and ,s' #j1 therefore also ,s - ,s' #j72 hence the equation ,s - ,s' #j is that of the line joining the two points of intersection of the two circles1 or say it is the equation of the common chord of the two circles4 ,considering then a third circle ,s1 x#b y#b ,ax ,by ,c #j1 the equations of the common chords are ,s-,s' #j1 ,s - ,s #j1 ,s' - ,s #j 7each of these a linear equation72 at the intersection of the first and second of these lines ,s ,s' and ,s ,s1 therefore also ,s' ,s1 or the equation of the third line is satisfied by the coordinates of the point in question2 that is1 the three chords intersect in a point ,o1 the coordinates of which are determined by the equations ,s ,s' ,s4 ,it further appears that if the two circles ,s #j1 ,s' #j do not intersect in any real points1 they must be regarded as intersecting in two imaginary points1 such that the line joining them is the real line represented by the equation ,s - ,s' #j2 or that two circles1 whether their intersections be real or imaginary1 have always a real common chord 7or radical axis71 and that for any three circles the common chords intersect in a point 7of course real7 which is the radical centre4 ,and by this very theorem1 given two circles with imaginary intersections1 we can1 by drawing circles which meet each of them in real points1 construct the radical axis of the first-mentioned two circles4 #ac4 ,the principle employed in showing that the equation of the common chord of two circles is ,s - ,s' #j is one of very extensive application1 and some more illustrations of it may be given4 ,suppose ,s #j1 ,s' #j are lines 7that is1 let ,s1 ,s' now denote linear functions ,ax ,by ,c1 ,a'x ,b'y ,c'71 then ,s - k,s' #j 7k an arbitrary constant7 is the equation of any line passing through the point of intersection of the two given lines4 ,such a line may be made to pass through any given point1 say the point 7x#j1 y#j72 if ,s#j1 ,s'#j are what ,s1 ,s' respectively become on writing for 7x1 y7 the values 7x#j1 y#j71 then the value of k is k ,s#j 3 ,s'#j4 ,the equation in fact is ,,ss'#j - ,s#j,s' #j2 and starting from this equation we at once verify it a posteriori2 the equation is a linear equation satisfied by the values of 7x1 y7 which make ,s #j1 ,s' #j2 and satisfied also by the values 7x#j1 y#j72 and it is thus the equation of the line in question4 ,if1 as before1 ,s #j1 ,s' #j represent circles1 then 7k being arbitrary7 ,s - k,s' #j is the equation of any circle passing through the two points of intersection of the two circles2 and to make this pass through a given point 7x#j1 y#j7 we have again k ,s#j 3 ,s'#j4 ,in the particular case k #a1 the circle becomes the common chord 7more accurately it becomes the common chord together with the line infinity2 see ,s #bc below74 ,if ,s denote the general quadric function1 ,s ax#b #b;hxy by#b #b;fy #b;gx c1 then the equation ,s #j represents a conic2 assuming this1 then1 if ,s' #j represents another conic1 the equation ,s - k,s' #j represents any conic through the four points of intersection of the two conics4 ,illustration3 ,,fig4 #eb4 #ad4 ,the object still being to illustrate the mode of working with coordinates for descriptive purposes1 we consider the theorem of the polar of a point in regard to a circle4 ,given a circle and a point ,o 7fig4 #eb71 we draw through ,o any two lines meeting the circle in the points ,a1 ,a' and ,b1 ,b' respectively1 and then taking ,q as the intersection of the lines ,,ab' and ,a',b1 the theorem is that the locus of the point ,q is a right line depending only upon ,o and the circle1 but independent of the particular lines ,,oaa' and ,,obb'4 ,taking ,o as the origin1 and for the axes any two lines through ,o at right angles to each other1 the equation of the circle will be x#b y#b #b;,ax #b;,by ,c #j2 and if the equation of the line ,,oaa' is taken to be y mx1 then the points ,a1 ,a' are found as the intersections of the straight line with the circle2 or to determine x we have x#b7a m#b7 #bx7,a ,bm7 ,c #j4 ,if7x#a1 y#a7 are the coordinates of ,a1 and 7x#b1 y#b7 of ,a'1 then the roots of this equation are x#a1 x#b1 whence easily #a #a ,a ,bm -- -- -#b ------4 x#a x#b ,c ,and similarly1 if the equation of the line ,,obb' is taken to be y m'x#a and the coordinates of ,b1 ,b' to be 7x#c1 y#c7 and 7x#d1 y#d7 respectively1 then #a #a ,a ,bm' -- -- -#b -------4 x#c x#d ,c' ,we have then by ,s #h x7y#a - y#d7 - y7x#a - x#d7 x#ay#d - x#dy#a #j1 x7y#b - y#c7 - y7x#b - x#c7 x#by#c - x#cy#b #j1 as the equations of the lines ,,ab' and ,a',b respectively4 ,reducing by means of the relations y#a - mx#a #j1 y#b - mx#b #j1 y#c - m'x#c #j1 y#d - m'x#d #j1 the two equations become x7mx#a - m'x#d7 - y7x#a - x#d7 7m'- m7x#ax#d #j1 x7mx#b - m'x#c7 - y7x#b - x#c7 7m'- m7x#bx#c #j1 and if we divide the first of these equations by x#ax#d1 and the second by x#bx#c and then add1 we obtain _/ #a #a _/ #a #a #a #a _/ #a #a x m7 -- -- 7 - m'7 -- -- 7 - y -- -- - 7 -- -- 7 x#c x#d _/ x#a x#b_/ x#c x#d x#a x#b_/ #bm' - #bm #j1 or1 what is the same thing1 _/ #a #a _/ #a #a 7 -- -- 77y - m'x7 - 7 -- -- 77y - mx7 #bm' - #bm #j1 x#a x#b _/ x#c x#d _/ which by what precedes is the equation of a line through the point ,q4 ,substituting herein for #a_/x#a #a_/x#b1 #a_/x#c #a_/x#d their foregoing values1 the equation becomes -7,a ,bm77y - m'x7 7,a ,bm'77y - mx7 ,c7m' - m7 #j2 that is1 7m - m'77,ax ,by ,c7 #j2 or finally it is ,ax ,by ,c #j1 showing that the point ,q lies in a line the position of which is independent of the particular lines ,,oaa'1 ,,obb' used in the construction4 ,it is proper to notice that there is no correspondence to each other of the points ,a1 ,a' and ,b1 ,b'2 the grouping might as well have been ,a1 ,a' and ,b'1 ,b2 and it thence appears that the line ,ax ,by ,c #j just obtained is in fact the line joining the point ,q with the point ,r which is the intersection of ,,ab and ,a',b'4 #ae4 ,in ,s #h it has been seen that two conditions determine the equation of a straight line1 because in ,ax ,by ,c #j one of the coefficients may be divided out1 leaving only two parameters to be determined4 ,similarly five conditions instead of six determine an equation of the second degree 7a1 b1 c1 f1 g1 h77x1 y1 #a7b #j1 and nine instead of ten determine a cubic 79977x1 y1 #a7c #j4 ,it thus appears that a cubic can be made to pass through #i given points1 and that the cubic so passing through #i given points is completely determined4 ,there is1 however1 a remarkable exception4 ,considering two given cubic curves ,s #j1 ,s' #j1 these intersect in #i points1 and through these #i points we have the whole series of cubics ,s - k,s' #j1 where k is an arbitrary constant3 k may be determined so that the cubic shall pass through a given tenth point 7k ,s#j 3 ,s'#j1 if the coordinates are 7x#j1 y#j71 and ,s#j1 ,s'#j denote the corresponding values of ,s1 ,s'74 ,the resulting curve ,,ss'#j - ,s',s#j #j may be regarded as the cubic determined by the conditions of passing through #h of the #i points and through the given point 7x#j1 y#j72 and from the equation it thence appears that the curve passes through the remaining one of the #i points4 ,in other words1 we thus have the theorem1 any cubic curve which passes through #h of the #i intersections of two given cubic curves passes through the #ith intersection4 ,the applications of this theorem are very numerous2 for instance1 we derive from it ,pascal's theorem of the inscribed hexagon4 ,consider a hexagon inscribed in a conic4 ,the three alternate sides constitute a cubic1 and the other three alternate sides another cubic4 ,the cubics intersect in #i points1 being the #f vertices of the hexagon1 and the #c ,pascalian points1 or intersections of the pairs of opposite sides of the hexagon4 ,drawing a line through two of the ,pascalian points1 the conic and this line constitute a cubic passing through #h of the #i points of intersection1 and it therefore passes through the remaining point of intersection--that is1 the third ,pascalian point2 and since obviously this does not lie on the conic1 it must lie on the line--that is1 we have the theorem that the three ,pascalian points 7or points of intersection of the pairs of opposite sides7 lie on a line4 #af4 ,metrical ,theory resumed4 ,projections and ,perpendiculars4--,it is a metrical fact of fundamental importance1 already used in ,s #h1 that1 if a finite line ,,pq be projected on any other line ,,oo' by perpendiculars ,,pp'1 ,,qq' to ,,oo'1 the length of the projection ,p',q' is equal to that of ,,pq multiplied by the cosine of the acute angle between the two lines4 ,also the algebraical sum of the projections of the sides of any closed polygon upon any line is zero1 because as a point goes round the polygon1 from any vertex ,a to ,a again1 the point which is its projection on the line passes from ,a' the projection of ,a to ,a' again1 i4e4 traverses equal distances along the line in positive and negative senses4 ,if we consider the polygon as consisting of two broken lines1 each extending from the same initial to the same terminal point1 the sum of the projections of the lines which compose the one is equal1 in sign and magnitude1 to the sum of the projections of the lines composing the other4 ,observe that the projection on a line of a length perpendicular to the line is zero4 ,let us hence find the equation of a straight line such that the perpendicular ,,od on it from the origin is of length rho taken as positive1 and is inclined to the axis of x at an angle x,o,d alpha1 measured counter-clockwise from ,ox4 ,take any point ,p 7x1 y7 on the line1 and construct ,,om and ,,mp as in fig4 #dh4 ,the sum of the projections of ,,om and ,,mp on ,,od is ,,od itself2 and this gives the equation of the line x cos alpha y sin alpha rho4 ,observe that cos alpha and sin alpha here are the sin alpha and -cos alpha1 or the -sin alpha and cos alpha of ,s #h according to circumstances4 ,we can write down an expression for the perpendicular distance from this line of any point 7x'1 y'7 which does not lie upon it4 ,if the parallel through 7x'1 y'7 to the line meet ,,od in ,e1 we have x' cos alpha y' sin alpha ,,oe1 and the perpendicular distance required is ,,od - ,,oe1 i4e4 rho - x' cos alpha - y' sin alpha2 it is the perpendicular distance taken positively or negatively according as 7x'1 y'7 lies on the same side of the line as the origin or not4 ,the general equation ,ax ,by ,c #j may be given the form x cos alpha y sin alpha - rho #j by dividing it by root7,a#b ,b#c74 ,thus 7,ax' ,by' ,c7 _/ root7,a#b ,b#b7 is in absolute value the perpendicular distance of 7x'1 y'7 from the line ,ax ,by ,c #j4 ,remember1 however1 that there is an essential ambiguity of sign attached to a square root4 ,the expression found gives the distance taken positively when 7x'1 y'7 is on the origin side of the line1 if the sign of ,c is given to root7,a#b ,b#b74 #ag4 ,transformation of ,coordinates4--,we often need to adopt new axes of reference in place of old ones2 and the above principle of projections readily expresses the old coordinates of any point in terms of the new4 ,illustration3 ,,fig4 #ec4 ,suppose1 for instance1 that we want to take for new origin the point ,o' of old coordinates ,,oa h1 ,,ao' k1 and for new axes of ,x and ,y lines through ,o' obtained by rotating parallels to the old axes of x and y through an angle theta counter-clockwise4 ,construct 7fig4 #ec7 the old and new coordinates of any point ,p4 ,expressing that the projections1 first on the old axis of x and secondly on the old axis of y1 of ,,op are equal to the sums of the projections1 on those axes respectively1 of the parts of the broken line ,,oo',m',p1 we obtain3 x h ,x cos theta ,y cos 7theta #a_/bpi7 h ,x cos theta - ,y sin theta1 and y k ,x cos 7#a_/bpi - theta7 ,y cos theta k ,x sin theta ,y cos theta4 ,be careful to observe that these formulae do not apply to every conceivable change of reference from one set of rectangular axes to another4 ,it might have been required to take ,o',x1 ,o',y' for the positive directions of the new axes1 so that the change of directions of the axes could not be effected by rotation4 ,we must then write -,y for ,y in the above4 ,were the new axes oblique1 making angles alpha1 beta respectively with the old axis of x1 and so inclined at the angle beta - alpha1 the same method would give the formulae x h ,x cos alpha ,y cos beta1 y k ,x sin alpha ,y sin beta4 #ah4 ,the ,conic ,sections4--,the conics1 as they are now called1 were at first defined as curves of intersection of planes and a cone2 but ,apollonius substituted a definition free from reference to space of three dimensions4 ,this1 in effect1 is that a conic is the locus of a point the distance of which from a given point1 called the focus1 has a given ratio to its distance from a given line1 called the directrix 7see ,,conic ,,section74 ,if e 3 #a is the ratio1 e is called the eccentricity4 ,the distances are considered signless4 ,take 7h1 k7 for the focus1 and x cos alpha y sin alpha - p #j for the directrix4 ,the absolute values of root 7x - h7#b 7y - k7#b and p - x cos alpha -y sin alpha are to have the ratio e 3 #a2 and this gives 7x - h7#b 7y - k7#b e#b7p - x cos alpha - y sin alpha7#b as the general equation1 in rectangular coordinates1 of a conic4 ,it is of the second degree1 and is the general equation of that degree4 ,if1 in fact1 we multiply it by an unknown lambda1 we can1 by solving six simultaneous equations in the six unknowns lambda1 h1 k1 e1 p1 alpha1 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 given4 ,there is no failure of this statement in the special case when the given equation represents two straight lines1 as in ,s #aj1 but there is speciality3 if the two lines intersect1 the intersection and either bisector of the angle between them are a focus and directrix2 if they are united in one line1 any point on the line and a perpendicular to it through the point are3 if they are parallel1 the case is a limiting one in which e and h#b k#b have become infinite while e7-#b77;h#b k#b7 remains finite4 ,in the case 7,s #i7 of an equation such as represents a circle there is another instance of proceeding to a limit3 e has to become #j1 while ep remains finite3 moreover alpha is indeterminate4 ,the centre of a circle is its focus1 and its directrix has gone to infinity1 having no special direction4 ,this last fact illustrates the necessity1 which is also forced on plane geometry by three-dimensional considerations1 of treating all points at infinity in a plane as lying on a single straight line4 ,sometimes1 in reducing an equation to the above focus and directrix form1 we find for h1 k1 e1 p1 tan alpha1 or some of them1 only imaginary values1 as quadratic equations have to be solved2 and we have in fact to contemplate the existence of entirely imaginary conics4 ,for instance1 no real values of x and y satisfy x#b #by#b #c #j4 ,even when the locus represented is real1 we obtain1 as a rule1 four sets of values of h1 k1 e1 p1 of which two sets are imaginary2 a real conic has1 besides two real foci and corresponding directrices1 two others that are imaginary4 ,in oblique as well as rectangular coordinates equations of the second degree represent conics4 #ai4 ,the three ,species of ,conics4--,a real conic1 which does not degenerate into straight lines1 is called an ellipse1 parabola or hyperbola according as e 1 1 or #a4 ,to trace the three forms it is best so to choose the axes of reference as to simplify their equations4 ,in the case of a parabola1 let #b;c be the distance between the given focus and directrix1 and take axes referred to which these are the point 7c1 #j7 and the line x - c4 ,the equation becomes 7x - c7#b y#b 7x c7#b1 i4e4 y#b #d;cx4 ,in the other cases1 take a such that a7e e7-#a77 is the distance of focus from directrix1 and so choose axes that these are 7ae1 #j7 and x ae7-#a71 thus getting the equation7x - ae7#b y#b e#b7x - ae7-#a77b1 i4e4 7#a - e#b7x#b y#b a#b7a - e#b74 ,when e #a1 i4e4 in the case of an ellipse1 this may be written x#b_/;a#b y#b_/;b#b #a1 where b#b a#b7a - e#b72 and when e #a1 i4e4 in the case of an hyperbola1 x#b_/;a#b - y#b_/;b#b #a1 where b#b a#b7;e#b - #a74 ,the axes thus chosen for the ellipse and hyperbola are called the principal axes4 ,in figs4 #ed1 #ee1 #ef in order1 conics of the three species1 thus referred1 are depicted4 ,illustration3 ,,fig4 #ed ,illustration3 ,,fig4 #ee ,illustration3 ,,fig4 #ef4 ,the oblique straight lines in fig4 #ef are the asymptotes x_/a -y_/b of the hyperbola1 lines to which the curve tends with unlimited closeness as it goes to infinity4 ,the hyperbola would have an equation of the form xy c if referred to its asymptotes as axes1 the coordinates being then oblique1 unless a b1 in which case the hyperbola is called rectangular4 ,an ellipse has two imaginary asymptotes4 ,in particular a circle x#b y#b a#b1 a particular ellipse1 has for asymptotes the imaginary lines x -y root-#a4 ,these run from the centre to the so-called circular points at infinity4 #bj4 ,tangents and ,curvature4--,let 7x'1 y'7 and 7x' h1 y' k7 be two neighbouring points ,p1 ,p' on a curve4 ,the equation of the line on which both lie is h7y - y'7 k7x - x'74 ,now keep ,p fixed1 and let ,p' move towards coincidence with it along the curve4 ,the connecting line will tend towards a limiting position1 to which it can never attain as long as ,p and ,p' are distinct4 ,the line which occupies this limiting position is the tangent at ,p4 ,now if we subtract the equation of the curve1 with 7x'1 y'7 for the coordinates in it1 from the like equation in 7x' h1 y' k71 we obtain a relation in h and k1 which will1 as a rule1 be of the form #j ,ah ,bk terms of higher degrees in h and k1 where ,a1 ,b and the other coefficients involve x' and y'4 ,this gives k_/h -,a_/,b terms which tend to vanish as h and k do1 so that -,a 3 ,b is the limiting value tended to by k 3 h4 ,hence the equation of the tangent is ,b7y - y'7 ,a7x - x'7 #j4 ,the normal at 7x'1 y'7 is the line through it at right angles to the tangent1 and its equation is ,a7y - y'7 - ,b7x - x'7 #j4 ,in the case of the conic 7a1 b1 c1 f1 g1 h7 7x1 y1 #a7b #j we find that ,a_/,b 7ax' hy' g7_/7hx' by' f74 ,we can obtain the coordinates of ,q1 the intersection of the normals ,,qp1 ,,qp' at 7x'1 y'7 and 7x' h1 y' k71 and then1 using the limiting value of k 3 h1 deduce those of its limiting position as ,p' moves up to ,p4 ,this is the centre of curvature of the curve at ,p 7x'1 y'71 and is so called because it is the centre of the circle of closest contact with the curve at that point4 ,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'1 and that the arc from ,p to ,p' of this circle lies between the circles of centre ,q and radii ,,qp1 ,,qp'1 which circles tend1 not to different limits as ,p' moves up to ,p1 but to one4 ,the distance from ,p to the centre of curvature is the radius of curvature4 #ba4 ,differential ,plane ,geometry4--,the language and notation of the differential calculus are very useful in the study of tangents and curvature4 ,denoting by 7xi1 eta7 the current coordinates1 we find1 as above1 that the tangent at a point 7x1 y7 of a curve is eta - y 7xi - x7dy_/dx1 where dy_/dx is found from the equation of the curve4 ,if this be f7x1 y7 #j the tangent is 7xi - x7 7d,pf_/d,px7 7eta - y7 7d,pf_/d,py7 #j4 ,if rho and 7alpha1 beta7 are the radius and centre of curvature at 7x1 y71 we find that q7alpha - x7 -p7#a p#b71 q7beta - y7 #a p#b1 q#brho#b 7#a p#b7c1 where p1 q denote dy_/dx1 d#by_/dx#b respectively4 7,see ,,infinitesimal ,,calculus47 ,in any given case we can1 at all events in theory1 eliminate x1 y between the above equations for alpha - x and beta - y1 and the equation of the curve4 ,the resulting equation in 7alpha1 beta7 represents the locus of the centre of curvature4 ,this is the evolute of the curve4 #bb4 ,polar ,coordinates4--,in plane geometry the distance of any point ,p from a fixed origin 7or pole7 ,o1 and the inclination x,o,p of ,,op to a fixed line ,ox1 determine the point3 r1 the numerical measure of ,,op1 the radius vector1 and theta1 the circular measure of x,o,p1 the inclination1 are called polar coordinates of ,p4 ,the formulae x r cos theta1 y r sin theta connect ,cartesian and polar coordinates1 and make transition from either system to the other easy4 ,in polar coordinates the equations of a circle through ,o1 and of a conic with ,o as focus1 take the simple forms r #b;a cos 7theta-alpha71 r #a - e cos 7theta-alpha7 l4 ,the use of polar coordinates is very convenient in discussing curves which have properties of symmetry akin to that of a regular polygon1 such curves for instance as r a cos m theta1 with m integral1 and also the curves called spirals1 which have equations giving r as functions of theta itself1 and not merely of sin theta and cos theta4 ,in the geometry of motion under central forces the advantage of working with polar coordinates is great4 #bc4 ,trilinear and ,areal ,coordinates4--,consider a fixed triangle ,,abc1 and regard its sides as produced without limit4 ,denote1 as in trigonometry1 by a1 b1 c the positive numbers of units of a chosen scale contained in the lengths ,,bc1 ,,ca1 ,,ab1 by ,a1 ,b1 ,c the angles1 and by ,delta the area1 of the triangle4 ,we might1 as in ,s #f1 take ,,ca1 ,,cb as axes of x and y1 inclined at an angle ,c4 ,any point ,p 7x1 y7 in the plane is at perpendicular distances y sin ,c and x sin ,c from ,,ca and ,,cb4 ,call these beta and alpha respectively4 ,the signs of beta and alpha are those of y and x1 i4e4 beta is positive or negative according as ,p lies on the same side of ,,ca as ,b does or the opposite1 and similarly for alpha4 ,an equation in 7x1 y7 of any degree may1 upon replacing in it x and y by alpha cosec ,c and beta cosec ,c1 be written as one of the same degree in 7alpha1 beta74 ,now let gamma be the perpendicular distance of ,p from the third side ,,ab1 taken as positive or negative as ,p is on the ,c side of ,,ab or not4 ,the geometry of the figure tells us that aalpha bbeta cgamma #b;,delta4 ,by means of this relation in alpha1 beta1 gamma we can give an equation considered countless other forms1 involving two or all of alpha1 beta1 gamma4 ,in particular we may make it homogeneous in alpha1 beta1 gamma3 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 7aalpha bbeta cgamma7_/#b;,delta just sufficient to raise them1 in each case1 to the highest degree4 ,we call 7alpha1 beta1 gamma7 trilinear coordinates1 and an equation in them the trilinear equation of the locus represented4 ,trilinear equations are1 as a rule1 dealt with in their homogeneous forms4 ,an advantage thus gained is that we need not mean by 7alpha1 beta1 gamma7 the actual measures of the perpendicular distances1 but any properly signed numbers which have the same ratio two and two as these distances4 ,in place of alpha1 beta1 gamma it is lawful to use1 as coordinates specifying the position of a point in the plane of a triangle of reference ,,abc1 any given multiples of these4 ,for instance1 we may use x aalpha_/#b;,delta1 y bbeta_/#b;,delta1 z cgamma_/#b;,delta1 the properly signed ratios of the triangular areas ,,pbc1 ,,pca1 ,,pab to the triangular area ,,abc4 ,these are called the areal coordinates of ,p4 ,in areal coordinates the relation which enables us to make any equation homogeneous takes the simple form x y z #a2 and1 as before1 we need mean by x1 y1 z1 in a homogeneous equation1 only signed numbers in the right ratios4 ,straight lines and conics are represented in trilinear and in areal1 because in ,cartesian1 coordinates by equations of the first and second degrees respectively1 and these degrees are preserved when the equations are made homogeneous4 ,what must be said about points infinitely far off in order to make universal the statement1 to which there is no exception as long as finite distances alone are considered1 that every homogeneous equation of the first degree represents a straight line8 ,let the point of areal coordinates 7x'1 y'1 z'7 move infinitely far off1 and mean by x1 y1 z finite quantities in the ratios which x'1 y'1 z' tend to assume as they become infinite4 ,the relation x' y' z' #a gives that the limiting state of things tended to is expressed by x y z #j4 ,this particular equation of the first degree is satisfied by no point at a finite distance2 but we see the propriety of saying that it has to be taken as satisfied by all the points conceived of as actually at infinity4 ,accordingly the special property of these points is expressed by saying that they lie on a special straight line1 of which the areal equation is x y z #j4 ,in trilinear coordinates this line at infinity has for equation aalpha bbeta cgamma #j4 ,on the one special line at infinity parallel lines are treated as meeting4 ,there are on it two special 7imaginary7 points1 the circular points at infinity of ,s #ai1 through which all circles pass in the same sense4 ,in fact if ,s ,o be one circle1 in areal coordinates1 ,s 7x y z77lx my nz7 #j may1 by proper choice of l1 m1 n1 be made any other2 since the added terms are once lx my nz1 and have the generality of any expression like a'x b'y c' in ,cartesian coordinates4 ,now these two circles intersect in the two points where either meets x y z #j as well as in two points on the radical axis lx my nz #j4 #bd4 ,let us consider the perpendicular distance of a point 7alpha'1 beta'1 gamma'7 from a line lalpha mbeta ngamma4 ,we can take rectangular axes of ,cartesian coordinates 7for clearness as to equalities of angle it is best to choose an origin inside ,,abc71 and refer to them1 by putting expressions p - x costheta - y sintheta1 c41 for alpha c42 we can then apply ,s #af to get the perpendicular distance2 and finally revert to the trilinear notation4 ,the result is to find that the required distance is 7lalpha' mbeta' ngamma'7_/l1 m1 n1 where l1 m1 n#b l#b m#b n#b - #bmn cos ,a - #bnl cos ,b - #blm cos ,c4 ,in areal coordinates the perpendicular distance from 7x'1 y'1 z'7 to lx my nz #j is #b;,delta7lx' my' nz'7_/al1 bm1 cn4 ,in both cases the coordinates are of course actual values4 ,now let xi1 eta1 zeta be the perpendiculars on the line from the vertices ,a1 ,b1 ,c1 i4e4 the points 7#a1 #j1 #j71 7#j1 #a1 #j71 7#j1 #j1 #a71 with signs in accord with a convention that oppositeness of sign implies distinction between one side of the line and the other4 ,three applications of the result above give xi_/l #b;,delta_/al1 bm1 cn eta_/m zeta_/n2 and we thus have the important fact that xix' etay' zetaz' is the perpendicular distance between a point of areal coordinates 7x'y'z'7 and a line on which the perpendiculars from ,a1 ,b1 ,c are xi1 eta1 zeta respectively4 ,we have also that xix etay zetaz #j is the areal equation of the line on which the perpendiculars are xi1 eta1 zeta2 and1 by equating the two expressions for the perpendiculars from 7x'1 y'1 z'7 on the line1 that in all cases axi1 beta1 czeta#b #d;,delta#b4 #be4 ,line-coordinates4 ,duality4--,a quite different order of ideas may be followed in applying analysis to geometry4 ,the notion of a straight line specified may precede that of a point1 and points may be dealt with as the intersections of lines4 ,the specification of a line may be by means of coordinates1 and that of a point by an equation1 satisfied by the coordinates of lines which pass through it4 ,systems of line-coordinates will here be only briefly considered4 ,every such system is allied to some system of point-coordinates2 and space will be saved by giving prominence to this fact1 and not recommencing ab initio4 ,suppose that any particular system of point-coordinates1 in which lx my nz #j may represent any straight line1 is before us3 notice that not only are trilinear and areal coordinates such systems1 but ,cartesian coordinates also1 since we may write x_/z1 y_/z for the ,cartesian x1 y1 and multiply through by z4 ,the line is exactly assigned if l1 m1 n1 or their mutual ratios1 are known4 ,call 7l1 m1 n7 the coordinates of the line4 ,now keep x1 y1 z constant1 and let the coordinates of the line vary1 but always so as to satisfy the equation4 ,this equation1 which we now write xl ym zn #j1 is satisfied by the coordinates of every line through a certain fixed point1 and by those of no other line2 it is the equation of that point in the line-coordinates l1 m1 n4 ,line-coordinates are also called tangential coordinates4 ,a curve is the envelope of lines which touch it1 as well as the locus of points which lie on it4 ,a homogeneous equation of degree above the first in l1 m1 n is a relation connecting the coordinates of every line which touches some curve1 and represents that curve1 regarded as an envelope4 ,for instance1 the condition that the line of coordinates 7l1 m1 n71 i4e4 the line of which the allied point-coordinate equation is lx my nz #j1 may touch a conic 7a1 b1 c1 f1 g1 h7 7x1 y1 z7#b #j1 is readily found to be of the form 7,a1 ,b1 ,c1 ,f1 ,g1 ,h7 7l1 m1 n7#b #j1 i4e4 to be of the second degree in the line-coordinates4 ,it is not hard to show that the general equation of the second degree in l1 m1 n thus represents a conic2 but the degenerate conics of line-coordinates are not line-pairs1 as in point-coordinates1 but point-pairs4 ,the degree of the point-coordinate equation of a curve is the order of the curve1 the number of points in which it cuts a straight line4 ,that of the line-coordinate equation is its class1 the number of tangents to it from a point4 ,the order and class of a curve are generally different when either exceeds two4 #bf4 ,the system of line-coordinates allied to the areal system of point-coordinates has special interest4 ,the l1 m1 n of this system are the perpendiculars xi1 eta1 zeta of ,s #bd2 and x'xi y'eta z'zeta #j is the equation of the point of areal coordinates 7x'1 y'1 z'71 i4e4 is a relation which the perpendiculars from the vertices of the triangle of reference on every line through the point1 but no other line1 satisfy4 ,notice that a non-homogeneous equation of the first degree in xi1 eta1 zeta does not1 as a homogeneous one does1 represent a point1 but a circle4 ,in fact x'xi y'eta z'zeta ,r expresses the constancy of the perpendicular distance of the fixed point x'xi y'eta z'zeta #j from the variable line 7xi1 eta1 zeta71 i4e4 the fact that 7xi1 eta1 zeta7 touches a circle with the fixed point for centre4 ,the relation in any xi1 eta1 zeta which enables us to make an equation homogeneous is not linear1 as in point-coordinates1 but quadratic1 viz4 it is the relation axi1 beta1 czeta#b #d;,delta#b of ,s #bd4 ,accordingly the homogeneous equation of the above circle is #d;,delta#b7x'xi y'eta z'zeta7#b ,r#b;axi1 beta1 czeta#b4 ,every circle has an equation of this form in the present system of line-coordinates4 ,notice that the equation of any circle is satisfied by those coordinates of lines which satisfy both x'xi y'eta z'zeta #j1 the equation of its centre1 and axi1 beta1 czeta#b #j4 ,this last equation1 of which the left-hand side satisfies the condition for breaking up into two factors1 represents the two imaginary circular points at infinity1 through which all circles and their asymptotes pass4 ,there is strict duality in descriptive geometry between point-line-locus and line-point-envelope theorems4 ,but in metrical geometry duality is encumbered by the fact that there is in a plane one special line only1 associated with distance1 while of special points1 associated with direction1 there are two3 moreover the line is real1 and the points both imaginary4 ,,ii4 ,solid ,analytical ,geometry4 #bg4 ,any point in space may be specified by three coordinates4 ,we consider three fixed planes of reference1 and generally1 as in all that follows1 three which are at right angles two and two4 ,they intersect1 two and two1 in lines x',ox1 y',oy1 z',oz1 called the axes of x1 y1 z respectively1 and divide all space into eight parts called octants4 ,if from any point ,p in space we draw ,,pn parallel to z,oz' to meet the plane x,oy in ,n1 and then from ,n draw ,,nm parallel to y,oy' to meet x',ox in ,m1 the coordinates 7x1 y1 z7 of ,p are the numerical measures of ,,om1 ,,mn1 ,,np2 in the case of rectangular coordinates these are the perpendicular distances of ,p from the three planes of reference4 ,the sign of each coordinate is positive or negative as ,p lies on one side or the other of the corresponding plane4 ,in the octant delineated the signs are taken all positive4 ,illustration3 ,,fig4 #eg4 ,illustration3 ,,fig4 #eh4 ,in fig4 #eg the delineation is on a plane of the paper taken parallel to the plane z,ox1 the points of a solid figure being projected on that plane by parallels to some chosen line through ,o in the positive octant4 ,sometimes it is clearer to delineate1 as in fig4 #eh1 by projection parallel to that line in the octant which is equally inclined to ,ox1 ,oy1 ,oz upon a plane of the paper perpendicular to it4 ,it is possible by parallel projection to delineate equal scales along ,ox1 ,oy1 ,oz by scales having any ratios we like along lines in a plane having any mutual inclinations we like4 ,illustration3 ,,fig4 #ei4 ,for the delineation of a surface of simple form it frequently suffices to delineate the sections by the coordinate planes2 and1 in particular1 when the surface has symmetry about each coordinate plane1 to delineate the quarter-sections belonging to a single octant4 ,thus fig4 #ei conveniently represents an octant of the wave surface1 which cuts each coordinate plane in a circle and an ellipse4 ,or we may delineate a series of contour lines1 i4e4 sections by planes parallel to x,oy1 or some other chosen plane2 of course other sections may be indicated too for greater clearness4 ,for the delineation of a curve a good method is to represent1 as above1 a series of points ,p thereof1 each accompanied by its ordinate ,,pn1 which serves to refer it to the plane of xy4 ,the employment of stereographic projection is also interesting4 #bh4 ,in plane geometry1 reckoning the line as a curve of the first order1 we have only the point and the curve4 ,in solid geometry1 reckoning a line as a curve of the first order1 and the plane as a surface of the first order1 we have the point1 the curve and the surface2 but the increase of complexity is far greater than would hence at first sight appear4 ,in plane geometry a curve is considered in connexion with lines 7its tangents72 but in solid geometry the curve is considered in connexion with lines and planes 7its tangents and osculating planes71 and the surface also in connexion with lines and planes 7its tangent lines and tangent planes72 there are surfaces arising out of the line--cones1 skew surfaces1 developables1 doubly and triply infinite systems of lines1 and whole classes of theories which have nothing analogous to them in plane geometry3 it is thus a very small part indeed of the subject which can be even referred to in the present article4 ,in the case of a surface we have between the coordinates 7x1 y1 z7 a single1 or say a onefold relation1 which can be represented by a single relation f7x1 y1 z7 #j2 or we may consider the coordinates expressed each of them as a given function of two variable parameters p1 q2 the form z f7x1 y7 is a particular case of each of these modes of representation2 in other words1 we have in the first mode f7x1 y1 z7 z - f7x1 y71 and in the second mode x p1 y q for the expression of two of the coordinates in terms of the parameters4 ,in the case of a curve we have between the coordinates 7x1 y1 z7 a twofold relation3 two equations f7x1 y1 z7 #j1 phi7x1 y1 z7 #j give such a relation2 i4e4 the curve is here considered as the intersection of two surfaces 7but the curve is not always the complete intersection of two surfaces1 and there are hence difficulties72 or1 again1 the coordinates may be given each of them as a function of a single variable parameter4 ,the form y phi7x71 z psi7x71 where two of the coordinates are given in terms of the third1 is a particular case of each of these modes of representation4 #bi4 ,the remarks under plane geometry as to descriptive and metrical propositions1 and as to the non-metrical character of the method of coordinates when used for the proof of a descriptive proposition1 apply also to solid geometry2 and they might be illustrated in like manner by the instance of the theorem of the radical centre of four spheres4 ,the proof is obtained from the consideration that ,s and ,s' being each of them a function of the form x#b y#b z#b ax by cz d1 the difference ,s-,s' is a mere linear function of the coordinates1 and consequently that ,s-,s' #j is the equation of the plane containing the circle of intersection of the two spheres ,s #j and ,s' #j4 ,illustration3 ,,fig4 #fj4 #cj4 ,metrical ,theory4--,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'1 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 7in the form in which it is now convenient to state the theorem7 the perpendicular distance ,p',q' of two parallel planes is equal to the inclined distance ,,pq into the cosine of the inclination4 ,the principle of ,s #af1 that the algebraical sum of the projections of the sides of any closed polygon on any line is zero1 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 line1 in sign and magnitude1 as we pass from ,a to ,b1 is applicable when the sides do not all lie in one plane4 #ca4 ,consider the skew quadrilateral ,,qmnp1 the sides ,,qm1 ,,mn1 ,,np being respectively parallel to the three rectangular axes ,ox1 ,oy1 ,oz2 let the lengths of these sides be xi1 eta1 zeta1 and that of the side ,,qp be rho2 and let the cosines of the inclinations 7or say the cosine-inclinations7 of rho to the three axes be alpha1 beta1 gamma2 then projecting successively on the three sides and on ,,qp we have xi1 eta1 zeta rhoalpha1 rhobeta1 rhogamma1 and rho alphaxi betaeta gammazeta1 whence rho#b xi#b eta#b zeta#b1 which is the relation between a distance rho and its projections xi1 eta1 zeta upon three rectangular axes4 ,and from the same equations we obtain alpha#b beta#b gamma#b #a1 which is a relation connecting the cosine-inclinations of a line to three rectangular axes4 ,suppose we have through ,q any other line ,,qt1 and let the cosine-inclinations of this to the axes be alpha'1 beta'1 gamma'1 and delta be its cosine-inclination to ,,qp2 also let rho be the length of the projection of ,,qp upon ,,qt2 then projecting on ,,qt we have rho alpha'xi beta'eta gamma'zeta rhodelta4 ,and in the last equation substituting for xi1 eta1 zeta their values rhoalpha1 rhobeta1 rhogamma we find delta alphaalpha' betabeta' gammagamma'1 which is an expression for the mutual cosine-inclination of two lines1 the cosine-inclinations of which to the axes are alpha1 beta1 gamma and alpha'1 beta'1 gamma' respectively4 ,we have of course alpha#b beta#b gamma#b #a and alpha'#b beta'#b gamma'#b #a2 and hence also #a - delta#b 7alpha#b beta#b gamma#b77;alpha'#b beta'#b gamma'#b7 - 7alphaalpha' betabeta' gammagamma'7#b1 7betagamma' - beta'gamma7#b 7gammaalpha' - gamma'alpha7#b 7alphabeta' - alpha'beta7#b2 so that the sine of the inclination can only be expressed as a square root4 ,these formulae are the foundation of spherical trigonometry4 #cb4 ,straight ,lines1 ,planes and ,spheres4--,the foregoing formulae give at once the equations of these loci4 ,for first1 taking ,q to be a fixed point1 coordinates 7a1 b1 c71 and the cosine-inclinations 7alpha1 beta1 gamma7 to be constant1 then ,p will be a point in the line through ,q in the direction thus determined2 or1 taking 7x1 y1 z7 for its coordinates1 these will be the current coordinates of a point in the line4 ,the values of xi1 eta1 zeta then are x - a1 y - b1 z - c1 and we thus have x - a y - b z - c ------- ----- ------- 7 rho71 alpha beta gamma which 7omitting the last equation1 rho7 are the equations of the line through the point 7a1 b1 c71 the cosine-inclinations to the axes being alpha1 beta1 gamma1 and these quantities being connected by the relation alpha#b beta#b gamma#b #a4 ,this equation may be omitted1 and then alpha1 beta1 gamma1 instead of being equal1 will only be proportional1 to the cosine-inclinations4 ,using the last equation1 and writing x1 y1 z a alpharho1 b betarho1 c gammarho1 these are expressions for the current coordinates in terms of a parameter rho1 which is in fact the distance from the fixed point 7a1 b1 c74 ,it is easy to see that1 if the coordinates 7x1 y1 z7 are connected by any two linear equations1 these equations can always be brought into the foregoing form1 and hence that the two linear equations represent a line4 ,secondly1 taking for greater simplicity the point ,q to be coincident with the origin1 and alpha'1 beta'1 gamma'1 p to be constant1 then p is the perpendicular distance of a plane from the origin1 and alpha'1 beta'1 gamma' are the cosine-inclinations of this distance to the axes 7alpha'#b beta'#b gamma'#b #a74 ,p is any point in this plane1 and taking its coordinates to be 7x1 y1 z7 then 7xi1 eta1 zeta7 are 7x1 y1 z71 and the foregoing equation p alpha'xi beta'eta gamma'zeta becomes alpha'x beta'y gamma'z p1 which is the equation of the plane in question4 ,if1 more generally1 ,q is not coincident with the origin1 then1 taking its coordinates to be 7a1 b1 c71 and writing p#a instead of p1 the equation is alpha'7x - a7 beta'7y - b7 gamma'7z - c7 p#a2 and we thence have p#a p - 7aalpha' bbeta' cgamma'71 which is an expression for the perpendicular distance of the point 7a1 b1 c7 from the plane in question4 ,it is obvious that any linear equation ,ax ,by ,cz ,d ,o between the coordinates can always be brought into the foregoing form1 and hence that such an equation represents a plane4 ,thirdly1 supposing ,q to be a fixed point1 coordinates 7a1 b1 c71 and the distance ,,qp rho1 to be constant1 say this is d1 then1 as before1 the values of xi1 eta1 zeta are x - a1 y - b1 z - c1 and the equation xi#b eta#b zeta#b rho#b becomes 7x - a7#b 7y - b7#b 7z - c7#b d#b1 which is the equation of the sphere1 coordinates of the centre 7a1 b1 c71 and radius d4 ,a quadric equation wherein the terms of the second order are x#b y#b z#b1 viz4 an equation x#b y#b z#b ,ax ,by ,cz ,d #j1 can always1 it is clear1 be brought into the foregoing form2 and it thus appears that this is the equation of a sphere1 coordinates of the centre -#a_/b;,a1 -#a_/b;,b1 -#a_/b;,c1 and squared radius #a_/d7;,a#b ,b#b ,c#b7 - ,d4 #cc4 ,cylinders1 ,cones1 ruled ,surfaces4--,if the two equations of a straight line involve a parameter to which any value may be given1 we have a singly infinite system of lines4 ,they cover a surface1 and the equation of the surface is obtained by eliminating the parameter between the two equations4 ,if the lines all pass through a given point1 then the surface is a cone2 and1 in particular1 if the lines are all parallel to a given line1 then the surface is a cylinder4 ,beginning with this last case1 suppose the lines are parallel to the line x mz1 y nz1 the equations of a line of the system are x mz a1 y nz b1--where a1 b are supposed to be functions of the variable parameter1 or1 what is the same thing1 there is between them a relation f7a1 b7 #j3 we have a x - mz1 b y - nz1 and the result of the elimination of the parameter therefore is f7x - mz1 y - nz7 #j1 which is thus the general equation of the cylinder the generating lines whereof are parallel to the line x mz1 y nz4 ,the equation of the section by the plane z #j is f7x1 y7 #j1 and conversely if the cylinder be determined by means of its curve of intersection with the plane z #j1 then1 taking the equation of this curve to be f7x1 y7 #j1 the equation of the cylinder is f7x - mz1 y - nz7 #j4 ,thus1 if the curve of intersection be the circle 7x - alpha7#b 7y - beta7#b gamma#b1 we have 7x - mz - alpha7#b 7y - nz - beta7#b gamma#b as the equation of an oblique cylinder on this base1 and thus also 7x - alpha7#b 7y - beta7#b gamma#b as the equation of the right cylinder4 ,if the lines all pass through a given point 7a1 b1 c71 then the equations of a line are x - a alpha7z - c71 y - b beta7z - c71 where alpha1 beta are functions of the variable parameter1 or1 what is the same thing1 there exists between them an equation f7alpha1 beta7 #j2 the elimination of the parameter gives1 therefore1 f7x - a7_/7x - c'71 7y - b7_/7z - c7 #j2 and this equation1 or1 what is the same thing1 any homogeneous equation f7x - a1 y - b1 z - c7 #j1 or1 taking f to be a rational and integral function of the order n1 say 79977x - a1 y - b1 z - c7n #j1 is the general equation of the cone having the point 7a1 b1 c7 for its vertex4 ,taking the vertex to be at the origin1 the equation is 79977x1 y1 z7n #j2 and1 in particular1 79977x1 y1 z7#b #j is the equation of a cone of the second order1 or quadricone1 having the origin for its vertex4 #cd4 ,in the general case of a singly infinite system of lines1 the locus is a ruled surface 7or regulus74 ,now1 when a line is changing its position in space1 it may be looked upon as in a state of turning about some point in itself1 while that point is1 as a rule1 in a state of moving out of the plane in which the turning takes place4 ,if instantaneously it is only in a state of turning1 it is usual1 though not strictly accurate1 to say that it intersects its consecutive position4 ,a regulus such that consecutive lines on it do not intersect1 in this sense1 is called a skew surface1 or scroll2 one on which they do is called a developable surface or torse4 ,suppose1 for instance1 that the equations of a line 7depending on the variable parameter theta7 are x_/a y_/c theta 7#a y_/b71 x_/a - z_/c #a_/theta 7#a - y_/b72 then1 eliminating theta we have x#b_/;a#b - z#b_/;c#b #a - y#b_/;b#b1 or say1 x#b_/;a#b z#b_/;b#b - z#b_/;c#b #a1 the equation of a quadric surface1 afterwards called the hyperboloid of one sheet2 this surface is consequently a scroll4 ,it is to be remarked that we have upon the surface a second singly infinite series of lines2 the equations of a line of this second system 7depending on the variable parameter phi7 are x z _/ y x z #a _/ y -- -- phi7 #a - -- 71 -- - -- ----- 7 #a -- 74 a c b _/ a c phi b _/ ,it is easily shown that any line of the one system intersects every line of the other system4 ,considering any curve 7of double curvature7 whatever1 the tangent lines of the curve form a singly infinite system of lines1 each line intersecting the consecutive line of the system1--that is1 they form a developable1 or torse2 the curve and torse are thus inseparably connected together1 forming a single geometrical figure4 ,an osculating plane of the curve 7see ,s #ch below7 is a tangent plane of the torse all along a generating line4 #ce4 ,transformation of ,coordinates4--,there is no difficulty in changing the origin1 and it is for brevity assumed that the origin remains unaltered4 ,we have1 then1 two sets of rectangular axes1 ,ox1 ,oy1 ,oz1 and ,ox#a1 ,oy#a1 ,ozx#a1 the mutual cosine-inclinations being shown by the diagram-- x y z ------------------------------ x#a alpha beta gamma ------------------------------ y#a alpha beta' gamma' ------------------------------ z#a alpha beta gamma ------------------------------ that is1 alpha1 beta1 gamma are the cosine-inclinations of ,ox#a to ,ox1 ,oy1 ,oz2 alpha'1 beta'1 gamma' those of ,oy#a1 c4 ,and this diagram gives also the linear expressions of the coordinates 7x#a1 y#a1 z#a7 or 7x1 y1 z7 of either set in terms of those of the other set2 we thus have x#a alpha x beta y gamma z1 x alphax#a alpha'y#a alphaz#a1 y#a alpha'x beta'y gamma'z1 y betax#a beta'y#a betaz#a1 z#a alphax betay gammaz1 z gammax#a gamma'y#a gammaz#a1 which are obtained by projection1 as above explained4 ,each of these equations is1 in fact1 nothing else than the before-mentioned equation p alpha'xi beta'eta gamma'zeta1 adapted to the problem in hand4 ,but we have to consider the relations between the nine coefficients4 ,by what precedes1 or by the consideration that we must have identically x#b y#b z#b x#ab y#ab z#ab1 it appears that these satisfy the relations-- a#b beta#b gamma#b #a1 alpha#b alpha'#b alpha#b #a1 alpha'#b beta'#b gamma'#b #a1 beta#b beta'#b beta#b #a1 alpha#b beta#b gamma#b #a1 gamma#b gamma'#b gamma#b #a1 a'a beta'beta gamma'gamma #j1 betagamma beta'gamma' betagamma #j1 alphaalpha betabeta gammagamma #j1 gammaalpha gamma'alpha' gammaalpha #j1 alphaalpha' betabeta' gammagamma' #j1 alphabeta alpha'beta' alphabeta #j1 either set of six equations being implied in the other set4 ,it follows that the square of the determinant alpha1 beta1 gamma alpha'1 beta'1 gamma' alpha1 beta1 gamma is #a2 and hence that the determinant itself is - #a4 ,the distinction of the two cases is an important one3 if the determinant is #a1 then the axes ,ox#a1 ,oy#a1 ,oz#a are such that they can by a rotation about ,o be brought to coincide with ,ox1 ,oy1 ,oz respectively2 if it is -#a1 then they cannot4 ,but in the latter case1 by measuring x#a1 y#a1 z#a in the opposite directions we change the signs of all the coefficients and so make the determinant to be #a2 hence the former case need alone be considered1 and it is accordingly assumed that the determinant is #a4 ,this being so1 it is found that we have the equality alpha beta'gamma - betagamma'1 and eight like ones1 obtained from this by cyclical interchanges of the letters alpha1 beta1 gamma1 and of unaccented1 singly and doubly accented letters4 #cf4 ,the nine cosine-inclinations above are1 as has been seen1 connected by six equations4 ,it ought then to be possible to express them all in terms of three parameters4 ,an elegant means of doing this has been given by ,rodrigues1 who has shown that the tabular expression of the formulae of transformation may be written x y z ---------------------------------------- ---------------------------------------- -------------------- x#a #a lambda#b - mu#b - nu#b #b7lambdamu - nu7 #b7nulambda mu7 ---------------------------------------- ---------------------------------------- ------------------- y#a #b7lambdamu nu7 #a - lambda#b mu#b - nu#b #b7munu lambda7 ---------------------------------------- ---------------------------------------- ------------------- z#a #b7nulambda - mu7 #b7munu lambda7 #a - lambda#b - mu#b nu#b ---------------------------------------- ---------------------------------------- -------------------- _/7#a lambda#b mu#b nu#b71 the meaning being that the coefficients in the transformation are fractions1 with numerators expressed as in the table1 and the common denominator4 #cg4 ,the ,species of ,quadric ,surfaces4--,surfaces represented by equations of the second degree are called quadric surfaces4 ,quadric surfaces are either proper or special4 ,the special ones arise when the coefficients in the general equation are limited to satisfy certain special equations2 they comprise 7#a7 plane-pairs1 including in particular one plane twice repeated1 and 7#b7 cones1 including in particular cylinders2 there is but one form of cone1 but cylinders may be elliptic1 parabolic or hyperbolic4 ,a discussion of the general equation of the second degree shows that the proper quadric surfaces are of five kinds1 represented respectively1 when referred to the most convenient axes of reference1 by equations of the five types 7a and b positive73 x#b y#b 7#a7 z --- ---1 elliptic paraboloid4 #b;a #b;b x#b y#b 7#b7 z --- - ---1 hyperbolic paraboloid4 #b;a #b;b x#b y#b z#b 7#c7 --- --- --- #a1 ellipsoid4 a#b b#b c#b x#b y#b z#b 7#d7 --- --- - --- #a1 hyperboloid of one sheet4 a#b b#b c#b x#b y#b z#b 7#e7 --- --- - --- -#a1 hyperboloid of two sheets4 a#b b#b c#b ,it is at once seen that these are distinct surfaces2 and the equations also show very readily the general form and mode of generation of the several surfaces4 ,illustration3 ,,fig4 #fa4 ,in the elliptic paraboloid 7fig4 #fa7 the sections by the planes of zx and zy are the parabolas x#b y#b z ---1 z --- #b;a #b;b having the common axes ,oz2 and the section by any plane z gamma parallel to that of xy is the ellipse x#b y#b gamma --- ---2 #b;a #b;b so that the surface is generated by a variable ellipse moving parallel to itself along the parabolas as directrices4 ,illustration3 ,,fig4 #fb4 ,illustration3 ,,fig4 #fc4 ,in the hyperbolic paraboloid 7figs4 #fb and #fc7 the sections by the planes of zx1 zy are the parabolas z x#b_/b;a1 z - y#b_/b;b1 having the opposite axes ,oz1 ,oz'1 and the section by a plane z gamma parallel to that of xy is the hyperbola gamma x#b_/b;a - y#b_/b;b1 which has its transverse axis parallel to ,ox or ,oy according as gamma is positive or negative4 ,the surface is thus generated by a variable hyperbola moving parallel to itself along the parabolas as directrices4 ,the form is best seen from fig4 #fc1 which represents the sections by planes parallel to the plane of xy1 or say the contour lines2 the continuous lines are the sections above the plane of xy1 and the dotted lines the sections below this plane4 ,the form is1 in fact1 that of a saddle4 ,illustration3 ,,fig4 #fd4 ,in the ellipsoid 7fig4 #fd7 the sections by the planes of zx1 zy1 and xy are each of them an ellipse1 and the section by any parallel plane is also an ellipse4 ,the surface may be considered as generated by an ellipse moving parallel to itself along two ellipses as directrices4 ,in the hyperboloid of one sheet 7fig4 #fe71 the sections by the planes of zx1 zy are the hyperbolas x#b z#b y#b z#b --- - --- #a1 --- - --- #a1 c#b c#b b#b c#b having a common conjugate axis z,oz'2 the section by the plane of x1 y1 and that by any parallel plane1 is an ellipse2 and the surface may be considered as generated by a variable ellipse moving parallel to itself along the two hyperbolas as directrices4 ,if we imagine two equal and parallel circular disks1 their points connected by strings of equal lengths1 so that these are the generators of a right circular cylinder1 and if we turn one of the disks about its centre through an angle in its plane1 the strings in their new positions will be one system of generators of a hyperboloid of one sheet1 for which a b2 and if we turn it through the same angle in the opposite direction1 we get in like manner the generators of the other system2 there will be the same general configuration when a b4 ,the hyperbolic paraboloid is also covered by two systems of rectilinear generators as a method like that used in ,s #cd establishes without difficulty4 ,the figures should be studied to see how they can lie4 ,illustration3 ,,fig4 #fe4 ,illustration3 ,,fig4 #ff4 ,in the hyperboloid of two sheets 7fig4 #ff7 the sections by the planes of zx and zy are the hyperbolas z#b x#b z#b y#b --- - --- #a1 --- - --- #a1 c#b a#b c#b b#b having a common transverse axis along z',oz2 the section by any plane z -gamma parallel to that of xy is the ellipse x#b y#b gamma#b --- --- --------- - #a1 a#b b#b c#b provided gamma#b c#b1 and the surface1 consisting of two distinct portions or sheets1 may be considered as generated by a variable ellipse moving parallel to itself along the hyperbolas as directrices4 #ch4 ,differential ,geometry of ,curves4--,for convenience consider the coordinates 7x1 y1 z7 of a point on a curve in space to be given as functions of a variable parameter theta1 which may in particular be one of themselves4 ,use the notation x'1 x for dx_/dtheta1 d#bx_/dtheta#b1 and similarly as to y and z4 ,only a few formulae will be given4 ,call the current coordinates 7xi1 eta1 zeta74 ,the tangent at 7x1 y1 z7 is the line tended to as a limit by the connector of 7x1 y1 z7 and a neighbouring point of the curve when the latter moves up to the former3 its equations are 7xi - x7_/x' 7eta - y7_/y' 7zeta - z7_/z'4 ,the osculating plane at 7x1 y1 z7 is the plane tended to as a limit by that through 7x1 y1 z7 and two neighbouring points of the curve as these1 remaining distinct1 both move up to 7x1 y1 z73 its one equation is 7xi - x77y'z - yz'7 7eta - y77z'x - zx'7 7zeta - z7 7x'y - xy'7 #j4 ,the normal plane is the plane through 7x1 y1 z7 at right angles to the tangent line1 i4e4 the plane x'7xi - x7 y'7eta - y7 z'7zeta - z7 #j4 ,it cuts the osculating plane in a line called the principal normal4 ,every line through 7x1 y1 z7 in the normal plane is a normal4 ,the normal perpendicular to the osculating plane is called the binormal4 ,a tangent1 principal normal1 and binormal are a convenient set of rectangular axes to use as those of reference1 when the nature of a curve near a point on it is to be discussed4 ,through 7x1 y1 z7 and three neighbouring points1 all on the curve1 passes a single sphere2 and as the three points all move up to 7x1 y1 z7 continuing distinct1 the sphere tends to a limiting size and position4 ,the limit tended to is the sphere of closest contact with the curve at 7x1 y1 z72 its centre and radius are called the centre and radius of spherical curvature4 ,it cuts the osculating plane in a circle1 called the circle of absolute curvature2 and the centre and radius of this circle are the centre and radius of absolute curvature4 ,the centre of absolute curvature is the limiting position of the point where the principal normal at 7x1 y1 z7 is cut by the normal plane at a neighbouring point1 as that point moves up to 7x1 y1 z74 #ci4 ,differential ,geometry of ,surfaces4--,let 7x1 y1 z7 be any chosen point on a surface f7x1 y1 z7 #j4 ,as a second point of the surface moves up to 7x1 y1 z71 its connector with 7x1 y1 z7 tends to a limiting position1 a tangent line to the surface at 7x1 y1 z74 ,all these tangent lines at 7x1 y1 z71 obtained by approaching 7x1 y1 z7 from different directions on a surface1 lie in one plane d,pf d,pf d,pf ----- 7xi - x7 ----- 7eta - y7 ----- 7zeta - z7 #j4 d,px d,py d,pz ,this plane is called the tangent plane at 7x1 y1 z74 ,one line through 7x1 y1 z7 is at right angles to the tangent plane4 ,this is the normal _/d,pf _/d,pf _/d,pf 7xi - x7 _/ ----- 7eta - y7 _/ ----- 7zeta - z7 _/ -----4 _/ d,px _/ d,py _/ d,pz ,the tangent plane is cut by the surface in a curve1 real or imaginary1 with a node or double point at 7x1 y1 z74 ,two of the tangent lines touch this curve at the node4 ,they are called the chief tangents 7,haupt-tangenten7 at 7x1 y1 z72 they have closer contact with the surface than any other tangents4 ,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 node1 it must therefore consist of two straight lines4 ,consequently a quadric surface is covered by two sets of straight lines1 a pair through every point on it2 these are imaginary for the ellipsoid1 hyperboloid of two sheets1 and elliptic paraboloid4 ,a surface of any order is covered by two singly infinite systems of curves1 a pair through every point1 the tangents to which are all chief tangents at their respective points of contact4 ,these are called chief-tangent curves2 on a quadric surface they are the above straight lines4 #dj4 ,the tangents at a point of a surface which bisect the angles between the chief tangents are called the principal tangents at the point4 ,they are at right angles1 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 discussion4 ,at a special point which is such that the chief tangents there run to the circular points at infinity in the tangent plane1 the principal tangents are indeterminate2 such a special point is called an umbilic of the surface4 ,there are two singly infinite systems of curves on a surface1 a pair cutting one another at right angles through every point upon it1 all tangents to which are principal tangents of the surface at their respective points of contact4 ,these are called lines of curvature1 because of a property next to be mentioned4 ,as a point ,q moves in an arbitrary direction on a surface from coincidence with a chosen point ,p1 the normal at it1 as a rule1 at once fails to meet the normal at ,p2 but1 if it takes the direction of a line of curvature through ,p1 this is instantaneously not the case4 ,we have thus on the normal two centres of curvature1 and the distances of these from the point on the surface are the two principal radii of curvature of the surface at that point2 these are also the radii of curvature of the sections of the surface by planes through the normal and the two principal tangents respectively2 or say they are the radii of curvature of the normal sections through the two principal tangents respectively4 ,take at the point the axis of z in the direction of the normal1 and those of x and y in the directions of the principal tangents respectively1 then1 if the radii of curvature be a1 b 7the signs being such that the coordinates of the two centres of curvature are z a and z b respectively71 the surface has in the neighbourhood of the point the form of the paraboloid x#b y#b z --- ---1 #b;a #b;b and the chief-tangents are determined by the equation #j x#b_/b;a y#b_/b;b4 ,the two centres of curvature may be on the same side of the point or on opposite sides2 in the former case a and b have the same sign1 the paraboloid is elliptic1 and the chief-tangents are imaginary2 in the latter case a and b have opposite signs1 the paraboloid is hyperbolic1 and the chief-tangents are real4 ,the normal sections of the surface and the paraboloid by the same plane have the same radius of curvature2 and it thence readily follows that the radius of curvature of a normal section of the surface by a plane inclined at an angle theta to that of zx is given by the equation #a cos#b theta sin#b theta ----- ------------- -------------4 rho a b ,the section in question is that by a plane through the normal and a line in the tangent plane inclined at an angle theta to the principal tangent along the axis of x4 ,to complete the theory1 consider the section by a plane having the same trace upon the tangent plane1 but inclined to the normal at an angle phi2 then it is shown without difficulty 7,meunier's theorem7 that the radius of curvature of this inclined section of the surface is rho cos phi4 ,,authorities4--,the above article is largely based on that by ,arthur ,cayley in the #ith edition of this work4 ,of early and important recent publications on analytical geometry1 special mention is to be made of ,r4 ,descartes1 ,geometrie 7,leyden1 #afcg72 ,john ,wallis1 ,tractatus de sectionibus conicis nova methodo expositis 7#afee1 ,opera mathematica1 i41 ,oxford1 #afie72 de l',hospital1 ,traite analytique des sections coniques 7,paris1 #agbj72 ,leonhard ,euler1 ,introductio in analysin infinitorum1 ii4 7,lausanne1 #agdh72 ,gaspard ,monge1 ,application d'algebre a la geometrie 7,journ4 ,ecole ,polytech41 #ahja72 ,julius ,plucker1 ,analytisch-geometrische ,entwickelungen1 #c ,bde4 7,essen1 #ahbh-ahca72 ,system der analytischen ,geometrie 7,berlin1 #ahce72 ,g4 ,salmon1 ,a ,treatise on ,conic ,sections 7,dublin1 #ahdh2 #fth ed41 ,london1 #ahgi72 ,ch4 ,briot and ,j4 ,bouquet1 ,lecons de geometrie analytique 7,paris1 #ahea2 #afth ed41 #ahig72 ,m4 ,chasles1 ,traite de geometrie superieure 7,paris1 #aheb72 ,wilhelm ,fiedler1 ,analytische ,geometrie der ,kegelschnitte nach ,g4 ,salmon frei bearbeitet 7,leipzig1 #ete ,aufl41 #ahhg-ahhh72 ,n4,m4 ,ferrers1 ,an ,elementary ,treatise on ,trilinear ,coordinates 7,london1 #ahfa72 ,otto ,hesse1 ,vorlesungen aus der analytischen ,geometrie 7,leipzig1 #ahfe1 #ahha72 ,w4,a4 ,whitworth1 ,trilinear ,coordinates and other ,methods of ,modern ,analytical ,geometry 7,cambridge1 #ahff72 ,j4 ,booth1 ,a ,treatise on ,some ,new ,geometrical ,methods 7,london1 i41 #ahgc2 ii41 #ahgg72 ,a4 ,clebsch-,f4 ,lindemann1 ,vorlesungen uber ,geometrie1 ,bd4 i4 7,leipzig1 #ahgf1 #bte ,aufl41 #ahia72 ,r4 ,baltser1 ,analytische ,geometrie 7,leipzig1 #ahhb72 ,charlotte ,a4 ,scott1 ,modern ,methods of ,analytical ,geometry 7,london1 #ahid72 ,g4 ,salmon1 ,a ,treatise on the ,analytical ,geometry of three ,dimensions 7,dublin1 #ahfb2 #dth ed41 #ahhb72 ,salmon-,fiedler1 ,analytische ,geometrie des ,raumes 7,leipzig1 #ahfc2 #dte ,aufl41 #ahih72 ,p4 ,frost1 ,solid ,geometry 7,london1 #crd ed41 #ahhf2 #ast ed41 ,frost and ,j4 ,wolstenholme74 ,see also ,e4 ,pascal1 ,repertorio di matematiche superiori1 ,,ii4 ,geometria 7,milan1 #aijj71 and articles now appearing in the ,encyklopadie der mathematischen ,wissenschaften1 ,bd4 iii4 #a1 #b4 7,e4 ,b4 ,el47 ,v4 ,,line ,,geometry ,line geometry is the name applied to those geometrical investigations in which the straight line replaces the point as element4 ,just as ordinary geometry deals primarily with points and systems of points1 this theory deals in the first instance with straight lines and systems of straight lines4 ,in two dimensions there is no necessity for a special line geometry1 inasmuch as the straight line and the point are interchangeable by the principle of duality2 but in three dimensions the straight line is its own reciprocal1 and for the better discussion of systems of lines we require some new apparatus1 e4g41 a system of coordinates applicable to straight lines rather than to points4 ,the essential features of the subject are most easily elucidated by analytical methods3 we shall therefore begin with the notion of line coordinates1 and in order to emphasize the merits of the system of coordinates ultimately adopted1 we first notice a system without these advantages1 but often useful in special investigations4 ,in ordinary ,cartesian coordinates the two equations of a straight line may be reduced to the form y rx s1 z tx u1 and r1 s1 t1 u may be regarded as the four coordinates of the line4 ,these coordinates lack symmetry3 moreover1 in changing from one base of reference to another the transformation is not linear1 so that the degree of an equation is deprived of real significance4 ,for purposes of the general theory we employ homogeneous coordinates2 if x#ay#az#aw#a and x#by#bz#bw#b are two points on the line1 it is easily verified that the six determinants of the array x#ay#az#aw#a x#by#bz#bw#b are in the same ratios for all point-pairs on the line1 and further1 that when the point coordinates undergo a linear transformation so also do these six determinants4 ,we therefore adopt these six determinants for the coordinates of the line1 and express them by the symbols l1 lambda1 m1 mu1 n1 nu where l x#aw#b - x#bw#a1 lambda y#az#b - y#bz#a1 c4 ,there is the further advantage that if a#a;b#a;c#a;d#a and a#b;b#b;c#b;d#b be two planes through the line1 the six determinants a#a;b#a;c#a;d#a a#b;b#b;c#b;d#b are in the same ratios as the foregoing1 so that except as regards a factor of proportionality we have lambda b#a;c#b - b#b;c#a1 l c#a;d#b - c#b;d#a1 c4 ,the identical relation llambda mmu nnu o reduces the number of independent constants in the six coordinates to four1 for we are only concerned with their mutual ratios2 and the quadratic character of this relation marks an essential difference between point geometry and line geometry4 ,the condition of intersection of two lines is llambda' l'lambda mmu' m'mu nnu' n'nu #j where the accented letters refer to the second line4 ,if the coordinates are ,cartesian and l1 m1 n are direction cosines1 the quantity on the left is the mutual moment of the two lines4 ,since a line depends on four constants1 there are three distinct types of configurations arising in line geometry--those containing a triply-infinite1 a doubly-infinite and a singly-infinite number of lines2 they are called ,complexes1 ,congruences1 and ,ruled ,surfaces or ,skews respectively4 ,a ,complex is thus a system of lines satisfying one condition--that is1 the coordinates are connected by a single relation2 and the degree of the complex is the degree of this equation supposing it to be algebraic4 ,the lines of a complex of the nth degree which pass through any point lie on a cone of the nth degree1 those which lie in any plane envelop a curve of the nth class and there are n lines of the complex in any plane pencil2 the last statement combines the former two1 for it shows that the cone is of the nth degree and the curve is of the nth class4 ,to find the lines common to four complexes of degrees n#a1 n#b1 n#c1 n#d1 we have to solve five equations1 viz4 the four complex equations together with the quadratic equation connecting the line coordinates1 therefore the number of common lines is #bn#an#bn#cn#d4 ,as an example of complexes we have the lines meeting a twisted curve of the nth degree1 which form a complex of the nth degree4 ,a ,congruence is the set of lines satisfying two conditions3 thus a finite number m of the lines pass through any point1 and a finite number n lie in any plane2 these numbers are called the degree and class respectively1 and the congruence is symbolically written 7m1 n74 ,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 planes2 through any other point there pass m of these lines1 and in any other plane there lie n1 therefore the congruence is of degree m and class n4 ,it has been shown by ,g4,h4 ,halphen that the number of lines common to two congruences is mm' nn'1 which may be verified by taking one of them to be of this simple type4 ,the lines meeting two fixed lines form the general 7#a1 #a7 congruence2 and the chords of a twisted cubic form the general type of a 7#a1 #c7 congruence2 ,halphen's result shows that two twisted cubics have in general ten common chords4 ,as regards the analytical treatment1 the difficulty is of the same nature as that arising in the theory of curves in space1 for a congruence is not in general the complete intersection of two complexes4 ,a ,ruled ,surface1 ,regulus or ,skew is a configuration of lines which satisfy three conditions1 and therefore depend on only one parameter4 ,such lines all lie on a surface1 for we cannot draw one through an arbitrary point2 only one line passes through a point of the surface2 the simplest example1 that of a quadric surface1 is really two skews on the same surface4 ,the degree of a ruled surface qua line geometry is the number of its generating lines contained in a linear complex4 ,now the number which meets a given line is the degree of the surface qua point geometry1 and as the lines meeting a given line form a particular case of linear complex1 it follows that the degree is the same from whichever point of view we regard it4 ,the lines common to three complexes of degrees1 n#an#bn#c1 form a ruled surface of degree #bn#an#bn#c2 but not every ruled surface is the complete intersection of three complexes4 ,linear complex4 ,in the case of a complex of the first degree 7or linear complex7 the lines through a fixed point lie in a plane called the polar plane or nul-plane of that point1 and those lying in a fixed plane pass through a point called the nul-point or pole of the plane4 ,if the nul-plane of ,a pass through ,b1 then the nul-plane of ,b will pass through ,a2 the nul-planes of all points on one line l#a pass through another line l#b4 ,the relation between l#a and l#b is reciprocal2 any line of the complex that meets one will also meet the other1 and every line meeting both belongs to the complex4 ,they are called conjugate or polar lines with respect to the complex4 ,on these principles can be founded a theory of reciprocation with respect to a linear complex4 ,this may be aptly illustrated by an elegant example due to ,a4 ,voss4 ,since a twisted cubic can be made to satisfy twelve conditions1 it might be supposed that a finite number could be drawn to touch four given lines1 but this is not the case4 ,for1 suppose one such can be drawn1 then its reciprocal with respect to any linear complex containing the four lines is a curve of the third class1 i4e4 another twisted cubic1 touching the same four lines1 which are unaltered in the process of reciprocation2 as there is an infinite number of complexes containing the four lines1 there is an infinite number of cubics touching the four lines1 and the problem is poristic4 ,the following are some geometrical constructions relating to the unique linear complex that can be drawn to contain five arbitrary lines3 ,to construct the nul-plane of any point ,o1 we observe that the two lines which meet any four of the given five are conjugate lines of the complex1 and the line drawn through ,o to meet them is therefore a ray of the complex2 similarly1 by choosing another four we can find another ray through ,o3 these rays lie in the nul-plane1 and there is clearly a result involved that the five lines so obtained all lie in one plane4 ,a reciprocal construction will enable us to find the nul-point of any plane4 ,proceeding now to the metrical properties and the statical and dynamical applications1 we remark that there is just one line such that the nul-plane of any point on it is perpendicular to it4 ,this is called the central axis2 if d be the shortest distance1 theta the angle between it and a ray of the complex1 then d tan theta p1 where p is a constant called the pitch or parameter4 ,any system of forces can be reduced to a force ,r along a certain line1 and a couple ,g perpendicular to that line2 the lines of nul-moment for the system form a linear complex of which the given line is the central axis and the quotient ,g_/,r is the pitch4 ,any motion of a rigid body can be reduced to a screw motion about a certain line1 i4e4 to an angular velocity omega about that line combined with a linear velocity u along the line4 ,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 axis1 and the quotient u_/omega for pitch 7cf4 ,sir ,r4,s4 ,ball1 ,theory of ,screws74 ,the following are some properties of a configuration of two linear complexes3 ,the lines common to the two-complexes also belong to an infinite number of linear complexes1 of which two reduce to single straight lines4 ,these two lines are conjugate lines with respect to each of the complexes1 but they may coincide1 and then some simple modifications are required4 ,the locus of the central axis of this system of complexes is a surface of the third degree called the cylindroid1 which plays a leading part in the theory of screws as developed synthetically by ,ball4 ,since a linear complex has an invariant of the second degree in its coefficients1 it follows that two linear complexes have a lineo-linear invariant4 ,this invariant is fundamental3 if the complexes be both straight lines1 its vanishing is the condition of their intersection as given above2 if only one of them be a straight line1 its vanishing is the condition that this line should belong to the other complex4 ,when it vanishes for any two complexes they are said to be in involution or apolar2 the nul-points ,p1 ,q of any plane then divide harmonically the points in which the plane meets the common conjugate lines1 and each complex is its own reciprocal with respect to the other4 ,as regards a configuration of these linear complexes1 the common lines from one system of generators of a quadric1 and the doubly infinite system of complexes containing the common lines1 include an infinite number of straight lines which form the other system of generators of the same quadric4 ,general line coordinates4 ,if the equation of a linear complex is ,al ,bm ,cn ,dlambda ,emu ,fnu #j1 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 complex1 the word moment being used in the statical sense2 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 complexes4 ,the essential features of this coordinate system are the same as those of the original one1 viz4 there are six coordinates connected by a quadratic equation1 but this relation has in general a different form4 ,by suitable choice of the six fundamental complexes1 as they may be called1 this connecting relation may be brought into other simple forms of which we mention two3 7i47 ,when the six are mutually in involution it can be reduced to x#ab x#bb x#cb x#db x#eb x#fb #j2 7ii47 ,when the first four are in involution and the other two are the lines common to the first four it is x#ab x#bb x#cb x#db - #bx#ex#f #j4 ,these generalized coordinates might be explained without reference to actual magnitude1 just as homogeneous point coordinates can be2 the essential remark is that the equation of any coordinate to zero represents a linear complex1 a point of view which includes our original system1 for the equation of a coordinate to zero represents all the lines meeting an edge of the fundamental tetrahedron4 ,the system of coordinates referred to six complexes mutually in involution was introduced by ,felix ,klein1 and in many cases is more useful than that derived directly from point coordinates2 e4g4 in the discussion of quadratic complexes3 by means of it ,klein has developed an analogy between line geometry and the geometry of spheres as treated by ,g4 ,darboux and others4 ,in fact1 in that geometry a point is represented by five coordinates1 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 degree4 ,extending this to four dimensions of space1 we obtain an exact analogue of line geometry1 in which 7i47 a point corresponds to a line2 7ii47 a linear complex to a hypersphere2 7iii47 two linear complexes in involution to two orthogonal hyperspheres2 7iv47 a linear complex and two conjugate lines to a hypersphere and two inverse points4 ,many results may be obtained by this principle1 and more still are suggested by trying to extend the properties of circles to spheres in three and four dimensions4 ,thus the elementary theorem1 that1 given four lines1 the circles circumscribed to the four triangles formed by them are concurrent1 may be extended to six hyperplanes in four dimensions2 and then we can derive a result in line geometry by translating the inverse of this theorem4 ,again1 just as there is an infinite number of spheres touching a surface at a given point1 two of them having contact of a closer nature1 so there is an infinite number of linear complexes touching a non-linear complex at a given line1 and three of these have contact of a closer nature 7cf4 ,klein1 ,math4 ,ann4 v474 ,sophus ,lie has pointed out a different analogy with sphere geometry4 ,suppose1 in fact1 that the equation of a sphere of radius r is x#b y#b z#b #b;ax #b;by #b;cz d #j1 so that r#b a#b b#b c#b - d2 then introducing the quantity e to make this equation homogeneous1 we may regard the sphere as given by the six coordinates a1 b1 c1 d1 e1 r connected by the equation a#b b#b c#b - r#b - de #j1 and it is easy to see that two spheres touch1 if the polar form #b;aa#a #b;bb#a #b;cc#a - #brr#a - de#a - d#a;e vanishes4 ,comparing this with the equation x#ab x#bb x#cb x#db - #bx#ex#f #j given above1 it appears that this sphere geometry and line geometry are identical1 for we may write a x#a1 b x#b1 c x#c1 r x#d7_/;delta - #a71 d x#e1 e #a_/bx#f2 but it is to be noticed that a sphere is really replaced by two lines whose coordinates only differ in the sign of x#d1 so that they are polar lines with respect to the complex x#d #j4 ,two spheres which touch correspond to two lines which intersect1 or more accurately to two pairs of lines 7p1 p'7 and 7q1 q'71 of which the pairs 7p1 q7 and 7p'1 q'7 both intersect4 ,by this means the problem of describing a sphere to touch four given spheres is reduced to that of drawing a pair of lines 7t1 t'7 7of which t intersects one line of the four pairs 7pp'71 7qq'71 7rr'71 7ss'71 and t' intersects the remaining four74 ,we may1 however1 ignore the accented letters in translating theorems1 for a configuration of lines and its polar with respect to a linear complex have the same projective properties4 ,in ,lie's transformation a linear complex corresponds to the totality of spheres cutting a given sphere at a given angle4 ,a most remarkable result is that lines of curvature in the sphere geometry become asymptotic lines in the line geometry4 ,some of the principles of line geometry may be brought into clearer light by admitting the ideas of space of four and five dimensions4 ,thus1 regarding the coordinates of a line as homogeneous coordinates in five dimensions1 we may say that line geometry is equivalent to geometry on a quadric surface in five dimensions4 ,a linear complex is represented by a hyperplane section2 and if two such complexes are in involution1 the corresponding hyperplanes are conjugate with respect to the fundamental quadric4 ,by projecting this quadric stereographically into space of four dimensions we obtain ,klein's analogy4 ,in the same way geometry in a linear complex is equivalent to geometry on a quadric in four dimensions2 when two lines intersect the representative points are on the same generator of this quadric4 ,stereographic projection1 therefore1 converts a curve in a linear complex1 i4e4 one whose tangents all belong to the complex1 into one whose tangents intersect a fixed conic3 when this conic is the imaginary circle at infinity the curve is what ,lie calls a minimal curve4 ,curves in a linear complex have been extensively studied4 ,the osculating plane at any point of such a curve is the nul-plane of the point with respect to the complex1 and points of superosculation always coincide in pairs at the points of contact of stationary tangents4 ,when a point of such a curve is given1 the osculating plane is determined1 hence all the curves through a given point with the same tangent have the same torsion4 ,non-linear complexes4 ,the lines through a given point that belong to a complex of the nth degree lie on a cone of the nth degree3 if this cone has a double line the point is said to be a singular point4 ,similarly1 a plane is said to be singular when the envelope of the lines in it has a double tangent4 ,it is very remarkable that the same surface is the locus of the singular points and the envelope of the singular planes3 this surface is called the singular surface1 and both its degree and class are in general #bn7n - #a7b1 which is equal to four for the quadratic complex4 ,the singular lines of a complex ,f #j are the lines common to ,f and the complex delta,f delta,f delta,f delta,f delta,f delta,f -------- --------------- -------- ----------- -------- ----------- #j4 deltal deltalambda deltam deltamu deltan deltanu ,as already mentioned1 at each line l of a complex there is an infinite number of tangent linear complexes1 and they all contain the lines adjacent to l4 ,if now l be a singular line1 these complexes all reduce to straight lines which form a plane pencil containing the line l4 ,suppose the vertex of the pencil is ,a1 its plane a1 and one of its lines xi1 then l' being a complex line near l1 meets xi1 or more accurately the mutual moment of l'1 and is of the second order of small quantities4 ,if ,p be a point on l1 a line through ,p quite near l in the plane a will meet xi and is therefore a line of the complex2 hence the complex-cones of all points on l touch a and the complex-curves of all planes through l touch l at ,a4 ,it follows that l is a double line of the complex-cone of ,a1 and a double tangent of the complex-curve of a4 ,conversely1 a double line of a cone or curve is a singular line1 and a singular line clearly touches the curves of all planes through it in the same point4 ,suppose now that the consecutive line l' is also a singular line1 ,a' being the allied singular point1 a' the singular plane and xi' any line of the pencil 7,a'1 a'7 so that xi' is a tangent line at l' to the complex3 the mutual moments of the pairs l'1 xi and l1 xi are each of the second order2 hence the plane a' meets the lines l and xi' in two points very near ,a4 ,this being true for all singular planes1 near a the point of contact of a with its envelope is in ,a1 i4e4 the locus of singular points is the same as the envelope of singular planes4 ,further1 when a line touches a complex it touches the singular surface1 for it belongs to a plane pencil like 7,aa71 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 surface4 ,the theory of cosingular complexes is thus brought into line with that of confocal surfaces in four dimensions1 and guided by these principles the existence of cosingular quadratic complexes can easily be established1 the analysis required being almost the same as that invented for confocal cyclides by ,darboux and others4 ,of cosingular complexes of higher degree nothing is known4 ,following ,j4 ,plucker1 we give an account of the lines of a quadratic complex that meet a given line4 ,the cones whose vertices are on the given line all pass through eight fixed points and envelop a surface of the fourth degree2 the conics whose planes contain the given line all lie on a surface of the fourth class and touch eight fixed planes4 ,it is easy to see by elementary geometry that these two surfaces are identical4 ,further1 the given line contains four singular points ,a#a1 ,a#b1 ,a#c1 ,a#d1 and the planes into which their cones degenerate are the eight common tangent planes mentioned above2 similarly1 there are four singular planes1 a#a1 a#b1 a#c1 a#d1 through the line1 and the eight points into which their conics degenerate are the eight common points above4 ,the locus of the pole of the line with respect to all the conics in planes through it is a straight line called the polar line of the given one2 and through this line passes the polar plane of the given line with respect to each of the cones4 ,the name polar is applied in the ordinary analytical sense2 any line has an infinite number of polar complexes with respect to the given complex1 for the equation of the latter can be written in an infinite number of ways2 one of these polars is a straight line1 and is the polar line already introduced4 ,the surface on which lie all the conics through a line l is called the ,plucker surface of that line3 from the known properties of 7#b1 #b7 correspondences it can be shown that the ,plucker surface of l cuts l#a in a range of the same cross ratio as that of the range in which the ,plucker surface of l#a cuts l4 ,applying this to the case in which l#a is the polar of l1 we find that the cross ratios of 7,a#a1 ,a#b1 ,a#c1 ,a#d7 and 7a#a1 a#b1 a#c1 a#d7 are equal4 ,the identity of the locus of the ,a's with the envelope of the a's follows at once2 moreover1 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 surface4 ,the ,plucker surface has eight nodes1 eight singular tangent planes1 and is a double line4 ,the relation between a line and its polar line is not a reciprocal one with respect to the complex2 but ,w4 ,stahl has pointed out that the relation is reciprocal as far as the singular surface is concerned4 ,quadratic complexes4 ,to facilitate the discussion of the general quadratic complex we introduce ,klein's canonical form4 ,we have1 in fact1 to deal with two quadratic equations in six variables2 and by suitable linear transformations these can be reduced to the form a#ax#ab a#bx#bb a#cx#cb a#dx#db a#ex#eb a#fx#fb #j x#ab x#bb x#cb x#db x#eb x#fb #j subject to certain exceptions1 which will be mentioned later4 ,taking the first equation to be that of the complex1 we remark that both equations are unaltered by changing the sign of any coordinate2 the geometrical meaning of this is1 that the quadratic complex is its own reciprocal with respect to each of the six fundamental complexes1 for changing the sign of a coordinate is equivalent to taking the polar of a line with respect to the corresponding fundamental complex4 ,it is easy to establish the existence of six systems of bitangent linear complexes1 for the complex l#ax#a l#bx#b l#cx#c l#dx#d l#ex#e l#fx#f #j is a bitangent when l#bb l#cb l#db l#eb l#fb l#a #j1 and ------- ------- ------- ------- ------- #j a#b - a#a a#c - a#a a#d - a#a a#e - a#a a#f - a#a and its lines of contact are conjugate lines with respect to the first fundamental complex4 ,we therefore infer the existence of six systems of bitangent lines of the complex1 of which the first is given by x#bb x#cb x#db x#eb x#fb x#a #j1 ------- ------- ------- ------- ------- #j4 a#b - a#a a#c - a#a a#d - a#a a#e - a#a a#f - a#a ,each of these lines is a bitangent of the singular surface1 which is therefore completely determined as being the focal surface of the 7#b1 #b7 congruence above4 ,it is thence easy to verify that the two complexes ,sigmaax#b #j and ,sigmabx#b #j are cosingular if br arlambda mu_/arnu rho4 ,the singular surface of the general quadratic complex is the famous quartic1 with sixteen nodes and sixteen singular tangent planes1 first discovered by ,e4,e4 ,kummer4 ,we cannot give a full account of its properties here1 but we deduce at once from the above that its bitangents break up into six 7#b1 #b7 congruences1 and the six linear complexes containing these are mutually in involution4 ,the nodes of the singular surface are points whose complex cones are coincident planes1 and the complex conic in a singular tangent plane consists of two coincident points4 ,this configuration of sixteen points and planes has many interesting properties2 thus each plane contains six points which lie on a conic1 while through each point there pass six planes which touch a quadric cone4 ,in many respects the ,kummer quartic plays a part in three dimensions analogous to the general quartic curve in two2 it further gives a natural representation of certain relations between hyperelliptic functions 7cf4 ,r4,w4,h4,t4 ,hudson1 ,kummer's ,quartic1 #aije74 ,classification of quadratic complexes4 ,as might be expected from the magnitude of a form in six variables1 the number of projectivally distinct varieties of quadratic complexes is very great2 and in fact ,adolf ,weiler1 by whom the question was first systematically studied on lines indicated by ,klein1 enumerated no fewer than forty-nine different types4 ,but the principle of the classification is so important1 and withal so simple1 that we give a brief sketch which indicates its essential features4 ,we have practically to study the intersection of two quadrics ,f and ,f' in six variables1 and to classify the different cases arising we make use of the results of ,karl ,weierstrass on the equivalence conditions of two pairs of quadratics4 ,as far as at present required1 they are as follows3 ,suppose that the factorized form of the determinantal equation ,disct 7,f lambda,f'7 #j is 7lambda - alpha77s#a s#b s#c 4447 7lambda - beta77t#a t#b t#c 4447 444 where the root alpha occurs s#a s#b s#c 444 times in the determinant1 s#b s#c 444 times in every first minor1 s#c 444 times in every second minor1 and so on2 the meaning of each exponent is then perfectly definite4 ,every factor of the type 7lambda - alpha7s is called an elementartheil 7elementary divisor7 of the determinant1 and the condition of equivalence of two pairs of quadratics is simply that their determinants have the same elementary divisors4 ,we write the pair of forms symbolically thus 7s#as#b 44471 7t#at#b 44471 4441 letters in the inner brackets referring to the same factor4 ,returning now to the two quadratics representing the complex1 the sum of the exponents will be six1 and two complexes are put in the same class if they have the same symbolical expression2 i4e4 the actual values of the roots of the determinantal equation need not be the same for both1 but their manner of occurrence1 as far as here indicated1 must be identical in the two4 ,the enumeration of all possible cases is thus reduced to a simple question in combinatorial analysis1 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 which1 of course1 represents the equation connecting line coordinates1 and the other the equation of the complex4 ,the general complex is naturally #aaaaaa2 the complex of tangents to a quadric is 7#aaa71 7#aaa7 and that of lines meeting a conic is 7#bbb74 ,full information will be found in ,weiler's memoir1 ,math4 ,ann4 vol4 vii4 ,the detailed study of each variety of complex opens up a vast subject2 we only mention two special cases1 the harmonic complex and the tetrahedral complex4 ,the harmonic complex1 first studied by ,battaglini1 is generated in an infinite number of ways by the lines cutting two quadrics harmonically4 ,taking the most general case1 and referring the quadrics to their common self-conjugate tetrahedron1 we can find its equation in a simple form1 and verify that this complex really depends only on seventeen constants1 so that it is not the most general quadratic complex4 ,it belongs to the general type in so far as it is discussed above1 but the roots of the determinant are in involution4 ,the singular surface is the tetrahedroid discussed by ,cayley4 ,as a particular case1 from a metrical point of view1 we have ,l4,f4 ,painvin's complex generated by the lines of intersection of perpendicular tangent planes of a quadric1 the singular surface now being ,fresnel's wave surface4 ,the tetrahedral or ,reye complex is the simplest and best known of proper quadratic complexes4 ,it is generated by the lines which cut the faces of a tetrahedron in a constant cross ratio1 and therefore by those subtending the same cross ratio at the four vertices4 ,the singular surface is made up of the faces or the vertices of the fundamental tetrahedron1 and each edge of this tetrahedron is a double line of the complex4 ,the complex was first discussed by ,k4,t4 ,reye as the assemblage of lines joining corresponding points in a homographic transformation of space1 and this point of view leads to many important and elegant properties4 ,a 7metrically7 particular case of great interest is the complex generated by the normals to a family of confocal quadrics1 and for many investigations it is convenient to deal with this complex referred to the principal axes4 ,for example1 ,lie has developed the theory of curves in a ,reye complex 7i4e4 curves whose tangents belong to the complex7 as solutions of a differential equation of the form 7b - c7xdydz 7c - a7ydzdx 7a - b7zdxdy #j1 and we can simplify this equation by a logarithmic transformation4 ,many theorems connecting complexes with differential equations have been given by ,lie and his school4 ,a line complex1 in fact1 corresponds to a ,mongian equation having oo#c line integrals4 ,congruences4 ,as the coordinates of a line belonging to a congruence are functions of two independent parameters1 the theory of congruences is analogous to that of surfaces1 and we may regard it as a fundamental inquiry to find the simplest form of surface into which a given congruence can be transformed4 ,most of those whose properties have been extensively discussed can be represented on a plane by a birational transformation4 ,but in addition to the difficulties of the theory of algebraic surfaces1 a subject still in its infancy1 the theory of congruences has other difficulties in that a congruence is seldom completely represented1 even by two equations4 ,a fundamental theorem is that the lines of a congruence are in general bitangents of a surface2 in fact1 since the condition of intersection of two consecutive straight lines is ldlambda dmdmu dndnu #j1 a line l of the congruence meets two adjacent lines1 say l#a and l#b4 ,suppose l1 l#a lie in the plane pencil 7,a#a;a#a7 and l1 l#b in the plane pencil 7,a#b;a#b71 then the locus of the ,a's is the same as the envelope of the a's1 but a#b is the tangent plane at ,a#a and a#a at ,a#b4 ,this surface is called the focal surface of the congruence1 and to it all the lines l are bitangent4 ,the distinctive property of the points ,a is that two of the congruence lines through them coincide1 and in like manner the planes a each contain two coincident lines4 ,the focal surface consists of two sheets1 but one or both may degenerate into curves2 thus1 for example1 the normals to a surface are bitangents of the surface of centres1 and in the case of ,dupin's cyclide this surface degenerates into two conics4 ,in the discussion of congruences it soon becomes necessary to introduce another number r1 called the rank1 which expresses the number of plane pencils each of which contains an arbitrary line and two lines of the congruence4 ,the order of the focal surface is #bm7n - #a7 - #br1 and its class is m7m - #a7 - #br4 ,our knowledge of congruences is almost exclusively confined to those in which either m or n does not exceed two4 ,we give a brief account of those of the second order without singular lines1 those of order unity not being especially interesting4 ,a congruence generally has singular points through which an infinite number of lines pass2 a singular point is said to be of order r when the lines through it lie on a cone of the rth degree4 ,by means of formulae connecting the number of singular points and their orders with the class m of quadratic congruence ,kummer proved that the class cannot exceed seven4 ,the focal surface is of degree four and class #bm2 this kind of quartic surface has been extensively studied by ,kummer1 ,cayley1 ,rohn and others4 ,the varieties 7#b1 #b71 7#b1 #c71 7#b1 #d71 7#b1 #e7 all belong to at least one ,reye complex2 and so also does the most important class of 7#b1 #f7 congruences which includes all the above as special cases4 ,the congruence 7#b1 #b7 belongs to a linear complex and forty different ,reye complexes2 as above remarked1 the singular surface is ,kummer's sixteen-nodal quartic1 and the same surface is focal for six different congruences of this variety4 ,the theory of 7#b1 #b7 congruences is completely analogous to that of the surfaces called cyclides in three dimensions4 ,further particulars regarding quadratic congruences will be found in ,kummer's memoir of #ahff1 and the second volume of ,sturm's treatise4 ,the properties of quadratic congruences having singular lines1 i4e4 degenerate focal surfaces1 are not so interesting as those of the above class2 they have been discussed by ,kummer1 ,sturm and others4 ,ruled surfaces4 ,since a ruled surface contains only infinity#a elements1 this theory is practically the same as that of curves4 ,if a linear complex contains more than n generators of a ruled surface of the nth degree1 it contains all the generators1 hence for n #b there are three linearly independent complexes1 containing all the generators1 and this is a well-known property of quadric surfaces4 ,in ruled cubics the generators all meet two lines which may or may not coincide2 these two cases correspond to the two main classes of cubics discussed by ,cayley and ,cremona4 ,as regards ruled quartics1 the generators must lie in one and may lie in two linear complexes4 ,the first class is equivalent to a quartic in four dimensions and is always rational1 but the latter class has to be subdivided into the elliptic and the rational1 just like twisted quartic curves4 ,a quintic skew may not lie in a linear complex1 and then it is unicursal1 while of sextics we have two classes not in a linear complex1 viz4 the elliptic variety1 having thirty-six places where a linear complex contains six consecutive generators1 and the rational1 having six such places4 ,the general theory of skews in two linear complexes is identical with that of curves on a quadric in three dimensions and is known4 ,but for skews lying in only one linear complex there are difficulties2 the curve now lies in four dimensions1 and we represent it in three by stereographic projection as a curve meeting a given plane in n points on a conic4 ,to find the maximum deficiency for a given degree would probably be difficult1 but as far as degree eight the space-curve theory of ,halphen and ,nother can be translated into line geometry at once4 ,when the skew does not lie in a linear complex at all the theory is more difficult still1 and the general theory clearly cannot advance until further progress is made in the study of twisted curves4 ,,references4--,the earliest works of a general nature are ,plucker1 ,neue ,geometrie des ,raumes 7,leipzig1 #ahfh72 and ,kummer1 ,uber die algebraischen ,strahlensysteme1 ,berlin ,academy 7#ahff74 ,systematic development on purely synthetic lines will be found in the three volumes of ,sturm1 ,liniengeometrie 7,leipzig1 #ahib1 #ahic1 #ahif72 vol4 i4 deals with the linear and ,reye complexes1 vols4 ii4 and iii4 with quadratic congruences and complexes respectively4 ,for a highly suggestive review by ,gino ,loria see ,bulletin des sciences mathematiques 7#ahic1 #ahig74 ,a shorter treatise1 giving a very interesting account of ,klein's coordinates1 is the work of ,koenigs1 ,la ,geometrie reglee et ses applications 7,paris1 #ahih74 ,english treatises are ,c4,m4 ,jessop1 ,treatise on the ,line ,complex 7#aijc72 ,r4,w4,h4,t4 ,hudson1 ,kummer's ,quartic 7#aije74 ,many references to memoirs on line geometry will be found in ,hagen1 ,synopsis der hoheren ,mathematik1 ii4 7,berlin1 #ahid72 ,loria1 ,il passato ed il presente delle principali teorie geometriche 7,milan1 #ahig72 a clear resume of the principal results is contained in the very elegant volume of ,pascal1 ,repertorio di mathematiche superiori1 ii4 7,milan1 #aijj74 ,another treatise dealing extensively with line geometry is ,lie1 ,geometrie der ,beruhrungstransformationen 7,leipzig1 #ahif74 ,many memoirs on the subject have appeared in the ,mathematische ,annalen2 a full list of these will be found in the index to the first fifty volumes1 p4 #aae4 ,perhaps the two memoirs which have left most impression on the subsequent development of the subject are ,klein1 ,zur ,theorie der ,liniencomplexe des ersten und zweiten ,grades1 ,math4 ,ann4 ii42 and ,lie1 ,uber ,complexe1 insbesondere ,linien- und ,kugelcomplexe1 ,math4 ,ann4 v4 7,j4 ,h4 ,gr47 ,,vi4 ,,non-,,euclidean ,,geometry ,the various metrical geometries are concerned with the properties of the various types of congruence-groups1 which are defined in the study of the axioms of geometry and of their immediate consequences4 ,but this point of view of the subject is the outcome of recent research1 and historically the subject has a different origin4 ,non-,euclidean geometry arose from the discussion1 extending from the ,greek period to the present day1 of the various assumptions which are implicit in the traditional ,euclidean system of geometry4 ,in the course of these investigations it became evident that metrical geometries1 each internally consistent but inconsistent in many respects with each other and with the ,euclidean system1 could be developed4 ,a short historical sketch will explain this origin of the subject1 and describe the famous and interesting progress of thought on the subject4 ,but previously a description of the chief characteristic properties of elliptic and of hyperbolic geometries will be given1 assuming the standpoint arrived at below under ,,vii4 ,axioms of ,geometry4 ,first assume the equation to the absolute 7cf4 loc4 cit47 to be w#b - x#b - y#b - z#b #j4 ,the absolute is then real1 and the geometry is hyberbolic4 ,the distance 7d#ab7 between the two points 7x#a1 y#a1 z#a1 w#a7 and 7x#b1 y#b1 z#b1 w#b7 is given by cosh 7d#ab_/;gamma7 7w#aw#b - x#ax#b - y#ay#b - z#az#b7_/7w#ab - x#ab - y#ab - ,z#ab7 7w#bb - x#bb - y#bb - z#bb7a_/b 7#a7 ,the only points to which the metrical geometry applies are those within the region enclosed by the quadric2 the other points are improper ideal points4 ,the angle 7theta#ab7 between two planes1 l#ax m#ay n#az r#aw #j and l#bx m#by n#bz r#bw #j1 is given by cos theta#ab 7l#al#b m#am#b n#an#b - r#ar#b7_/7l#ab m#ab n#ab - r#ab7 7l#bb m#bb n#bb - r#bb7a_/b 7#b7 ,these planes only have a real angle of inclination if they possess a line of intersection within the actual space1 i4e4 if they intersect4 ,planes which do not intersect possess a shortest distance along a line which is perpendicular to both of them4 ,if this shortest distance is delta#ab1 we have cosh 7delta#ab_/;gamma7 7l#al#b m#am#b n#an#b - r#ar#b7_/7l#ab m#ab n#ab - r#ab7 7l#bb m#bb n#bb - r#bb7a_/b 7#c7 ,illustration3 ,,fig4 #fg4 ,thus in the case of the two planes one and only one of the two1 theta#ab and delta#ab1 is real4 ,the same considerations hold for coplanar straight lines 7see ,,vii4 ,axioms of ,geometry74 ,let ,o 7fig4 #fg7 be the point 7#j1 #j1 #j1 #a71 ,,ox the line y #j1 z #j1 ,,oy the line z #j1 x #j1 and ,,oz the line x #j1 y #j4 ,these are the coordinate axes and are at right angles to each other4 ,let ,p be any point1 and let rho be the distance ,,op1 theta the angle ,,poz1 and phi the angle between the planes ,,zox and ,,zop4 ,then the coordinates of ,p can be taken to be sinh 7rho_/gamma7 sin theta cos phi1 sinh 7rho_/gamma7 sin theta sin phi1 sinh 7rho_/gamma7 costheta1 cosh 7rho_/gamma74 ,illustration3 ,,fig4 #fh4 ,if ,,abc is a triangle1 and the sides and angles are named according to the usual convention1 we have sinh 7a_/gamma7_/sin ,a sinh 7b_/gamma7_/sin ,b sinh 7c_/gamma7_/sin ,c1 7#d7 and also cosh 7a_/gamma7 cosh 7b_/gamma7 cosh 7c_/gamma7 - sinh 7b_/gamma7 sinh 7c_/gamma7 cos ,a1 7#e7 with two similar equations4 ,the sum of the three angles of a triangle is always less than two right angles4 ,the area of the triangle ,,abc is lambda#b7pi - ,a - ,b - ,c74 ,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 constant1 then the locus of ,a is a line of equal distance from ,,bc4 ,this locus is not a straight line4 ,the whole theory of similarity is inapplicable2 two triangles are either congruent1 or their angles are not equal two by two4 ,thus the elements of a triangle are determined when its three angles are given4 ,by keeping ,a and ,b and the line ,,bc fixed1 but by making ,c move off to infinity along ,,bc1 the lines ,,bc and ,,ac become parallel1 and the sides a and b become infinite4 ,hence from equation 7#e7 above1 it follows that two parallel lines 7cf4 ,section ,,vii4 ,axioms of ,geometry7 must be considered as making a zero angle with each other4 ,also if ,b be a right angle1 from the equation 7#e71 remembering that1 in the limit1 cosh 7a_/gamma7_/cosh 7b_/gamma7 cosh 7a_/gamma7_/sinh 7b_/gamma7 #a1 we have cos ,a tanh 7c_/#b;gamma7 4444 7#f74 ,the angle ,a is called by ,n4,i4 ,lobatchewsky the angle of parallelism4 ,the whole theory of lines and planes at right angles to each other is simply the theory of conjugate elements with respect to the absolute1 where ideal lines and planes are introduced4 ,thus if l and l' be any two conjugate lines with respect to the absolute 7of which one of the two must be improper1 say l'71 then any plane through l' and containing proper points is perpendicular to l4 ,also if p is any plane containing proper points1 and ,p is its pole1 which is necessarily improper1 then the lines through ,p are the normals to ,p4 ,the equation of the sphere1 centre 7x#a1 y#a1 z#a1 w#a7 and radius rho1 is 7w#ab- x#ab- y#ab- z#ab77w#b - x#b - y#b - z#b7 cosh#b7rho_/gamma7 7w#aw - x#ax - y#ay -z#az7#b 7#g74 ,the equation of the surface of equal distance 7sigma7 from the plane lx my nz rw #j is 7l#b m#b n#b - r#b77w#b - x#b - y#b - z#b7 sinh#b7sigma_/gamma7 7rw lx my nz7#b 7#h74 ,a surface of equal distance is a sphere whose centre is improper2 and both types of surface are included in the family k#b7w#b - x#b - y#b - z#b7 7ax by cz dw7#b 7#i74 ,but this family also includes a third type of surfaces1 which can be looked on either as the limits of spheres whose centres have approached the absolute1 or as the limits of surfaces of equal distance whose central planes have approached a position tangential to the absolute4 ,these surfaces are called limit-surfaces4 ,thus 7#i7 denotes a limit-surface1 if d#b - a#b - b#b - c#b #j4 ,two limit-surfaces only differ in position4 ,thus the two limit-surfaces which touch the plane ,,yoz at ,o1 but have their concavities turned in opposite directions1 have as their equations w#b - x#b - y#b - z#b 7w - x7#b4 ,the geodesic geometry of a sphere is elliptic1 that of a surface of equal distance is hyperbolic1 and that of a limit-surface is parabolic 7i4e4 ,euclidean74 ,the equation of the surface 7cylinder7 of equal distance 7delta7 from the line ,,ox is 7w#b - x#b7 tanh#b7;delta_/gamma7 - y#b - z#b #j4 ,this is not a ruled surface4 ,hence in this geometry it is not possible for two straight lines to be at a constant distance from each other4 ,secondly1 let the equation of the absolute be x#b y#b z#b w#b #j4 ,the absolute is now imaginary and the geometry is elliptic4 ,the distance 7d#ab7 between the two points 7x#a1 y#a1 z#a1 w#a7 and 7x#b1 y#b1 z#b1 w#b7 is given by cos 7d#ab_/;gamma7 -7x#ax#b y#ay#b z#az#b w#aw#b7 _/ 7x#ab y#ab z#ab w#ab7 7x#bb y#bb z#bb w#bb7a_/b 7#aj74 ,thus there are two distances between the points1 and if one is d#ab1 the other is pigamma-d#ab4 ,every straight line returns into itself1 forming a closed series4 ,thus there are two segments between any two points1 together forming the whole line which contains them2 one distance is associated with one segment1 and the other distance with the other segment4 ,the complete length of every straight line is pigamma4 ,the angle between the two planes l#ax m#ay n#az r #aw #j and l#bx m#by n#bz r#bw #j is cos theta#ab 7l#al#b m#am#b n#an#b r#ar#b7_/ 7l#ab m#ab n#ab r#ab7 7l#bb m#bb n#bb r#bb7a_/b 7#aa74 ,the polar plane with respect to the absolute of the point 7x#a1 y#a1 z#a1 w#a7 is the real plane x#ax y#ay z#az w#aw #j1 and the pole of the plane l#ax m#ay n#az r#aw #j is the point 7l#a1 m#a1 n#a1 r#a74 ,thus 7from equations #aj and #aa7 it follows that the angle between the polar planes of the points 7x#a1 4447 and 7x#b1 4447 is d#ab_/;gamma1 and that the distance between the poles of the planes 7l#a1 4447 and 7l#b1 4447 is gammatheta#ab4 ,thus there is complete reciprocity between points and planes in respect to all properties4 ,this complete reign of the principle of duality is one of the great beauties of this geometry4 ,the theory of lines and planes at right angles is simply the theory of conjugate elements with respect to the absolute4 ,a tetrahedron self-conjugate with respect to the absolute has all its intersecting elements 7edges and planes7 at right angles4 ,if l and l' are two conjugate lines1 the planes through one are the planes perpendicular to the other4 ,if ,p is the pole of the plane p1 the lines through ,p are the normals to the plane p4 ,the distance from ,p to p is #a_/bpigamma4 ,thus every sphere is also a surface of equal distance from the polar of its centre1 and conversely4 ,a plane does not divide space2 for the line joining any two points ,p and ,q only cuts the plane once1 in ,l say1 then it is always possible to go from ,p to ,q by the segment of the line ,,pq which does not contain ,l4 ,but ,p and ,q may be said to be separated by a plane p1 if the point in which ,,pq cuts p lies on the shortest segment between ,p and ,q4 ,with this sense of separation1 it is possible#b to find three points ,p1 ,q1 ,r such that ,p and ,q are separated by the plane p1 but ,p and ,r are not separated by p1 nor are ,q and ,r4 ,let ,a1 ,b1 ,c be any three non-collinear points1 then four triangles are defined by these points4 ,thus if a1 b1 c and ,a1 ,b1 ,c are the elements of any one triangle1 then the four triangles have as their elements3 7#a7 a1 b1 c1 ,a1 ,b1 ,c4 7#b7 a1 pigamma - b1 pigamma - c1 ,a1 pi - ,b1 pi - ,c4 7#c7 pigamma - a1 b1 pigamma - c1 pi - ,a1 ,b1 pi - ,c4 7#d7 pigamma - a1 pigamma - b1 c1 pi - ,a1 pi - ,b1 ,c4 ,the formulae connecting the elements are sin ,a_/sin 7a_/gamma7 sin ,b_/sin 7b_/gamma7 sin ,c_/sin 7c_/gamma71 7#ab7 and cos 7a_/gamma7 cos 7b_/gamma7 cos 7c_/gamma7 sin 7b_/gamma7 sin 7c_/gamma7 cos ,a1 7#ac7 with two similar equations4 ,two cases arise1 namely 7,i47 according as one of the four triangles has as its sides the shortest segments between the angular points1 or 7,,ii47 according as this is not the case4 ,when case ,i4 holds there is said to be a principal triangle4#c ,if all the figures considered lie within a sphere of radius #a_/dpigamma only case ,i4 can hold1 and the principal triangle is the triangle wholly within this sphere1 also the peculiarities in respect to the separation of points by a plane cannot then arise4 ,the sum of the three angles of a triangle ,,abc is always greater than two right angles1 and the area of the triangle is gamma#b7;,a ,b ,c--pi74 ,thus as in hyperbolic geometry the theory of similarity does not hold1 and the elements of a triangle are determined when its three angles are given4 ,the coordinates of a point can be written in the form sin 7rho_/gamma7 sin ,phi cos phi1 sin 7rho_/gamma7 sin ,phi sin phi1 sin 7rho_/gamma7 cos ,phi1 cos 7rho_/gamma71 where rho1 ,phi and phi have the same meanings as in the corresponding formulae in hyperbolic geometry4 ,again1 suppose a watch is laid on the plane ,,oxy1 face upwards with its centre at ,o1 and the line #ab to #f 7as marked on dial7 along the line ,,yoy4 ,let the watch be continually pushed along the plane along the line ,,ox1 that is1 in the direction #i to #c4 ,then the line ,,xox being of finite length1 the watch will return to ,o1 but at its first return it will be found to be face downwards on the other side of the plane1 with the line #ab to #f reversed in direction along the line ,,yoy4 ,this peculiarity was first pointed out by ,felix ,klein4 ,the theory of parallels as it exists in hyperbolic space has no application in elliptic geometry4 ,but another property of ,euclidean parallel lines holds in elliptic geometry1 and by the use of it parallel lines are defined4 ,for the equation of the surface 7cylinder7 of equal distance 7delta7 from the line ,,xox is 7x#b w#b7 tan#b7;delta_/gamma7 - 7y#b z#b7 #j4 ,this is also the surface of equal distance1 #a_/bpigamma-delta1 from the line conjugate to ,,xox4 ,now from the form of the above equation this is a ruled surface1 and through every point of it two generators pass4 ,but these generators are lines of equal distance from ,,xox4 ,thus throughout every point of space two lines can be drawn which are lines of equal distance from a given line l4 ,this property was discovered by ,w4,k4 ,clifford4 ,the two lines are called ,clifford's right and left parallels to l through the point4 ,this property of parallelism is reciprocal1 so that if m is a left parallel to l1 then l is a left parallel to m4 ,note also that two parallel lines l and m are not coplanar4 ,many of those properties of ,euclidean parallels1 which do not hold for ,lobatchewsky's parallels in hyperbolic geometry1 do hold for ,clifford's parallels in elliptic geometry4 ,the geodesic geometry of spheres is elliptic1 the geodesic geometry of surfaces of equal distance from lines 7cylinders7 is ,euclidean1 and surfaces of revolution can be found#d of which the geodesic geometry is hyperbolic4 ,but it is to be noticed that the connectivity of these surfaces is different to that of a ,euclidean plane4 ,for instance there are only infin#b congruence transformations of the cylindrical surfaces of equal distance into themselves1 instead of the infin#c for the ordinary plane4 ,it would obviously be possible to state axioms which these geodesics satisfy1 and thus to define independently1 and not as loci1 quasi-spaces of these peculiar types4 ,the existence of such ,euclidean quasi-geometries was first pointed out by ,clifford4#e ,in both elliptic and hyperbolic geometry the spherical geometry1 i4e4 the relations between the angles formed by lines and planes passing through the same point1 is the same as the spherical trigonometry in ,euclidean geometry4 ,the constant gamma1 which appears in the formulae both of hyperbolic and elliptic geometry1 does not by its variation produce different types of geometry4 ,there is only one type of elliptic geometry and one type of hyperbolic geometry2 and the magnitude of the constant gamma in each case simply depends upon the magnitude of the arbitrary unit of length in comparison with the natural unit of length which each particular instance of either geometry presents4 ,the existence of a natural unit of length is a peculiarity common both to hyperbolic and elliptic geometries1 and differentiates them from ,euclidean geometry4 ,it is the reason for the failure of the theory of similarity in them4 ,if gamma is very large1 that is1 if the natural unit is very large compared to the arbitrary unit1 and if the lengths involved in the figures considered are not large compared to the arbitrary unit1 then both the elliptic and hyperbolic geometries approximate to the ,euclidean4 ,for from formulae 7#d7 and 7#e7 and also from 7#ab7 and 7#ac7 we find1 after retaining only the lowest powers of small quantities1 as the formulae for any triangle ,,abc1 a_/ sin ,a b_/ sin ,b c_/ sin ,c1 and a#b b#b c#b - #b;bc cos ,a1 with two similar equations4 ,thus the geometries of small figures are in both types ,euclidean4 ,theory of parallels before ,gauss4 ,history4--,in pulcherrimo ,geometriae corpore1 wrote ,sir ,henry ,savile in #afba1 duo sunt naevi1 duae labes nec quod sciam plures1 in quibus eluendis et emaculendis cum veterum tum recentiorum 444 vigilavit industria4 ,these two blemishes are the theory of parallels and the theory of proportion4 ,the industry of the moderns1 in both respects1 has given rise to important branches of mathematics1 while at the same time showing that ,euclid is in these respects more free from blemish than had been previously credible4 ,it was from endeavours to improve the theory of parallels that non-,euclidean geometry arose2 and though it has now acquired a far wider scope1 its historical origin remains instructive and interesting4 ,euclid's axiom of parallels appears as ,postulate ,v4 to the first book of his ,elements1 and is stated thus1 ,and that1 if a straight line falling on two straight lines make the angles1 internal and on the same side1 less than two right angles1 the two straight lines1 being produced indefinitely1 meet on the side on which are the angles less than two right angles4 ,the original ,greek is ,greek3 kai ean eis duo eutheias eutheia empiptousa tas entos kai epi ta auta mere gonias duo orthon elassonas poie1 ekballomenas tas duo eutheias ep' apeiron sympiptein1 eph' ha mere eisin hai ton duo orthon elassones4 ,to ,euclid's successors this axiom had signally failed to appear self-evident1 and had failed equally to appear indemonstrable4 ,without the use of the postulate its converse is proved in ,euclid's #bhth proposition1 and it was hoped that by further efforts the postulate itself could be also proved4 ,the first step consisted in the discovery of equivalent axioms4 ,christoph ,clavius in #aegd deduced the axiom from the assumption that a line whose points are all equidistant from a straight line is itself straight4 ,john ,wallis in #affc showed that the postulate follows from the possibility of similar triangles on different scales4 ,girolamo ,saccheri 7#agcc7 showed that it is sufficient to have a single triangle1 the sum of whose angles is two right angles4 ,other equivalent forms may be obtained1 but none shows any essential superiority to ,euclid's4 ,indeed plausibility1 which is chiefly aimed at1 becomes a positive demerit where it conceals a real assumption4 ,saccheri4 ,a new method1 which1 though it failed to lead to the desired goal1 proved in the end immensely fruitful1 was invented by ,saccheri1 in a work entitled ,euclides ab omni naevo vindicatus 7,milan1 #agcc74 ,if the postulate of parallels is involved in ,euclid's other assumptions1 contradictions must emerge when it is denied while the others are maintained4 ,this led ,saccheri to attempt a reductio ad absurdum1 in which he mistakenly believed himself to have succeeded4 ,what is interesting1 however1 is not his fallacious conclusion1 but the non-,euclidean results which he obtains in the process4 ,saccheri distinguishes three hypotheses 7corresponding to what are now known as ,euclidean or parabolic1 elliptic and hyperbolic geometry71 and proves that some one of the three must be universally true4 ,his three hypotheses are thus obtained3 equal perpendiculars ,,ac1 ,,bd are drawn from a straight line ,,ab1 and ,,cd are joined4 ,it is shown that the angles ,,acd1 ,,bdc are equal4 ,the first hypothesis is that these are both right angles2 the second1 that they are both obtuse2 and the third1 that they are both acute4 ,many of the results afterwards obtained by ,lobatchewsky and ,bolyai are here developed4 ,saccheri fails to be the founder of non-,euclidean geometry only because he does not perceive the possible truth of his non-,euclidean hypotheses4 ,lambert4 ,some advance is made by ,johann ,heinrich ,lambert in his ,theorie der ,parallellinien 7written #agff2 posthumously published #aghf74 ,though he still believed in the necessary truth of ,euclidean geometry1 he confessed that1 in all his attempted proofs1 something remained undemonstrated4 ,he deals with the same three hypotheses as ,saccheri1 showing that the second holds on a sphere1 while the third would hold on a sphere of purely imaginary radius4 ,the second hypothesis he succeeds in condemning1 since1 like all who preceded ,bernhard ,riemann1 he is unable to conceive of the straight line as finite and closed4 ,but the third hypothesis1 which is the same as ,lobatchewsky's1 is not even professedly refuted4#f ,three periods of non-,euclidean geometry4 ,non-,euclidean geometry proper begins with ,karl ,friedrich ,gauss4 ,the advance which he made was rather philosophical than mathematical3 it was he 7probably7 who first recognized that the postulate of parallels is possibly false1 and should be empirically tested by measuring the angles of large triangles4 ,the history of non-,euclidean geometry has been aptly divided by ,felix ,klein into three very distinct periods4 ,the first--which contains only ,gauss1 ,lobatchewsky and ,bolyai--is characterized by its synthetic method and by its close relation to ,euclid4 ,the attempt at indirect proof of the disputed postulate would seem to have been the source of these three men's discoveries2 but when the postulate had been denied1 they found that the results1 so far from showing contradictions1 were just as self-consistent as ,euclid4 ,they inferred that the postulate1 if true at all1 can only be proved by observations and measurements4 ,only one kind of non-,euclidean space is known to them1 namely1 that which is now called hyperbolic4 ,the second period is analytical1 and is characterized by a close relation to the theory of surfaces4 ,it begins with ,riemann's inaugural dissertation1 which regards space as a particular case of a manifold2 but the characteristic standpoint of the period is chiefly emphasized by ,eugenio ,beltrami4 ,the conception of measure of curvature is extended by ,riemann from surfaces to spaces1 and a new kind of space1 finite but unbounded 7corresponding to the second hypothesis of ,saccheri and ,lambert71 is shown to be possible4 ,as opposed to the second period1 which is purely metrical1 the third period is essentially projective in its method4 ,it begins with ,arthur ,cayley1 who showed that metrical properties are projective properties relative to a certain fundamental quadric1 and that different geometries arise according as this quadric is real1 imaginary or degenerate4 ,klein1 to whom the development of ,cayley's work is due1 showed further that there are two forms of ,riemann's space1 called by him the elliptic and the spherical4 ,finally1 it has been shown by ,sophus ,lie1 that if figures are to be freely movable throughout all space in oo#f ways1 no other three-dimensional spaces than the above four are possible4 ,gauss4 ,gauss published nothing on the theory of parallels1 and it was not generally known until after his death that he had interested himself in that theory from a very early date4 ,in #agii he announces that ,euclidean geometry would follow from the assumption that a triangle can be drawn greater than any given triangle4 ,though unwilling to assume this1 we find him in #ahjd still hoping to prove the postulate of parallels4 ,in #ahcj he announces his conviction that geometry is not an a priori science2 in the following year he explains that non-,euclidean geometry is free from contradictions1 and that1 in this system1 the angles of a triangle diminish without limit when all the sides are increased4 ,he also gives for the circumference of a circle of radius r the formula pik7e7r_/k7 - e7r-_/k771 where k is a constant depending upon the nature of the space4 ,in #ahcb1 in reply to the receipt of ,bolyai's ,appendix1 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 triangle4 ,from these and a few other remarks it appears that ,gauss possessed the foundations of hyperbolic geometry1 which he was probably the first to regard as perhaps true4 ,it is not known with certainty whether he influenced ,lobatchewsky and ,bolyai1 but the evidence we possess is against such a view4#g ,lobatchewsky4 ,the first to publish a non-,euclidean geometry was ,nicholas ,lobatchewsky1 professor of mathematics in the new university of ,kazan4#h ,in the place of the disputed postulate he puts the following3 ,all straight lines which1 in a plane1 radiate from a given point1 can1 with respect to any other straight line in the same plane1 be divided into two classes1 the intersecting and the non-intersecting4 ,the boundary line of the one and the other class is called parallel to the given line4 ,it follows that there are two parallels to the given line through any point1 each meeting the line at infinity1 like a ,euclidean parallel4 7,hence a line has two distinct points at infinity1 and not one only as in ordinary geometry47 ,the two parallels to a line through a point make equal acute angles with the perpendicular to the line through the point4 ,if p be the length of the perpendicular1 either of these angles is denoted by ,pi7p74 ,the determination of ,pi7p7 is the chief problem 7cf4 equation 7#f7 above72 it appears finally that1 with a suitable choice of the unit of length1 tan #a_/b ,pi7p7 e7-p74 ,before obtaining this result it is shown that spherical trigonometry is unchanged1 and that the normals to a circle or a sphere still pass through its centre4 ,when the radius of the circle or sphere becomes infinite all these normals become parallel1 but the circle or sphere does not become a straight line or plane4 ,it becomes what ,lobatchewsky calls a limit-line or limit-surface4 ,the geometry on such a surface is shown to be ,euclidean1 limit-lines replacing ,euclidean straight lines4 7,it is1 in fact1 a surface of zero measure of curvature47 ,by the help of these propositions ,lobatchewsky obtains the above value of ,pi7p71 and thence the solution of triangles4 ,he points out that his formulae result from those of spherical trigonometry by substituting ia1 ib1 ic1 for the sides a1 b1 c4 ,bolyai4 ,john ,bolyai1 a ,hungarian1 obtained results closely corresponding to those of ,lobatchewsky4 ,these he published in an appendix to a work by his father1 entitled ,appendix ,scientiam spatii absolute veram exhibens3 a veritate aut falsitate ,axiomatis ,,xi4 ,euclidei 7a priori haud unquam decidenda7 independentem3 adjecta ad casum falsitatis1 quadratura circuli geometrica4#i ,this work was published in #ahca1 but its conception dates from #ahbc4 ,it reveals a profounder appreciation of the importance of the new ideas1 but otherwise differs little from ,lobatchewsky's4 ,both men point out that ,euclidean geometry as a limiting case of their own more general system1 that the geometry of very small spaces is always approximately ,euclidean1 that no a priori grounds exist for a decision1 and that observation can only give an approximate answer4 ,bolyai gives also1 as his title indicates1 a geometrical construction1 in hyperbolic space1 for the quadrature of the circle1 and shows that the area of the greatest possible triangle1 which has all its sides parallel and all its angles zero1 is piiota#b1 where i is what we should now call the space-constant4 ,riemann4 ,the works of ,lobatchewsky and ,bolyai1 though known and valued by ,gauss1 remained obscure and ineffective until1 in #ahff1 they were translated into ,french by ,j4 ,houel4 ,but at this time ,riemann's dissertation1 ,uber die ,hypothesen1 welche der ,geometrie zu ,grunde liegen1#aj was already about to be published4 ,in this work ,riemann1 without any knowledge of his predecessors in the same field1 inaugurated a far more profound discussion1 based on a far more general standpoint2 and by its publication in #ahfg the attention of mathematicians and philosophers was at last secured4 7,the dissertation dates from #ahed1 but owing to changes which ,riemann wished to make in it1 it remained unpublished until after his death47 ,definition of a manifold4 ,riemann's work contains two fundamental conceptions1 that of a manifold and that of the measure of curvature of a continuous manifold possessed of what he calls flatness in the smallest parts4 ,by means of these conceptions space is made to appear at the end of a gradual series of more and more specialized conceptions4 ,conceptions of magnitude1 he explains1 are only possible where we have a general conception capable of determination in various ways4 ,the manifold consists of all these various determinations1 each of which is an element of the manifold4 ,the passage from one element to another may be discrete or continuous2 the manifold is called discrete or continuous accordingly4 ,where it is discrete two portions of it can be compared1 as to magnitude1 by counting2 where continuous1 by measurement4 ,but measurement demands superposition1 and consequently some magnitude independent of its place in the manifold4 ,in passing1 in a continuous manifold1 from one element to another in a determinate way1 we pass through a series of intermediate terms1 which form a one-dimensional manifold4 ,if this whole manifold be similarly caused to pass over into another1 each of its elements passes through a one-dimensional manifold1 and thus on the whole a two-dimensional manifold is generated4 ,in this way we can proceed to n dimensions4 ,conversely1 a manifold of n dimensions can be analysed into one of one dimension and one of 7n - #a7 dimensions4 ,by repetitions of this process the position of an element may be at last determined by n magnitudes4 ,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 7#ahdg7 of his ,ausdehnungslehre4#aa ,measure of curvature4 ,both concepts have been elaborated and superseded by the modern procedure in respect to the axioms of geometry1 and by the conception of abstract geometry involved therein4 ,riemann proceeds to specialize the manifold by considerations as to measurement4 ,if measurement is to be possible1 some magnitude1 we saw1 must be independent of position2 let us consider manifolds in which lengths of lines are such magnitudes1 so that every line is measurable by every other4 ,the coordinates of a point being x#a1 x#b1 444 xn1 let us confine ourselves to lines along which the ratios dx#a 3 dx#b 3 444 3 dxn alter continuously4 ,let us also assume that the element of length1 ds1 is unchanged 7to the first order7 when all its points undergo the same infinitesimal motion4 ,then if all the increments dx be altered in the same ratio1 ds is also altered in this ratio4 ,hence ds is a homogeneous function of the first degree of the increments dx4 ,moreover1 ds must be unchanged when all the dx change sign4 ,the simplest possible case is1 therefore1 that in which ds is the square root of a quadratic function of the dx4 ,this case includes space1 and is alone considered in what follows4 ,it is called the case of flatness in the smallest parts4 ,its further discussion depends upon the measure of curvature1 the second of ,riemann's fundamental conceptions4 ,this conception1 derived from the theory of surfaces1 is applied as follows4 ,any one of the shortest lines which issue from a given point 7say the origin7 is completely determined by the initial ratios of the dx4 ,two such lines1 defined by dx and deltax say1 determine a pencil1 or one-dimensional series1 of shortest lines1 any one of which is defined by lambdadx mudeltax1 where the parameter lambda 3 mu may have any value4 ,this pencil generates a two-dimensional series of points1 which may be regarded as a surface1 and for which we may apply ,gauss's formula for the measure of curvature at any point4 ,thus at every point of our manifold there is a measure of curvature corresponding to every such pencil2 but all these can be found when n4_/7n-#a7_/b of them are known4 ,if figures are to be freely movable1 it is necessary and sufficient that the measure of curvature should be the same for all points and all directions at each point4 ,where this is the case1 if alpha be the measure of curvature1 the linear element can be put into the form ds root7,sigmadx#b7_/7a #a_/d;alpha,sigmax#b74 ,if alpha be positive1 space is finite1 though still unbounded1 and every straight line is closed--a possibility first recognized by ,riemann4 ,it is pointed out that1 since the possible values of a form a continuous series1 observations cannot prove that our space is strictly ,euclidean4 ,it is also regarded as possible that1 in the infinitesimal1 the measure of curvature of our space should be variable4 ,there are four points in which this profound and epoch-making work is open to criticism or development--7#a7 the idea of a manifold requires more precise determination2 7#b7 the introduction of coordinates is entirely unexplained and the requisite presuppositions are unanalysed2 7#c7 the assumption that ds is the square root of a quadratic function of dx#a1 dx#b1 444 is arbitrary2 7#d7 the idea of superposition1 or congruence1 is not adequately analysed4 ,the modern solution of these difficulties is properly considered in connexion with the general subject of the axioms of geometry4 ,helmholtz4 ,the publication of ,riemann's dissertation was closely followed by two works of ,hermann von ,helmholtz1#ab again undertaken in ignorance of the work of predecessors4 ,in these a proof is attempted that ds must be a rational integral quadratic function of the increments of the coordinates4 ,this proof has since been shown by ,lie to stand in need of correction 7see ,,vii4 ,axioms of ,geometry74 ,helmholtz's remaining works on the subject#ac are of almost exclusively philosophical interest4 ,we shall return to them later4 ,beltrami4 ,the only other writer of importance in the second period is ,eugenio ,beltrami1 by whom ,riemann's work was brought into connexion with that of ,lobatchewsky and ,bolyai4 ,as he gave1 by an elegant method1 a convenient ,euclidean interpretation of hyperbolic plane geometry1 his results will be stated at some length#ad4 ,the ,saggio shows that ,lobatchewsky's plane geometry holds in ,euclidean geometry on surfaces of constant negative curvature1 straight lines being replaced by geodesics4 ,such surfaces are capable of a conformal representation on a plane1 by which geodesics are represented by straight lines4 ,hence if we take1 as coordinates on the surface1 the ,cartesian coordinates of corresponding points on the plane1 the geodesics must have linear equations4 ,hence it follows that ds#b ,r#bw7-#d77;alpha#b - v#b7;du#b #buvdudv 7alpha#b - u#b7;dv#b where w#b alpha#b - u#b - v#b1 and 7-#a7_/,r#b is the measure of curvature of our surface 7note that k gamma as used above74 ,the angle between two geodesics u const41 v const4 is theta1 where cos theta uv_/root7alpha#b - u#b77;alpha#b - v#b71 sin theta aw_/root7a#b - u#b77;a#b - v#b74 ,thus u #j is orthogonal to all geodesies v const41 and vice versa4 ,in order that sin theta may be real1 w#b must be positive2 thus geodesics have no real intersection when the corresponding straight lines intersect outside the circle u#b v#b alpha#b4 ,when they intersect on this circle1 theta #j4 ,thus ,lobatchewsky's parallels are represented by straight lines intersecting on the circle4 ,again1 transforming to polar coordinates u r cos mu1 v r sin mu1 and calling rho the geodesic distance of u1 v from the origin1 we have1 for a geodesic through the origin1 drho ,radr_/7a#b - r#b71 rho #a_/b,r log7a r7_/7a - r71 r a tan h 7rho_/,r74 ,thus points on the surface corresponding to points in the plane on the limiting circle r a1 are all at an infinite distance from the origin4 ,again1 considering r constant1 the arc of a geodesic circle subtending an angle mu at the origin is sigma ,rrmu_/root7a#b - r#b7 mu,r sin h 7rho_/,r71 whence the circumference of a circle of radius rho is #bpi,r sin h 7rho_/,r74 ,again1 if alpha be the angle between any two geodesics ,v - v m7,u - u71 ,v - v n7,u - u71 then tan alpha a7n - m7w_/7#a mn7a#b - 7v - mu7 7v - nu74 ,thus alpha is imaginary when u1 v is outside the limiting circle1 and is zero when1 and only when1 u1 v is on the limiting circle4 ,all these results agree with those of ,lobatchewsky and ,bolyai4 ,the maximum triangle1 whose angles are all zero1 is represented in the auxiliary plane by a triangle inscribed in the limiting circle4 ,the angle of parallelism is also easily obtained4 ,the perpendicular to v #j at a distance delta from the origin is u a tan h 7delta_/,r71 and the parallel to this through the origin is u v sin h 7delta_/,r74 ,hence ,pi 7delta71 the angle which this parallel makes with v #j1 is given by tan ,pi7delta7 4 sin h 7delta_/,r7 #a1 or tan #a_/b,pi7delta7 e7-delta_/,r7 which is ,lobatchewsky's formula4 ,we also obtain easily for the area of a triangle the formula ,r#b7pi - ,a - ,b - ,c74 ,beltrami's treatment connects two curves which1 in the earlier treatment1 had no connexion4 ,these are limit-lines and curves of constant distance from a straight line4 ,both may be regarded as circles1 the first having an infinite1 the second an imaginary radius4 ,the equation to a circle of radius rho and centre u#jv#j is 7a#b - uu#j - vv#j7b cos h#b 7rho_/,r7w#jbw#b ,c#bw#b 7say74 ,this equation remains real when rho is a pure imaginary1 and remains finite when w#j #j1 provided rho becomes infinite in such a way that w#j cos h 7rho_/,r7 remains finite4 ,in the latter case the equation represents a limit-line4 ,in the former case1 by giving different values to ,c1 we obtain concentric circles with the imaginary centre u#jv#j4 ,one of these1 obtained by putting ,c #j1 is the straight line a#b - uu#j - vv#j #j4 ,hence the others are each throughout at a constant distance from this line4 7,it may be shown that all motions in a hyperbolic plane consist1 in a general sense1 of rotations2 but three types must be distinguished according as the centre is real1 imaginary or at infinity4 ,all points describe1 accordingly1 one of the three types of circles47 ,the above ,euclidean interpretation fails for three or more dimensions4 ,in the ,teoria fondamentale1 accordingly1 where n dimensions are considered1 ,beltrami treats hyperbolic space in a purely analytical spirit4 ,the paper shows that ,lobatchewsky's space of any number of dimensions has1 in ,riemann's sense1 a constant negative measure of curvature4 ,beltrami starts with the formula 7analogous to that of the ,saggio7 ds#b ,r#bx7-#b77;dx#b dx#ab dx#bb 444 dxn#b7 where x#b x#ab x#bb 444 xn#b a#b4 ,he shows that geodesics are represented by linear equations between x#a1 x#b1 4441 xn1 and that the geodesic distance rho between two points x and x' is given by rho a#b - x#ax'#a - x#bx'#b - 444 - xn x'n cosh ----- ---------------------------------------- ----------------------------------- ,r 7a#b - x#ab - x#bb - 444 - xn#b77;a#b - x'#ab - x'#bb - 444 - x'n#b7a_/b 7a formula practically identical with ,cayley's1 though obtained by a very different method74 ,in order to show that the measure of curvature is constant1 we make the substitutions x#a rlambda#a1 x#b rlambda#b 444 xn rlambdan1 where ,sigmalambda#b #a4 ,hence ds#b 7,radr_/7a#b - r#b77b ,r#br#b;d,delta#b_/7;a#b - r#b74 where d,delta#b ,sigmadlambda#b4 ,also calling rho the geodesic distance from the origin1 we have rho a rho r cosh ----- -----------------1 sinh ----- -----------------4 ,r root7a#b - r#b7 ,r root7a#b - r#b7 ,hence ds#b drho#b 7,r sin h 7rho_/,r77#b;d,delta#b4 ,putting z#a rholambda#a1 z#b rholambda#b1 444 zn rholambdan1 we obtain #a _/ ,r rho#b ds#b ,sigmadz#b ------ 7 ----- sinh ----- 7 - #a ,sigma7zi dzk - zk dzi7#b4 rho#b rho ,r _/ ,hence when rho is small1 we have approximately #a ds#b ,sigmadz#b ----,sigma7zi dzk - zk dzi7#b 7#a74 #c,r#b ,considering a surface element through the origin1 we may choose our axes so that1 for this element1 z#c ,z#d 444 zn #j4 ,thus #a ds#b dz#ab dz#bb ----7z#a;dz#b - z#b;dz#a7b 7#b74 #c,r#b ,now the area of the triangle whose vertices are 7#j1 #j71 7z#a1 z#b71 7dz#a1 dz#b7 is #a_/b7z#a1 dz#b - z#b;dz#a74 ,hence the quotient when the terms of the fourth order in 7#b7 are divided by the square of this triangle is #d_/c,r#b2 hence1 returning to general axes1 the same is the quotient when the terms of the fourth order in 7#a7 are divided by the square of the triangle whose vertices are 7#j1 #j1 444 #j71 7z#a1 z#b1 z#c1 444 zn71 7dz#a1 dz#b1 dz#c 444 dzn74 ,but -#c_/d of this quotient is defined by ,riemann as the measure of curvature4#ae ,hence the measure of curvature is -#a_/,r#b1 i4e4 is constant and negative4 ,the properties of parallels1 triangles1 c41 are as in the ,saggio4 ,it is also shown that the analogues of limit surfaces have zero curvature2 and that spheres of radius rho have constant positive curvature #a_/,r#b sinh#b 7rho_/,r71 so that spherical geometry may be regarded as contained in the pseudo-spherical 7as ,beltrami calls ,lobatchewsky's system74 ,transition to the projective method4 ,the ,saggio1 as we saw1 gives a ,euclidean interpretation confined to two dimensions4 ,but a consideration of the auxiliary plane suggests a different interpretation1 which may be extended to any number of dimensions4 ,if1 instead of referring to the pseudosphere1 we merely define distance and angle1 in the ,euclidean plane1 as those functions of the coordinates which gave us distance and angle on the pseudosphere1 we find that the geometry of our plane has become ,lobatchewsky's4 ,all the points of the limiting circle are now at infinity1 and points beyond it are imaginary4 ,if we give our circle an imaginary radius the geometry on the plane becomes elliptic4 ,replacing the circle by a sphere1 we obtain an analogous representation for three dimensions4 ,instead of a circle or sphere we may take any conic or quadric4 ,with this definition1 if the fundamental quadric be ,sigma7xx7 #j1 and if ,sigma7xx7' be the polar form of ,sigma7xx71 the distance rho between x and x' is given by the projective formula cos7rho_/k7 ,sigmaxx'_/,sigma7xx74,sigmax'x'#a_/b4 ,that this formula is projective is rendered evident by observing that e7-#b;irho_/k7 is the anharmonic ratio of the range consisting of the two points and the intersections of the line joining them with the fundamental quadric4 ,with this we are brought to the third or projective period4 ,the method of this period is due to ,cayley2 its application to previous non-,euclidean geometry is due to ,klein4 ,the projective method contains a generalization of discoveries already made by ,laguerre#af in #ahec as regards ,euclidean geometry4 ,the arbitrariness of this procedure of deriving metrical geometry from the properties of conics is removed by ,lie's theory of congruence4 ,we then arrive at the stage of thought which finds its expression in the modern treatment of the axioms of geometry4 ,the two kinds of elliptic space4 ,the projective method leads to a discrimination1 first made by ,klein1#ag of two varieties of ,riemann's space2 ,klein calls these elliptic and spherical4 ,they are also called the polar and antipodal forms of elliptic space4 ,the latter names will here be used4 ,the difference is strictly analogous to that between the diameters and the points of a sphere4 ,in the polar form two straight lines in a plane always intersect in one and only one point2 in the antipodal form they intersect always in two points1 which are antipodes4 ,according to the definition of geometry adopted in section ,,vii4 7,axioms of ,geometry71 the antipodal form is not to be termed geometry1 since any pair of coplanar straight lines intersect each other in two points4 ,it may be called a quasi-geometry4 ,similarly in the antipodal form two diameters always determine a plane1 but two points on a sphere do not determine a great circle when they are antipodes1 and two great circles always intersect in two points4 ,again1 a plane does not form a boundary among lines through a point3 we can pass from any one such line to any other without passing through the plane4 ,but a great circle does divide the surface of a sphere4 ,so1 in the polar form1 a complete straight line does not divide a plane1 and a plane does not divide space1 and does not1 like a ,euclidean plane1 have two sides4#ah ,but1 in the antipodal form1 a plane is1 in these respects1 like a ,euclidean plane4 ,it is explained in section ,,vii4 in what sense the metrical geometry of the material world can be considered to be determinate and not a matter of arbitrary choice4 ,the scientific question as to the best available evidence concerning the nature of this geometry is one beset with difficulties of a peculiar kind4 ,we are obstructed by the fact that all existing physical science assumes the ,euclidean hypothesis4 ,this hypothesis has been involved in all actual measurements of large distances1 and in all the laws of astronomy and physics4 ,the principle of simplicity would therefore lead us1 in general1 where an observation conflicted with one or more of those laws1 to ascribe this anomaly1 not to the falsity of ,euclidean geometry1 but to the falsity of the laws in question4 ,this applies especially to astronomy4 ,on the earth our means of measurement are many and direct1 and so long as no great accuracy is sought they involve few scientific laws4 ,thus we acquire1 from such direct measurements1 a very high degree of probability that the space-constant1 if not infinite1 is yet large as compared with terrestrial distances4 ,but astronomical distances and triangles can only be measured by means of the received laws of astronomy and optics1 all of which have been established by assuming the truth of the ,euclidean hypothesis4 ,it therefore remains possible 7until a detailed proof of the contrary is forthcoming7 that a large but finite space-constant1 with different laws of astronomy and optics1 would have equally explained the phenomena4 ,we cannot1 therefore1 accept the measurements of stellar parallaxes1 c41 as conclusive evidence that the space-constant is large as compared with stellar distances4 ,for the present1 on grounds of simplicity1 we may rightly adopt this view2 but it must remain possible that1 in view of some hitherto undiscovered discrepancy1 a slight correction of the sort suggested might prove the simplest alternative4 ,but conversely1 a finite parallax for very distant stars1 or a negative parallax for any star1 could not be accepted as conclusive evidence that our geometry is non-,euclidean1 unless it were shown--and this seems scarcely possible--that no modification of astronomy or optics could account for the phenomenon4 ,thus although we may admit a probability that the space-constant is large in comparison with stellar distances1 a conclusive proof or disproof seems scarcely possible4 ,finally1 it is of interest to note that1 though it is theoretically possible to prove1 by scientific methods1 that our geometry is non-,euclidean1 it is wholly impossible to prove by such methods that it is accurately ,euclidean4 ,for the unavoidable errors of observation must always leave a slight margin in our measurements4 ,a triangle might be found whose angles were certainly greater1 or certainly less1 than two right angles2 but to prove them exactly equal to two right angles must always be beyond our powers4 ,if1 therefore1 any man cherishes a hope of proving the exact truth of ,euclid1 such a hope must be based1 not upon scientific1 but upon philosophical considerations4 ,,bibliography4--,the bibliography appended to section ,,vii4 should be consulted in this connexion4 ,also1 in addition to the citations already made1 the following works may be mentioned4 ,for ,lobatchewsky's writings1 cf4 ,urkunden zur ,geschichte der nichteuklidischen ,geometrie1 i41 ,nikolaj ,iwanowitsch ,lobatschefsky1 by ,f4 ,engel and ,p4 ,stackel 7,leipzig1 #ahih74 ,for ,john ,bolyai's ,appendix1 cf4 ,absolute ,geometrie nach ,johann ,bolyai1 by ,j4 ,frischauf 7,leipzig1 #ahgb71 and also the new edition of his father's large work1 ,tentamen 4441 published by the ,mathematical ,society of ,budapest2 the second volume contains the appendix4 ,cf4 also ,j4 ,frischauf1 ,elemente der absoluten ,geometrie 7,leipzig1 #ahgf72 ,m4,l4 ,gerard1 ,sur la geometrie non-,euclidienne 7thesis for doctorate7 7,paris1 #ahib72 de ,tilly1 ,essai sur les principes fondamentales de la geometrie et de la mecanique 7,bordeaux1 #ahgi72 ,sir ,r4,s4 ,ball1 ,on the ,theory of ,content1 ,trans4 ,roy4 ,irish ,acad4 vol4 xxix4 7#ahhi72 ,f4 ,lindemann1 ,mechanik bei projectiver ,maasbestimmung1 ,math4 ,annal4 vol4 vii42 ,w4,k4 ,clifford1 ,preliminary ,sketch of ,biquaternions1 ,proc4 of ,lond4 ,math4 ,soc4 7#ahgc71 and ,coll4 ,works2 ,a4 ,buchheim1 ,on the ,theory of ,screws in ,elliptic ,space1 ,proc4 ,lond4 ,math4 ,soc4 vols4 xv41 xvi41 xvii42 ,h4 ,cox1 ,on the ,application of ,quaternions and ,grassmann's ,algebra to different ,kinds of ,uniform ,space1 ,trans4 ,camb4 ,phil4 ,soc4 7#ahhb72 ,m4 ,dehn1 ,die ,legendarischen ,satze uber die ,winkelsumme im ,dreieck1 ,math4 ,ann4 vol4 #ec 7#aijj71 and ,uber den ,rauminhalt1 ,math4 ,annal4 vol4 #ee 7#aijb74 ,for expositions of the whole subject1 cf4 ,f4 ,klein1 ,nicht-,euklidische ,geometrie 7,gottingen1 #ahic72 ,r4 ,bonola1 ,la ,geometria non-,euclidea 7,bologna1 #aijf72 ,p4 ,barbarin1 ,la ,geometrie non-,euclidienne 7,paris1 #aijb72 ,w4 ,killing1 ,die nicht-,euklidischen ,raumformen in analytischer ,behandlung 7,leipzig1 #ahhe74 ,the last-named work also deals with geometry of more than three dimensions2 in this connexion cf4 also ,g4 ,veronese1 ,fondamenti di geometria a piu dimensioni ed a piu specie di unita rettilinee 444 7,padua1 #ahia1 ,german translation1 ,leipzig1 #ahid72 ,g4 ,fontene1 ,l',hyperespace a 7n-#a7 dimensions 7,paris1 #ahib72 and ,a4,n4 ,whitehead1 loc4 cit4 ,cf4 also ,e4 ,study1 ,uber nicht-,euklidische und ,liniengeometrie1 ,jahr4 d4 ,deutsch4 ,math4 ,ver4 vol4 xv4 7#aijf72 ,w4 ,burnside1 ,on the ,kinematics of non-,euclidean ,space1 ,proc4 ,lond4 ,math4 ,soc4 vol4 xxvi4 7#ahid74 ,a bibliography on the subject up to #ahgh has been published by ,g4,b4 ,halsted1 ,amer4 ,journ4 of ,math4 vols4 i4 and ii42 and one up to #aijj by ,r4 ,bonola1 ,index operum ad geometriam absolutam spectantium 444 7#aijb1 and ,leipzig1 #aijc74 7,b4 ,a4 ,w4 ,r42 ,a4 ,n4 ,w47 ,,vii4 ,,axioms ,,of ,,geometry ,theories of space4 ,until the discovery of the non-,euclidean geometries 7,lobatchewsky1 #ahbf and #ahbi2 ,j4 ,bolyai1 #ahcb2 ,b4 ,riemann1 #ahed71 geometry was universally considered as being exclusively the science of existent space4 7,see section ,,vi4 ,non-,euclidean ,geometry47 ,in respect to the science1 as thus conceived1 two controversies may be noticed4 ,first1 there is the controversy respecting the absolute and relational theories of space4 ,according to the absolute theory1 which is the traditional view 7held explicitly by ,newton71 space has an existence1 in some sense whatever it may be1 independent of the bodies which it contains4 ,the bodies occupy space1 and it is not intrinsically unmeaning to say that any definite body occupies this part of space1 and not that part of space1 without reference to other bodies occupying space4 ,according to the relational theory of space1 of which the chief exponent was ,leibnitz1#ai space is nothing but a certain assemblage of the relations between the various particular bodies in space4 ,the idea of space with no bodies in it is absurd4 ,accordingly there can be no meaning in saying that a body is here and not there1 apart from a reference to the other bodies in the universe4 ,thus1 on this theory1 absolute motion is intrinsically unmeaning4 ,it is admitted on all hands that in practice only relative motion is directly measurable4 ,newton1 however1 maintains in the ,principia 7scholium to the #hth definition7 that it is indirectly measurable by means of the effects of centrifugal force as it occurs in the phenomena of rotation4 ,this irrelevance of absolute motion 7if there be such a thing7 to science has led to the general adoption of the relational theory by modern men of science4 ,but no decisive argument for either view has at present been elaborated4#bj ,kant's view of space as being a form of perception at first sight appears to cut across this controversy4 ,but he1 saturated as he was with the spirit of the ,newtonian physics1 must 7at least in both editions of the ,critique7 be classed with the upholders of the absolute theory4 ,the form of perception has a type of existence proper to itself independently of the particular bodies which it contains4 ,for example he writes3#ba ,space does not represent any quality of objects by themselves1 or objects in their relation to one another1 i4e4 space does not represent any determination which is inherent in the objects themselves1 and would remain1 even if all subjective conditions of intuition were removed4 ,axioms4 ,the second controversy is that between the view that the axioms applicable to space are known only from experience1 and the view that in some sense these axioms are given a priori4 ,both these views1 thus broadly stated1 are capable of various subtle modifications1 and a discussion of them would merge into a general treatise on epistemology4 ,the cruder forms of the a priori view have been made quite untenable by the modern mathematical discoveries4 ,geometers now profess ignorance in many respects of the exact axioms which apply to existent space1 and it seems unlikely that a profound study of the question should thus obliterate a priori intuitions4 ,another question irrelevant to this article1 but with some relevance to the above controversy1 is that of the derivation of our perception of existent space from our various types of sensation4 ,this is a question for psychology4#bb ,definition of ,abstract ,geometry4--,existent space is the subject matter of only one of the applications of the modern science of abstract geometry1 viewed as a branch of pure mathematics4 ,geometry has been defined#bc as the study of series of two or more dimensions4 ,it has also been defined#bd as the science of cross classification4 ,these definitions are founded upon the actual practice of mathematicians in respect to their use of the term ,geometry4 ,either of them brings out the fact that geometry is not a science with a determinate subject matter4 ,it is concerned with any subject matter to which the formal axioms may apply4 ,geometry is not peculiar in this respect4 ,all branches of pure mathematics deal merely with types of relations4 ,thus the fundamental ideas of geometry 7e4g4 those of points and of straight lines7 are not ideas of determinate entities1 but of any entities for which the axioms are true4 ,and a set of formal geometrical axioms cannot in themselves be true or false1 since they are not determinate propositions1 in that they do not refer to a determinate subject matter4 ,the axioms are propositional functions4#be ,when a set of axioms is given1 we can ask 7#a7 whether they are consistent1 7#b7 whether their existence theorem is proved1 7#c7 whether they are independent4 ,axioms are consistent when the contradictory of any axiom cannot be deduced from the remaining axioms4 ,their existence theorem is the proof that they are true when the fundamental ideas are considered as denoting some determinate subject matter1 so that the axioms are developed into determinate propositions4 ,it follows from the logical law of contradiction that the proof of the existence theorem proves also the consistency of the axioms4 ,this is the only method of proof of consistency4 ,the axioms of a set are independent of each other when no axiom can be deduced from the remaining axioms of the set4 ,the independence of a given axiom is proved by establishing the consistency of the remaining axioms of the set1 together with the contradictory of the given axiom4 ,the enumeration of the axioms is simply the enumeration of the hypotheses#bf 7with respect to the undetermined subject matter7 of which some at least occur in each of the subsequent propositions4 ,any science is called a geometry if it investigates the theory of the classification of a set of entities 7the points7 into classes 7the straight lines71 such that 7#a7 there is one and only one class which contains any given pair of the entities1 and 7#b7 every such class contains more than two members4 ,in the two geometries1 important from their relevance to existent space1 axioms which secure an order of the points on any line also occur4 ,these geometries will be called ,projective ,geometry and ,descriptive ,geometry4 ,in projective geometry any two straight lines in a plane intersect1 and the straight lines are closed series which return into themselves1 like the circumference of a circle4 ,in descriptive geometry two straight lines in a plane do not necessarily intersect1 and a straight line is an open series without beginning or end4 ,ordinary ,euclidean geometry is a descriptive geometry2 it becomes a projective geometry when the so-called points at infinity are added4 ,projective ,geometry4 ,projective geometry may be developed from two undefined fundamental ideas1 namely1 that of a point and that of a straight line4 ,these undetermined ideas take different specific meanings for the various specific subject matters to which projective geometry can be applied4 ,the number of the axioms is always to some extent arbitrary1 being dependent upon the verbal forms of statement which are adopted4 ,they will be presented#bg here as twelve in number1 eight being axioms of classification1 and four being axioms of order4 ,axioms of ,classification4--,the eight axioms of classification are as follows3 #a4 ,points form a class of entities with at least two members4 #b4 ,any straight line is a class of points containing at least three members4 #c4 ,any two distinct points lie in one and only one straight line4 #d4 ,there is at least one straight line which does not contain all the points4 #e4 ,if ,a1 ,b1 ,c are non-collinear points1 and ,a' is on the straight line ,,bc1 and ,b' is on the straight line ,,ca1 then the straight lines ,,aa' and ,,bb' possess a point in common4 ,definition4--,if ,a1 ,b1 ,c are any three non-collinear points1 the plane ,,abc is the class of points lying on the straight lines joining ,a with the various points on the straight line ,,bc4 #f4 ,there is at least one plane which does not contain all the points4 #g4 ,there exists a plane alpha1 and a point ,a not incident in alpha1 such that any point lies in some straight line which contains both ,a and a point in alpha4 ,definition4--,harm4 7,,abcd7 symbolizes the following conjoint statements3 7#a7 that the points ,a1 ,b1 ,c1 ,d are collinear1 and 7#b7 that a quadrilateral can be found with one pair of opposite sides intersecting at ,a1 with the other pair intersecting at ,c1 and with its diagonals passing through ,b and ,d respectively4 ,then ,b and ,d are said to be harmonic conjugates with respect to ,a and ,c4 #h4 ,harm4 7,,abcd7 implies that ,b and ,d are distinct points4 ,in the above axioms #d secures at least two dimensions1 axiom #e is the fundamental axiom of the plane1 axiom #f secures at least three dimensions1 and axiom #g secures at most three dimensions4 ,from axioms #a-e it can be proved that any two distinct points in a straight line determine that line1 that any three non-collinear points in a plane determine that plane1 that the straight line containing any two points in a plane lies wholly in that plane1 and that any two straight lines in a plane intersect4 ,from axioms #a-f ,desargue's well-known theorem on triangles in perspective can be proved4 ,the enunciation of this theorem is as follows3 ,if ,,abc and ,a',b',c' are two coplanar triangles such that the lines ,,aa'1 ,,bb'1 ,,cc' are concurrent1 then the three points of intersection of ,,bc and ,b',c' of ,,ca and ,c',a'1 and of ,,ab and ,a',b' are collinear2 and conversely if the three points of intersection are collinear1 the three lines are concurrent4 ,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 two1 but in perspective with each of them4 ,it has been proved#bh that ,desargues's theorem cannot be deduced from axioms #a-e1 that is1 if the geometry be confined to two dimensions4 ,all the proofs proceed by the method of producing a specification of points and straight lines which satisfies axioms #a-e1 and such that ,desargues's theorem does not hold4 ,it follows from axioms #a-e that ,harm4 7,,abcd7 implies ,harm4 7,,adcb7 and ,harm4 7,,cbad71 and that1 if ,a1 ,b1 ,c be any three distinct collinear points1 there exists at least one point ,d such that ,harm4 7,,abcd74 ,but it requires ,desargues's theorem1 and hence axiom #f1 to prove that ,harm4 7,,abcd7 and ,harm4 7,,abcd'7 imply the identity of ,d and ,d'4 ,the necessity for axiom #h has been proved by ,g4 ,fano1#bi who has produced a three dimensional geometry of fifteen points1 i4e4 a method of cross classification of fifteen entities1 in which each straight line contains three points1 and each plane contains seven straight lines4 ,in this geometry axiom #h does not hold4 ,also from axioms #a-f and #h it follows that ,harm4 7,,abcd7 implies ,harm4 7,,bcda74 ,definitions4--,when two plane figures can be derived from one another by a single projection1 they are said to be in perspective4 ,when two plane figures can be derived one from the other by a finite series of perspective relations between intermediate figures1 they are said to be projectively related4 ,any property of a plane figure which necessarily also belongs to any projectively related figure1 is called a projective property4 ,the following theorem1 known from its importance as the fundamental theorem of projective geometry1 cannot be proved#cj from axioms #a-h4 ,the enunciation is3 ,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 other4 ,this theorem is equivalent#ca 7assuming axioms #a-h7 to another theorem1 known as ,pappus's ,theorem1 namely3 ,if l and l' are two distinct coplanar lines1 and ,a1 ,b1 ,c are three distinct points on l1 and ,a'1 ,b'1 ,c' are three distinct points on l'1 then the three points of intersection of ,,aa' and ,b',c1 of ,a',b and ,,cc'1 of ,,bb' and ,c',a1 are collinear4 ,this theorem is obviously ,pascal's well-known theorem respecting a hexagon inscribed in a conic1 for the special case when the conic has degenerated into the two lines l and l'4 ,another theorem also equivalent 7assuming axioms #a-h7 to the fundamental theorem is the following3#cb ,if the three collinear pairs of points1 ,a and ,a'1 ,b and ,b'1 ,c and ,c'1 are such that the three pairs of opposite sides of a complete quadrangle pass respectively through them1 i4e4 one pair through ,a and ,a' respectively1 and so on1 and if also the three sides of the quadrangle which pass through ,a1 ,b1 and ,c1 are concurrent in one of the corners of the quadrangle1 then another quadrangle can be found with the same relation to the three pairs of points1 except that its three sides which pass through ,a1 ,b1 and ,c1 are not concurrent4 ,thus1 if we choose to take any one of these three theorems as an axiom1 all the theorems of projective geometry which do not require ordinal or metrical ideas for their enunciation can be proved4 ,also a conic can be defined as the locus of the points found by the usual construction1 based upon ,pascal's theorem1 for points on the conic through five given points4 ,but it is unnecessary to assume here any one of the suggested axioms2 for the fundamental theorem can be deduced from the axioms of order together with axioms #a-h4 ,axioms of ,order4--,it is possible to define 7cf4 ,pieri1 loc4 cit47 the property upon which the order of points on a straight line depends4 ,but to secure that this property does in fact range the points in a serial order1 some axioms are required4 ,a straight line is to be a closed series2 thus1 when the points are in order1 it requires two points on the line to divide it into two distinct complementary segments1 which do not overlap1 and together form the whole line4 ,accordingly the problem of the definition of order reduces itself to the definition of these two segments formed by any two points on the line2 and the axioms are stated relatively to these segments4 ,definition4--,if ,a1 ,b1 ,c are three collinear points1 the points on the segment ,,abc are defined to be those points such as ,x1 for which there exist two points ,y and ,y' with the property that ,harm4 7,,aycy'7 and ,harm4 7,,byxy'7 both hold4 ,the supplementary segment ,,abc is defined to be the rest of the points on the line4 ,this definition is elucidated by noticing that with our ordinary geometrical ideas1 if ,b and ,x are any two points between ,a and ,c1 then the two pairs of points1 ,a and ,c1 ,b and ,x1 define an involution with real double points1 namely1 the ,y and ,y' of the above definition4 ,the property of belonging to a segment ,,abc is projective1 since the harmonic relation is projective4 ,the first three axioms of order 7cf4 ,pieri1 loc4 cit47 are3 #i4 ,if ,a1 ,b1 ,c are three distinct collinear points1 the supplementary segment ,,abc is contained within the segment ,,bca4 #aj4 ,if ,a1 ,b1 ,c are three distinct collinear points1 the common part of the segments ,,bca and ,,cab is contained in the supplementary segment ,,abc4 #aa4 ,if ,a1 ,b1 ,c are three distinct collinear points1 and ,d lies ,in the segment ,,abc1 then the segment ,,adc is contained within the segment ,,abc4 ,from these axioms all the usual properties of a closed order follow4 ,it will be noticed that1 if ,a1 ,b1 ,c are any three collinear points1 ,c is necessarily traversed in passing from ,a to ,b by one route along the line1 and is not traversed in passing from ,a to ,b along the other route4 ,thus there is no meaning1 as referred to closed straight lines1 in the simple statement that ,c lies between ,a and ,b4 ,but there may be a relation of separation between two pairs of collinear points1 such as ,a and ,c1 and ,b and ,d4 ,the couple ,b and ,d is said to separate ,a and ,c1 if the four points are collinear and ,d lies in the segment complementary to the segment ,,abc4 ,the property of the separation of pairs of points by pairs of points is projective4 ,also it can be proved that ,harm4 7,,abcd7 implies that ,b and ,d separate ,a and ,c4 ,definitions4--,a series of entities arranged in a serial order1 open or closed1 is said to be compact1 if the series contains no immediately consecutive entities1 so that in traversing the series from any one entity to any other entity it is necessary to pass through entities distinct from either4 ,it was the merit of ,r4 ,dedekind and of ,g4 ,cantor explicitly to formulate another fundamental property of series4 ,the ,dedekind property#cc as applied to an open series can be defined thus3 ,an open series possesses the ,dedekind property1 if1 however1 it be divided into two mutually exclusive classes u and v1 which 7#a7 contain between them the whole series1 and 7#b7 are such that every member of u precedes in the serial order every member of v1 there is always a member of the series1 belonging to one of the two1 u or v1 which precedes every member of v 7other than itself if it belong to v71 and also succeeds every member of u 7other than itself if it belong to u74 ,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 v4 ,an open series is continuous if it is compact and possesses the ,dedekind property4 ,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 series4 ,thus the definitions of compactness and of the ,dedekind property can be at once transferred to a closed series4 #ab4 ,the last axiom of order is that there exists at least one straight line for which the point order possesses the ,dedekind property4 ,it follows from axioms #a-ab by projection that the ,dedekind property is true for all lines4 ,again the harmonic system ,,abc1 where ,a1 ,b1 ,c are collinear points1 is defined#cd thus3 take the harmonic conjugates ,a'1 ,b'1 ,c' of each point with respect to the other two1 again take the harmonic conjugates of each of the six points ,a1 ,b1 ,c1 ,a'1 ,b'1 ,c' with respect to each pair of the remaining five1 and proceed in this way by an unending series of steps4 ,the set of points thus obtained is called the harmonic system ,,abc4 ,it can be proved that a harmonic system is compact1 and that every segment of the line containing it possesses members of it4 ,furthermore1 it is easy to prove that the fundamental theorem holds for harmonic systems1 in the sense that1 if ,a1 ,b1 ,c are three points on a line l1 and ,a'1 ,b'1 ,c' are three points on a line l'1 and if by any two distinct series of projections ,a1 ,b1 ,c are projected into ,a'1 ,b'1 ,c'1 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'4 ,it now follows immediately that the fundamental theorem must hold for all the points on the lines l and l'1 since 7as has been pointed out7 harmonic systems are everywhere dense on their containing lines4 ,thus the fundamental theorem follows from the axioms of order4 ,a system of numerical coordinates can now be introduced1 possessing the property that linear equations represent planes and straight lines4 ,the outline of the argument by which this remarkable problem 7in that distance is as yet undefined7 is solved1 will now be given4 ,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 numbers1 so as to form with them a one-one correspondence4 ,the arbitrary elements in the establishment of this relation are the points on the line associated with #j1 #a and oo4 ,this association#ce is most easily effected by considering a class of projective relations of the line with itself1 called by ,f4 ,schur 7loc4 cit47 prospectivities4 ,let l 7fig4 #fi7 be the given line1 m and n any two lines intersecting at ,u on l1 ,s and ,s' two points on n4 ,then a projective relation between l and itself is formed by projecting l from ,s on to m1 and then by projecting m from ,s' back on to l4 ,all such projective relations1 however m1 n1 ,s and ,s' be varied1 are called prospectivities1 and ,u is the double point of the prospectivity4 ,if a point ,o on l is related to ,a by a prospectivity1 then all prospectivities1 which 7#a7 have the same double point ,u1 and 7#b7 relate ,o to ,a1 give the same correspondent 7,q1 in figure7 to any point ,p on the line l2 in fact they are all the same prospectivity1 however m1 n1 ,s1 and ,s' may have been varied subject to these conditions4 ,such a prospectivity will be denoted by 7,,oau#b74 ,illustration3 ,,fig4 #fi4 ,the sum of two prospectivities1 written 7,,oau#b7 7,,obu#b71 is defined to be that transformation of the line l into itself which is obtained by first applying the prospectivity 7,,oau#b7 and then applying the prospectivity 7,,obu#b74 ,such a transformation1 when the two summands have the same double point1 is itself a prospectivity with that double point4 ,illustration3 ,,fig4 #gj ,with this definition of addition it can be proved that prospectivities with the same double point satisfy all the axioms of magnitude4 ,accordingly they can be associated in a one-one correspondence with the positive and negative real numbers4 ,let ,e 7fig4 #gj7 be any point on l1 distinct from ,o and ,u4 ,then the prospectivity 7,,oeu#b7 is associated with unity1 the prospectivity 7,,oou#b7 is associated with zero1 and 7,,ouu#b7 with infinity4 ,the prospectivities of the type 7,,opu#b71 where ,p is any point on the segment ,,oeu1 correspond to the positive numbers2 also if ,p' is the harmonic conjugate of ,p with respect to ,o and ,u1 the prospectivity 7,,op',u#b7 is associated with the corresponding negative number4 7,the subjoined figure explains this relation of the positive and negative prospectivities47 ,then any point ,p on l is associated with the same number as is the prospectivity 7,,opu#b74 ,illustration3 ,,fig4 #ga4 ,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 points4 ,let the numbers1 assigned according to the preceding specification1 be said to be associated with the points according to the numeration-system 7,,oeu74 ,the introduction of a coordinate system for a plane is now managed as follows3 ,take any triangle ,,ouv in the plane1 and on the lines ,,ou and ,,ov establish the numeration systems 7,,oe#a,u7 and 7,,oe#b,v71 where ,e#a and ,e#b are arbitrarily chosen4 ,then 7cf4 fig4 #ga7 if ,m and ,n are associated with the numbers x and y according to these systems1 the coordinates of ,p are x and y4 ,it then follows that the equation of a straight line is of the form ax by c #j4 ,both coordinates of any point on the line ,,uv are infinite4 ,this can be avoided by introducing homogeneous coordinates ,x1 ,y1 ,z1 where x ,x_/,z1 and y ,y_/,z1 and ,z #j is the equation of ,,uv4 ,illustration3 ,,fig4 #gb4 ,the procedure for three dimensions is similar4 ,let ,,ouvw 7fig4 #gb7 be any tetrahedron1 and associate points on ,,ou1 ,,ov1 ,,ow with numbers according to the numeration systems 7,,oe#a,u71 7,,oe#b,v71 and 7,,oe#c,w74 ,let the planes ,,vwp1 ,,wup1 ,,uvp cut ,,ou1 ,,ov1 ,,ow in ,l1 ,m1 ,n respectively2 and let x1 y1 z be the numbers associated with ,l1 ,m1 ,n respectively4 ,then ,p is the point 7x1 y1 z74 ,also homogeneous coordinates can be introduced as before1 thus avoiding the infinities on the plane ,,uvw4 ,the cross ratio of a range of four collinear points can now be defined as a number characteristic of that range4 ,let the coordinates of any point ,pr of the range ,p#a ,p#b ,p#c ,p#d be lambdar a mur a' lambdar b mur b' -------------------------1 ------------------------1 lambdar mur lambdar mur lambdar c mur c' ------------------------1 7r #a1 #b1 #c1 #d7 lambdar mur and let 7lambdar mus7 be written for lambdar mus -lambdas mur4 ,then the cross ratio ,p#a ,p#b ,p#c ,p#d is defined to be the number 7lambda#amu#b7 7lambda#cmu#d7 _/ 7lambda#amu#d7 7lambda#cmu#b74 ,the equality of the cross ratios of the ranges 7,p#a ,p#b ,p#c ,p#d7 and 7,q#a ,q#b ,q#c ,q#d7 is proved to be the necessary and sufficient condition for their mutual projectivity4 ,the cross ratios of all harmonic ranges are then easily seen to be all equal to -#a1 by comparing with the range 7,,oe#a,,ue'#a7 on the axis of x4 ,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 proved4 ,accordingly the greater part of the analytical theory of conics and quadrics belongs to geometry at this stage ,the theory of distance will be considered after the principles of descriptive geometry have been developed4 ,descriptive ,geometry4 ,descriptive geometry is essentially the science of multiple order for open series4 ,the first satisfactory system of axioms was given by ,m4 ,pasch4#cf ,an improved version is due to ,g4 ,peano4#cg ,both these authors treat the idea of the class of points constituting the segment lying between two points as an undefined fundamental idea4 ,thus in fact there are in this system two fundamental ideas1 namely1 of points and of segments4 ,it is then easy enough to define the prolongations of the segments1 so as to form the complete straight lines4 ,d4 ,hilbert's#ch formulation of the axioms is in this respect practically based on the same fundamental ideas4 ,his work is justly famous for some of the mathematical investigations contained in it1 but his exposition of the axioms is distinctly inferior to that of ,peano4 ,descriptive geometry can also be considered#ci as the science of a class of relations1 each relation being a two-termed serial relation1 as considered in the logic of relations1 ranging the points between which it holds into a linear open order4 ,thus the relations are the straight lines1 and the terms between which they hold are the points4 ,but a combination of these two points of view yields#dj the simplest statement of all4 ,descriptive geometry is then conceived as the investigation of an undefined fundamental relation between three terms 7points72 and when the relation holds between three points ,a1 ,b1 ,c1 the points are said to be in the linear order ,,abc4 ,o4 ,veblen's axioms and definitions1 slightly modified1 are as follows3-- #a4 ,if the points ,a1 ,b1 ,c are in the order ,,abc1 they are in the order ,,cba4 #b4 ,if the points ,a1 ,b1 ,c are in the order ,,abc1 they are not in the order ,,bca4 #c4 ,if the points ,a1 ,b1 ,c are in the order ,,abc1 ,a is distinct from ,c4 #d4 ,if ,a and ,b are any two distinct points1 there exists a point ,c such that ,a1 ,b1 ,c are in the order ,,abc4 ,definition4--,the line ,,ab 7,a ,b7 consists of ,a and ,b1 and of all points ,x in one of the possible orders1 ,,abx1 ,,axb1 ,,xab4 ,the points ,x in the order ,,axb constitute the segment ,,ab4 #e4 ,if points ,c and ,d 7,c ,d7 lie on the line ,,ab1 then ,a lies on the line ,,cd4 #f4 ,there exist three distinct points ,a1 ,b1 ,c not in any of the orders ,,abc1 ,,bca1 ,,cab4 #g4 ,if three distinct points ,a1 ,b1 ,c 7fig4 #gc7 do not lie on the same line1 and ,d and ,e are two distinct points in the orders ,,bcd and ,,cea1 then a point ,f exists in the order ,,afb1 and such that ,d1 ,e1 ,f are collinear4 ,illustration3 ,,fig4 #gc4 ,definition4--,if ,a1 ,b1 ,c are three non-collinear points1 the plane ,,abc is the class of points which lie on any one of the lines joining any two of the points belonging to the boundary of the triangle ,,abc1 the boundary being formed by the segments ,,bc1 ,,ca and ,,ab4 ,the interior of the triangle ,,abc is formed by the points in segments such as ,,pq1 where ,p and ,q are points respectively on two of the segments ,,bc1 ,,ca1 ,,ab4 #h4 ,there exists a plane ,,abc1 which does not contain all the points4 ,definition4--,if ,a1 ,b1 ,c1 ,d are four non-coplanar points1 the space ,,abcd is the class of points which lie on any of the lines containing two points on the surface of the tetrahedron ,,abcd1 the surface being formed by the interiors of the triangles ,,abc1 ,,bcd1 ,,dca1 ,,dab4 #i4 ,there exists a space ,,abcd which contains all the points4 #aj4 ,the ,dedekind property holds for the order of the points on any straight line4 ,it follows from axioms #a-i that the points on any straight line are arranged in an open serial order4 ,also all the ordinary theorems respecting a point dividing a straight line into two parts1 a straight line dividing a plane into two parts1 and a plane dividing space into two parts1 follow4 ,again1 in any plane alpha consider a line l and a point ,a 7fig4 #gd74 ,illustration3 ,,fig4 #gd4 ,let any point ,b divide l into two half-lines l#a and l#b4 ,then it can be proved that the set of half-lines1 emanating from ,a and intersecting l#a 7such as m71 are bounded by two half-lines1 of which ,,abc is one4 ,let r be the other4 ,then it can be proved that r does not intersect l#a4 ,similarly for the half-line1 such as n1 intersecting l#b4 ,let s be its bounding half-line4 ,then two cases are possible4 7#a7 ,the half-lines r and s are collinear1 and together form one complete line4 ,in this case1 there is one and only one line 7viz4 r s7 through ,a and lying in alpha which does not intersect l4 ,this is the ,euclidean case1 and the assumption that this case holds is the ,euclidean parallel axiom4 ,but 7#b7 the half-lines r and s may not be collinear4 ,in this case there will be an infinite number of lines1 such as k for instance1 containing ,a and lying in alpha1 which do not intersect l4 ,then the lines through ,a in alpha are divided into two classes by reference to l1 namely1 the secant lines which intersect l1 and the non-secant lines which do not intersect l4 ,the two boundary non-secant lines1 of which r and s are respectively halves1 may be called the two parallels to l through ,a4 ,the perception of the possibility of case #b constituted the starting-point from which ,lobatchewsky constructed the first explicit coherent theory of non-,euclidean geometry1 and thus created a revolution in the philosophy of the subject4 ,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 axiom4#da ,associated ,projective and ,descriptive ,spaces4--,a region of a projective space1 such that one1 and only one1 of the two supplementary segments between any pair of points within it lies entirely within it1 satisfies the above axioms 7#a-aj7 of descriptive geometry1 where the points of the region are the descriptive points1 and the portions of straight lines within the region are the descriptive lines4 ,if the excluded part of the original projective space is a single plane1 the ,euclidean parallel axiom also holds1 otherwise it does not hold for the descriptive space of the limited region4 ,again1 conversely1 starting from an original descriptive space an associated projective space can be constructed by means of the concept of ideal points4#db ,these are also called projective points1 where it is understood that the simple points are the points of the original descriptive space4 ,an ideal point is the class of straight lines which is composed of two coplanar lines a and b1 together with the lines of intersection of all pairs of intersecting planes which respectively contain a and b1 together with the lines of intersection with the plane ab of all planes containing any one of the lines 7other than a or b7 already specified as belonging to the ideal point4 ,it is evident that1 if the two original lines a and b intersect1 the corresponding ideal point is nothing else than the whole class of lines which are concurrent at the point ab4 ,but the essence of the definition is that an ideal point has an existence when the lines a and b do not intersect1 so long as they are coplanar4 ,an ideal point is termed proper1 if the lines composing it intersect2 otherwise it is improper4 ,a theorem essential to the whole theory is the following3 if any two of the three lines a1 b1 c are coplanar1 but the three lines are not all coplanar1 and similarly for the lines a1 b1 d1 then c and d are coplanar4 ,it follows that any two lines belonging to an ideal point can be used as the pair of guiding lines in the definition4 ,an ideal point is said to be coherent with a plane1 if any of the lines composing it lie in the plane4 ,an ideal line is the class of ideal points each of which is coherent with two given planes4 ,if the planes intersect1 the ideal line is termed proper1 otherwise it is improper4 ,it can be proved that any two planes1 with which any two of the ideal points are both coherent1 will serve as the guiding planes used in the definition4 ,the ideal planes are defined as in projective geometry1 and all the other definitions 7for segments1 order1 c47 of projective geometry are applied to the ideal elements4 ,if an ideal plane contains some proper ideal points1 it is called proper1 otherwise it is improper4 ,every ideal plane contains some improper ideal points4 ,it can now be proved that all the axioms of projective geometry hold of the ideal elements as thus obtained2 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 geometry4 ,thus a projective space has been constructed out of the ideal elements1 and the proper ideal elements correspond element by element with the associated descriptive elements4 ,thus the proper ideal elements form a region in the projective space within which the descriptive axioms hold4 ,accordingly1 by substituting ideal elements1 a descriptive space can always be considered as a region within a projective space4 ,this is the justification for the ordinary use of the points at infinity in the ordinary ,euclidean geometry2 the reasoning has been transferred from the original descriptive space to the associated projective space of ideal elements2 and with the ,euclidean parallel axiom the improper ideal elements reduce to the ideal points on a single improper ideal plane1 namely1 the plane at infinity4#dc ,congruence and ,measurement4--,the property of physical space which is expressed by the term measurability has now to be considered4 ,this property has often been considered as essential to the very idea of space4 ,for example1 ,kant writes1#dd ,space is represented as an infinite given quantity4 ,this quantitative aspect of space arises from the measurability of distances1 of angles1 of surfaces and of volumes4 ,these four types of quantity depend upon the two first among them as fundamental4 ,the measurability of space is essentially connected with the idea of congruence1 of which the simplest examples are to be found in the proofs of equality by the method of superposition1 as used in elementary plane geometry4 ,the mere concepts of part and of whole must of necessity be inadequate as the foundation of measurement1 since we require the comparison as to quantity of regions of space which have no portions in common4 ,the idea of congruence1 as exemplified by the method of superposition in geometrical reasoning1 appears to be founded upon that of the rigid body1 which moves from one position to another with its internal spatial relations unchanged4 ,but unless there is a previous concept of the metrical relations between the parts of the body1 there can be no basis from which to deduce that they are unchanged4 ,it would therefore appear as if the idea of the congruence1 or metrical equality1 of two portions of space 7as empirically suggested by the motion of rigid bodies7 must be considered as a fundamental idea incapable of definition in terms of those geometrical concepts which have already been enumerated4 ,this was in effect the point of view of ,pasch4#de ,it has1 however1 been proved by ,sophus ,lie#df that congruence is capable of definition without recourse to a new fundamental idea4 ,this he does by means of his theory of finite continuous groups 7see ,,groups1 ,,theory ,,of71 of which the definition is possible in terms of our established geometrical ideas1 remembering that coordinates have already been introduced4 ,the displacement of a rigid body is simply a mode of defining to the senses a one-one transformation of all space into itself4 ,for at any point of space a particle may be conceived to be placed1 and to be rigidly connected with the rigid body2 and thus there is a definite correspondence of any point of space with the new point occupied by the associated particle after displacement4 ,again two successive displacements of a rigid body from position ,a to position ,b1 and from position ,b to position ,c1 are the same in effect as one displacement from ,a to ,c4 ,but this is the characteristic group property4 ,thus the transformations of space into itself defined by displacements of rigid bodies form a group4 ,call this group of transformations a congruence-group4 ,now according to ,lie a congruence-group is defined by the following characteristics3-- #a4 ,a congruence-group is a finite continuous group of one-one transformations1 containing the identical transformation4 #b4 ,it is a sub-group of the general projective group1 i4e4 of the group of which any transformation converts planes into planes1 and straight lines into straight lines4 #c4 ,an infinitesimal transformation can always be found satisfying the condition that1 at least throughout a certain enclosed region1 any definite line and any definite point on the line are latent1 i4e4 correspond to themselves4 #d4 ,no infinitesimal transformation of the group exists1 such that1 at least in the region for which 7#c7 holds1 a straight line1 a point on it1 and a plane through it1 shall all be latent4 ,the property enunciated by conditions 7#c7 and 7#d71 taken together1 is named by ,lie ,free mobility in the infinitesimal4 ,lie proves the following theorems for a projective space3-- #a4 ,if the above four conditions are only satisfied by a group throughout part of projective space1 this part either 7alpha7 must be the region enclosed by a real closed quadric1 or 7beta7 must be the whole of the projective space with the exception of a single plane4 ,in case 7alpha7 the corresponding congruence group is the continuous group for which the enclosing quadric is latent2 and in case 7beta7 an imaginary conic 7with a real equation7 lying in the latent plane is also latent1 and the congruence group is the continuous group for which the plane and conic are latent4 #b4 ,if the above four conditions are satisfied by a group throughout the whole of projective space1 the congruence group is the continuous group for which some imaginary quadric 7with a real equation7 is latent4 ,by a proper choice of non-homogeneous co-ordinates the equation of any quadrics of the types considered1 either in theorem #a 7alpha71 or in theorem #b1 can be written in the form #a c7x#b y#b z#b7 #j1 where c is negative for a real closed quadric1 and positive for an imaginary quadric4 ,then the general infinitesimal transformation is defined by the three equations3 dx_/dt u - omega#cy omega#bz cx7ux vy wz71 dy_/dt v - omega#az omega#cx cy7ux vy wz71 7,a7 dz_/dt w - omega#bx omega#ay cz7ux vy wz74 _/ ,in the ease considered in theorem #a 7beta71 with the proper choice of co-ordinates the three equations defining the general infinitesimal transformation are3 dx_/dt u - omega#cy omega#bz1 dy_/dt v - omega#az omega#cx1 7,b7 dz_/dt w - omega#bx omega#ay4 _/ ,in this case the latent plane is the plane for which at least one of x1 y1 z are infinite1 that is1 the plane #j4x #j4y #j4z a #j2 and the latent conic is the conic in which the cone x#b y#b z#b #j intersects the latent plane4 ,it follows from theorems #a and #b that there is not one unique congruence-group1 but an indefinite number of them4 ,there is one congruence-group corresponding to each closed real quadric1 one to each imaginary quadric with a real equation1 and one to each imaginary conic in a real plane and with a real equation4 ,the quadric thus associated with each congruence-group is called the absolute for that group1 and in the degenerate case of #a 7beta7 the absolute is the latent plane together with the latent imaginary conic4 ,if the absolute is real1 the congruence-group is hyperbolic2 if imaginary1 it is elliptic2 if the absolute is a plane and imaginary conic1 the group is parabolic4 ,metrical geometry is simply the theory of the properties of some particular congruence-group selected for study4 ,the definition of distance is connected with the corresponding congruence-group by two considerations in respect to a range of five points 7,a#a1 ,a#b1 ,p#a1 ,p#b1 ,p#c71 of which ,a#a and ,a#b are on the absolute4 ,let ,a#a,p#a;,a#b,p#b stand for the cross ratio 7as defined above7 of the range 7,a#a,p#a;,a#b,p#b71 with a similar notation for the other ranges4 ,then 7#a7 log,a#a,p#a;,a#b,p#b log,a#a,p#b;,a#b,p#c log,a#a,p#a;,a#b,p#c1 and 7#b71 if the points ,a#a1 ,a#b1 ,p#a1 ,p#b are transformed into ,a'#a1 ,a'#b1 ,p'#a1 ,p'#b by any transformation of the congruence-group1 7alpha7 ,a#a,p7#a;,a#b,p#b ,a'#a,p'#a;,a'#b,p'#b1 since the transformation is projective1 and 7beta7 ,a'#a1 ,a'#b are on the absolute since ,a#a and ,a#b are on it4 ,thus if we define the distance ,p#a,p#b to be #a_/bk log ,a#a,p#a;,a#b,p#b1 where ,a#a and ,a#b are the points in which the line ,p#a,p#b cuts the absolute1 and k is some constant1 the two characteristic properties of distance1 namely1 7#a7 the addition of consecutive lengths on a straight line1 and 7#b7 the invariability of distances during a transformation of the congruence-group1 are satisfied4 ,this is the well-known ,cayley-,klein projective definition#dg of distance1 which was elaborated in view of the addition property alone1 previously to ,lie's discovery of the theory of congruence-groups4 ,for a hyperbolic group when ,p#a and ,p#b are in the region enclosed by the absolute1 log ,a#a,p#a;,a#b,p#b is real1 and therefore k must be real4 ,for an elliptic group ,a#a and ,a#b are conjugate imaginaries1 and log ,a#a,p#a;,a#b,p#b is a pure imaginary1 and k is chosen to be kappa_/iota1 where kappa is real and iota root-4 ,similarly the angle between two planes1 p#a and p#b1 is defined to be 7#a_/b;iota7 log 7t#ap#at#bp#b71 where t#a and t#b are tangent planes to the absolute through the line p#ap#b4 ,the planes t#a and t#b are imaginary for an elliptic group1 and also for an hyperbolic group when the planes p#a and p#b intersect at points within the region enclosed by the absolute4 ,the development of the consequences of these metrical definitions is the subject of non-,euclidean geometry4 ,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 vanish4 ,it is also obvious that1 if ,p#a and ,p#b be the points 7x#a1 y#a1 z#a7 and 7x#b1 y#b1 z#b71 it follows from equations 7,b7 above that 7x#a - x#b7b 7y#a - y#b7b 7z#a - z#b7ba_/b is unaltered by a congruence transformation and also satisfies the addition property for collinear distances4 ,also the previous definition of an angle can be adapted to this case1 by making t#a and t#b to be the tangent planes through the line p#ap#b to the imaginary conic4 ,similarly if p#a and p#b are intersecting lines1 the same definition of an angle holds1 where t#a and t#b are now the lines from the point p#ap#b to the two points where the plane p#ap#b cuts the imaginary conic4 ,these points are in fact the circular points at infinity on the plane4 ,the development of the consequences of these definitions for the parabolic case gives the ordinary ,euclidean metrical geometry4 ,thus the only metrical geometry for the whole of projective space is of the elliptic type4 ,but the actual measure-relations 7though not their general properties7 differ according to the elliptic congruence-group selected for study4 ,in a descriptive space a congruence-group should possess the four characteristics of such a group throughout the whole of the space4 ,then form the associated ideal projective space4 ,the associated congruence-group for this ideal space must satisfy the four conditions throughout the region of the proper ideal points4 ,thus the boundary of this region is the absolute4 ,accordingly there can be no metrical geometry for the whole of a descriptive space unless its boundary 7in the associated ideal space7 is a closed quadric or a plane4 ,if the boundary is a closed quadric1 there is one possible congruence-group of the hyperbolic type4 ,if the boundary is a plane 7the plane at infinity71 the possible congruence-groups are parabolic2 and there is a congruence-group corresponding to each imaginary conic in this plane1 together with a ,euclidean metrical geometry corresponding to each such group4 ,owing to these alternative possibilities1 it would appear to be more accurate to say that systems of quantities can be found in a space1 rather than that space is a quantity4 ,lie has also deduced#dh the same results with respect to congruence-groups from another set of defining properties1 which explicitly assume the existence of a quantitative relation 7the distance7 between any two points1 which is invariant for any transformation of the congruence-group4#di ,the above results1 in respect to congruence and metrical geometry1 considered in relation to existent space1 have led to the doctrine#ej that it is intrinsically unmeaning to ask which system of metrical geometry is true of the physical world4 ,any one of these systems can be applied1 and in an indefinite number of ways4 ,the only question before us is one of convenience in respect to simplicity of statement of the physical laws4 ,this point of view seems to neglect the consideration that science is to be relevant to the definite perceiving minds of men2 and that 7neglecting the ambiguity introduced by the invariable slight inexactness of observation which is not relevant to this special doctrine7 we have1 in fact1 presented to our senses a definite set of transformations forming a congruence-group1 resulting in a set of measure relations which are in no respect arbitrary4 ,accordingly our scientific laws are to be stated relevantly to that particular congruence-group4 ,thus the investigation of the type 7elliptic1 hyperbolic or parabolic7 of this special congruence-group is a perfectly definite problem1 to be decided by experiment4 ,the consideration of experiments adapted to this object requires some development of non-,euclidean geometry 7see section ,,vi41 ,non-,euclidean ,geometry74 ,but if the doctrine means that1 assuming some sort of objective reality for the material universe1 beings can be imagined1 to whom either all congruence-groups are equally important1 or some other congruence-group is specially important1 the doctrine appears to be an immediate deduction from the mathematical facts4 ,assuming a definite congruence-group1 the investigation of surfaces 7or three-dimensional loci in space of four dimensions7 with geodesic geometries of the form of metrical geometries of other types of congruence-groups forms an important chapter of non-,euclidean geometry4 ,arising from this investigation there is a widely-spread fallacy1 which has found its way into many philosophic writings1 namely1 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 dimensions4 ,the foregoing exposition shows the baselessness of this idea4 ,,bibliography4--,for an account of the investigations on the axioms of geometry during the ,greek period1 see ,m4 ,cantor1 ,vorlesungen uber die ,geschichte der ,mathematik1 ,bd4 i4 and iii42 ,t4,l4 ,heath1 ,the ,thirteen ,books of ,euclid's ,elements1 a ,new ,translation from the ,greek1 with ,introductory ,essays and ,commentary1 ,historical1 ,critical1 and ,explanatory 7,cambridge1 #aijh7--this work is the standard source of information2 ,w4,b4 ,frankland1 ,euclid1 ,book ,i41 with a ,commentary 7,cambridge1 #aije7--the commentary contains copious extracts from the ancient commentators4 ,the next period of really substantive importance is that of the #ahth century4 ,the leading authors are3 ,g4 ,saccheri1 ,s4,j41 ,euclides ab omni naevo vindicatus 7,milan1 #agcc74 ,saccheri was an ,italian ,jesuit who unconsciously discovered non-,euclidean geometry in the course of his efforts to prove its impossibility4 ,j4,h4 ,lambert1 ,theorie der ,parallellinien 7#agff72 ,a4,m4 ,legendre1 ,elements de geometrie 7#agid74 ,an adequate account of the above authors is given by ,p4 ,stackel and ,f4 ,engel1 ,die ,theorie der ,parallellinien von ,euklid bis auf ,gauss 7,leipzig1 #ahie74 ,the next period of time 7roughly from #ahjj to #ahgj7 contains two streams of thought1 both of which are essential to the modern analysis of the subject4 ,the first stream is that which produced the discovery and investigation of non-,euclidean geometries1 the second stream is that which has produced the geometry of position1 comprising both projective and descriptive geometry not very accurately discriminated4 ,the leading authors on non-,euclidean geometry are ,k4,f4 ,gauss1 in private letters to ,schumacher1 cf4 ,stackel and ,engel1 loc4 cit42 ,n4 ,lobatchewsky1 rector of the university of ,kazan1 to whom the honour of the effective discovery of non-,euclidean geometry must be assigned4 ,his first publication was at ,kazan in #ahbf4 ,his various memoirs have been re-edited by ,engel2 cf4 ,urkunden zur ,geschichte der nichteuklidischen ,geometrie by ,stackel and ,engel1 vol4 i4 ,lobatchewsky4 ,j4 ,bolyai discovered non-,euclidean geometry apparently in independence of ,lobatchewsky4 ,his memoir was published in #ahca as an appendix to a work by his father ,w4 ,bolyai1 ,tentamen juventutem4444 ,this memoir has been separately edited by ,j4 ,frischauf1 ,absolute ,geometrie nach ,j4 ,bolyai 7,leipzig1 #ahgb72 ,b4 ,riemann1 ,uber die ,hypothesen1 welche der ,geometrie zu ,grunde liegen 7#ahed72 cf4 ,gesamte ,werke1 a translation in ,the ,collected ,papers of ,w4,k4 ,clifford4 ,this is a fundamental memoir on the subject and must rank with the work of ,lobatchewsky4 ,riemann discovered elliptic metrical geometry1 and ,lobatchewsky hyperbolic geometry4 ,a full account of ,riemann's ideas1 with the subsequent developments due to ,clifford1 ,f4 ,klein and ,w4 ,killing1 will be found in ,the ,boston ,colloquium for #aijc 7,new ,york1 #aije71 article ,forms of ,non-,euclidean ,space1 by ,f4,s4 ,woods4 ,a4 ,cayley1 loc4 cit4 7#ahei71 and ,f4 ,klein1 ,uber die sogenannte nichteuklidische ,geometrie1 ,math4 ,annal4 vols4 iv4 and vi4 7#ahga and #ahgb71 between them elaborated the projective theory of distance2 ,h4 ,helmholtz1 ,uber die thatsachlichen ,grundlagen der ,geometrie 7#ahff71 and ,uber die ,thatsachen1 die der ,geometrie zu ,grunde liegen 7#ahfh71 both in his ,wissenschaftliche ,abhandlungen1 vol4 ii41 and ,s4 ,lie1 loc4 cit4 7#ahij and #ahic71 between them elaborated the group theory of congruence4 ,the numberless works which have been written to suggest equivalent alternatives to ,euclid's parallel axioms may be neglected as being of trivial importance1 though many of them are marvels of geometric ingenuity4 ,the second stream of thought confined itself within the circle of ideas of ,euclidean geometry4 ,its origin was mainly due to a succession of great ,french mathematicians1 for example1 ,g4 ,monge1 ,geometrie descriptive 7#ahjj72 ,j4,v4 ,poncelet1 ,traite des proprietes projectives des figures 7#ahbb72 ,m4 ,chasles1 ,apercu historique sur l'origine et le developpement des methodes en geometrie 7,bruxelles1 #ahcg71 and ,traite de geometrie superieure 7,paris1 #aheb72 and many others4 ,but the works which have been1 and are still1 of decisive influence on thought as a store-house of ideas relevant to the foundations of geometry are ,k4,g4,c4 von ,staudt's two works1 ,geometrie der ,lage 7,nurnberg1 #ahdg72 and ,beitrage zur ,geometrie der ,lage 7,nurnberg1 #ahef1 #crd ed4 #ahfj74 ,the final period is characterized by the successful production of exact systems of axioms1 and by the final solution of problems which have occupied mathematicians for two thousand years4 ,the successful analysis of the ideas involved in serial continuity is due to ,r4 ,dedekind1 ,stetigkeit und irrationale ,zahlen 7#ahgb71 and to ,g4 ,cantor1 ,grundlagen einer allgemeinen ,mannigfaltigkeitslehre 7,leipzig1 #ahhc71 and ,acta math4 vol4 #b4 ,complete systems of axioms have been stated by ,m4 ,pasch1 loc4 cit42 ,g4 ,peano1 loc4 cit42 ,m4 ,pieri1 loc4 cit42 ,b4 ,russell1 ,principles of ,mathematics2 ,o4 ,veblen1 loc4 cit42 and by ,g4 ,veronese in his treatise1 ,fondamenti di geometria 7,padua1 #ahia2 ,german transl4 by ,a4 ,schepp1 ,grundzuge der ,geometrie1 ,leipzig1 #ahid74 ,most of the leading memoirs on special questions involved have been cited in the text2 in addition there may be mentioned ,m4 ,pieri1 ,nuovi principii di geometria projettiva complessa1 ,trans4 ,accad4 ,r4 d4 ,sci4 7,turin1 #aije72 ,e4,h4 ,moore1 ,on the ,projective ,axioms of ,geometry1 ,trans4 ,amer4 ,math4 ,soc41 #aijb2 ,o4 ,veblen and ,w4,h4 ,bussey1 ,finite ,projective ,geometries1 ,trans4 ,amer4 ,math4 ,soc41 #aije2 ,a4,b4 ,kempe1 ,on the ,relation between the ,logical ,theory of ,classes and the ,geometrical ,theory of ,points1 ,proc4 ,lond4 ,math4 ,soc41 #ahij2 ,j4 ,royce1 ,the ,relation of the ,principles of ,logic to the ,foundations of ,geometry1 ,trans4 of ,amer4 ,math4 ,soc41 #aije2 ,a4 ,schoenflies1 ,uber die ,moglichkeit einer projectiven ,geometrie bei transfiniter 7nichtarchimedischer7 ,massbestimmung1 ,deutsch4 ,m4-,v4 ,jahresb41 #aijf4 ,for general expositions of the bearings of the above investigations1 cf4 ,hon4 ,bertrand ,russell1 loc4 cit42 ,l4 ,couturat1 ,les ,principes des mathematiques 7,paris1 #aije72 ,h4 ,poincare1 loc4 cit42 ,russell and ,whitehead1 ,principia mathematica 7,cambridge1 ,univ4 ,press74 ,the philosophers whose views on space and geometric truth deserve especial study are ,descartes1 ,leibnitz1 ,hume1 ,kant and ,j4,s4 ,mill4 7,a4 ,n4 ,w47 ,,footnotes3 #a ,for ,egyptian geometry see ,,egypt1 ,s ,science and ,mathematics4 #b ,cf4 ,a4,n4 ,whitehead1 ,universal ,algebra1 ,bk4 vi4 7,cambridge1 #ahih74 #c ,cf4 ,a4,n4 ,whitehead1 loc4 cit4 #d ,cf4 ,a4,n4 ,whitehead1 ,the ,geodesic ,geometry of ,surfaces in non-,euclidean ,space1 ,proc4 ,lond4 ,math4 ,soc4 vol4 xxix4 #e ,cf4 ,klein1 ,zur nicht-,euklidischen ,geometrie1 ,math4 ,annal4 vol4 xxxvii4 #f ,on the theory of parallels before ,lobatchewsky1 see ,stackel und ,engel1 ,theorie der ,parallellinien von ,euklid bis auf ,gauss 7,leipzig1 #ahie74 ,the foregoing remarks are based upon the materials collected in this work4 #g ,see ,stackel und ,engel1 op4 cit41 and ,gauss1 die beiden ,bolyai1 und die nicht-,euklidische ,geometrie1 ,math4 ,annalen1 ,bd4 xlix42 also ,engel's translation of ,lobatchewsky 7,leipzig1 #ahih71 pp4 #cgh ff4 #h ,lobatchewsky's works on the subject are the following3--,on the ,foundations of ,geometry1 ,kazan ,messenger1 #ahbi-ahcj2 ,new ,foundations of ,geometry1 with a complete ,theory of ,parallels1 ,proceedings of the ,university of ,kazan1 #ahce 7both in ,russian1 but translated into ,german by ,engel1 ,leipzig1 #ahih72 ,geometrie imaginaire1 ,crelle's ,journal1 #ahcg2 ,theorie der ,parallellinien 7,berlin1 #ahdj2 #bnd ed41 #ahhg2 translated by ,halsted1 ,austin1 ,texas1 #ahia74 ,his results appear to have been set forth in a paper 7now lost7 which he read at ,kazan in #ahbf4 #i ,translated by ,halsted 7,austin1 ,texas1 #dth ed41 #ahif47 #aj ,abhandlungen d4 ,konigl4 ,ges4 d4 ,wiss4 zu ,gottingen1 ,bd4 xiii42 ,ges4 math4 ,werke1 pp4 #bed-bfi2 translated by ,clifford1 ,collected ,mathematical ,papers4 #aa ,cf4 ,gesamm4 math4 und phys4 ,werke1 vol4 i4 7,leipzig1 #ahid74 #ab ,wiss4 ,abh4 vol4 ii4 pp4 #faj1 #fah 7#ahff1 #ahfh74 #ac ,mind1 ,o4,s41 vols4 i4 and iii42 ,vortrage und ,reden1 vol4 ii4 pp4 #a1 #bef4 #ad ,his papers are ,saggio di interpretazione della geometria non-,euclidea1 ,giornale di matematiche1 vol4 vi4 7#ahfh72 ,teoria fondamentale degli spazii di curvatura costante1 ,annali di matematica1 vol4 ii4 7#ahfh-ahfi74 ,both were translated into ,french by ,j4 ,houel1 ,annales scientifiques de l',ecole ,normale superieure1 vol4 vi4 7#ahfi74 #ae ,beltrami shows also that this definition agrees with that of ,gauss4 #af ,sur la theorie des foyers1 ,nouv4 ,ann4 vol4 xii4 #ag ,math4 ,annalen1 iv4 vi41 #ahga-ahgb4 #ah ,for an investigation of these and similar properties1 see ,whitehead1 ,universal ,algebra 7,cambridge1 #ahih71 bk4 vi4 ch4 ii4 ,the polar form was independently discovered by ,simon ,newcomb in #ahgg4 #ai ,for an analysis of ,leibnitz's ideas on space1 cf4 ,b4 ,russell1 ,the ,philosophy of ,leibnitz1 chs4 viii4-x4 #bj ,cf4 ,hon4 ,bertrand ,russell1 ,is ,position in ,time and ,space ,absolute or ,relative8 ,mind1 n4s4 vol4 #aj 7#aija71 and ,a4,n4 ,whitehead1 ,mathematical ,concepts of the ,material ,world1 ,phil4 ,trans4 7#aijf71 p4 #bje4 #ba ,cf4 ,critique of ,pure ,reason1 #ast section3 ,of ,space1 conclusion ,a1 ,max ,muller's translation4 #bb ,cf4 ,ernst ,mach1 ,erkenntniss und ,irrtum 7,leipzig72 the relevant chapters are translated by ,t4,j4 ,mc,cormack1 ,space and ,geometry 7,london1 #aijf72 also ,a4 ,meinong1 ,uber die ,stellung der ,gegenstandstheorie im ,system der ,wissenschaften 7,leipzig1 #aijg74 #bc ,cf4 ,russell1 ,principles of ,mathematics1 ,s #ceb 7,cambridge1 #aijc74 #bd ,cf4 ,a4,n4 ,whitehead1 ,the ,axioms of ,projective ,geometry1 ,s #c 7,cambridge1 #aijf74 #be ,cf4 ,russell1 ,princ4 of ,math41 ch4 i4 #bf ,cf4 ,russell1 loc4 cit41 and ,g4 ,frege1 ,uber die ,grundlagen der ,geometrie1 ,jahresber4 der ,deutsch4 ,math4 ,ver4 7#aijf74 #bg ,this formulation--though not in respect to number--is in all essentials that of ,m4 ,pieri1 cf4 ,i principii della ,geometria di ,posizione1 ,accad4 ,r4 di ,torino 7#ahih72 also cf4 ,whitehead1 loc4 cit4 #bh ,cf4 ,g4 ,peano1 ,sui fondamenti della ,geometria1 p4 #gc1 ,rivista di matematica1 vol4 iv4 7#ahid71 and ,d4 ,hilbert1 ,grundlagen der ,geometrie 7,leipzig1 #ahii72 and ,r4,f4 ,moulton1 ,a ,simple non-,desarguesian ,plane ,geometry1 ,trans4 ,amer4 ,math4 ,soc41 vol4 iii4 7#aijb74 #bi ,cf4 ,sui postulati fondamentali della geometria projettiva1 ,giorn4 di matematica1 vol4 xxx4 7#ahia72 also of ,pieri1 loc4 cit41 and ,whitehead1 loc4 cit4 #cj ,cf4 ,hilbert1 loc4 cit42 for a fuller exposition of ,hilbert's proof cf4 ,k4,t4 ,vahlen1 ,abstrakte ,geometrie 7,leipzig1 #aije71 also ,whitehead1 loc4 cit4 #ca ,cf4 ,h4 ,wiener1 ,jahresber4 der ,deutsch4 ,math4 ,ver4 vol4 i4 7#ahij72 and ,f4 ,schur1 ,uber den ,fundamentalsatz der projectiven ,geometrie1 ,math4 ,ann4 vol4 li4 7#ahii74 #cb ,cf4 ,hilbert1 loc4 cit41 and ,whitehead1 loc4 cit4 #cc ,cf4 ,dedekind1 ,stetigkeit und irrationale ,zahlen 7#ahgb74 #cd ,cf4 v4 ,staudt1 ,geometrie der ,lage 7#ahdg74 #ce ,cf4 ,pasch1 ,vorlesungen uber neuere ,geometrie 7,leipzig1 #ahhb71 a classic work2 also ,fiedler1 ,die darstellende ,geometrie 7#ast ed41 #ahga1 #crd ed41 #ahhh72 ,clebsch1 ,vorlesungen uber ,geometrie1 vol4 iii42 ,hilbert1 loc4 cit42 ,f4 ,schur1 ,math4 ,ann4 ,bd4 lv4 7#aijb72 ,vahlen1 loc4 cit42 ,whitehead1 loc4 cit4 #cf ,cf4 loc4 cit4 #cg ,cf4 ,i ,principii di geometria 7,turin1 #ahhi7 and ,sui fondamenti della geometria1 ,rivista di mat4 vol4 iv4 7#ahid74 #ch ,cf4 loc4 cit4 #ci ,cf4 ,vailati1 ,rivista di mat4 vol4 iv4 and ,russell1 loc4 cit4 ,s #cgf4 #dj ,cf4 ,o4 ,veblen1 ,on the ,projective ,axioms of ,geometry1 ,trans4 ,amer4 ,math4 ,soc4 vol4 iii4 7#aijb74 #da ,cf4 ,p4 ,stackel and ,f4 ,engel1 ,die ,theorie der ,parallellinien von ,euklid bis auf ,gauss 7,leipzig1 #ahie74 #db ,cf4 ,pasch1 loc4 cit41 and ,r4 ,bonola1 ,sulla introduzione degli enti improprii in geometria projettive1 ,giorn4 di mat4 vol4 xxxviii4 7#aijj72 and ,whitehead1 ,axioms of ,descriptive ,geometry 7,cambridge1 #aijg74 #dc ,the original idea 7confined to this particular case7 of ideal points is due to von ,staudt 7loc4 cit474 #dd ,cf4 ,critique1 ,trans4 ,aesth4 ,sect4 ,i4 #de ,cf4 loc4 cit4 #df ,cf4 ,uber die ,grundlagen der ,geometrie 7,leipzig1 ,ber41 #ahij72 and ,theorie der ,transformationsgruppen 7,leipzig1 #ahic71 vol4 iii4 #dg ,cf4 ,a4 ,cayley1 ,a ,sixth ,memoir on ,quantics1 ,trans4 ,roy4 ,soc41 #ahei1 and ,coll4 ,papers1 vol4 ii42 and ,f4 ,klein1 ,math4 ,ann4 vol4 iv41 #ahga4 #dh ,cf4 loc4 cit4 #di ,for similar deductions from a third set of axioms1 suggested in essence by ,peano1 ,riv4 mat4 vol4 iv4 loc4 cit4 cf4 ,whitehead1 ,desc4 ,geom4 loc4 cit4 #ej ,cf4 ,h4 ,poincare1 ,la ,science et l'hypothese1 ch4 iii4