,,transcriber',s ,,note ,italic text is denoted by underscores4 ,bold text is denoted by equal signs4 ,a superscript is denoted by x4 ,for example1 y#d indicates y to the power #d4 ,a subscript is denoted by x4 ,for example1 aik indicates the ik-th element of array a4 ,algebraic expressions in the original text usually had the variables in italic4 ,the italic underscore markup has been removed for readability1 so for example3 dxi dxi dxi has been changed to3 dxi dxi dxi ,obvious typographical errors and punctuation errors have been corrected after careful comparison with other occurrences within the text and consultation of external sources4 ,more detail can be found at the end of the book4 ,,the ,,foundations ,,of ,,geometry4 ,london3 ,c4 ,j4 ,,clay ,,and ,,sons1 ,,cambridge ,,university ,,press ,,warehouse1 ,,ave ,,maria ,,lane4 ,glasgow3 #bfc1 ,,argyle ,,street4 ,illustration ,leipzig3 ,f4 ,a4 ,,brockhaus4 ,new ,york3 ,,the ,,macmillan ,,company4 ,bombay3 ,,george ,,bell ,,and ,,sons4 ,,an ,,essay ,,on ,,the ,,foundations ,,of ,,geometry ,,by ,,bertrand ,a4 ,w4 ,,russell4 ,m4,a4 ,,fellow ,,of ,,trinity ,,college1 ,,cambridge4 ,,cambridge3 ,,at ,,the ,,university ,,press4 #ahig ,all ,rights reserved4 ,cambridge3 ,,printed ,,by ,j4 ,,and ,c4 ,f4 ,,clay1 ,,at ,,the ,,university ,,press4 ,,preface4 ,the present work is based on a dissertation submitted at the ,fellowship ,examination of ,trinity ,college1 ,cambridge1 in the year #ahie4 ,section ,b of the third chapter is in the main a reprint1 with some serious alterations1 of an article in ,mind 7,new ,series1 ,no4 #ag74 ,the substance of the book has been given in the form of lectures at the ,johns ,hopkins ,university1 ,baltimore1 and at ,bryn ,mawr ,college1 ,pennsylvania4 ,my chief obligation is to ,professor ,klein4 ,throughout the first chapter1 ,i have found his ,lectures on non-,euclidean ,geometry an invaluable guide2 ,i have accepted from him the division of ,metageometry into three periods1 and have found my historical work much lightened by his references to previous writers4 ,in ,logic1 ,i have learnt most from ,mr ,bradley1 and next to him1 from ,sigwart and ,dr ,bosanquet4 ,on several important points1 ,i have derived useful suggestions from ,professor ,james's ,principles of ,psychology4 ,my thanks are due to ,mr ,g4 ,f4 ,stout and ,mr ,a4 ,n4 ,whitehead for kindly reading my proofs1 and helping me by many useful criticisms4 ,to ,mr ,whitehead ,i owe1 also1 the inestimable assistance of constant criticism and suggestion throughout the course of construction1 especially as regards the philosophical importance of projective ,geometry4 ,,haslemere4 ,may1 #ahig4 ,,to ,,john ,mc,t,a,g,g,a,r,t ,,ellis ,mc,t,a,g,g,a,r,t ,,to ,,whose ,,discourse ,,and ,,friendship ,,is ,,owing ,,the ,,existence ,,of ,,this ,,book4 ,,table ,,of ,,contents4 ,,introduction4 ,,our ,,problem ,,defined ,,by ,,its ,,relations ,,to ,,logic1 ,,psychology ,,and ,,mathematics4 ,,page #a4 ,the problem first received a modern form through ,kant1 who connected the priori with the subjective #a #b4 ,a mental state is subjective1 for ,psychology1 when its immediate cause does not lie in the outer world #b #c4 ,a piece of knowledge is priori1 for ,epistemology1 when without it knowledge would be impossible #b #d4 ,the subjective and the priori belong respectively to ,psychology and to ,epistemology4 ,the latter alone will be investigated in this essay #c #e4 ,my test of the priori will be purely logical3 what knowledge is necessary for experience8 #c #f4 ,but since the necessary is hypothetical1 we must include1 in the priori1 the ground of necessity #d #g4 ,this may be the essential postulate of our science1 or the element1 in the subject-matter1 which is necessary to experience2 #d #h4 ,which1 however1 are both at bottom the same ground #e #i4 ,forecast of the work #e ,,chapter ,i4 ,a ,,short ,,history ,,of ,,metageometry4 #aj4 ,metageometry began by rejecting the axiom of parallels #g #aa4 ,its history may be divided into three periods3 the synthetic1 the metrical and the projective #g #ab4 ,the first period was inaugurated by ,gauss1 #aj #ac4 ,whose suggestions were developed independently by ,lobatchewsky #aj #ad4 ,and ,bolyai #aa #ae4 ,the purpose of all three was to show that the axiom of parallels could not be deduced from the others1 since its denial did not lead to contradictions #ab #af4 ,the second period had a more philosophical aim1 and was inspired chiefly by ,gauss and ,herbart #ac #ag4 ,the first work of this period1 that of ,riemann1 invented two new conceptions3 #ad #ah4 ,the first1 that of a manifold1 is a class-conception1 containing space as a species1 #ad #ai4 ,and defined as such that its determinations form a collection of magnitudes #ae #bj4 ,the second1 the measure of curvature of a manifold1 grew out of curvature in curves and surfaces #af #ba4 ,by means of ,gauss's analytical formula for the curvature of surfaces1 #ai #bb4 ,which enables us to define a constant measure of curvature of a three-dimensional space without reference to a fourth dimension #bj #bc4 ,the main result of ,riemann's mathematical work was to show that1 if magnitudes are independent of place1 the measure of curvature of space must be constant #ba #bd4 ,helmholtz1 who was more of a philosopher than a mathematician1 #bb #be4 ,gave a new but incorrect formulation of the essential axioms1 #bc #bf4 ,and deduced the quadratic formula for the infinitesimal arc1 which ,riemann had assumed #bd #bg4 ,beltrami gave ,lobatchewsky's planimetry a ,euclidean interpretation1 #be #bh4 ,which is analogous to ,cayley's theory of distance2 #bf #bi4 ,and dealt with n-dimensional spaces of constant negative curvature #bg #cj4 ,the third period abandons the metrical methods of the second1 and extrudes the notion of spatial quantity #bg #ca4 ,cayley reduced metrical properties to projective properties1 relative to a certain conic or quadric1 the ,absolute2 #bh #cb4 ,and ,klein showed that the ,euclidean or non-,euclidean systems result1 according to the nature of the ,absolute2 #bi #cc4 ,hence ,euclidean space appeared to give rise to all the kinds of ,geometry1 and the question1 which is true1 appeared reduced to one of convention #cj #cd4 ,but this view is due to a confusion as to the nature of the coordinates employed #cj #ce4 ,projective coordinates have been regarded as dependent on distance1 and thus really metrical #ca #cf4 ,but this is not the case1 since anharmonic ratio can be projectively defined #cb #cg4 ,projective coordinates1 being purely descriptive1 can give no information as to metrical properties1 and the reduction of metrical to projective properties is purely technical #cc #ch4 ,the true connection of ,cayley's measure of distance with non-,euclidean ,geometry is that suggested by ,beltrami's ,saggio1 and worked out by ,sir ,r4 ,ball1 #cf #ci4 ,which provides a ,euclidean equivalent for every non-,euclidean proposition1 and so removes the possibility of contradictions in ,metageometry #ch #dj4 ,klein's elliptic ,geometry has not been proved to have a corresponding variety of space #ci #da4 ,the geometrical use of imaginaries1 of which ,cayley demanded a philosophical discussion1 #da #db4 ,has a merely technical validity1 #db #dc4 ,and is capable of giving geometrical results only when it begins and ends with real points and figures #de #dd4 ,we have now seen that projective ,geometry is logically prior to metrical ,geometry1 but cannot supersede it #df #de4 ,sophus ,lie has applied projective methods to ,helmholtz's formulation of the axioms1 and has shown the axiom of ,monodromy to be superfluous #df #df4 ,metageometry has gradually grown independent of philosophy1 but has grown continually more interesting to philosophy #ej #dg4 ,metrical ,geometry has three indispensable axioms1 #ej #dh4 ,which we shall find to be not results1 but conditions1 of measurement1 #ea #di4 ,and which are nearly equivalent to the three axioms of projective ,geometry #eb #ej4 ,both sets of axioms are necessitated1 not by facts1 but by logic #eb ,,chapter ,,ii4 ,,critical ,,account ,,of ,,some ,,previous ,,philosophical ,,theories ,,of ,,geometry4 #ea4 ,a criticism of representative modern theories need not begin before ,kant #ed #eb4 ,kant's doctrine must be taken1 in an argument about ,geometry1 on its purely logical side #ee #ec4 ,kant contends that since ,geometry is apodeictic1 space must be priori and subjective1 while since space is priori and subjective1 ,geometry must be apodeictic #ee #ed4 ,metageometry has upset the first line of argument1 not the second #ef #ee4 ,the second may be attacked by criticizing either the distinction of synthetic and analytic judgments1 or the first two arguments of the metaphysical deduction of space #eg #ef4 ,modern ,logic regards every judgment as both synthetic and analytic1 #eg #eg4 ,but leaves the priori1 as that which is presupposed in the possibility of experience #ei #eh4 ,kant's first two arguments as to space suffice to prove some form of externality1 but not necessarily ,euclidean space1 a necessary condition of experience #fj #ei4 ,among the successors of ,kant1 ,herbart alone advanced the theory of ,geometry1 by influencing ,riemann #fb #fj4 ,riemann regarded space as a particular kind of manifold1 i4e4 wholly quantitatively #fc #fa4 ,he therefore unduly neglected the qualitative adjectives of space #fd #fb4 ,his philosophy rests on a vicious disjunction #fe #fc4 ,his definition of a manifold is obscure1 #ff #fd4 ,and his definition of measurement applies only to space #fg #fe4 ,though mathematically invaluable1 his view of space as a manifold is philosophically misleading #fi #ff4 ,helmholtz attacked ,kant both on the mathematical and on the psychological side2 #gj #fg4 ,but his criterion of apriority is changeable and often invalid2 #ga #fh4 ,his proof that non-,euclidean spaces are imaginable is inconclusive2 #gb #fi4 ,and his assertion of the dependence of measurement on rigid bodies1 which may be taken in three senses1 #gd #gj4 ,is wholly false if it means that the axiom of ,congruence actually asserts the existence of rigid bodies1 #ge #ga4 ,is untrue if it means that the necessary reference of geometrical propositions to matter renders pure ,geometry empirical1 #gf #gb4 ,and is inadequate to his conclusion if it means1 what is true1 that actual measurement involves approximately rigid bodies #gh #gc4 ,geometry deals with an abstract matter1 whose physical properties are disregarded2 and ,physics must presuppose ,geometry #hj #gd4 ,erdmann accepted the conclusions of ,riemann and ,helmholtz1 #ha #ge4 ,and regarded the axioms as necessarily successive steps in classifying space as a species of manifold #hb #gf4 ,his deduction involves four fallacious assumptions1 namely3 #hb #gg4 ,that conceptions must be abstracted from a series of instances2 #hc #gh4 ,that all definition is classification2 #hc #gi4 ,that conceptions of magnitude can be applied to space as a whole2 #hd #hj4 ,and that if conceptions of magnitude could be so applied1 all the adjectives of space would result from their application #hf #ha4 ,erdmann regards ,geometry alone as incapable of deciding on the truth of the axiom of ,congruence1 #hf #hb4 ,which he affirms to be empirically proved by ,mechanics4 #hh #hc4 ,the variety and inadequacy of ,erdmann's tests of apriority #hi #hd4 ,invalidate his final conclusions on the theory of ,geometry #ij #he4 ,lotze has discussed two questions in the theory of ,geometry3 #ic #hf4 7#a7 ,he regards the possibility of non-,euclidean spaces as suggested by the subjectivity of space1 #ic #hg4 ,and rejects it owing to a mathematical misunderstanding1 #if #hh4 ,having missed the most important sense of their possibility1 #if #hi4 ,which is that they fulfil the logical conditions to which any form of externality must conform #ig #ij4 7#b7 ,he attacks the mathematical procedure of ,metageometry #ih #ia4 ,the attack begins with a question-begging definition of parallels #ii #ib4 ,lotze maintains that all apparent departures from ,euclid could be physically explained1 a view which really makes ,euclid empirical #ii #ic4 ,his criticism of ,helmholtz's analogies rests wholly on mathematical mistakes #aja #id4 ,his proof that space must have three dimensions rests on neglect of different orders of infinity #ajd #ie4 ,he attacks non-,euclidean spaces on the mistaken ground that they are not homogeneous #ajg #if4 ,lotze's objections fall under four heads #ajh #ig4 ,two other semi-philosophical objections may be urged1 #aji #ih4 ,one of which1 the absence of similarity1 has been made the basis of attack by ,delbuf1 #aaj #ii4 ,but does not form a valid ground of objection #aaa #ajj4 ,recent ,french speculation on the foundations of ,geometry has suggested few new views #aab #aja4 ,all homogeneous spaces are priori possible1 and the decision between them is empirical #aad ,,chapter ,,iii4 ,,section ,a4 ,,the ,,axioms ,,of ,,projective ,,geometry4 #ajb4 ,projective ,geometry does not deal with magnitude1 and applies to all spaces alike #aag #ajc4 ,it will be found wholly priori #aag #ajd4 ,its axioms have not yet been formulated philosophically #aah #aje4 ,coordinates1 in projective ,geometry1 are not spatial magnitudes1 but convenient names for points #aah #ajf4 ,the possibility of distinguishing various points is an axiom #aai #ajg4 ,the qualitative relations between points1 dealt with by projective ,geometry1 are presupposed by the quantitative treatment #aai #ajh4 ,the only qualitative relation between two points is the straight line1 and all straight lines are qualitatively similar #abj #aji4 ,hence follows1 by extension1 the principle of projective transformation #aba #aaj4 ,by which figures qualitatively indistinguishable from a given figure are obtained #abb #aaa4 ,anharmonic ratio may and must be descriptively defined #abb #aab4 ,the quadrilateral construction is essential to the projective definition of points1 #abc #aac4 ,and can be projectively defined1 #abd #aad4 ,by the general principle of projective transformation #abf #aae4 ,the principle of duality is the mathematical form of a philosophical circle1 #abg #aaf4 ,which is an inevitable consequence of the relativity of space1 and makes any definition of the point contradictory #abh #aag4 ,we define the point as that which is spatial1 but contains no space1 whence other definitions follow #abh #aah4 ,what is meant by qualitative equivalence in ,geometry8 #abi #aai4 ,two pairs of points on one straight line1 or two pairs of straight lines through one point1 are qualitatively equivalent #abi #abj4 ,this explains why four collinear points are needed1 to give an intrinsic relation by which the fourth can be descriptively defined when the first three are given #acj #aba4 ,any two projectively related figures are qualitatively equivalent1 i4e4 differ in no non-quantitative conceptual property #aca #abb4 ,three axioms are used by projective ,geometry1 #acb #abc4 ,and are required for qualitative spatial comparison1 #acb #abd4 ,which involves the homogeneity1 relativity and passivity of space #acc #abe4 ,the conception of a form of externality1 #acd #abf4 ,being a creature of the intellect1 can be dealt with by pure mathematics #acd #abg4 ,the resulting doctrine of extension will be1 for the moment1 hypothetical #ace #abh4 ,but is rendered assertorical by the necessity1 for experience1 of some form of externality #acf #abi4 ,any such form must be relational #acf #acj4 ,and homogeneous #acg #aca4 ,and the relations constituting it must appear infinitely divisible #acg #acb4 ,it must have a finite integral number of dimensions1 #aci #acc4 ,owing to its passivity and homogeneity #adj #acd4 ,and to the systematic unity of the world #adj #ace4 ,a one-dimensional form alone would not suffice for experience #ada #acf4 ,since its elements would be immovably fixed in a series #adb #acg4 ,two positions have a relation independent of other positions1 #adc #ach4 ,since positions are wholly defined by mutually independent relations #adc #aci4 ,hence projective ,geometry is wholly priori1 #adf #adj4 ,though metrical ,geometry contains an empirical element #adf ,,section ,b4 ,,the ,,axioms ,,of ,,metrical ,,geometry4 #ada4 ,metrical ,geometry is distinct from projective1 but has the same fundamental postulate #adg #adb4 ,it introduces the new idea of motion1 and has three priori axioms #adh ,i4 ,the ,axiom of ,free ,mobility4 #adc4 ,measurement requires a criterion of spatial equality #adi #add4 ,which is given by superposition1 and involves the axiom of ,free ,mobility #aej #ade4 ,the denial of this axiom involves an action of empty space on things #aea #adf4 ,there is a mathematically possible alternative to the axiom1 #aeb #adg4 ,which1 however1 is logically and philosophically untenable #aec #adh4 ,though ,free ,mobility is priori1 actual measurement is empirical #aed #adi4 ,some objections remain to be answered1 concerning-- #aed #aej4 7#a7 ,the comparison of volumes and of ,kant's symmetrical objects #aed #aea4 7#b7 ,the measurement of time1 where congruence is impossible #aef #aeb4 7#c7 ,the immediate perception of spatial magnitude2 and #aeg #aec4 7#d7 ,the ,geometry of non-congruent surfaces #aeh #aed4 ,free ,mobility includes ,helmholtz's ,monodromy #aei #aee4 ,free ,mobility involves the relativity of space #aei #aef4 ,from which1 reciprocally1 it can be deduced #afj #aeg4 ,our axiom is therefore priori in a double sense #afj ,,ii4 ,the ,axiom of ,dimensions4 #aeh4 ,space must have a finite integral number of dimensions #afa #aei4 ,but the restriction to three is empirical #afb #afj4 ,the general axiom follows from the relativity of position #afb #afa4 ,the limitation to three dimensions1 unlike most empirical knowledge1 is accurate and certain #afc ,,iii4 ,the ,axiom of ,distance4 #afb4 ,the axiom of distance corresponds1 here1 to that of the straight line in projective ,geometry #afd #afc4 ,the possibility of spatial measurement involves a magnitude uniquely determined by two points1 #afd #afd4 ,since two points must have some relation1 and the passivity of space proves this to be independent of external reference #afe #afe4 ,there can be only one such relation #aff #aff4 ,this must be measured by a curve joining the two points1 #aff #afg4 ,and the curve must be uniquely determined by the two points #afg #afh4 ,spherical ,geometry contains an exception to this axiom1 #afh #afi4 ,which1 however1 is not quite equivalent to ,euclid's #afh #agj4 ,the exception is due to the fact that two points1 in spherical space1 may have an external relation unaltered by motion1 #afi #aga4 ,which1 however1 being a relation of linear magnitude1 presupposes the possibility of linear magnitude #agj #agb4 ,a relation between two points must be a line joining them #agj #agc4 ,conversely1 the existence of a unique line between two points can be deduced from the nature of a form of externality1 #aga #agd4 ,and necessarily leads to distance1 when quantity is applied to it #agb #age4 ,hence the axiom of distance1 also1 is priori in a double sense #agb #agf4 ,no metrical coordinate system can be set up without the straight line #agd #agg4 ,no axioms besides the above three are necessary to metrical ,geometry #age #agh4 ,but these three are necessary to the direct measurement of any continuum #agf #agi4 ,two philosophical questions remain for a final chapter #agg ,,chapter ,,iv4 ,,philosophical ,,consequences4 #ahj4 ,what is the relation to experience of a form of externality in general8 #agh #aha4 ,this form is the class-conception1 containing every possible intuition of externality2 and some such intuition is necessary to experience #agh #ahb4 ,what relation does this view bear to ,kant's8 #agi #ahc4 ,it is less psychological1 since it does not discuss whether space is given in sensation1 #ahj #ahd4 ,and maintains that not only space1 but any form of externality which renders experience possible1 must be given in sense-perception #aha #ahe4 ,externality should mean1 not externality to the ,self1 but the mutual externality of presented things #aha #ahf4 ,would this be unknowable without a given form of externality8 #ahb #ahg4 ,bradley has proved that space and time preclude the existence of mere particulars1 #ahb #ahh4 ,and that knowledge requires the ,this to be neither simple nor self-subsistent #ahc #ahi4 ,to prove that experience requires a form of externality1 ,i assume that all knowledge requires the recognition of identity in difference #ahd #aij4 ,such recognition involves time #ahd #aia4 ,and some other form giving simultaneous diversity #ahe #aib4 ,the above argument has not deduced sense-perception from the categories1 but has shown the former1 unless it contains a certain element1 to be unintelligible to the latter #ahf #aic4 ,how to account for the realization of this element1 is a question for metaphysics #ahg #aid4 ,what are we to do with the contradictions in space8 #ahh #aie4 ,three contradictions will be discussed in what follows #ahh #aif4 7#a7 ,the antinomy of the ,point proves the relativity of space1 #ahi #aig4 ,and shows that ,geometry must have some reference to matter1 #aij #aih4 ,by which means it is made to refer to spatial order1 not to empty space #aia #aii4 ,the causal properties of matter are irrelevant to ,geometry1 which must regard it as composed of unextended atoms1 by which points are replaced #aia #bjj4 7#b7 ,the circle in defining straight lines and planes is overcome by the same reference to matter #aib #bja4 7#c7 ,the antinomy that space is relational and yet more than relational1 #aic #bjb4 ,seems to depend on the confusion of empty space with spatial order #aic #bjc4 ,kant regarded empty space as the subject-matter of ,geometry1 #aid #bjd4 ,but the arguments of the ,aesthetic are inconclusive on this point1 #aie #bje4 ,and are upset by the mathematical antinomies1 which prove that spatial order should be the subject-matter of ,geometry #aif #bjf4 ,the apparent thinghood of space is a psychological illusion1 due to the fact that spatial relations are immediately given #aif #bjg4 ,the apparent divisibility of spatial relations is either an illusion1 arising out of empty space1 or the expression of the possibility of quantitatively different spatial relations #aig #bjh4 ,externality is not a relation1 but an aspect of relations4 ,spatial order1 owing to its reference to matter1 is a real relation #aih #bji4 ,conclusion #aii ,,introduction4 ,,our ,,problem ,,defined ,,by ,,its ,,relations ,,to ,,logic1 ,,psychology ,,and ,,mathematics4 #a4 ,geometry1 throughout the #agth and #ahth centuries1 remained1 in the war against empiricism1 an impregnable fortress of the idealists4 ,those who held--as was generally held on the ,continent--that certain knowledge1 independent of experience1 was possible about the real world1 had only to point to ,geometry3 none but a madman1 they said1 would throw doubt on its validity1 and none but a fool would deny its objective reference4 ,the ,english ,empiricists1 in this matter1 had1 therefore1 a somewhat difficult task2 either they had to ignore the problem1 or if1 like ,hume and ,mill1 they ventured on the assault1 they were driven into the apparently paradoxical assertion that ,geometry1 at bottom1 had no certainty of a different kind from that of ,mechanics--only the perpetual presence of spatial impressions1 they said1 made our experience of the truth of the axioms so wide as to seem absolute certainty4 ,here1 however1 as in many other instances1 merciless logic drove these philosophers1 whether they would or no1 into glaring opposition to the common sense of their day4 ,it was only through ,kant1 the creator of modern ,epistemology1 that the geometrical problem received a modern form4 ,he reduced the question to the following hypotheticals3 ,if ,geometry has apodeictic certainty1 its matter1 i4e4 space1 must be priori1 and as such must be purely subjective2 and conversely1 if space is purely subjective1 ,geometry must have apodeictic certainty4 ,the latter hypothetical has more weight with ,kant1 indeed it is ineradicably bound up with his whole ,epistemology2 nevertheless it has1 ,i think1 much less force than the former4 ,let us accept1 however1 for the moment1 the ,kantian formulation1 and endeavour to give precision to the terms priori and subjective4 #b4 ,one of the great difficulties1 throughout this controversy1 is the extremely variable use to which these words1 as well as the word empirical1 are put by different authors4 ,to ,kant1 who was nothing of a psychologist1 priori and subjective were almost interchangeable terms#a2 in modern usage there is1 on the whole1 a tendency to confine the word subjective to ,psychology1 leaving priori to do duty for ,epistemology4 ,if we accept this differentiation1 we may set up1 corresponding to the problems of these two sciences1 the following provisional definitions3 priori applies to any piece of knowledge which1 though perhaps elicited by experience1 is logically presupposed in experience3 subjective applies to any mental state whose immediate cause lies1 not in the external world1 but within the limits of the subject4 ,the latter definition1 of course1 is framed exclusively for ,psychology3 from the point of view of physical ,science all mental states are subjective4 ,but for a ,science whose matter1 strictly speaking1 is only mental states1 we require1 if we are to use the word to any purpose1 some differentia among mental states1 as a mark of a more special subjectivity on the part of those to which this term is applied4 ,now the only mental states whose immediate causes lie in the external world are sensations4 ,a pure sensation is1 of course1 an impossible abstraction--we are never wholly passive under the action of an external stimulus--but for the purposes of ,psychology the abstraction is a useful one4 ,whatever1 then1 is not sensation1 we shall1 in ,psychology1 call subjective4 ,it is in sensation alone that we are directly affected by the external world1 and only here does it give us direct information about itself4 #c4 ,let us now consider the epistemological question1 as to the sort of knowledge which can be called priori4 ,here we have nothing to do--in the first instance1 at any rate--with the cause or genesis of a piece of knowledge2 we accept knowledge as a datum to be analysed and classified4 ,such analysis will reveal a formal and a material element in knowledge4 ,the formal element will consist of postulates which are required to make knowledge possible at all1 and of all that can be deduced from these postulates2 the material element1 on the other hand1 will consist of all that comes to fill in the form given by the formal postulates--all that is contingent or dependent on experience1 all that might have been otherwise without rendering knowledge impossible4 ,we shall then call the formal element priori1 the material element empirical4 #d4 ,now what is the connection between the subjective and the priori8 ,it is a connection1 obviously--if it exists at all--from the outside1 i4e4 not deducible directly from the nature of either1 but provable--if it can be proved--only by a general view of the conditions of both4 ,the question1 what knowledge is priori1 must1 on the above definition1 depend on a logical analysis of knowledge1 by which the conditions of possible experience may be revealed2 but the question1 what elements of a cognitive state are subjective1 is to be investigated by pure ,psychology1 which has to determine what1 in our perceptions1 belongs to sensation1 and what is the work of thought or of association4 ,since1 then1 these two questions belong to different sciences1 and can be settled independently1 will it not be wise to conduct the two investigations separately8 ,to decree that the priori shall always be subjective1 seems dangerous1 when we reflect that such a view places our results1 as to the priori1 at the mercy of empirical psychology4 ,how serious this danger is1 the controversy as to ,kant's pure intuition sufficiently shows4 #e4 ,i shall1 therefore1 throughout the present ,essay1 use the word priori without any psychological implication4 ,my test of apriority will be purely logical3 ,would experience be impossible1 if a certain axiom or postulate were denied8 ,or1 in a more restricted sense1 which gives apriority only within a particular science3 ,would experience as to the subject-matter of that science be impossible1 without a certain axiom or postulate8 ,my results also1 therefore1 will be purely logical4 ,if ,psychology declares that some things1 which ,i have declared priori1 are not subjective1 then1 failing an error of detail in my proofs1 the connection of the priori and the subjective1 so far as those things are concerned1 must be given up4 ,there will be no discussion1 accordingly1 throughout this ,essay1 of the relation of the priori to the subjective--a relation which cannot determine what pieces of knowledge are priori1 but rather depends on that determination1 and belongs1 in any case1 rather to ,metaphysics than to ,epistemology4 #f4 ,as ,i have ventured to use the word priori in a slightly unconventional sense1 ,i will give a few elucidatory remarks of a general nature4 ,the priori1 since ,kant at any rate1 has generally stood for the necessary or apodeictic element in knowledge4 ,but modern logic has shown that necessary propositions are always1 in one aspect at least1 hypothetical4 ,there may be1 and usually is1 an implication that the connection1 of which necessity is predicated1 has some existence1 but still1 necessity always points beyond itself to a ground of necessity1 and asserts this ground rather than the actual connection4 ,as ,bradley points out1 arsenic poisons remains true1 even if it is poisoning no one4 ,if1 therefore1 the priori in knowledge be primarily the necessary1 it must be the necessary on some hypothesis1 and the ground of necessity must be included as priori4 ,but the ground of necessity is1 so far as the necessary connection in question can show1 a mere fact1 a merely categorical judgment4 ,hence necessity alone is an insufficient criterion of apriority4 ,to supplement this criterion1 we must supply the hypothesis or ground1 on which alone the necessity holds1 and this ground will vary from one science to another1 and even1 with the progress of knowledge1 in the same science at different times4 ,for as knowledge becomes more developed and articulate1 more and more necessary connections are perceived1 and the merely categorical truths1 though they remain the foundation of apodeictic judgments1 diminish in relative number4 ,nevertheless1 in a fairly advanced science such as ,geometry1 we can1 ,i think1 pretty completely supply the appropriate ground1 and establish1 within the limits of the isolated science1 the distinction between the necessary and the merely assertorical4 #g4 ,there are two grounds1 ,i think1 on which necessity may be sought within any science4 ,these may be 7very roughly7 distinguished as the ground which ,kant seeks in the ,prolegomena1 and that which he seeks in the ,pure ,reason4 ,we may start from the existence of our science as a fact1 and analyse the reasoning employed with a view to discovering the fundamental postulate on which its logical possibility depends2 in this case1 the postulate1 and all which follows from it alone1 will be priori4 ,or we may accept the existence of the subject-matter of our science as our basis of fact1 and deduce dogmatically whatever principles we can from the essential nature of this subject-matter4 ,in this latter case1 however1 it is not the whole empirical nature of the subject-matter1 as revealed by the subsequent researches of our science1 which forms our ground2 for if it were1 the whole science would1 of course1 be priori4 ,rather it is that element1 in the subject-matter1 which makes possible the branch of experience dealt with by the science in question#b4 ,the importance of this distinction will appear more clearly as we proceed#c4 #h4 ,these two grounds of necessity1 in ultimate analysis1 fall together4 ,the methods of investigation in the two cases differ widely1 but the results cannot differ4 ,for in the first case1 by analysis of the science1 we discover the postulate on which alone its reasonings are possible4 ,now if reasoning in the science is impossible without some postulate1 this postulate must be essential to experience of the subject-matter of the science1 and thus we get the second ground4 ,nevertheless1 the two methods are useful as supplementing one another1 and the first1 as starting from the actual science1 is the safest and easiest method of investigation1 though the second seems the more convincing for exposition4 #i4 ,the course of my argument1 therefore1 will be as follows3 ,in the first chapter1 as a preliminary to the logical analysis of ,geometry1 ,i shall give a brief history of the rise and development of non-,euclidean systems4 ,the second chapter will prepare the ground for a constructive theory of ,geometry1 by a criticism of some previous philosophical views2 in this chapter1 ,i shall endeavour to exhibit such views as partly true1 partly false1 and so to establish1 by preliminary polemics1 the truth of such parts of my own theory as are to be found in former writers4 ,a large part of this theory1 however1 cannot be so introduced1 since the whole field of projective ,geometry1 so far as ,i am aware1 has been hitherto unknown to philosophers4 ,passing1 in the third chapter1 from criticism to construction1 ,i shall deal first with projective ,geometry4 ,this1 ,i shall maintain1 is necessarily true of any form of externality1 and is1 since some such form is necessary to experience1 completely priori4 ,in metrical ,geometry1 however1 which ,i shall next consider1 the axioms will fall into two classes3 7#a7 ,those common to ,euclidean and non-,euclidean spaces4 ,these will be found1 on the one hand1 essential to the possibility of measurement in any continuum1 and on the other hand1 necessary properties of any form of externality with more than one dimension4 ,they will1 therefore1 be declared priori4 7#b7 ,those axioms which distinguish ,euclidean from non-,euclidean spaces4 ,these will be regarded as wholly empirical4 ,the axiom that the number of dimensions is three1 however1 though empirical1 will be declared1 since small errors are here impossible1 exactly and certainly true of our actual world2 while the two remaining axioms1 which determine the value of the space-constant1 will be regarded as only approximately known1 and certain only within the errors of observation#d4 ,the fourth chapter1 finally1 will endeavour to prove1 what was assumed in ,chapter ,,iii41 that some form of externality is necessary to experience1 and will conclude by exhibiting the logical impossibility1 if knowledge of such a form is to be freed from contradictions1 of wholly abstracting this knowledge from all reference to the matter contained in the form4 ,i shall hope to have touched1 with this discussion1 on all the main points relating to the ,foundations of ,geometry4 ,,footnotes3 #a ,cf4 ,erdmann1 ,axiome der ,geometrie1 p4 #aaa3 ,fr ,kant sind ,aprioritt und ausschliessliche ,subjectivitt allerdings ,wechselbegriffe4 #b ,i use experience here in the widest possible sense1 the sense in which the word is used by ,bradley4 #c ,where the branch of experience in question is essential to all experience1 the resulting apriority may be regarded as absolute2 where it is necessary only to some special science1 as relative to that science4 #d ,i have given no account of these empirical proofs1 as they seem to be constituted by the whole body of physical science4 ,everything in physical science1 from the law of gravitation to the building of bridges1 from the spectroscope to the art of navigation1 would be profoundly modified by any considerable inaccuracy in the hypothesis that our actual space is ,euclidean4 ,the observed truth of physical science1 therefore1 constitutes overwhelming empirical evidence that this hypothesis is very approximately correct1 even if not rigidly true4 ,,chapter ,i4 ,a ,,short ,,history ,,of ,,metageometry4 #aj4 ,when a long established system is attacked1 it usually happens that the attack begins only at a single point1 where the weakness of the established doctrine is peculiarly evident4 ,but criticism1 when once invited1 is apt to extend much further than the most daring1 at first1 would have wished4 ,first cut the liquefaction1 what comes last1 ,but ,fichte's clever cut at ,god himself8 ,so it has been with ,geometry4 ,the liquefaction of ,euclidean orthodoxy is the axiom of parallels1 and it was by the refusal to admit this axiom without proof that ,metageometry began4 ,the first effort in this direction1 that of ,legendre#e1 was inspired by the hope of deducing this axiom from the others--a hope which1 as we now know1 was doomed to inevitable failure4 ,parallels are defined by ,legendre as lines in the same plane1 such that1 if a third line cut them1 it makes the sum of the interior and opposite angles equal to two right angles4 ,he proves without difficulty that such lines would not meet1 but is unable to prove that non-parallel lines in a plane must meet4 ,similarly he can prove that the sum of the angles of a triangle cannot exceed two right angles1 and that if any one triangle has a sum equal to two right angles1 all triangles have the same sum2 but he is unable to prove the existence of this one triangle4 #aa4 ,thus ,legendre's attempt broke down2 but mere failure could prove nothing4 ,a bolder method1 suggested by ,gauss1 was carried out by ,lobatchewsky and ,bolyai#f4 ,if the axiom of parallels is logically deducible from the others1 we shall1 by denying it and maintaining the rest1 be led to contradictions4 ,these three mathematicians1 accordingly1 attacked the problem indirectly3 they denied the axiom of parallels1 and yet obtained a logically consistent ,geometry4 ,they inferred that the axiom was logically independent of the others1 and essential to the ,euclidean system4 ,their works1 being all inspired by this motive1 may be distinguished as forming the first period in the development of ,metageometry4 ,the second period1 inaugurated by ,riemann1 had a much deeper import3 it was largely philosophical in its aims and constructive in its methods4 ,it aimed at no less than a logical analysis of all the essential axioms of ,geometry1 and regarded space as a particular case of the more general conception of a manifold4 ,taking its stand on the methods of analytical metrical ,geometry1 it established two non-,euclidean systems1 the first that of ,lobatchewsky1 the second--in which the axiom of the straight line1 in ,euclid's form1 was also denied--a new variety1 by analogy called spherical4 ,the leading conception in this period is the measure of curvature1 a term invented by ,gauss1 but applied by him only to surfaces4 ,gauss had shown that free mobility on surfaces was only possible when the measure of curvature was constant2 ,riemann and ,helmholtz extended this proposition to n dimensions1 and made it the fundamental property of space4 ,in the third period1 which begins with ,cayley1 the philosophical motive1 which had moved the first pioneers1 is less apparent1 and is replaced by a more technical and mathematical spirit4 ,this period is chiefly distinguished from the second1 in a mathematical point of view1 by its method1 which is projective instead of metrical4 ,the leading mathematical conception here is the ,absolute 7,grundgebild71 a figure by relation to which all metrical properties become projective4 ,cayley's work1 which was very brief1 and attracted little attention1 has been perfected and elaborated by ,f4 ,klein1 and through him has found general acceptance4 ,klein has added to the two kinds of non-,euclidean ,geometry already known1 a third1 which he calls elliptic2 this third kind closely resembles ,helmholtz's spherical ,geometry1 but is distinguished by the important difference that1 in it1 two straight lines meet in only one point#g4 ,the distinctive mark of the spaces represented by both is that1 like the surface of a sphere1 they are finite but unbounded4 ,the reduction of metrical to projective properties1 as will be proved hereafter1 has only a technical importance2 at the same time1 projective ,geometry is able to deal directly with those purely descriptive or qualitative properties of space which are common to ,euclid and ,metageometry alike4 ,the third period has1 therefore1 great philosophical importance1 while its method has1 mathematically1 much greater beauty and unity than that of the second2 it is able to treat all kinds of space at once1 so that every symbolic proposition is1 according to the meaning given to the symbols1 a proposition in whichever ,geometry we choose4 ,this has the advantage of proving that further research cannot lead to contradictions in non-,euclidean systems1 unless it at the same moment reveals contradictions in ,euclid4 ,these systems1 therefore1 are logically as sound as that of ,euclid himself4 ,after this brief sketch of the characteristics of the three periods1 ,i will proceed to a more detailed account4 ,it will be my aim to avoid1 as far as possible1 all technical mathematics1 and bring into relief only those fundamental points in the mathematical development1 which seem of logical or philosophical importance4 ,first ,period4 #ab4 ,the originator of the whole system1 ,gauss1 does not appear1 as regards strictly non-,euclidean ,geometry1 in any of his hitherto published papers1 to have given more than results2 his proofs remain unknown to us4 ,nevertheless he was the first to investigate the consequences of denying the axiom of parallels#h1 and in his letters he communicated these consequences to some of his friends1 among whom was ,wolfgang ,bolyai4 ,the first mention of the subject in his letters occurs when he was only #ah2 four years later1 in #agii1 writing to ,w4 ,bolyai1 he enunciates the important theorem that1 in hyperbolic ,geometry1 there is a maximum to the area of a triangle4 ,from later writings it appears that he had worked out a system nearly1 if not quite1 as complete as those of ,lobatchewsky and ,bolyai#i4 ,it is important to remember1 however1 that ,gauss's work on curvature1 which was published1 laid the foundation for the whole method of the second period1 and was undertaken1 according to ,riemann and ,helmholtz#aj1 with a view to an 7unpublished7 investigation of the foundations of ,geometry4 ,his work in this direction will1 owing to its method1 be better treated of under the second period1 but it is interesting to observe that he stood1 like many pioneers1 at the head of two tendencies which afterwards diverged4 #ac4 ,lobatchewsky1 a professor in the ,university of ,kasan1 first published his results1 in their native ,russian1 in the proceedings of that learned body for the years #ahbi-ahcj4 ,owing to this double obscurity of language and place1 they attracted little attention1 until he translated them into ,french#aa and ,german#ab3 even then1 they do not appear to have obtained the notice they deserved1 until1 in #ahfh1 ,beltrami unearthed the article in ,crelle1 and made it the theme of a brilliant interpretation4 ,in the introduction to his little ,german book1 ,lobatchewsky laments the slight interest shown in his writings by his compatriots1 and the inattention of mathematicians1 since ,legendre's abortive attempt1 to the difficulties in the theory of parallels4 ,the body of the work begins with the enunciation of several important propositions which hold good in the system proposed as well as in ,euclid3 of these1 some are in any case independent of the axiom of parallels1 while others are rendered so by substituting1 for the word parallel1 the phrase not intersecting1 however far produced4 ,then follows a definition1 intentionally framed so as to contradict ,euclid's3 ,with respect to a given straight line1 all others in the same plane may be divided into two classes1 those which cut the given straight line1 and those which do not cut it2 a line which is the limit between the two classes is called parallel to the given straight line4 ,it follows that1 from any external point1 two parallels can be drawn1 one in each direction4 ,from this starting-point1 by the ,euclidean synthetic method1 a series of propositions are deduced2 the most important of these is1 that in a triangle the sum of the angles is always less than1 or always equal to two right angles1 while in the latter case the whole system becomes orthodox4 ,a certain analogy with spherical ,geometry--whose meaning and extent will appear later--is also proved1 consisting roughly in the substitution of hyperbolic for circular functions4 #ad4 ,very similar is the system of ,johann ,bolyai1 so similar1 indeed1 as to make the independence of the two works1 though a well-authenticated fact1 seem all but incredible4 ,johann ,bolyai first published his results in #ahcb1 in an appendix to a work by his father ,wolfgang1 entitled2 ,appendix1 scientiam spatii absolute veram exhibens3 a veritate aut falsitate ,axiomatis ,,xi4 ,euclidei 7a priori haud unquam decidenda7 independentem2 adjecta ad casum falsitatis1 quadratura circuli geometrica4 ,gauss1 whose bosom friend he became at college and remained through life1 was1 as we have seen1 the inspirer of ,wolfgang ,bolyai1 and used to say that the latter was the only man who appreciated his philosophical speculations on the axioms of ,geometry2 nevertheless1 ,wolfgang appears to have left to his son ,johann the detailed working out of the hyperbolic system4 ,the works of both the ,bolyai are very rare1 and their method and results are known to me only through the renderings of ,frischauf and ,halsted#ac4 ,both as to method and as to results1 the system is very similar to ,lobatchewsky's1 so that neither need detain us here4 ,only the initial postulates1 which are more explicit than ,lobatchewsky's1 demand a brief attention4 ,frischauf's introduction1 which has a philosophical and ,newtonian air1 begins by setting forth that ,geometry deals with absolute 7empty7 space1 obtained by abstracting from the bodies in it1 that two figures are called congruent when they differ only in position1 and that the axiom of ,congruence is indispensable in all determination of spatial magnitudes4 ,congruence was to refer to geometrical bodies1 with none of the properties of ordinary bodies except impenetrability 7,erdmann1 ,axiome der ,geometrie1 p4 #bf74 ,a straight line is defined as determined by two of its points#ad1 and a plane as determined by three4 ,these premisses1 with a slight exception as to the straight line1 we shall hereafter find essential to every ,geometry4 ,i have drawn attention to them1 as it is often supposed that non-,euclideans deny the axiom of ,congruence1 which1 here and elsewhere1 is never the case4 ,the stress laid on this axiom by ,bolyai is probably due to the influence of ,gauss1 whose work on the curvature of surfaces laid the foundation for the use made of congruence by ,helmholtz4 #ae4 ,it is important to remember that1 throughout the period we have just reviewed1 the purpose of hyperbolic ,geometry is indirect3 not the truth of the latter1 but the logical independence of the axiom of parallels from the rest1 is the guiding motive of the work4 ,if1 by denying the axiom of parallels while retaining the rest1 we can obtain a system free from logical contradictions1 it follows that the axiom of parallels cannot be implicitly contained in the others4 ,if this be so1 attempts to dispense with the axiom1 like ,legendre's1 cannot be successful2 ,euclid must stand or fall with the suspected axiom4 ,of course1 it remained possible that1 by further development1 latent contradictions might have been revealed in these systems4 ,this possibility1 however1 was removed by the more direct and constructive work of the second period1 to which we must now turn our attention4 ,second ,period4 #af4 ,the work of ,lobatchewsky and ,bolyai remained1 for nearly a quarter of a century1 without issue--indeed1 the investigations of ,riemann and ,helmholtz1 when they came1 appear to have been inspired1 not by these men1 but rather by ,gauss#ae and ,herbart4 ,we find1 accordingly1 very great difference1 both of aim and method1 between the first period and the second4 ,the former1 beginning with a criticism of one point in ,euclid's system1 preserved his synthetic method1 while it threw over one of his axioms4 ,the latter1 on the contrary1 being guided by a philosophical rather than a mathematical spirit1 endeavoured to classify the conception of space as a species of a more general conception3 it treated space algebraically1 and the properties it gave to space were expressed in terms1 not of intuition1 but of algebra4 ,the aim of ,riemann and ,helmholtz was to show1 by the exhibition of logically possible alternatives1 the empirical nature of the received axioms4 ,for this purpose1 they conceived space as a particular case of a manifold1 and showed that various relations of magnitude 7,massverhltnisse7 were mathematically possible in an extended manifold4 ,their philosophy1 which seems to me not always irreproachable1 will be discussed in ,chapter ,,ii42 here1 while it is important to remember the philosophical motive of ,riemann and ,helmholtz1 we shall confine our attention to the mathematical side of their work4 ,in so doing1 while we shall1 ,i fear1 somewhat maim the system of their thoughts1 we shall secure a closer unity of subject1 and a more compact account of the purely mathematical development4 ,but there is1 in my opinion1 a further reason for separating their philosophy from their mathematics4 ,while their philosophical purpose was1 to prove that all the axioms of ,geometry are empirical1 and that a different content of our experience might have changed them all1 the unintended result of their mathematical work was1 if ,i am not mistaken1 to afford material for an priori proof of certain axioms4 ,these axioms1 though they believed them to be unnecessary1 were always introduced in their mathematical works1 before laying the foundations of non-,euclidean systems4 ,i shall contend1 in ,chapter ,,iii41 that this retention was logically inevitable1 and was not merely due1 as they supposed1 to a desire for conformity with experience4 ,if ,i am right in this1 there is a divergence between ,riemann and ,helmholtz the philosophers1 and ,riemann and ,helmholtz the mathematicians4 ,this divergence makes it the more desirable to trace the mathematical development apart from the accompanying philosophy4 #ag4 ,riemann's epoch-making work1 ,ueber die ,hypothesen1 welche der ,geometrie zu ,grande liegen#af1 was written1 and read to a small circle1 in #ahed2 owing1 however1 to some changes which he desired to make in it1 it remained unpublished till #ahfg1 when it was published by his executors4 ,the two fundamental conceptions1 on whose invention rests the historic importance of this dissertation1 are that of a manifold1 and that of the measure of curvature of a manifold4 ,the former conception serves a mainly philosophical purpose1 and is designed1 principally1 to exhibit space as an instance of a more general conception4 ,on this aspect of the manifold1 ,i shall have much to say in ,chapter ,,ii42 its mathematical aspect1 which alone concerns us here1 is less complicated and less fruitful of controversy4 ,the latter conception also serves a double purpose1 but its mathematical use is the more prominent4 ,we will consider these two conceptions successively4 #ah4 7#a7 ,conception of a manifold#ag4 ,the general purpose of ,riemann's dissertation is1 to exhibit the axioms as successive steps in the classification of the species space4 ,the axioms of ,geometry1 like the marks of a scholastic definition1 appear as successive determinations of class-conceptions1 ending with ,euclidean space4 ,we have thus1 from the analytical point of view1 about as logical and precise a formulation as can be desired--a formulation in which1 from its classificatory character1 we seem certain of having nothing superfluous or redundant1 and obtain the axioms explicitly in the most desirable form1 namely as adjectives of the conception of space4 ,at the same time1 it is a pity that ,riemann1 in accordance with the metrical bias of his time1 regarded space as primarily a magnitude#ah1 or assemblage of magnitudes1 in which the main problem consists in assigning quantities to the different elements or points1 without regard to the qualitative nature of the quantities assigned4 ,considerable obscurity thus arises as to the whole nature of magnitude#ai4 ,this view of ,geometry underlies the definition of the manifold1 as the general conception of which space forms a special case4 ,this definition1 which is not very clear1 may be rendered as follows4 #ai4 ,conceptions of magnitude1 according to ,riemann1 are possible there only1 where we have a general conception1 capable of various determinations 7,bestimmungsweisen74 ,the various determinations of such a conception together form a manifold1 which is continuous or discrete1 according as the passage from one determination to another is continuous or discrete4 ,particular bits of a manifold1 or quanta1 can be compared by counting when discrete1 and by measurement when continuous4 ,measurement consists in a superposition of the magnitudes to be compared4 ,if this be absent1 magnitudes can only be compared when one is part of another1 and then only the more or less1 not the how much1 can be decided 7p4 #bef74 ,we thus reach the general conception of a manifold of several dimensions1 of which space and colours are mentioned as special cases4 ,to the absence of this conception ,riemann attributes the obscurity which1 on the subject of the axioms1 lasted from ,euclid to ,legendre 7p4 #bed74 ,and ,riemann certainly has succeeded1 from an algebraic point of view1 in exhibiting1 far more clearly than any of his predecessors1 the axioms which distinguish spatial quantity from other quantities with which mathematics is conversant4 ,but by the assumption1 from the start1 that space can be regarded as a quantity1 he has been led to state the problem as3 ,what sort of magnitude is space8 rather than3 ,what must space be in order that we may be able to regard it as a magnitude at all8 ,he does not realise1 either--indeed in his day there were few who realized--that an elaborate ,geometry is possible which does not deal with space as a quantity at all4 ,his definition of space as a species of manifold1 therefore1 though for analytical purposes it defines1 most satisfactorily1 the nature of spatial magnitudes1 leaves obscure the true ground for this nature1 which lies in the nature of space as a system of relations1 and is anterior to the possibility of regarding it as a system of magnitudes at all4 ,but to proceed with the mathematical development of ,riemann's ideas4 ,we have seen that he declared measurement to consist in a superposition of the magnitudes to be compared4 ,but in order that this may be a possible means of determining magnitudes1 he continues1 these magnitudes must be independent of their position in the manifold 7p4 #bei74 ,this can occur1 he says1 in several ways1 as the simplest of which1 he assumes that the lengths of lines are independent of their position4 ,one would be glad to know what other ways are possible3 for my part1 ,i am unable to imagine any other hypothesis on which magnitude would be independent of place4 ,setting this aside1 however1 the problem1 owing to the fact that measurement consists in superposition1 becomes identical with the determination of the most general manifold in which magnitudes are independent of place4 ,this brings us to ,riemann's other fundamental conception1 which seems to me even more fruitful than that of a manifold4 #bj4 7#b7 ,measure of curvature4 ,this conception is due to ,gauss1 but was applied by him only to surfaces2 the novelty in ,riemann's dissertation was its extension to a manifold of n dimensions4 ,this extension1 however1 is rather briefly and obscurely expressed1 and has been further obscured by ,helmholtz's attempts at popular exposition4 ,the term curvature1 also1 is misleading1 so that the phrase has been the source of more misunderstanding1 even among mathematicians1 than any other in ,pangeometry4 ,it is often forgotten1 in spite of ,helmholtz's explicit statement#bj1 that the measure of curvature of an n-dimensional manifold is a purely analytical expression1 which has only a symbolic affinity to ordinary curvature4 ,as applied to three-dimensional space1 the implication of a four-dimensional plane space is wholly misleading2 ,i shall1 therefore1 generally use the term space-constant instead#ba4 ,nevertheless1 as the conception grew1 historically1 out of that of curvature1 ,i will give a very brief exposition of the historical development of theories of curvature4 ,just as the notion of length was originally derived from the straight line1 and extended to other curves by dividing them into infinitesimal straight lines1 so the notion of curvature was derived from the circle1 and extended to other curves by dividing them into infinitesimal circular arcs4 ,curvature may be regarded1 originally1 as a measure of the amount by which a curve departs from a straight line2 in a circle1 which is similar throughout1 this amount is evidently constant1 and is measured by the reciprocal of the radius4 ,but in all other curves1 the amount of curvature varies from point to point1 so that it cannot be measured without infinitesimals4 ,the measure which at once suggests itself is1 the curvature of the circle most nearly coinciding with the curve at the point considered4 ,since a circle is determined by three points1 this circle will pass through three consecutive points of the curve4 ,we have thus defined the curvature of any curve1 plane or tortuous2 for1 since any three points lie in a plane1 such a circle can always be described4 ,if we now pass to a surface1 what we want is1 by analogy1 a measure of its departure from a plane4 ,the curvature1 as above defined1 has become indeterminate1 for through any point of the surface we can draw an infinite number of arcs1 which will not1 in general1 all have the same curvature4 ,let us1 then1 draw all the geodesics joining the point in question to neighbouring points of the surface in all directions4 ,since these arcs form a singly infinite manifold1 there will be among them1 if they have not all the same curvature1 one arc of maximum1 and one of minimum curvature#bb4 ,the product of these maximum and minimum curvatures is called the measure of curvature of the surface at the point under consideration4 ,to illustrate by a few simple examples3 on a sphere1 the curvatures of all such lines are equal to the reciprocal of the radius of the sphere1 hence the measure of curvature everywhere is the square of the reciprocal of the radius of the sphere4 ,on any surface1 such as a cone or a cylinder1 on which straight lines can be drawn1 these have no curvature1 so that the measure of curvature is everywhere zero--this is the case1 in particular1 with the plane4 ,in general1 however1 the measure of curvature of a surface varies from point to point4 ,gauss1 the inventor of this conception#bc1 proved that1 in order that two surfaces may be developable upon each other--i4e4 may be such that one can be bent into the shape of the other without stretching or tearing--it is necessary that the two surfaces should have equal measures of curvature at corresponding points4 ,when this is the case1 every figure which is possible on the one is1 in general1 possible on the other1 and the two have practically the same ,geometry#bd4 ,as a corollary1 it follows that a necessary condition1 for the free mobility of figures on any surface1 is the constancy of the measure of curvature#be4 ,this condition was proved to be sufficient1 as well as necessary1 by ,minding#bf4 #ba4 ,so far1 all has been plain sailing--we have been dealing with purely geometrical ideas in a purely geometrical manner--but we have not1 as yet1 found any sense of the measure of curvature1 in which it can be extended to space1 still less to an n-dimensional manifold4 ,for this purpose1 we must examine ,gauss's method1 which enables us to determine the measure of curvature of a surface at any point as an inherent property1 quite independent of any reference to the third dimension4 ,the method of determining the measure of curvature from within is1 briefly1 as follows3 ,if any point on the surface be determined by two coordinates1 u1 v1 then small arcs of the surface are given by the formula ds#b ,edu#b #b;,fdu dv ,gdv#b1 where ,e1 ,f1 ,g are1 in general1 functions of u1 v4#bg ,from this formula alone1 without reference to any space outside the surface1 we can determine the measure of curvature at the point u1 v1 as a function of ,e1 ,f1 ,g and their differentials with respect to u and v4 ,thus we may regard the measure of curvature of a surface as an inherent property1 and the above geometrical definition1 which involved a reference to the third dimension1 may be dropped4 ,but at this point a caution is necessary4 ,it will appear in ,chap4 ,,iii4 7 #agf71 that it is logically impossible to set up a precise coordinate system1 in which the coordinates represent spatial magnitudes1 without the axiom of ,free ,mobility1 and this axiom1 as we have just seen1 holds on surfaces only when the measure of curvature is constant4 ,hence our definition of the measure of curvature will only be really free from reference to the third dimension1 when we are dealing with a surface of constant measure of curvature--a point which ,riemann entirely overlooks4 ,this caution1 however1 applies only in space1 and if we take the coordinate system as presupposed in the conception of a manifold1 we may neglect the caution altogether--while remembering that the possibility of a coordinate system in space involves axioms to be investigated later4 ,we can thus see how a meaning might be found1 without reference to any higher dimension1 for a constant measure of curvature of three-dimensional space1 or for any measure of curvature of an n-dimensional manifold in general4 #bb4 ,such a meaning is supplied by ,riemann's dissertation1 to which1 after this long digression1 we can now return4 ,we may define a continuous manifold as any continuum of elements1 such that a single element is defined by n continuously variable magnitudes4 ,this definition does not really include space1 for coordinates in space do not define a point1 but its relations to the origin1 which is itself arbitrary4 ,it includes1 however1 the analytical conception of space with which ,riemann deals1 and may1 therefore1 be allowed to stand for the moment4 ,riemann then assumes that the difference--or distance1 as it may be loosely called--between any two elements is comparable1 as regards magnitude1 to the difference between any other two4 ,he assumes further1 what it is ,helmholtz's merit to have proved1 that the difference ds between two consecutive elements can be expressed as the square root of a quadratic function of the differences of the coordinates3 i4e4 ds#b #an #an aik dxi4dxk1 where the coefficients aik are1 in general1 functions of the coordinates x#a x#b 444 xn4#bh ,the question is3 ,how are we to obtain a definition of the measure of curvature out of this formula8 ,it is noticeable1 in the first place1 that1 just as in a surface we found an infinite number of radii of curvature at a point1 so in a manifold of three or more dimensions we must find an infinite number of measures of curvature at a point1 one for every two-dimensional manifold passing through the point1 and contained in the higher manifold4 ,what we have first to do1 therefore1 is to define such two-dimensional manifolds4 ,they must consist1 as we saw on the surface1 of a singly infinite series of geodesics through the point4 ,now a geodesic is completely determined by one point and its direction at that point1 or by one point and the next consecutive point4 ,hence a geodesic through the point considered is determined by the ratios of the increments of coordinates1 dx#a dx#b 444 dxn4 ,suppose we have two such geodesics1 in which the ith increments are respectively dxi and dxi4 ,then all the geodesics given by dxi dxi dxi ,illustration form a singly infinite series1 since they contain one parameter1 namely 3 4 ,such a series of geodesics1 therefore1 must form a two-dimensional manifold1 with a measure of curvature in the ordinary ,gaussian sense4 ,this measure of curvature can be determined from the above formula for the elementary arc1 by the help of ,gauss's general formula alluded to above4 ,we thus obtain an infinite number of measures of curvature at a point1 but from n47n - #a7_/b of these1 the rest can be deduced 7,riemann1 ,gesammelte ,werke1 p4 #bfb74 ,when all the measures of curvature at a point are constant1 and equal to all the measures of curvature at any other point1 we get what ,riemann calls a manifold of constant curvature4 ,in such a manifold free mobility is possible1 and positions do not differ intrinsically from one another4 ,if a be the measure of curvature1 the formula for the arc becomes1 in this case1 ds#b dx#b_/7a a_/#d x#b7b4 ,in this case only1 as ,i pointed out above1 can the term measure of curvature be properly applied to space without reference to a higher dimension1 since free mobility is logically indispensable to the existence of quantitative or metrical ,geometry4 #bc4 ,the mathematical result of ,riemann's dissertation may be summed up as follows4 ,assuming it possible to apply magnitude to space1 i4e4 to determine its elements and figures by means of algebraical quantities1 it follows that space can be brought under the conception of a manifold1 as a system of quantitatively determinable elements4 ,owing1 however1 to the peculiar nature of spatial measurement1 the quantitative determination of space demands that magnitudes shall be independent of place--in so far as this is not the case1 our measurement will be necessarily inaccurate4 ,if we now assume1 as the quantitative relation of distance between two elements1 the square root of a quadratic function of the coordinates--a formula subsequently proved by ,helmholtz and ,lie--then it follows1 since magnitudes are to be independent of place1 that space must1 within the limits of observation1 have a constant measure of curvature1 or must1 in other words1 be homogeneous in all its parts4 ,in the infinitesimal1 ,riemann says 7p4 #bfg71 observation could not detect a departure from constancy on the part of the measure of curvature2 but he makes no attempt to show how ,geometry could remain possible under such circumstances1 and the only ,geometry he has constructed is based entirely on ,free ,mobility4 ,i shall endeavour to prove1 in ,chapter ,,iii41 that any metrical ,geometry1 which should endeavour to dispense with this axiom1 would be logically impossible4 ,at present ,i will only point out that ,riemann1 in spite of his desire to prove that all the axioms can be dispensed with1 has nevertheless1 in his mathematical work1 retained three fundamental axioms1 namely1 ,free ,mobility1 the finite integral number of dimensions1 and the axiom that two points have a unique relation1 namely distance4 ,these1 as we shall see hereafter1 are retained1 in actual mathematical work1 by all metrical ,metageometers1 even when they believe1 like ,riemann and ,helmholtz1 that no axioms are philosophically indispensable4 #bd4 ,helmholtz1 the historically nearest follower of ,riemann1 was guided by a similar empirical philosophy1 and arrived independently at a very similar method of formulating the axioms4 ,although ,helmholtz published nothing on the subject until after ,riemann's death1 he had then only just seen ,riemann's dissertation 7which was published posthumously71 and had worked out his results1 so far as they were then completed1 in entire independence both of ,riemann and of ,lobatchewsky4 ,helmholtz is by far the most widely read of all writers on ,metageometry1 and his writings1 almost alone1 represent to philosophers the modern mathematical standpoint on this subject4 ,but his importance is much greater1 in this domain1 as a philosopher than as a mathematician2 almost his only original mathematical result1 as regards ,geometry1 is his proof of ,riemann's formula for the infinitesimal arc1 and even this proof was far from rigid1 until ,lie reformed it by his method of continuous groups4 ,in this chapter1 therefore1 only two of his writings need occupy us1 namely the two articles in the ,wissenschaftliche ,abhandlungen1 ,vol4 ,,ii41 entitled respectively ,ueber die thatschlichen ,grundlagen der ,geometrie1 #ahff 7p4 #faj ff471 and ,ueber die ,thatsachen1 die der ,geometrie zum ,grunde liegen1 #ahfh 7p4 #fah ff474 #be4 ,in the first of these1 which is chiefly philosophical1 ,helmholtz gives hints of his then uncompleted mathematical work1 but in the main contents himself with a statement of results4 ,he announces that he will prove ,riemann's quadratic formula for the infinitesimal arc2 but for this purpose1 he says1 we have to start with ,congruence1 since without it spatial measurement is impossible4 ,nevertheless1 he maintains that ,congruence is proved by experience4 ,how we could1 without the help of measurement1 discover lapses from ,congruence1 is a point which he leaves undiscussed4 ,he then enunciates the four axioms which he considers essential to ,geometry1 as follows3 7#a7 ,as regards continuity and dimensions4 ,in a space of n dimensions1 a point is uniquely determined by the measurement of n continuous variables 7coordinates74 7#b7 ,as regards the existence of moveable rigid bodies4 ,between the #bn coordinates of any point-pair of a rigid body1 there exists an equation which is the same for all congruent point-pairs4 ,by considering a sufficient number of point-pairs1 we get more equations than unknown quantities3 this gives us a method of determining the form of these equations1 so as to make it possible for them all to be satisfied4 7#c7 ,as regards free mobility4 ,every point can pass freely and continuously from one position to another4 ,from 7#b7 and 7#c7 it follows1 that if two systems ,a and ,b can be brought into congruence in any one position1 this is also possible in every other position4 7#d7 ,as regards independence of rotation in rigid bodies 7,monodromy74 ,if 7n - #a7 points of a body remain fixed1 so that every other point can only describe a certain curve1 then that curve is closed4 ,these axioms1 says ,helmholtz1 suffice to give1 with the axiom of three dimensions1 the ,euclidean and non-,euclidean systems as the only alternatives4 ,that they suffice1 mathematically1 cannot be denied1 but they seem1 in some respects1 to go too far4 ,in the first place1 there is no necessity to make the axiom of ,congruence apply to actual rigid bodies--on this subject ,i have enlarged in ,chapter ,,ii4#bi ,again1 ,free ,mobility1 as distinct from ,congruence1 hardly needs to be specially formulated3 what barrier could empty space offer to a point's progress8 ,the axiom is involved in the homogeneity of space1 which is the same thing as the axiom of ,congruence4 ,monodromy1 also1 has been severely criticized2 not only is it evident that it might have been included in ,congruence1 but even from the purely analytical point of view1 ,sophus ,lie has proved it to be superfluous#cj4 ,thus the axiom of ,congruence1 rightly formulated1 includes ,helmholtz's third and fourth axioms and part of his second axiom4 ,all the four1 or rather1 as much of them as is relevant to ,geometry1 are consequences1 as we shall see hereafter1 of the one fundamental principle of the relativity of position4 #bf4 ,the second article1 which is mainly mathematical1 supplies the promised proof of the arc-formula1 which is ,helmholtz's most important contribution to ,geometry4 ,riemann had assumed this formula1 as the simplest of a number of alternatives3 ,helmholtz proved it to be a necessary consequence of his axioms4 ,the present paper begins with a short repetition of the first1 including the statement of the axioms1 to which1 at the end of the paper1 two more are added1 7#e7 that space has three dimensions1 and 7#f7 that space is infinite4 ,it is supposed in the text1 as also in the first paper1 that the measure of curvature cannot be negative1 and1 consequently1 that an infinite space must be ,euclidean4 ,this error in both papers is corrected in notes1 added after the appearance of ,beltrami's paper on negative curvature4 ,it is a sample of the slightly unprofessional nature of ,helmholtz's mathematical work on this subject1 which elicits from ,klein the following remarks#ca3 ,helmholtz is not a mathematician by profession1 but a physicist and physiologist4444 ,from this non-mathematical quality of ,helmholtz1 it follows naturally that he does not treat the mathematical portion of his work with the thoroughness which one would demand of a mathematician by trade 7von ,fach74 ,he tells us himself that it was the physiological study of vision which led him to the question of the axioms1 and it is as a physicist that he makes his axioms refer to actual rigid bodies4 ,accordingly1 we find errors in his mathematics1 such as the axiom of ,monodromy1 and the assumption that the measure of curvature must be positive4 ,nevertheless1 the proof of ,riemann's arc-formula is extremely able1 and has1 on the whole1 been substantiated by ,lie's more thorough investigations4 #bg4 ,helmholtz's other writings on ,geometry are almost wholly philosophical1 and will be discussed at length in ,chapter ,,ii4 ,for the present1 we may pass to the only other important writer of the second period1 ,beltrami4 ,as his work is purely mathematical1 and contains few controverted points1 it need not1 despite its great importance1 detain us long4 ,the ,saggio di ,interpretazione della ,geometria non-,euclidea#cb1 which is principally confined to two dimensions1 interprets ,lobatchewsky's results by the characteristic method of the second period4 ,it shows1 by a development of the work of ,gauss and ,minding#cc1 that all the propositions in plane ,geometry1 which ,lobatchewsky had set forth1 hold1 within ordinary ,euclidean space1 on surfaces of constant negative curvature4 ,it is strange1 as ,klein points out#cd1 that this interpretation1 which was known to ,riemann and perhaps even to ,gauss1 should have remained so long without explicit statement4 ,this is the more strange1 as ,lobatchewsky's ,gomtrie ,imaginaire had appeared in ,crelle1 ,vol4 ,,xvii4#ce1 and ,minding's article1 from which the interpretation follows at once1 had appeared in ,crelle1 ,vol4 ,,xix4 ,minding had shewn that the ,geometry of surfaces of constant negative curvature1 in particular as regards geodesic triangles1 could be deduced from that of the sphere by giving the radius a purely imaginary value ia#cf4 ,this result1 as we have seen1 had also been obtained by ,lobatchewsky for his ,geometry1 and yet it took thirty years for the connection to be brought to general notice4 #bh4 ,in ,beltrami's ,saggio1 straight lines are1 of course1 replaced by geodesics2 his coordinates are obtained through a point-by-point correspondence with an auxiliary plane1 in which straight lines correspond to geodesics on the surface4 ,thus geodesics have linear equations1 and are always uniquely determined by two points4 ,distances on the surface1 however1 are not equal to distances on the plane2 thus while the surface is infinite1 the corresponding portion of the plane is contained within a certain finite circle4 ,the distance of two points on the surface is a certain function of the coordinates1 not the ordinary function of elementary ,geometry4 ,these relations of plane and surface are important in connection with ,cayley's theory of distance1 which we shall have to consider next4 ,if we were to define distance on the plane as that function of the coordinates which gives the corresponding distance on the surface1 we should obtain what ,klein calls a plane with a hyperbolic system of measurement 7,massbestimmung7 in which ,cayley's theory of distance would hold4 ,it is evident1 however1 that the ordinary notion of distance has been presupposed in setting up the coordinate system1 so that we do not really get alternative ,geometries on one and the same plane4 ,the bearing of these remarks will appear more fully when we come to consider ,cayley and ,klein4 #bi4 ,the value of ,beltrami's ,saggio1 in his own eyes1 lies in the intelligible ,euclidean sense which it gives to ,lobatchewsky's planimetry3 the corresponding system of ,solid ,geometry1 since it has no meaning for ,euclidean space1 is barely mentioned in this work4 ,in a second paper#cg1 however1 almost contemporaneous with the first1 he proceeds to consider the general theory of n-dimensional manifolds of constant negative curvature4 ,this paper is greatly influenced by ,riemann's dissertation2 it begins with the formula for the linear element1 and proves from this first1 that ,congruence holds for such spaces1 and next1 that they have1 according to ,riemann's definition1 a constant negative measure of curvature4 7,it is instructive to observe1 that both in this and in the former ,essay1 great stress is laid on the necessity of the ,axiom of ,congruence47 ,this work has less philosophical interest than the former1 since it does little more than repeat1 in a general form1 the results which the ,saggio had obtained for two dimensions--results which sink1 when extended to n dimensions1 to the level of mere mathematical constructions4 ,nevertheless1 the paper is important1 both as a restoration of negative curvature1 which had been overlooked by ,helmholtz1 and as an analytical treatment of ,lobatchewsky's results--a treatment which1 together with the ,saggio1 at last restored to them the prominence they deserved4 ,third ,period4 #cj4 ,the third period differs radically1 alike in its methods and aims1 and in the underlying philosophical ideas1 from the period which it replaced4 ,whereas everything1 in the second period1 turned on measurement1 with its apparatus of ,congruence1 ,free ,mobility1 ,rigid ,bodies1 and the rest1 these vanish completely in the third period1 which1 swinging to the opposite extreme1 regards quantity as a perfectly irrelevant category in ,geometry1 and dispenses with congruence and the method of superposition4 ,the ideas of this period1 unfortunately1 have found no exponent so philosophical as ,riemann or ,helmholtz1 but have been set forth only by technical mathematicians4 ,moreover the change of fundamental ideas1 which is immense1 has not brought about an equally great change in actual procedure2 for though spatial quantity is no longer a part of projective ,geometry1 quantity is still employed1 and we still have equations1 algebraic transformations1 and so on4 ,this is apt to give rise to confusion1 especially in the mind of the student1 who fails to realise that the quantities used1 so far as the propositions are really projective1 are mere names for points1 and not1 as in metrical ,geometry1 actual spatial magnitudes4 ,nevertheless1 the fundamental difference between this period and the former must strike any one at once4 ,whereas ,riemann and ,helmholtz dealt with metrical ideas1 and took1 as their foundations1 the measure of curvature and the formula for the linear element--both purely metrical--the new method is erected on the formulae for transformation of coordinates required to express a given collineation4 ,it begins by reducing all so-called metrical notions--distance1 angle1 etc4--to projective forms1 and obtains1 from this reduction1 a methodological unity and simplicity before impossible4 ,this reduction depends1 however1 except where the space-constant is negative1 upon imaginary figures--in ,euclid1 the circular points at infinity2 it is moreover purely symbolic and analytical1 and must be regarded as philosophically irrelevant4 ,as the question concerning the import of this reduction is of fundamental importance to our theory of ,geometry1 and as ,cayley1 in his ,presidential ,address to the ,british ,association in #ahhc1 formally challenged philosophers to discuss the use of imaginaries1 on which it depends1 ,i will treat this question at some length4 ,but first let us see how1 as a matter of mathematics1 the reduction is effected4 #ca4 ,we shall find1 throughout this period1 that almost every important proposition1 though misleading in its obvious interpretation1 has nevertheless1 when rightly interpreted1 a wide philosophical bearing4 ,so it is with the work of ,cayley1 the pioneer of the projective method4 ,the projective formula for angles1 in ,euclidean ,geometry1 was first obtained by ,laguerre1 in #ahec4 ,this formula had1 however1 a perfectly ,euclidean character1 and it was left for ,cayley to generalize it so as to include both angles and distances in ,euclidean and non-,euclidean systems alike#ch4 ,cayley was1 to the last1 a staunch supporter of ,euclidean space1 though he believed that non-,euclidean ,geometries could be applied1 within ,euclidean space1 by a change in the definition of distance#ci4 ,he has thus1 in spite of his ,euclidean orthodoxy1 provided the believers in the possibility of non-,euclidean spaces with one of their most powerful weapons4 ,in his ,sixth ,memoir upon ,quantics 7#ahei71 he set himself the task of establishing the notion of distance upon purely descriptive principles4 ,he showed that1 with the ordinary notion of distance1 it can be rendered projective by reference to the circular points and the line at infinity1 and that the same is true of angles#dj4 ,not content with this1 he suggested a new definition of distance1 as the inverse sine or cosine of a certain function of the coordinates2 with this definition1 the properties usually known as metrical become projective properties1 having reference to a certain conic1 called by ,cayley the ,absolute4 7,the circular points are1 analytically1 a degenerate conic1 so that ordinary ,geometry forms a particular case of the above47 ,he proves that1 when the ,absolute is an imaginary conic1 the ,geometry so obtained for two dimensions is spherical ,geometry4 ,the correspondence with ,lobatchewsky1 in the case where the ,absolute is real1 is not worked out3 indeed there is1 throughout1 no evidence of acquaintance with non-,euclidean systems4 ,the importance of the memoir1 to ,cayley1 lies entirely in its proof that metrical is only a branch of descriptive ,geometry4 #cb4 ,the connection of ,cayley's ,theory of ,distance with ,metageometry was first pointed out by ,klein#da4 ,klein showed in detail that1 if the ,absolute be real1 we get ,lobatchewsky's 7hyperbolic7 system2 if it be imaginary1 we get either spherical ,geometry or a new system1 analogous to that of ,helmholtz1 called by ,klein elliptic2 if the ,absolute be an imaginary point-pair1 we get parabolic ,geometry1 and if1 in particular1 the point-pair be the circular points1 we get ordinary ,euclid4 ,in elliptic ,geometry1 two straight lines in the same plane meet in only one point1 not two as in ,helmholtz's system4 ,the distinction between the two kinds of ,geometry is difficult1 and will be discussed later4 #cc4 ,since these systems are all obtained from a ,euclidean plane1 by a mere alteration in the definition of distance1 ,cayley and ,klein tend to regard the whole question as one1 not of the nature of space1 but of the definition of distance4 ,since this definition1 on their view1 is perfectly arbitrary1 the philosophical problem vanishes--,euclidean space is left in undisputed possession1 and the only problem remaining is one of convention and mathematical convenience#db4 ,this view has been forcibly expressed by ,poincar3 ,what ought one to think1 he says1 of this question3 ,is the ,euclidean ,geometry true8 ,the question is nonsense4 ,geometrical axioms1 according to him1 are mere conventions3 they are definitions in disguise#dc4 ,thus ,klein blames ,beltrami for regarding his auxiliary plane as merely auxiliary1 and remarks that1 if he had known ,cayley's ,memoir1 he would have seen the relation between the plane and the pseudosphere to be far more intimate than he supposed#dd4 ,a view which removes the problem entirely from the arena of philosophy demands1 plainly1 a full discussion4 ,to this discussion we will now proceed4 #cd4 ,the view in question has arisen1 it would seem1 from a natural confusion as to the nature of the coordinates employed4 ,those who hold the view have not adequately realised1 ,i believe1 that their coordinates are not spatial quantities1 as in metrical ,geometry1 but mere conventional signs1 by which different points can be distinctly designated4 ,there is no reason1 therefore1 until we already have metrical ,geometry1 for regarding one function of the coordinates as a better expression of distance than another1 so long as the fundamental addition-equation#de is preserved4 ,hence1 if our coordinates are regarded as adequate for all ,geometry1 an indeterminateness arises in the expression of distance1 which can only be avoided by a convention4 ,but projective coordinates--so our argument will contend--though perfectly adequate for all projective properties1 and entirely free from any metrical presupposition1 are inadequate to express metrical properties1 just because they have no metrical presupposition4 ,thus where metrical properties are in question1 ,beltrami remains justified as against ,klein2 the reduction of metrical to projective properties is only apparent1 though the independence of these last1 as against metrical ,geometry1 is perfectly real4 #ce4 ,but what are projective coordinates1 and how are they introduced8 ,this question was not touched upon in ,cayley's ,memoir1 and it seemed1 therefore1 as if a logical error were involved in using coordinates to define distance4 ,for coordinates1 in all previous systems1 had been deduced from distance2 to use any existing coordinate system in defining distance was1 accordingly1 to incur a vicious circle4 ,cayley mentions this difficulty in a note1 where he only remarks1 however1 that he had regarded his coordinates as numbers arbitrarily assigned1 on some system not further investigated1 to different points4 ,the difficulty has been treated at length by ,sir ,r4 ,ball 7,theory of the ,content1 ,trans4 ,r4 ,i4 ,a4 #ahhi71 who urges that if the values of our coordinates already involve the usual measure of distance1 then to give a new definition1 while retaining the usual coordinates1 is to incur a contradiction4 ,he says 7op4 cit4 p4 #a73 ,in the study of non-,euclidean ,geometry ,i have often felt a difficulty which has1 ,i know1 been shared by others4 ,in that theory it seems as if we try to replace our ordinary notion of distance between two points by the logarithm of a certain anharmonic ratio#df4 ,but this ratio itself involves the notion of distance measured in the ordinary way4 ,how1 then1 can we supersede our old notion of distance by the non-,euclidean notion1 inasmuch as the very definition of the latter involves the former8 #cf4 ,this objection is valid1 we must admit1 so long as anharmonic ratio is defined in the ordinary metrical manner4 ,it would be valid1 for example1 against any attempt to found a new definition of distance on ,cremona's account of anharmonic ratio#dg1 in which it appears as a metrical property unaltered by projective transformation4 ,if a logical error is to be avoided1 in fact1 all reference to spatial magnitude of any kind must be avoided2 for all spatial magnitude1 as will be shown hereafter#dh1 is logically dependent on the fundamental magnitude of distance4 ,anharmonic ratio and coordinates must alike be defined by purely descriptive properties1 if the use afterwards made of them is to be free from metrical presuppositions1 and therefore from the objections of ,sir ,r4 ,ball4 ,such a definition has been satisfactorily given by ,klein#di1 who appeals1 for the purpose1 to v4 ,staudt's quadrilateral construction#ej4 ,by this construction1 which ,i have reproduced in outline in ,chapter ,,iii4 ,section ,a1 #aab ff41 we obtain a purely descriptive definition of harmonic and anharmonic ratio1 and1 given a pair of points1 we can obtain the harmonic conjugate to any third point on the same straight line4 ,on this construction1 the introduction of projective coordinates is based4 ,starting with any three points on a straight line1 we assign to them arbitrarily the numbers #j1 #a1 4 ,we then find the harmonic conjugate to the first with respect to #a1 1 and assign to it the number #b4 ,the object of assigning this number rather than any other1 is to obtain the value -#a for the anharmonic ratio of the four numbers corresponding to the four points#ea4 ,we then find the harmonic conjugate to the point #a1 with respect to #b1 1 and assign to it the number #c2 and so on4 ,klein has shown that by this construction1 we can obtain any number of points1 and can construct a point corresponding to any given number1 fractional or negative4 ,moreover1 when two sets of four points have the same anharmonic ratio1 descriptively defined#eb1 the corresponding numbers also have the same anharmonic ratio4 ,by introducing such a numerical system on two straight lines1 or on three1 we obtain the coordinates of any point in a plane1 or in space4 ,by this construction1 which is of fundamental importance to projective ,geometry1 the logical error1 upon which ,sir ,r4 ,ball bases his criticism1 is satisfactorily avoided4 ,our coordinates are introduced by a purely descriptive method1 and involve no presupposition whatever as to the measurement of distance4 #cg4 ,with this coordinate system1 then1 to define distance as a certain function of the coordinates is not to be guilty of a vicious circle4 ,but it by no means follows that the definition of distance is arbitrary4 ,all reference to distance has been hitherto excluded1 to avoid metrical ideas2 but when distance is introduced1 metrical ideas inevitably reappear1 and we have to remember that our coordinates give no information1 prim facie1 as to any of these metrical ideas4 ,it is open to us1 of course1 if we choose1 to continue to exclude distance in the ordinary sense1 as the quantity of a finite straight line1 and to define the word distance in any way we please4 ,but the conception1 for which the word has hitherto stood1 will then require a new name1 and the only result will be a confusion between the apparent meaning of our propositions1 to those who retain the associations belonging to the old sense of the word1 and the real meaning1 resulting from the new sense in which the word is used4 ,this confusion1 ,i believe1 has actually occurred1 in the case of those who regard the question between ,euclid and ,metageometry as one of the definition of distance4 ,distance is a quantitative relation1 and as such presupposes identity of quality4 ,but projective ,geometry deals only with quality--for which reason it is called descriptive--and cannot distinguish between two figures which are qualitatively alike4 ,now the meaning of qualitative likeness1 in ,geometry1 is the possibility of mutual transformation by a collineation#ec4 ,any two pairs of points on the same straight line1 therefore1 are qualitatively alike2 their only qualitative relation is the straight line1 which both pairs have in common2 and it is exactly the qualitative identity of the relations of the two pairs1 which enables the difference of their relations to be exhaustively dealt with by quantity1 as a difference of distance4 ,but where quantity is excluded1 any two pairs of points on the same straight line appear as alike1 and even any two sets of three3 for any three points on a straight line can be projectively transformed into any other three4 ,it is only with four points in a line that we acquire a projective property distinguishing them from other sets of four1 and this property is anharmonic ratio1 descriptively defined4 ,the projective ,geometer1 therefore1 sees no reason to give a name to the relation between two points1 in so far as this relation is anything over and above the unlimited straight line on which they lie2 and when he introduces the notion of distance1 he defines it1 in the only way in which projective principles allow him to define it1 as a relation between four points4 ,as he nevertheless wishes the word to give him the power of distinguishing between different pairs of points1 he agrees to take two out of the four points as fixed4 ,in this way1 the only variables in distance are the two remaining points1 and distance appears1 therefore1 as a function of two variables1 namely the coordinates of the two variable points4 ,when we have further defined our function so that distance may be additive1 we have a function with many of the properties of distance in the ordinary sense4 ,this function1 therefore1 the projective ,geometer regards as the only proper definition of distance4 ,we can see1 in fact1 from the manner in which our projective coordinates were introduced1 that some function of these coordinates must express distance in the ordinary sense4 ,for they were introduced serially1 so that1 as we proceeded from the zero-point towards the infinity-point1 our coordinates continually grew4 ,to every point1 a definite coordinate corresponded3 to the distance between two variable points1 therefore1 as a function dependent on no other variables1 must correspond some definite function of the coordinates1 since these are themselves functions of their points4 ,the function discussed above1 therefore1 must certainly include distance in the ordinary sense4 ,but the arbitrary and conventional nature of distance1 as maintained by ,poincar and ,klein1 arises from the fact that the two fixed points1 required to determine our distance in the projective sense1 may be arbitrarily chosen1 and although1 when our choice is once made1 any two points have a definite distance1 yet1 according as we make that choice1 distance will become a different function of the two variable points4 ,the ambiguity thus introduced is unavoidable on projective principles2 but are we to conclude1 from this1 that it is really unavoidable8 ,must we not rather conclude that projective ,geometry cannot adequately deal with distance8 ,if ,a1 ,b1 ,c1 be three different points on a line1 there must be some difference between the relation of ,a to ,b and of ,a to ,c1 for otherwise1 owing to the qualitative identity of all points1 ,b and ,c could not be distinguished4 ,but such a difference involves a relation1 between ,a and ,b1 which is independent of other points on the line2 for unless we have such a relation1 the other points cannot be distinguished as different4 ,before we can distinguish the two fixed points1 therefore1 from which the projective definition starts1 we must already suppose some relation1 between any two points on our line1 in which they are independent of other points2 and this relation is distance in the ordinary sense#ed4 ,when we have measured this quantitative relation by the ordinary methods of metrical ,geometry1 we can proceed to decide what base-points must be chosen1 on our line1 in order that the projective function discussed above may have the same value as ordinary distance4 ,but the choice of these base-points1 when we are discussing distance in the ordinary sense1 is not arbitrary1 and their introduction is only a technical device4 ,distance1 in the ordinary sense1 remains a relation between two points1 not between four2 and it is the failure to perceive that the projective sense differs from1 and cannot supersede1 the ordinary sense1 which has given rise to the views of ,klein and ,poincar4 ,the question is not one of convention1 but of the irreducible metrical properties of space4 ,to sum up3 ,quantities1 as used in projective ,geometry1 do not stand for spatial magnitudes1 but are conventional symbols for purely qualitative spatial relations4 ,but distance1 qu quantity1 presupposes identity of quality1 as the condition of quantitative comparison4 ,distance in the ordinary sense is1 in short1 that quantitative relation1 between two points on a line1 by which their difference from other points can be defined4 ,the projective definition1 however1 being unable to distinguish a collection of less than four points from any other on the same straight line1 makes distance depend on two other points besides those whose relation it defines4 ,no name remains1 therefore1 for distance in the ordinary sense1 and many projective ,geometers1 having abolished the name1 believe the thing to be abolished also1 and are inclined to deny that two points have a unique relation at all4 ,this confusion1 in projective ,geometry1 shows the importance of a name1 and should make us chary of allowing new meanings to obscure one of the fundamental properties of space4 #ch4 ,it remains to discuss the manner in which non-,euclidean ,geometries result from the projective definition of distance1 as also the true interpretation to be given to this view of ,metageometry4 ,it is to be observed that the projective methods which follow ,cayley deal throughout with a ,euclidean plane1 on which they introduce different measures of distance4 ,hence arises1 in any interpretation of these methods1 an apparent subordination of the non-,euclidean spaces1 as though these were less self-subsistent than ,euclid's4 ,this subordination is not intended in what follows2 on the contrary1 the correlation with ,euclidean space is regarded as valuable1 first1 because ,euclidean space has been longer studied and is more familiar1 but secondly1 because this correlation proves1 when truly interpreted1 that the other spaces are self-subsistent4 ,we may confine ourselves chiefly1 in discussing this interpretation1 to distances measured along a single straight line4 ,but we must be careful to remember that the metrical definition of distance--which1 according to the view here advocated1 is the only adequate definition--is the same in ,euclidean and in non-,euclidean spaces2 to argue in its favour is not1 therefore1 to argue in favour of ,euclid4 ,the projective scheme of coordinates consists of a series of numbers1 of which each represents a certain anharmonic ratio and denotes one and only one point1 and which increase uniformly with the distance from a fixed origin1 until they become infinite on reaching a certain point4 ,now ,cayley showed that1 in ,euclidean ,geometry1 distance may be expressed as the limit of the logarithm of the anharmonic ratio of the two points and the 7coincident7 points at infinity on their straight line2 while1 if we assumed that the points at infinity were distinct1 we obtained the formula for distance in hyperbolic or spherical ,geometry1 according as these points were real or imaginary4 ,hence it follows that1 with the projective definition of distance1 we shall obtain precisely the formulae of hyperbolic1 parabolic or spherical ,geometry1 according as we choose the point1 to which the value is assigned1 at a finite1 infinite or imaginary distance 7in the ordinary sense7 from the point to which we assign the value #j4 ,our straight line remains1 all the while1 an ordinary ,euclidean straight line4 ,but we have seen that the projective definition of distance fits with the true definition only when the two fixed points to which it refers are suitably chosen4 ,now the ordinary meaning of distance is required in non-,euclidean as in ,euclidean ,geometries--indeed1 it is only in metrical properties that these ,geometries differ4 ,hence our ,euclidean straight line1 though it may serve to illustrate other ,geometries than ,euclid's1 can only be dealt with correctly by ,euclid4 ,where we give a different definition of distance from ,euclid's1 we are still in the domain of purely projective properties1 and derive no information as to the metrical properties of our straight line4 ,but the importance1 to ,metageometry1 of this new interpretation1 lies in the fact that1 having independently established the metrical formulae of non-,euclidean spaces1 we find1 as in ,beltrami's ,saggio1 that these spaces can be related1 by a homographic correspondence1 with the points of ,euclidean space2 and that this can be effected in such a manner as to give1 for the distance between two points of our non-,euclidean space1 the hyperbolic or spherical measure of distance for the corresponding points of ,euclidean space4 #ci4 ,on the whole1 then1 a modification of ,sir ,r4 ,ball's view1 which is practically a generalized statement of ,beltrami's method1 seems the most tenable4 ,he imagines what1 with ,grassmann1 he calls a ,content1 i4e4 a perfectly general three-dimensional manifold1 and then correlates its elements1 one by one1 with points in ,euclidean space4 ,thus every element of the ,content acquires1 as its coordinates1 the ordinary ,euclidean coordinates of the corresponding point in ,euclidean space4 ,by means of this correlation1 our calculations1 though they refer to the ,content1 are carried on1 as in ,beltrami's ,saggio1 in ordinary ,euclidean space4 ,thus the confusion disappears1 but with it1 the supposed ,euclidean interpretation also disappears4 ,sir ,r4 ,ball's ,content1 if it is to be a space at all1 must be a space radically different from ,euclid's#ee2 to speak1 as ,klein does1 of ordinary planes with hyperbolic or elliptic measures of distance1 is either to incur a contradiction1 or to forego any metrical meaning of distance4 ,instead of ordinary planes1 we have surfaces like ,beltrami's1 of constant measure of curvature2 instead of ,euclid's space1 we have hyperbolic or spherical space4 ,at the same time1 it remains true that we can1 by ,klein's method1 give a ,euclidean meaning to every symbolic proposition in non-,euclidean ,geometry4 ,for by substituting1 for distance1 the logarithm above alluded to1 we obtain1 from the non-,euclidean result1 a result which follows from the ordinary ,euclidean axioms4 ,this correspondence removes1 once for all1 the possibility of a lurking contradiction in ,metageometry1 since1 to a proposition in the one1 corresponds one and only one proposition in the other1 and contradictory results in one system1 therefore1 would correspond to contradictory results in the other4 ,hence ,metageometry cannot lead to contradictions1 unless ,euclidean ,geometry1 at the same moment1 leads to corresponding contradictions4 ,thus the ,euclidean plane with hyperbolic or elliptic measure of distance1 though either contradictory or not metrical as an independent notion1 has1 as a help in the interpretation of non-,euclidean results1 a very high degree of utility4 #dj4 ,we have still to discuss ,klein's third kind of non-,euclidean ,geometry1 which he calls elliptic4 ,the difference between this and spherical ,geometry is difficult to grasp1 but it may be illustrated by a simpler example4 ,a plane1 as every one knows1 can be wrapped1 without stretching1 on a cylinder1 and straight lines in the plane become1 by this operation1 geodesics on the cylinder4 ,the ,geometries of the plane and the cylinder1 therefore1 have much in common4 ,but since the generating circle of the cylinder1 which is one of its geodesics1 is finite1 only a portion of the plane is used up in wrapping it once round the cylinder4 ,hence1 if we endeavour to establish a point-to-point correspondence between the plane and the cylinder1 we shall find an infinite series of points on the plane for a single point on the cylinder4 ,thus it happens that geodesics1 though on the plane they have only one point in common1 may on the cylinder have an infinite number of intersections4 ,somewhat similar to this is the relation between the spherical and elliptic ,geometries4 ,to any one point in elliptic space1 two points correspond in spherical space4 ,thus geodesics1 which in spherical space may have two points in common1 can never1 in elliptic space1 have more than one intersection4 ,but ,klein's method can only prove that elliptic ,geometry holds of the ordinary ,euclidean plane with elliptic measure of distance4 ,klein has made great endeavours to enforce the distinction between the spherical and elliptic ,geometries#ef1 but it is not immediately evident that the latter1 as distinct from the former1 is valid4 ,in the first place1 ,klein's elliptic ,geometry1 which arises as one of the alternative metrical systems on a ,euclidean plane or in a ,euclidean space1 does not by itself suffice1 if the above discussion has been correct1 to prove the possibility of an elliptic space1 i4e4 of a space having a point-to-point correspondence with the ,euclidean space1 and having as the ordinary distance between two of its points the elliptic definition of the distance between corresponding points of the ,euclidean space4 ,to prove this possibility1 we must adopt the direct method of ,newcomb 7,crelle's ,journal1 ,vol4 #hc74 ,now in the first place ,newcomb has not proved that his postulates are self-consistent2 he has only failed to prove that they are contradictory#eg4 ,this would leave elliptic space in the same position in which ,lobatchewsky and ,bolyai left hyperbolic space4 ,but further there seems to be1 at first sight1 in two-dimensional elliptic space1 a positive contradiction4 ,to explain this1 however1 some account of the peculiarities of the elliptic plane will be necessary4 ,illustration ,the elliptic plane1 regarded as a figure in three-dimensional elliptic space1 is what is called a double surface#eh1 i4e4 as ,newcomb says 7loc4 cit4 p4 #bih73 ,the two sides of a complete plane are not distinct1 as in a ,euclidean surface4444 ,if 444 a being should travel to distance #b;,d1 he would1 on his return1 find himself on the opposite surface to that on which he started1 and would have to repeat his journey in order to return to his original position without leaving the surface4 ,now if we imagine a two-dimensional elliptic space1 the distinction between the sides of a plane becomes unmeaning1 since it only acquires significance by reference to the third dimension4 ,nevertheless1 some such distinction would be forced upon us4 ,suppose1 for example1 that we took a small circle provided with an arrow1 as in the figure1 and moved this circle once round the universe4 ,then the sense of the arrow would be reversed4 ,we should thus be forced1 either to regard the new position as distinct from the former1 which transforms our plane into a spherical plane1 or to attribute the reversal of the arrow to the action of a motion which restores our circle to its original place4 ,it is to be observed that nothing short of moving round the universe would suffice to reverse the sense of the arrow4 ,this reversal seems like an action of empty space1 which would force us to regard the points which1 from a three-dimensional point of view1 are coincident though opposite1 as really distinct1 and so reduce the elliptic to the spherical plane4 ,but motion1 not space1 really causes the change1 and the elliptic plane is therefore not proved to be impossible4 ,the question is not1 however1 of any great philosophic importance4 #da4 ,in connection with the reduction of metrical to projective ,geometry1 we have one more topic for discussion4 ,this is the geometrical use of imaginaries1 by means of which1 except in the case of hyperbolic space1 the reduction is effected4 ,i have already contended1 on other grounds1 that this reduction1 in spite of its immense technical importance1 and in spite of the complete logical freedom of projective ,geometry from metrical ideas1 is purely technical1 and is not philosophically valid4 ,the same conclusion will appear1 if we take up ,cayley's challenge at the ,british ,association1 in his ,presidential ,address of #ahhc4 ,in this address1 ,professor ,cayley devoted most of his time to non-,euclidean systems4 ,non-,euclidean spaces1 he declared1 seemed to him mistaken priori#ei2 but non-,euclidean ,geometries1 here as in his mathematical works1 were accepted as flowing from a change in the definition of distance4 ,this view has been already discussed1 and need not1 therefore1 be further criticised here4 ,what ,i wish to speak about1 is the question with which ,cayley himself opened his address1 namely1 the geometrical use and meaning of imaginary quantities4 ,from the manner in which he spoke of this question1 it becomes imperative to treat it somewhat at length4 ,for he said 7pp4 #h-i73 444 ,the notion which is the really fundamental one 7and ,i cannot too strongly emphasize the assertion7 underlying and pervading the whole notion of modern analysis and ,geometry1 is that of imaginary magnitude in analysis1 and of imaginary space 7or space as the locus in quo of imaginary points and figures7 in ,geometry3 ,i use in each case the word imaginary as including real4444 ,say even the conclusion were that the notion belongs to mere technical mathematics1 or has reference to nonentities in regard to which no science is possible1 still it seems to me that 7as a subject of philosophical discussion7 the notion ought not to be thus ignored2 it should at least be shown that there is a right to ignore it4 #db4 ,this right it is now my purpose to demonstrate4 ,but for fear non-mathematicians should miss the point of ,cayley's remark 7which has sometimes been erroneously supposed to refer to non-,euclidean spaces71 ,i may as well explain1 at the outset1 that this question is radically distinct from1 and only indirectly connected with1 the validity or import of ,metageometry4 ,an imaginary quantity is one which involves -#a3 its most general form is a -#a;b where a and b are real2 ,cayley uses the word imaginary so as to include real1 in order to cover the special case where b #j4 ,it will be convenient1 in what follows1 to exclude this wider meaning1 and assume that b is not zero4 ,an imaginary point is one whose coordinates involve -#a1 i4e4 whose coordinates are imaginary quantities4 ,an imaginary curve is one whose points are imaginary--or1 in some special uses1 one whose equation contains imaginary coefficients4 ,the mathematical subtleties to which this notion leads need not be here discussed2 the reader who is interested in them will find an excellent elementary account of their geometrical uses in ,klein's ,nicht-,euklid1 ,,ii4 pp4 #ch-df4 ,but for our present purpose1 we may confine ourselves to imaginary points4 ,if these are found to have a merely technical import1 and to be destitute of any philosophical meaning1 then the same will hold of any collection of imaginary points1 i4e4 of any imaginary curve or surface4 ,that the notion of imaginary points is of supreme importance in ,geometry1 will be seen by any one who reflects that the circular points are imaginary1 and that the reduction of metrical to projective ,geometry1 which is one of ,cayley's greatest achievements1 depends on these points4 ,but to discuss adequately their philosophical import is difficult to me1 since ,i am unacquainted with any satisfactory philosophy of imaginaries in pure ,algebra4 ,i will therefore adopt the most favourable hypothesis1 and assume that no objection can be successfully urged against this use4 ,even on this hypothesis1 ,i think1 no case can be made out for imaginary points in ,geometry4 ,in the first place1 we must exclude1 from the imaginary points considered1 those whose coordinates are only imaginary with certain special systems of coordinates4 ,for example1 if one of a point's coordinates be the tangent from it to a sphere1 this coordinate will be imaginary for any point inside the sphere1 and yet the point is perfectly real4 ,a point1 then1 is only to be called imaginary1 when1 whatever real system of coordinates we adopt1 one or more of the quantities expressing these coordinates remains imaginary4 ,for this purpose1 it is mathematically sufficient to suppose our coordinates ,cartesian--a point whose ,cartesian coordinates are imaginary1 is a true imaginary point in the above sense4 ,to discuss the meaning of such a point1 it is necessary to consider briefly the fundamental nature of the correspondence between a point and its coordinates4 ,assuming that elementary ,geometry has proved--what ,i think it does satisfactorily prove--that spatial relations are susceptible of quantitative measurement1 then a given point will have1 with a suitable system of coordinates1 in a space of n dimensions1 n quantitative relations to the fixed spatial figure forming the axes of coordinates1 and these n quantitative relations will1 under certain reservations1 be unique--i4e41 no other point will have the same quantities assigned to it4 7,with many possible coordinate systems1 this latter condition is not realized3 but for that very reason they are inconvenient1 and employed only in special problems47 ,thus given a coordinate system1 and given any set of quantities1 these quantities1 if they determine a point at all1 determine it uniquely4 ,but1 by a natural extension of the method1 the above reservation is dropped1 and it is assumed that to every set of quantities some point must correspond4 ,for this assumption there seems to me no vestige of evidence4 ,as well might a postman assume that1 because every house in a street is uniquely determined by its number1 therefore there must be a house for every imaginable number4 ,we must know1 in fact1 that a given set of quantities can be the coordinates of some point in space1 before it is legitimate to give any spatial significance to these quantities3 and this knowledge1 obviously1 cannot be derived from operations with coordinates alone1 on pain of a vicious circle4 ,we must1 to return to the above analogy1 know the number of houses in ,piccadilly1 before we know whether a given number has a corresponding house or not2 and arithmetic alone1 however subtly employed1 will never give us this information4 ,thus the distinction which is important is1 not the distinction between real and imaginary quantities1 but between quantities to which points correspond and quantities to which no points correspond4 ,we can conventionally agree to denote real points by imaginary coordinates1 as in the ,gaussian method of denoting by the single quantity 7a -#a;b7 the point whose ordinary coordinates are a1 b4 ,but this does not touch ,cayley's meaning4 ,cayley means that it is of great utility in mathematics to regard1 as points with a real existence in space1 the assumed spatial correlates of quantities which1 with the coordinate system employed1 have no correlates in every-day space2 and that this utility is supposed1 by many mathematicians1 to indicate the validity of so fruitful an assumption4 ,to fix our ideas1 let us consider ,cartesian axes in three-dimensional ,euclidean space4 ,then it appears1 by inspection1 that a point may be situated at any distance to right or left of any of the three coordinate planes2 taking this distance as a coordinate1 therefore1 it appears that real points correspond to all quantities from - to 4 ,the same appears for the other two coordinates2 and since elementary ,geometry proves their variations mutually independent1 we know that one and only one real point corresponds to any three real quantities4 ,but we also know1 from the exhaustive method pursued1 that all space is covered by the range of these three variable quantities3 a fresh set of quantities1 therefore1 such as is introduced by the use of imaginaries1 possesses no spatial correlate1 and can be supposed to possess one only by a convenient fiction4 #dc4 ,the fact that the fiction is convenient1 however1 may be thought to indicate that it is more than a fiction4 ,but this presumption1 ,i think1 can be easily explained away4 ,for all the fruitful uses of imaginaries1 in ,geometry1 are those which begin and end with real quantities1 and use imaginaries only for the intermediate steps4 ,now in all such cases1 we have a real spatial interpretation at the beginning and end of our argument1 where alone the spatial interpretation is important3 in the intermediate links1 we are dealing in a purely algebraical manner with purely algebraical quantities1 and may perform any operations which are algebraically permissible4 ,if the quantities with which we end are capable of spatial interpretation1 then1 and only then1 our result may be regarded as geometrical4 ,to use geometrical language1 in any other case1 is only a convenient help to the imagination4 ,to speak1 for example1 of projective properties which refer to the circular points1 is a mere memoria technica for purely algebraical properties2 the circular points are not to be found in space1 but only in the auxiliary quantities by which geometrical equations are transformed4 ,that no contradictions arise from the geometrical interpretation of imaginaries1 is not wonderful3 for they are interpreted solely by the rules of ,algebra1 which we may admit as valid in their application to imaginaries4 ,the perception of space being wholly absent1 ,algebra rules supreme1 and no inconsistency can arise4 ,wherever1 for a moment1 we allow our ordinary spatial notions to intrude1 the grossest absurdities do arise--every one can see that a circle1 being a closed curve1 cannot get to infinity4 ,the metaphysician1 who should invent anything so preposterous as the circular points1 would be hooted from the field4 ,but the mathematician may steal the horse with impunity4 ,finally1 then1 only a knowledge of space1 not a knowledge of ,algebra1 can assure us that any given set of quantities will have a spatial correlate1 and in the absence of such a correlate1 operations with these quantities have no geometrical import4 ,this is the case with imaginaries in ,cayley's sense1 and their use in ,geometry1 great as are its technical advantages1 and rigid as is its technical validity1 is wholly destitute of philosophical importance4 #dd4 ,we have now1 ,i think1 discussed most of the questions concerning the scope and validity of the projective method4 ,we have seen that it is independent of all metrical presuppositions1 and that its use of coordinates does not involve the assumption that spatial magnitudes are measured or expressed by them4 ,we have seen that it is able to deal1 by its own methods alone1 with the question of the qualitative likeness of geometrical figures1 which is logically prior to any comparison as to quantity1 since quantity presupposes qualitative likeness4 ,we have seen also that1 so far as its legitimate use extends1 it applies equally to all homogeneous spaces1 and that its criterion of an independently possible space--the determination of a straight line by two points#fj--;is not subject to the qualifications and limitations which belong1 as we have seen in the case of the cylinder1 to the metrical criterion of constant curvature4 ,but we have also seen that1 when projective ,geometry endeavours to grapple with spatial magnitude1 and bring distance and the measurement of angles beneath its sway1 its success1 though technically valid and important1 is philosophically an apparent success only4 ,metrical ,geometry1 therefore1 if quantity is to be applied to space at all1 remains a separate1 though logically subsequent branch of ,mathematics4 #de4 ,it only remains to say a few words about ,sophus ,lie4 ,as a mathematician1 as the inventor of a new and immensely powerful method of analysis1 he cannot be too highly praised4 ,geometry is only one of the numerous subjects to which his theory of continuous groups applies1 but its application to ,geometry has made a revolution in method1 and has rendered possible1 in such problems as ,helmholtz's1 a treatment infinitely more precise and exhaustive than any which was possible before4 ,the general definition of a group is as follows3 ,if we have any number of independent variables x#ax#b444xn1 and any series of transformations of these into new variables--the transformations being defined by equations of specified forms1 with parameters varying from one transformation to another--then the series of transformations form a group1 if the successive application of any two is equivalent to a single member of the original series of transformations4 ,the group is continuous1 when we can pass1 by infinitesimal gradations within the group1 from any one of the transformations to any other4 ,now1 in ,geometry1 the result of two successive motions or collineations of a figure can always be obtained by a single motion or collineation1 and any motion or collineation can be built up of a series of infinitesimal motions or collineations4 ,moreover the analytical expression of either is a certain transformation of the coordinates of all the points of the figure#fa4 ,hence the transformations determining a motion or a collineation are such as to form a continuous group4 ,but the question of the projective equivalence of two figures1 to which all projective ,geometry is reducible1 must always be dealt with by a collineation2 and the question of the equality of two figures1 to which all metrical ,geometry is reducible1 must always be decided by a motion such as to cause superposition2 hence the whole subject of ,geometry may be regarded as a theory of the continuous groups which define all possible collineations and motions4 ,now ,sophus ,lie has developed1 at great length1 the purely analytical theory of groups2 he has therefore1 by this method of formulating the problem1 a very powerful weapon ready for the attack4 ,in two papers ,on the foundations of ,geometry#fb1 undertaken at ,klein's urgent request1 he takes premisses which roughly correspond to those of ,helmholtz1 omitting ,monodromy1 and applies the theory of groups to the deduction of their consequences#fc4 ,helmholtz's work1 he says1 can hardly be looked upon as proving its conclusions1 and indeed the more searching analysis of the group-theory reveals several possibilities unknown to ,helmholtz4 ,nevertheless1 as a pioneer1 devoid of ,lie's machinery1 ,helmholtz deserves1 ,i think1 more praise than ,lie is willing to give him#fd4 ,lie's method is perfectly exhaustive2 omitting the premiss of ,monodromy1 the others show that a body has six degrees of freedom1 i4e4 that the group giving all possible motions of a body will have six independent members2 if we keep one point fixed1 the number of independent members is reduced to three4 ,he then1 from his general theory1 enumerates all the groups which satisfy this condition4 ,in order that such a group should give possible motions1 it is necessary1 by ,helmholtz's second axiom1 that it should leave invariant some function of the coordinates of any two points4 ,this eliminates several of the groups previously enumerated1 each of which he discusses in turn4 ,he is thus led to the following results3 ,i4 ,in two dimensions1 if free mobility is to hold universally1 there are no groups satisfying ,helmholtz's first three axioms1 except those which give the ordinary ,euclidean and non-,euclidean motions2 but if it is to hold only within a certain region1 there is also a possible group in which the curve described by any point in a rotation is not closed1 but an equiangular spiral4 ,to exclude this possibility1 ,helmholtz's axiom of ,monodromy is required4 ,,ii4 ,in three dimensions1 the results go still more against ,helmholtz4 ,assuming free mobility only within a certain region1 we have to distinguish two cases3 ,either free mobility holds1 within that region1 absolutely without exception1 i4e4 when one point is held fast1 every other point within the region can move freely over a surface3 in this case the axiom of ,monodromy is unnecessary1 and the first three axioms suffice to define our group as that of ,euclidean and non-,euclidean motions4 ,or free mobility1 within the specified region1 holds only of every point of general position1 while the points of a certain line1 when one point is fixed1 are only able to move on that line1 not on a surface3 when this is the case1 other groups are possible1 and can only be excluded by ,helmholtz's fourth axiom4 ,having now stated the purely mathematical results of ,lie's investigations1 we may return to philosophical considerations1 by which ,helmholtz's work was mainly motived4 ,it becomes obvious1 not only that exceptions within a certain region1 but also that limitation to a certain region1 of the axiom of ,free ,mobility1 are philosophically quite impossible and inconceivable4 ,how can a certain line1 or a certain surface1 form an impassable barrier in space1 or have any mobility different in kind from that of all other lines or surfaces8 ,the notion cannot1 in philosophy1 be permitted for a moment1 since it destroys that most fundamental of all the axioms1 the homogeneity of space4 ,we not only may1 therefore1 but must take ,helmholtz's axiom of ,free ,mobility in its very strictest sense2 the axiom of ,monodromy thus becomes mathematically1 as well as philosophically1 superfluous4 ,this is1 from a philosophical standpoint1 the most important of ,lie's results4 #df4 ,i have now come to the end of my history of ,metageometry4 ,it has not been my aim to give an exhaustive account of even the important works on the subject--in the third period1 especially1 the names of ,poincar1 ,pasch1 ,cremona1 ,veronese1 and others who might be mentioned1 would have cried shame upon me1 had ,i had any such object4 ,but ,i have tried to set forth1 as clearly as ,i could1 the principles at work in the various periods1 the motives and results of successive theories4 ,we have seen how the philosophical motive1 at first predominant1 has been gradually extruded by the purely mathematical and technical spirit of most recent ,geometers4 ,at first1 to discredit the ,transcendental ,aesthetic seemed1 to ,metageometers1 as important as to advance their science2 but from the works of ,cayley1 ,klein or ,lie1 no reader could gather that ,kant had ever lived4 ,we have also seen1 however1 that as the interest in philosophy waned1 the interest for philosophy increased3 as the mathematical results shook themselves free from philosophical controversies1 they assumed gradually a stable form1 from which further development1 we may reasonably hope1 will take the form of growth1 rather than transformation4 ,the same gradual development out of philosophy might1 ,i believe1 be traced in the infancy of most branches of mathematics2 when philosophical motives cease to operate1 this is1 in general1 a sign that the stage of uncertainty as to premisses is past1 so that the future belongs entirely to mathematical technique4 ,when this stable stage has been attained1 it is time for ,philosophy to borrow of ,science1 accepting its final premisses as those imposed by a real necessity of fact or logic4 #dg4 ,now in discussing the systems of ,metageometry1 we have found two kinds1 radically distinct and subject to different axioms4 ,the historically prior kind1 which deals with metrical ideas1 discusses1 to begin with1 the conditions of ,free ,mobility1 which is essential to all measurement of space4 ,it finds the analytical expression of these conditions in the existence of a space-constant1 or constant measure of curvature1 which is equivalent to the homogeneity of space4 ,this is its first axiom4 ,its second axiom states that space has a finite integral number of dimensions1 i4e4 in metrical terms1 that the position of a point1 relative to any other figure in space1 is uniquely determined by a finite number of spatial magnitudes1 called coordinates4 ,the third axiom of metrical ,geometry may be called1 to distinguish it from the corresponding projective axiom1 the axiom of distance4 ,there exists one relation1 it says1 between any two points1 which can be preserved unaltered in a combined motion of both points1 and which1 in any motion of a system as one rigid body1 is always unaltered4 ,this relation we call distance4 ,the above statement of the three essential axioms of metrical ,geometry is taken from ,helmholtz as amended by ,lie4 ,lie's own statement of the axioms1 as quoted above1 has been too much influenced by projective methods to give a historically correct rendering of the spirit of the second period2 ,helmholtz's statement1 on the other hand1 requires1 as ,lie has shewn1 very considerable modifications4 ,the above compromise may1 therefore1 ,i hope be taken as accepting ,lie's corrections while retaining ,helmholtz's spirit4 #dh4 ,but metrical ,geometry1 though it is historically prior1 is logically subsequent to projective ,geometry4 ,for projective ,geometry deals directly with that qualitative likeness1 which the judgment of quantitative comparison requires as its basis4 ,now the above three axioms of metrical ,geometry1 as we shall see in ,chapter ,,iii4 ,section ,b1 do not presuppose measurement1 but are1 on the contrary1 the conditions presupposed by measurement4 ,without these axioms1 which are common to all three spaces1 measurement would be impossible2 with them1 so ,i shall contend1 measurement is able1 though only empirically1 to decide approximately which of the three spaces is valid of our actual world4 ,but if these three axioms themselves express1 not results1 but conditions1 of measurement1 must they not be equivalent to the statement of that qualitative likeness on which quantitative comparison depends8 ,and if so1 must we not expect to find the same axioms1 though perhaps under a different form1 in projective ,geometry8 #di4 ,this expectation will not be disappointed4 ,the above three axioms1 as we shall see hereafter1 are one and all philosophically equivalent to the homogeneity of space1 and this in turn is equivalent to the axioms of projective ,geometry4 ,the axioms of projective ,geometry1 in fact1 may be roughly stated thus3 ,i4 ,space is continuous and infinitely divisible2 the zero of extension1 resulting from infinite division1 is called a ,point4 ,all points are qualitatively similar1 and distinguished by the mere fact that they lie outside one another4 ,,ii4 ,any two points determine a unique figure1 the straight line2 two straight lines1 like two points1 are qualitatively similar1 and distinguished by the mere fact that they are mutually external4 ,,iii4 ,three points not in one straight line determine a unique figure1 the plane1 and four points not in one plane determine a figure of three dimensions4 ,this process may1 so far as can be seen priori1 be continued1 without in any way interfering with the possibility of projective ,geometry1 to five or to n points4 ,but projective ,geometry requires1 as an axiom1 that the process should stop with some positive integral number of points1 after which1 any fresh point is contained in the figure determined by those already given4 ,if the process stops with 7n #a7 points1 our space is said to have n dimensions4 ,these three axioms1 it will be seen1 are the equivalents of the three axioms of metrical ,geometry#fe1 expressed without reference to quantity4 ,we shall find them to be deducible1 as before1 from the homogeneity of space1 or1 more generally still1 from the possibility of experiencing externality4 ,they will therefore appear as priori1 as essential to the existence of any ,geometry and to experience of an external world as such4 #ej4 ,that some logical necessity is involved in these axioms might1 ,i think1 be inferred as probable1 from their historical development alone4 ,for the systems of ,metageometry have not1 in general1 been set up as more likely to fit facts than the system of ,euclid2 with the exception of ,zllner1 for example1 ,i know of no one who has regarded the fourth dimension as required to explain phenomena4 ,as regards the space-constant again1 though a small space-constant is regarded as empirically possible1 it is not usually regarded as probable2 and the finite space-constants1 with which ,metageometry is equally conversant1 are not usually thought even possible1 as explanations of empirical fact#ff4 ,thus the motive has been throughout not one of fact1 but one of logic4 ,does not this give a strong presumption1 that those axioms which are retained1 are retained because they are logically indispensable8 ,if this be so1 the axioms common to ,euclid and ,metageometry will be priori1 while those peculiar to ,euclid will be empirical4 ,after a criticism of some differing theories of ,geometry1 ,i shall proceed1 in ,chapters ,,iii4 and ,,iv41 to the proof and consequences of this thesis1 which will form the remainder of the present work4 ,,footnotes3 #e ,v4 ,mmoires de l',acadmie royale des ,sciences de l'lnstitut de ,france1 ,t4 ,,xii4 #ahcc1 for a full statement of his results1 with references to former writings4 #f ,this bolder method1 it appears1 had been suggested1 nearly a century earlier1 by an ,italian1 ,saccheri4 ,his work1 which seems to have remained completely unknown until ,beltrami rediscovered it in #ahhi1 is called ,euclides ab omni naevo vindicatus1 etc4 ,mediolani1 #agcc4 7,see ,veronese1 ,grundzge der ,geometrie1 ,german translation1 ,leipzig1 #ahid1 p4 #fcf47 ,his results included spherical as well as hyperbolic space2 but they alarmed him to such an extent that he devoted the last half of his book to disproving them4 #g ,klein's first account of elliptic ,geometry1 as a result of ,cayley's projective theory of distance1 appeared in two articles entitled ,ueber die sogenannte ,nicht-,euklidische ,geometrie1 ,i1 ,,ii1 ,math4 ,annalen #d1 #f 7#ahga-b74 ,it was afterwards independently discovered by ,newcomb1 in an article entitled ,elementary ,theorems relating to the geometry of a space of three dimensions1 and of uniform positive curvature in the fourth dimension1 ,crelle's ,journal fr die reine und angewandte ,mathematik1 ,vol4 #hc 7#ahgg74 ,for an account of the mathematical controversies concerning elliptic ,geometry1 see ,klein's ,vorlesungen ber ,nicht-,euklidische ,geometrie1 ,gttingen #ahic1 ,i4 p4 #bhd ff4 ,a bibliography of the relevant literature up to the year #ahgh was given by ,halsted in the ,american ,journal of ,mathematics1 ,vols4 #a1 #b4 #h ,veronese 7op4 cit4 p4 #fch7 denies the priority of ,gauss in the invention of a non-,euclidean system1 though he admits him to have been the first to regard the axiom of parallels as indemonstrable4 ,his grounds for the former assertion seem scarcely adequate3 on the evidence against it1 see ,klein1 ,nicht-,euklid1 ,i4 pp4 #aga-agd4 #i ,v4 ,briefwechsel mit ,schumacher1 ,bd4 ,,ii4 p4 #bfh4 #aj f4 ,helmholtz1 ,wiss4 ,abh4 ,,ii4 p4 #faa4 #aa ,crelle's ,journal1 #ahcg4 #ab ,theorie der ,parallellinien1 ,berlin1 #ahdj4 ,republished1 ,berlin1 #ahhg4 ,translated by ,halsted1 ,austin1 ,texas1 ,u4,s4,a4 #dth edition1 #ahib4 #ac ,frischauf1 ,absolute ,geometrie1 nach ,johann ,bolyai1 ,leipzig1 #ahgb4 ,halsted1 ,the ,science ,absolute of ,space1 translated from the ,latin1 #dth edition1 ,austin1 ,texas1 ,u4,s4,a4 #ahif4 #ad ,both ,lobatchewsky and ,bolyai1 as ,veronese remarks1 start rather from the point-pair than from distance4 ,see ,frischauf1 ,absolute ,geometrie1 ,anhang4 #ae ,compare ,stallo1 ,concepts of ,modern ,physics1 p4 #bdh4 #af ,gesammelte ,werke1 pp4 #bee-bfh4 #ag ,on the history of this word1 see ,stallo1 ,concepts of ,modern ,physics1 p4 #beh4 ,it was used by ,kant1 and adapted by ,herbart to almost the same meaning as it bears in ,riemann4 ,herbart1 however1 also uses the word ,reihenform to express a similar idea4 ,see ,psychologie als ,wissenschaft1 ,i4 #ajj and ,,ii4 #aci1 where ,riemann's analogy with colours is also suggested4 #ah ,compare ,erdmann's ,grssenbegriff vom ,raum4 #ai ,compare ,veronese1 op4 cit4 p4 #fdb3 ,riemann ist in seiner ,definition des ,begriffs ,grsse dunkel4 ,see also ,veronese's whole following criticism4 #bj ,vortrge und ,reden1 ,vol4 ,,ii4 p4 #ah4 #ba ,cf4 ,klein1 ,nicht-,euklid1 ,i4 p4 #afj4 #bb ,since we are considering the curvature at a point1 we are only concerned with the first infinitesimal elements of the geodesics that start from such a point4 #bc ,disquisitiones generales circa superficies curvas1 ,werke1 ,bd4 ,,iv4 ,,ss4 #bai-beh1 #ahbg4 #bd ,nevertheless1 the ,geometries of different surfaces of equal curvature are liable to important differences4 ,for example1 the cylinder is a surface of zero curvature1 but since its lines of curvature in one direction are finite1 its ,geometry coincides with that of the plane only for lengths smaller than the circumference of its generating circle 7see ,veronese1 op4 cit4 p4 #fdd74 ,two geodesics on a cylinder may meet in many points4 ,for surfaces of zero curvature on which this is not possible1 the identity with the plane may be allowed to stand4 ,otherwise1 the identity extends only to the properties of figures not exceeding a certain size4 #be ,for we may consider two different parts of the same surface as corresponding parts of different surfaces2 the above proposition then shows that a figure can be reproduced in one part when it has been drawn in another1 if the measures of curvature correspond in the two parts4 #bf ,crelle1 ,vols1 ,,xix41 ,,xx41 #ahci-dj4 #bg ,in this formula1 u1 v may be the lengths of lines1 or the angles between lines1 drawn on the surface1 and having thus no necessary reference to a third dimension4 #bh ,in what follows1 ,i have given rather ,klein's exposition of ,riemann1 than ,riemann's own account4 ,the former is much clearer and fuller1 and not substantially different in any way4 ,v4 ,klein1 ,nicht-,euklid1 ,i4 pp4 #bjf ff4 #bi ,see #fi-gc4 #cj ,grundlagen der ,geometrie1 ,i4 and ,,ii41 ,leipziger ,berichte1 #ahij2 v4 end of present chapter1 #de4 #ca ,nicht-,euklid1 ,i4 pp4 #beh-i4 #cb ,giornale di ,matematiche1 ,vol4 ,,vi41 #ahfh4 ,translated into ,french by ,j4 ,hoel in the ,annales ,scientifiques de l'cole ,normale ,suprieure1 ,vol4 ,,vi4 #ahfi4 #cc ,crelle's ,journal1 ,vols4 ,,xix4 ,,xx41 #ahci-dj4 #cd ,nicht-,euklid1 ,i4 p4 #aij4 #ce ,this article is more trigonometrical and analytical than the ,german book1 and therefore makes the above interpretation peculiarly evident4 #cf ,such surfaces are by no means particularly remote4 ,one of them1 for example1 is formed by the revolution of the common ,tractrix x asin 1 y a7log tan _/#b cos 74 #cg ,teoria fondamentale degli spazii di curvatura costanta1 ,annali di ,matematica1 ,,ii4 ,vol4 #b1 #ahfh-i4 ,also translated by ,j4 ,hoel1 loc4 cit4 #ch ,see ,klein1 ,nicht-,euklid1 ,i4 p4 #dg ff41 and the references there given4 #ci ,see quotation below1 from his ,british ,association ,address4 #dj ,compare the opening sentence1 due to ,cayley1 of ,salmon's ,higher ,plane ,curves4 #da ,v4 ,nicht-,euklid1 ,i4 ,chaps4 ,i4 and ,,ii4 #db ,see p4 #i of ,cayley's address to the ,brit4 ,ass4 #ahhc4 ,also a quotation from ,klein in ,erdmann's ,axiome der ,geometrie1 p4 #abd note4 #dc ,nature1 ,vol4 ,,xlv4 p4 #djg4 #dd ,nicht-,euklid1 ,i4 p4 #bjj4 #de ,i4e4 the equation ,,ab ,,bc ,,ac1 for three points in one straight line4 #df ,the formula substituted by ,klein for ,cayley's inverse sine or cosine4 ,the two are equivalent1 but ,klein's is mathematically much the more convenient4 #dg ,elements of ,projective ,geometry1 ,second ,edition1 ,oxford1 #ahic1 ,chap4 ,,ix4 #dh ,chap4 ,,iii4 ,section ,b4 #di ,see ,nicht-,euklid1 ,i4 p4 #cch ff4 #ej ,see his ,geometrie der ,lage1 #h1 ,harmonische ,gebilde4 #ea ,the anharmonic ratio of four numbers1 p1 q1 r1 s1 is defined as 7p - q747r - s7 _/ 7p - r747q - s74 #eb ,i4e4 as transformable into each other by a collineation4 ,see ,chap4 ,,iii4 ,sec4 ,a1 #aaj4 #ec ,see ,chap4 ,,iii4 ,sec4 ,a4 #ed ,it follows from this1 that the reduction of metrical to projective properties1 even when1 as in hyperbolic ,geometry1 the ,absolute is real1 is only apparent1 and has a merely technical validity4 #ee ,sir ,r4 ,ball does not regard his non-,euclidean content as a possible space 7v4 op4 cit4 p4 #aea74 ,in this important point ,i disagree with his interpretation1 holding such a content to be a space as possible1 priori1 as ,euclid's1 and perhaps actually true within the margin due to errors of observation4 #ef ,see ,nicht-,euklid1 ,i4 p4 #ig ff4 and p4 #bib ff4 #eg ,newcomb says 7loc4 cit4 p4 #bic73 ,the system here set forth is founded on the following three postulates4 #a4 ,i assume that space is triply extended1 unbounded1 without properties dependent either on position or direction1 and possessing such planeness in its smallest parts that both the postulates of the ,euclidean ,geometry1 and our common conceptions of the relations of the parts of space are true for every indefinitely small region in space4 #b4 ,i assume that this space is affected with such curvature that a right line shall always return into itself at the end of a finite and real distance #b;,d without losing1 in any part of its course1 that symmetry with respect to space on all sides of it which constitutes the fundamental property of our conception of it4 #c4 ,i assume that if two right lines emanate from the same point1 making the indefinitely small angle a with each other1 their distance apart at the distance r from the point of intersection will be given by the equation s #b;a,d_/ sin r_/#b;,d4 ,the right line thus has this property in common with the ,euclidean right line that two such lines intersect only in a single point4 ,it may be that the number of points in which two such lines can intersect admit of being determined from the laws of curvature1 but not being able so to determine it1 ,i assume as a postulate the fundamental property of the ,euclidean right line4 ,it is plain that in the absence of the determination spoken of1 the possibility of elliptic space is not established4 ,it may be possible1 for example1 to prove that1 in a space where there is a maximum to distance1 there must be an infinite number of straight lines joining two points of maximum distance4 ,in this event1 elliptic space would become impossible4 #eh ,for an elucidation of this term1 see ,klein1 ,nicht-,euklid1 ,i4 p4 #ii ff4 #ei ,cf4 p4 #i of ,report3 ,my own view is that ,euclid's twelfth axiom1 in ,playfair's form of it1 does not need demonstration1 but is part of our notion of space1 of the physical space of our experience1 but which is the representation lying at the bottom of all external experience4 #fj ,the exception to this axiom1 in spherical space1 presupposes metrical ,geometry1 and does not destroy the validity of the axiom for projective ,geometry4 ,see ,chap4 ,,iii4 ,sec4 ,b1 #aga4 #fa ,mathematicians of ,lie's school have a habit1 at first somewhat confusing1 of speaking of motions of space instead of motions of bodies1 as though space as a whole could move4 ,all that is meant is1 of course1 the equivalent motion of the coordinate axes1 i4e4 a change of axes in the usual elementary sense4 #fb ,ueber die ,grundlagen der ,geometrie1 ,leipziger ,berichte1 #ahij4 ,the problem of these two papers is really metrical1 since it is concerned1 not with collineations in general1 but with motions4 ,the problem1 however1 is dealt with by the projective method1 motions being regarded as collineations which leave the ,absolute unchanged4 ,it seemed impossible1 therefore1 to discuss ,lie's work1 until some account had been given of the projective method4 #fc ,lie's premisses1 to be accurate1 are the following3 ,let x#a f7x1 y1 z1 a#a1 a#b4447 x#b 7x1 y1 z1 a#a1 a#b4447 x#c 7x1 y1 z1 a#a1 a#b4447 give an infinite family of real transformations of space1 as to which we make the following hypotheses3 ,a4 ,the functions f1 1 1 are analytical functions of x1 y1 z1 a#a1 a#b4444 ,b4 ,two points x#ay#az#a1 x#by#bz#b possess an invariant1 i4e4 7x#a1 y#a1 z#a1 x#b1 y#b1 z#b7 7x#a1 y#a1 z#a1 x#b1 y#b1 z#b7 where x#a4441 x#b4441 are the transformed coordinates of the two points4 ,c4 ,free ,mobility3 i4e41 any point can be moved into any other position2 when one point is fixed1 any other point of general position can take up #b positions2 when two points are fixed1 any other of general position can take up #a positions2 when three1 no motion is possible--these limitations being results of the equations given by the invariant 4 #fd ,on this point1 cf4 ,klein1 ,hhere ,geometrie1 ,gttingen1 #ahic1 ,,ii4 pp4 #bbe-bdd1 especially pp4 #bcj-a4 #fe ,axiom ,,ii4 of the metrical triad corresponds to ,axiom ,,iii4 of the projective1 and vice vers4 #ff ,cf4 ,helmholtz1 ,wiss4 ,abh4 ,vol4 ,,ii4 p4 #fdj1 note3 ,die ,bearbeiter der ,nicht-,euklidischen ,geometrie 7haben7 deren objective ,wahrheit nie behauptet4 ,,chapter ,,ii4 ,,critical ,,account ,,of ,,some ,,previous ,,philosophical ,,theories ,,of ,,geometry4 #ea4 ,we have now traced the mathematical development of the theory of geometrical axioms1 from the first revolt against ,euclid to the present day4 ,we may hope1 therefore1 to have at our command the technical knowledge required for the philosophy of the subject4 ,the importance of ,geometry1 in the theories of knowledge which have arisen in the past1 can scarcely be exaggerated4 ,in ,descartes1 we find the whole theory of method dominated by analytical ,geometry1 of whose fruitfulness he was justly proud4 ,in ,spinoza1 the paramount influence of ,geometry is too obvious to require comment4 ,among mathematicians1 ,newton's belief in absolute space was long supreme1 and is still responsible for the current formulation of the laws of motion4 ,against this belief on the one hand1 and against ,leibnitz's theory of space on the other1 and not1 as ,caird has pointed out#fg1 against ,hume's empiricism1 was directed that keystone of the ,critical ,philosophy1 the ,kantian doctrine of space4 ,thus ,geometry has been1 throughout1 of supreme importance in the theory of knowledge4 ,but in a criticism of representative modern theories of ,geometry1 which is designed to be1 not a history of the subject1 but an introduction to1 and defence of1 the views of the author1 it will not be necessary to discuss any more ancient theory than that of ,kant4 ,kant's views on this subject1 true or false1 have so dominated subsequent thought1 that whether they were accepted or rejected1 they seemed equally potent in forming the opinions1 and the manner of exposition1 of almost all later writers4 ,kant4 #eb4 ,it is not my purpose1 in this chapter1 to add to the voluminous literature of ,kantian criticism1 but only to discuss the bearing of ,metageometry on the argument of the ,transcendental ,aesthetic1 and the aspect under which this argument must be viewed in a discussion of ,geometry#fh4 ,on this point several misunderstandings seem to me to have had wide prevalence1 both among friends and foes1 and these misunderstandings ,i shall endeavour1 if ,i can1 to remove4 ,in the first place1 what does ,kant's doctrine mean for ,geometry8 ,obviously not the aspect of the doctrine which has been attacked by psychologists1 the ,kantian machine-shop as ,james calls it--at any rate1 if this can be clearly separated from the logical aspect4 ,the question whether space is given in sensation1 or whether1 as ,kant maintained1 it is given by an intuition to which no external matter corresponds1 may for the present be disregarded4 ,if1 indeed1 we held the view which seems crudely to sum up the standpoint of the ,critique1 the view that all certain knowledge is self-knowledge1 then we should be committed1 if we had decided that ,geometry was apodeictic1 to the view that space is subjective4 ,but even then1 the psychological question could only arise when the epistemological question had been solved1 and could not1 therefore1 be taken into account in our first investigation4 ,the question before us is precisely the question whether1 or how far1 ,geometry is apodeictic1 and for the moment we have only to investigate this question1 without fear of psychological consequences4 #ec4 ,now on this question1 as on almost all questions in the ,aesthetic or the ,analytic1 ,kant's argument is twofold4 ,on the one hand1 he says1 ,geometry is known to have apodeictic certainty3 therefore space must be priori and subjective4 ,on the other hand1 it follows1 from grounds independent of ,geometry1 that space is subjective and priori2 therefore ,geometry must have apodeictic certainty4 ,these two arguments are not clearly distinguished in the ,aesthetic1 but a little analysis1 ,i think1 will disentangle them4 ,thus in the first edition1 the first two arguments deduce1 from non-geometrical grounds1 the apriority of space2 the third deduces the apodeictic certainty of ,geometry1 and maintains1 conversely1 that no other view can account for this certainty#fi2 the last two arguments only maintain that space is an intuition1 not a concept4 ,in the second edition1 the double argument is clearer1 the apriority of space being proved independently of ,geometry in the metaphysical deduction1 and deduced from the certainty of ,geometry1 as the only possible explanation of this1 in the transcendental deduction4 ,in the ,prolegomena1 the latter argument alone is used1 but in the ,critique both are employed4 #ed4 ,now it must be admitted1 ,i think1 that ,metageometry has destroyed the legitimacy of the argument from ,geometry to space2 we can no longer affirm1 on purely geometrical grounds1 the apodeictic certainty of ,euclid4 ,but unless ,metageometry has done more than this--unless it has proved1 what ,i believe it alone cannot prove1 that ,euclid has not apodeictic certainty--then ,kant's other line of argument retains what force it may ever have had4 ,the actual space we know1 it may say1 is admittedly ,euclidean1 and is proved1 without any reference to ,geometry1 to be priori2 hence ,euclid has apodeictic certainty1 and non-,euclid stands condemned4 ,to this it is no answer to urge1 with the ,metageometers1 that non-,euclidean systems are logically self-consistent2 for ,kant is careful to argue that geometrical reasoning1 by virtue of our intuition of space1 is synthetic1 and cannot1 though priori1 be upheld by the principle of contradiction alone#gj4 ,unless non-,euclideans can prove1 what they have certainly failed to prove up to the present1 that we can frame an intuition of non-,euclidean spaces1 ,kant's position cannot be upset by ,metageometry alone1 but must also be attacked1 if it is to be successfully attacked1 on its purely philosophical side4 #ee4 ,for such an attack1 two roads lie open3 either we may disprove the first two arguments of the ,aesthetic1 or we may criticize1 from the standpoint of general logic1 the ,kantian doctrine of synthetic priori judgments and their connection with subjectivity4 ,both these attacks1 ,i believe1 could be conducted with some success2 but if we are to disprove the apodeictic certainty of ,geometry1 one or other is essential1 and both1 ,i believe1 will be found only partially successful4 ,it will be my aim to prove1 in discussing these two lines of attack1 7#a7 that the distinction of synthetic and analytic judgments is untenable1 and further1 that the principle of contradiction can only give fruitful results on the assumption that experience in general1 or1 in a particular science1 some special branch of experience1 is to be formally possible2 7#b7 that the first two arguments of the ,transcendental ,aesthetic suffice to prove1 not ,euclidean space1 but some form of externality--which may be sensational or intuitional1 but not merely conceptual--a necessary prerequisite of experience of an external world4 ,in the third and fourth chapters1 ,i shall contend1 as a result of these conclusions1 that those axioms1 which ,euclid and ,metageometry have in common1 coincide with those properties of any form of externality which are deducible1 by the principle of contradiction1 from the possibility of experience of an external world4 ,these properties1 then1 may be said1 though not quite in the ,kantian sense1 to be priori properties of space1 and as to these1 ,i think1 a modified ,kantian position may be maintained4 ,but the question of the subjective or objective nature of space may be left wholly out of account during the course of this discussion1 which will gain by dealing exclusively with logical1 as opposed to psychological points of view4 #ef4 7#a7 ,kant's logical position4 ,the doctrine of synthetic and analytic judgments--at any rate if this is taken as the corner-stone of ,epistemology--has been so completely rejected by most modern logicians#ga1 that it would demand little attention here1 but for the fact that an enthusiastic ,french ,kantian1 ,m4 ,renouvier1 has recently appealed to it1 with perfect confidence1 on the very question of ,geometry#gb4 ,and it must be owned1 with ,m4 ,renouvier1 that if such judgments existed1 in the ,kantian sense1 non-,euclidean ,geometry1 which makes no appeal to intuition1 could have nothing to say against them4 ,m4 ,renouvier's contention1 therefore1 forces us briefly to review the arguments against ,kant's doctrine1 and briefly to discuss what logical canon is to replace it4 ,every judgment--so modern logic contends--is both synthetic and analytic2 it combines parts into a whole1 and analyses a whole into parts#gc4 ,if this be so1 the distinction of analysis and synthesis1 whatever may be its importance in pure ,logic1 can have no value in ,epistemology4 ,but such a doctrine1 it must be observed1 allows full scope to the principle of contradiction3 this criterion1 since all judgments1 in one aspect at least1 are analytic1 is applicable to all judgments alike4 ,on the other hand1 the whole which is analysed must be supposed already given1 before the parts can be mutually contradictory3 for only by connection in a given whole can two parts or adjectives be incompatible4 ,thus the principle of contradiction remains barren until we already have some judgments1 and even some inference3 for the parts may be regarded1 to some extent1 as an inference from the whole1 or vice vers4 ,when once the arch of knowledge is constructed1 the parts support one another1 and the principle of contradiction is the keystone3 but until the arch is built1 the keystone remains suspended1 unsupported and unsupporting1 in the empty air4 ,in other words1 knowledge once existent can be analysed1 but knowledge which should have to win every inch of the way against a critical scepticism1 could never begin1 and could never attain that circular condition in which alone it can stand4 ,but ,kant's doctrine1 if true1 is designed to restrain a critical scepticism even where it might be effective4 ,certain fundamental propositions1 he says1 are not deducible from logic1 i4e4 their contradictories are not self-contradictory2 they combine a subject and predicate which cannot1 in any purely logical way1 be shewn to have any connection1 and yet these judgments have apodeictic certainty4 ,but concerning such judgments1 ,kant is generally careful not to rely upon the mere subjective conviction that they are undeniable3 he proves1 with every precaution1 that without them experience would be impossible4 ,experience consists in the combination of terms which formal logic leaves apart1 and presupposes1 therefore1 certain judgments by which a framework is made for bringing such terms together4 ,without these judgments--so ,kant contends--all synthesis and all experience would be impossible4 ,if1 therefore1 the detail of the ,kantian reasoning be sound1 his results may be obtained by the principle of contradiction plus the possibility of experience1 as well as by his distinction of synthetic and analytic judgments4 ,logic1 at the present day1 arrogates to itself at once a wider and a narrower sphere than ,kant allowed to it4 ,wider1 because it believes itself capable of condemning any false principle or postulate2 narrower1 because it believes that its law of contradiction1 without a given whole or a given hypothesis1 is powerless1 and that two terms1 per se1 though they may be different1 cannot be contradictories1 but acquire this relation only by combination in a whole about which something is known1 or by connection with a postulate which1 for some reason1 must be preserved4 ,thus no judgment1 per se1 is either analytic or synthetic1 for the severance of a judgment from its context robs it of its vitality1 and makes it not truly a judgment at all4 ,but in its proper context it is neither purely synthetic nor purely analytic2 for while it is the further determination of a given whole1 and thus in so far analytic1 it also involves the emergence of new relations within this whole1 and is so far synthetic4 #eg4 ,we may retain1 however1 a distinction roughly corresponding to the ,kantian priori and posteriori1 though less rigid1 and more liable to change with the degree of organisation of knowledge4 ,kant usually endeavoured to prove1 as observed above1 that his synthetic priori propositions were necessary prerequisites of experience2 now although we cannot retain the term synthetic1 we can retain the term priori1 for those assumptions1 or those postulates1 from which alone the possibility of experience follows4 ,whatever can be deduced from these postulates1 without the aid of the matter of experience1 will also1 of course1 be priori4 ,from the standpoint of general logic1 the laws of thought and the categories1 with the indispensable conditions of their applicability1 will be alone priori2 but from the standpoint of any special science1 we may call priori whatever renders possible the experience which forms the subject-matter of our science4 ,in ,geometry1 to particularize1 we may call priori whatever renders possible experience of externality as such4 ,it is to be observed that this use of the term is at once more rationalistic and less precise than that of ,kant4 ,kant would seem to have supposed himself immediately aware1 by inspection1 that some knowledge was apodeictic1 and its subject-matter1 therefore1 priori3 but he did not always deduce its apriority from any further principle4 ,here1 however1 it is to be shown1 before admitting apriority1 that the falsehood of the judgment in question would not be effected by a mere change in the matter of experience1 but only by a change which should render some branch of experience formally impossible1 i4e4 inaccessible to our methods of cognition4 ,the above use is also less precise1 for it varies according to the specialization of the experience we are assuming possible1 and with every progress of knowledge some new connection is perceived1 two previously isolated judgments are brought into logical relation1 and the priori may thus1 at any moment1 enlarge its sphere1 as more is found deducible from fundamental postulates4 #eh4 7#b7 ,kant's arguments for the apriority of space4 ,having now discussed the logical canon to be used as regards the priori1 we may proceed to test ,kant's arguments as regards space4 ,the argument from ,geometry1 as remarked above1 is upset by ,metageometry1 at least so far as those properties are concerned1 which belong to ,euclid but not to non-,euclidean spaces2 as regards the common properties of both kinds of space1 we cannot decide on their apriority till we have discussed the consequences of denying them1 which will be done in ,chapter ,,iii4 ,as regards the two arguments which prove that space is an intuition1 not a concept1 they would call for much discussion in a special criticism of ,kant1 but here they may be passed by with the obvious comment that infinite homogeneous ,euclidean space is a concept1 not an intuition--a concept invented to explain an intuition1 it is true1 but still a pure concept#gd4 ,and it is this pure concept which1 in all discussions of ,geometry1 is primarily to be dealt with2 the intuition need only be referred to where it throws light on the functions or the nature of the concept4 ,the second of ,kant's arguments1 that we can imagine empty space1 though not the absence of space1 is false if it means a space without matter anywhere1 and irrelevant if it merely means a space between matters and regarded as empty#ge4 ,the only argument of importance1 then1 is the first argument4 ,but ,i must insist1 at the outset1 that our problem is purely logical1 and that all psychological implications must be excluded to the utmost possible extent4 ,moreover1 as will be proved in ,chapter ,,iv41 the proper function of space is to distinguish between different presented things1 not between the ,self and the object of sensation or perception4 ,the argument then becomes the following3 consciousness of a world of mutually external things demands1 in presentations1 a cognitive but non-inferential element leading to the discrimination of the objects presented4 ,this element must be non-inferential1 for from whatever number or combination of presentations1 which did not of themselves demand diversity in their objects1 ,i could never be led to infer the mutual externality of their objects4 ,kant says3 ,in order that sensations may be ascribed to something external to me 444 and similarly in order that ,i may be able to present them as outside and beside one another1 444 the presentation of space must be already present4 ,but this goes rather too far3 in the first place1 the question should be only as to the mutual externality of presented things1 not as to their externality to the ,self#gf2 and in the second place1 things will appear mutually external if ,i have the presentation of any form of externality1 whether ,euclidean or non-,euclidean4 ,whatever may be true of the psychological scope of this argument--whose validity is here irrelevant--the logical scope extends1 not to ,euclidean space1 but only to any form of externality which could exist intuitively1 and permit knowledge1 in beings with our laws of thought1 of a world of diverse but interrelated things4 ,moreover externality1 to render the scope of the argument wholly logical1 must not be left with a sensational or intuitional meaning1 though it must be supposed given in sensation or intuition4 ,it must mean1 in this argument1 the fact of ,otherness#gg1 the fact of being different from some other thing3 it must involve the distinction between different things1 and must be that element1 in a cognitive state1 which leads us to discriminate constituent parts in its object4 ,so much1 then1 would appear to result from ,kant's argument1 that experience of diverse but interrelated things demands1 as a necessary prerequisite1 some sensational or intuitional element1 in perception1 by which we are led to attribute complexity to objects of perception#gh2 that this element1 in its isolation may be called a form of externality2 and that those properties of this form1 if any such be found1 which can be deduced from its mere function of rendering experience of interrelated diversity possible1 are to be regarded as priori4 ,what these properties are1 and how the various lines of argument here suggested converge to a single result1 we shall see in ,chapters ,,iii4 and ,,iv4 #ei4 ,in the philosophers who followed ,kant1 ,metaphysics1 for the most part1 so predominated over ,epistemology1 that little was added to the theory of ,geometry4 ,what was added1 came indirectly from the one philosopher who stood out against the purely ontological speculations of his time1 namely ,herbart4 ,herbart's actual views on ,geometry1 which are to be found chiefly in the first section of his ,synechologie1 are not of any great value1 and have borne no great fruit in the development of the subject4 ,but his psychological theory of space1 his construction of extension out of series of points1 his comparison of space with the tone and colour-series1 his general preference for the discrete above the continuous1 and finally his belief in the great importance of classifying space with other forms of series 7,reihenformen#gi71 gave rise to many of ,riemann's epoch-making speculations1 and encouraged the attempt to explain the nature of space by its analytical and quantitative aspect alone#hj4 ,through his influence on ,riemann1 he acquired1 indirectly1 a great importance in geometrical philosophy4 ,to ,riemann's dissertation1 which we have already discussed in its mathematical aspect1 we must now return1 considering1 this time1 only its philosophical views4 ,riemann4 #fj4 ,the aim of ,riemann's dissertation1 as we saw in ,chapter ,i41 was to define space as a species of manifold1 i4e4 as a particular kind of collection of magnitudes4 ,it was thus assumed1 to begin with1 that spatial figures could be regarded as magnitudes1 and the axioms which emerged1 accordingly1 determined only the particular place of these among the many algebraically possible varieties of magnitudes4 ,the resulting formulation of the axioms--while1 from the mathematical standpoint of metrical ,geometry1 it was almost wholly laudable--must1 from the standpoint of philosophy1 be regarded1 in my opinion1 as a petitio principii4 ,for when we have arrived at regarding spatial figures as magnitudes1 we have already traversed the most difficult part of the ground4 ,the axioms of metrical ,geometry--and it is metrical ,geometry1 exclusively1 which is considered in ,riemann's ,essay--will appear1 in ,chapter ,,iii41 to be divisible into two classes4 ,of these1 the first class--which contains the axioms common to ,euclid and ,metageometry1 the only axioms seriously discussed by ,riemann--are not the results of measurement1 nor of any conception of magnitude1 but are conditions to be fulfilled before measurement becomes possible4 ,the second class only--those which express the difference between ,euclidean and non-,euclidean spaces--can be deduced as results of measurement or of conceptions of magnitude4 ,as regards the first class1 on the contrary1 we shall see that the relativity of position--by which space is distinguished from all other known manifolds1 except time--leads logically to the necessity of three of the most distinctive axioms of ,geometry1 and yet this relativity cannot be called a deduction from conceptions of magnitude4 ,in analytical ,geometry1 owing to the fact that coordinate systems start from points1 and hence build up lines and surfaces1 it is easy to suppose that points can be given independently of lines and of each other1 and thus the relativity of position is lost sight of4 ,the error thus suggested by mathematics was probably reinforced by ,herbart's theory of space1 which1 by its serial character1 as we have seen1 appeared to him to facilitate a construction out of successive points1 and to which ,riemann acknowledges his indebtedness both in his ,dissertation and elsewhere4 ,the same error reappears in ,helmholtz1 in whom it is probably due wholly to the methods of analytical ,geometry4 ,it is a striking fact that1 throughout the writings of these two men1 there is not1 so far as ,i know1 one allusion to the relativity of position1 that property of space from which1 as our next chapter will shew1 the richest quarry of consequences can be extracted4 ,this is not a result of any conception of magnitude1 but follows from the nature of our space-intuition2 yet no one1 surely1 could call it empirical1 since it is bound up in the very possibility of locating things there as opposed to here4 #fa4 ,indeed we can see1 from a purely logical consideration of the judgment of quantity1 that ,riemann's manner of approaching the problem can never1 by legitimate methods1 attain to a philosophically sound formulation of the axioms4 ,for quantity is a result of comparison of two qualitatively similar objects1 and the judgment of quantity neglects altogether the qualitative aspect of the objects compared4 ,hence a knowledge of the essential properties of space can never be obtained from judgments of quantity1 which neglect these properties1 while they yet presuppose them4 ,as well might one hope to learn the nature of man from a census4 ,moreover1 the judgment of quantity is the result of comparison1 and therefore presupposes the possibility of comparison4 ,to know whether1 or by what means1 comparison is possible1 we must know the qualities of the things compared and of the medium in which comparison is effected2 while to know that quantitative comparison is possible1 we must know that there is a qualitative identity between the things compared1 which again involves a previous qualitative knowledge4 ,when spatial figures have once been reduced to quantity1 their quality has already been neglected1 as known and similar to the quality of other figures4 ,to hope1 therefore1 for the qualities of space1 from a comparison of its expression as pure quantity with other pure quantities1 is an error natural to an analytical geometer1 but an error1 none the less1 from which there is no return to the qualitative basis of spatial quantity4 #fb4 ,we must entirely dissent1 therefore1 from the disjunction which underlies ,riemann's philosophy of space4 ,either the axioms must be consequences of general conceptions of magnitude1 he thinks1 or else they can only be proved by experience 7p4 #bee74 ,whatever can be derived from general conceptions of magnitude1 we may retort1 cannot be an priori adjective of space3 for all the necessary adjectives of space are presupposed in any judgment of spatial quantity1 and cannot1 therefore1 be consequences of such a judgment4 ,riemann's disjunction1 accordingly1 since one of its alternatives is obviously impossible1 really begs the question4 ,in formulating the axioms of metrical ,geometry1 our question should be3 ,what axioms1 i4e4 what adjectives of space1 must be presupposed1 in order that quantitative comparison of the parts of space may be possible at all8 ,and only when we have determined these conditions1 which are priori necessary to any quantitative science of space1 does the second question arise3 what inferences can we draw1 as to space1 from the observed results of this quantitative science1 i4e4 of this measurement of spatial figures8 ,the conditions of measurement themselves1 though not results of any conception of magnitude1 will be priori1 if it can be shown that1 without them1 experience of externality would be impossible4 ,after this initial protest against ,riemann's general philosophical position1 let us proceed to examine1 in detail1 his use of the notion of a manifold4 #fc4 ,in the first place there is1 if ,i am not mistaken1 considerable obscurity in the definition of a manifold1 of which an almost verbal rendering was given in ,chapter ,i4 ,what is meant1 to begin with1 by a general conception capable of various determinations8 ,does not this property belong to all conceptions8 ,it affords1 certainly1 a basis for counting1 but if continuous quantity is to arise1 we must1 surely1 have some less discrete formulation4 ,it might afford a basis1 for example1 for the distinction of points in projective ,geometry1 but projective ,geometry has nothing to do with quantity4 ,something more fluid and flexible than a conception1 one would think1 is necessary as the basis of continua4 ,then1 again1 what is meant by a quantum of a manifold8 ,in space1 the answer is obvious3 what is meant is a piece of volume4 ,but how about ,riemann's other continuous manifold1 colour8 ,does a quantum of colour mean a single line in the spectrum1 or a band of finite thickness8 ,in either case1 what are the magnitudes to be compared8 ,and how is superposition necessary1 or even possible8 ,a colour is fixed by its position in the spectrum3 two lines in the same spectrum cannot be superposed1 and two lines in different spectra need not be--their positions in their respective spectra suffice1 or even1 roughly1 their immediate sense-quality4 ,the fact is1 ,riemann had space in his mind from the start1 and many of the properties1 which he enunciates as belonging to all manifolds1 belong1 as a matter of fact1 only to space4 ,it is far from clear what the magnitudes are which the various determinations make possible4 ,do these magnitudes measure the elements of the manifold1 or the relations between elements8 ,this is surely a very fundamental point1 but it is one which ,riemann never touches on4 ,in the former case1 the superposition which he speaks of becomes unnecessary1 since the magnitude is inherent in the element considered4 ,we do not require superposition to measure quantities corresponding to different tones or colours2 these can be discovered by analysis of single tones or colours4 ,with space1 on the other hand1 if we seek for elements1 we can find none except points1 and no analysis of a point will find magnitudes inherent in it--such magnitudes are a fiction of coordinate ,geometry4 ,the magnitudes which space deals with1 as we shall see in ,chapter ,,iii41 are relations between points1 and it is for this reason that superposition is essential to space-measurement4 ,there is no inherent quality in a single point1 as there is in a single colour1 by which it can be quantitatively distinguished from another4 ,thus the conception of a manifold1 as defined by ,riemann1 either does not include colours1 or does not involve superposition as the only means of measurement4 ,from this dilemma there is no escape4 #fd4 ,but if measurement consists in a superposition of the magnitudes compared 7p4 #bef71 does it not follow immediately that measurement is logically possible only where such superposition leaves the magnitudes unchanged8 ,and therefore that measurement1 as above defined1 involves1 as an priori condition1 that magnitudes are unchanged by motion8 ,this consequence is not drawn by ,riemann2 indeed he proceeds immediately 7pp4 #bef-g7 to consider what he calls a general portion of the doctrine of magnitude 7,grssenlehre71 independent of measurement4 ,but how is any doctrine of magnitude possible1 in which the magnitudes cannot be measured8 ,the reason of the confusion is1 that ,riemann's definition of measurement is applicable to no single manifold except space1 since it depends on the noteworthy property that what we measure in ,geometry is not points1 but relations between points1 and the latter1 though not the former1 may of course be unaltered by motion4 ,let us try1 in illustration1 to apply ,riemann's definition of measurement to colours4 ,we must remember that motion1 in dealing with the colour manifold1 means--not motion in space but--motion in the colour manifold itself4 ,now since every point of the colour manifold is completely determined by three magnitudes1 which are given in fact1 and cannot be arbitrarily chosen1 it is plain that measurement by superposition--involving1 as it does1 motion1 and therefore change in these determining magnitudes--is totally out of the question4 ,the superposition of one colour on another1 as a means of measurement1 is sheer nonsense4 ,and yet measurement is possible in the colour-manifold1 by means of ,helmholtz's law of mixture 7,mischungsgesetz72 but the measurement is of every separate element1 not of the relations between elements1 and is thus radically different from space-measurement#ha4 ,the elements are not1 like points in space1 qualitatively alike1 and distinguished by the mere fact of their mutual externality4 ,what we have1 in colours1 is three fundamental qualitatively distinct elements1 out of certain proportions of which we can build up all the other elements of the manifold--each of the resulting elements having the same combination of qualitative diversity and similarity as the three original elements4 ,but in space1 what could we make of such a procedure8 ,given three points1 how are we to combine them in certain proportions8 ,the phrase is meaningless4 ,if some one makes the obvious retort1 that we have to combine lines1 not points1 my rejoinder is equally obvious4 ,to begin with1 lines are not elements4 ,metaphysically1 space has no elements1 being1 as the sequel will show1 mere relations between non-spatial elements4 ,mathematically1 this fact exhibits itself in the self-contradictory notion of the point1 or zero magnitude in space1 as the limit in our vain search for spatial elements4 ,but even if we allow the line to pass as the spatial element1 what does the combination of three lines in definite proportions give us8 ,it gives us1 simply1 the coordinates of a point4 ,here again we see a great difference between the colour and space-manifolds4 ,in colours1 the combination of magnitudes gives a new magnitude of the same kind2 in space1 it defines1 not a magnitude at all1 but a would-be element of a different kind from the defining magnitudes4 ,in the tone-manifold1 we should find still different conditions4 ,here1 no one of the measuring magnitudes can vanish without the tone vanishing too1 and all three are so bound up together1 in the single resulting sensation1 that none can exist without a finite quantity of the others4 ,they are all qualitatively different1 both from each other1 and from any possible tone1 being constituents of it1 as mass and velocity are constituents of momentum4 ,all these different conditions require to be examined1 before a manifold can be completely defined2 and until we have conducted such an examination in detail1 we cannot pronounce as to the priori or empirical nature of the laws of the manifold4 ,as regards space1 ,i have attempted such an examination in the third and fourth chapters of this ,essay4 #fe4 ,i do not wish to deny1 however1 the great value of the conception of space as a manifold4 ,on the contrary1 this conception seems to have become essential to any treatment of the question4 ,i only wish to urge that the purely algebraical treatment of any manifold1 important as it may be in deducing fresh consequences from known premisses1 tends rather to conceal than to make clear the basis of the premisses themselves1 and is therefore misleading in a philosophical investigation4 ,for mathematics1 where quantity reigns supreme1 ,riemann's conception has proved itself abundantly fruitful2 for philosophy1 on the contrary1 where quantity appears rather as a cloak to conceal the qualities it abstracts from1 the conception seems to me more productive of error and confusion than of sound doctrine4 ,we are thus brought back to the point from which we started1 namely1 the falsity of ,riemann's initial disjunction1 and the consequent fallacy in his proof of the empirical nature of the axioms4 ,his philosophy is chiefly vitiated1 to my mind1 by this fallacy1 and by the uncritical assumption that a metrical coordinate system can be set up independently of any axioms as to space-measurement#hb4 ,riemann has failed to observe1 what ,i have endeavoured to prove in the next chapter1 that1 unless space had a strictly constant measure of curvature1 ,geometry would become impossible2 also that the absence of constant measure of curvature involves absolute position1 which is an absurdity4 ,hence he is led to the conclusion that all geometrical axioms are empirical1 and may not hold in the infinitesimal1 where observation is impossible4 ,thus he says 7p4 #bfg73 ,now the empirical conceptions1 on which spatial measurements are based1 the conceptions of the rigid body and the light-ray1 appear to lose their validity in the infinitesimal3 it is therefore quite conceivable that the relations of spatial magnitudes in the infinitesimal do not correspond to the presuppositions of ,geometry1 and this would1 in fact1 have to be assumed1 as soon as it would enable us to explain the phenomena more simply4 ,from this conclusion ,i must entirely dissent4 ,in very large spaces1 there might be a departure from ,euclid2 for they depend upon the axiom of parallels1 which is not contained in the axiom of ,free ,mobility2 but in the infinitesimal1 departures from ,euclid could only be due to the absence of ,free ,mobility1 which1 as ,i hope my third chapter will show1 is once for all impossible4 ,helmholtz4 #ff4 ,helmholtz1 like ,riemann1 was important both in the mathematics and in the philosophy of ,geometry4 ,from the mathematical point of view1 his work has been already considered in ,chapter ,i42 the consideration of his philosophy1 which must occupy us here1 will be a more serious task4 ,like ,riemann1 he endeavoured to prove that all the axioms are empirical1 and like ,riemann1 he based his proof chiefly on ,metageometry4 ,he had an additional resource1 however1 in the physiology of the senses1 which first led him to reject the ,transcendental ,aesthetic1 and enabled him to attack ,kant from the psychological as well as the mathematical side#hc4 ,the principal topics1 for a criticism of ,helmholtz1 are three3 ,first1 his criterion of the priori2 second1 his discussion with ,land as to the imaginability of non-,euclidean spaces2 third--and this is by far the most important of the three--his theory of the dependence of ,geometry on ,mechanics4 ,let us discuss these three points successively4 #fg4 ,helmholtz's criterion of apriority is difficult to discover1 as he never1 to my knowledge1 gives a precise statement of it4 ,from his discussion of physical and transcendental ,geometry#hd1 however1 it would appear that he regards as empirical whatever applies to empirical matter4 ,for he there maintains1 that even if space were an priori form1 yet any ,geometry1 which aimed at an application to ,physics1 would1 since the actual places of bodies are not known priori1 be necessarily empirical#he4 ,it seems the more probable that he regards this as a possible criterion1 as it is adopted1 in several passages1 by his disciple ,erdmann#hf1 and so strange a test could hardly be accepted by a philosopher1 unless he had found it in his master4 ,i have called this a strange test1 because it seems to me completely to ignore the work of the ,critical ,philosophy4 ,for if there is one thing which1 one might have hoped1 had been made sufficiently clear by ,kant's ,critique1 it is this1 that knowledge which is priori1 being itself the condition of possible experience1 applies--and in ,kant's view1 applies only--to empirical matter4 ,helmholtz and ,erdmann1 therefore1 in setting up this test without discussion1 simply ignore the existence of ,kant and the possibility of a transcendental argument4 ,helmholtz assumes always that empirical knowledge must be wholly empirical1 that there can be no priori conditions of the experience in question1 that experience will always be possible1 and may give any kind of result4 ,thus in discussing physical ,geometry1 he assumes that the possibility of empirical measurement involves no priori axioms1 and that no priori element can be contained in the process4 ,this assumption1 as we shall see in ,chapter ,,iii41 is quite unwarrantable3 certain properties of space1 in fact1 are involved in the possibility of measuring matter4 ,in spite of the fact1 therefore1 that we apply measurement to empirical matter1 and that our results are therefore empirical1 there may well be an priori element in measurement1 which is presupposed in its possibility4 ,such a criterion1 therefore1 must pronounce everything empirical1 but must itself be pronounced worthless4 ,another and a better criterion1 it is true1 is also to be found in ,helmholtz1 and has also been adopted by ,erdmann4 ,whatever might1 by a different experience1 have been rendered different--so this criterion contends--must itself be dependent on experience1 and so empirical4 ,this criterion seems perfectly sound1 but ,helmholtz's use of it is usually vitiated by his neglecting to prove the possibility of the different experience in question4 ,he says1 for example1 that if our experience showed us only bodies which changed their shapes in motion1 we should not arrive at the axiom of ,congruence1 which he pronounces accordingly to be empirical4 ,but ,i shall endeavour to prove1 in ,chapter ,,iii41 that without the axiom of ,congruence1 experience of spatial magnitude would be impossible4 ,if my proof be correct1 it follows that no experience can ever reveal spatial magnitudes which contradict this axiom--a possibility which ,helmholtz nowhere discusses1 in setting up his hypothetical experience4 ,thus this second criterion1 though perfectly sound1 requires always an accompanying transcendental argument1 as to the conditions of possible experience4 ,but this accompaniment is seldom to be found in ,helmholtz4 #fh4 ,one of the few cases1 in which ,helmholtz has attempted such an accompaniment1 occurs in connection with our second point1 the imaginability of non-,euclidean spaces4 ,the argument on this point was elicited by ,helmholtz's ,kantian opponents1 who maintained that the merely logical possibility of these spaces was irrelevant1 since the basis of ,geometry was not logic1 but intuition4 ,the axioms1 they said1 are synthetic propositions1 and their contraries are1 therefore1 not self-contradictory2 they are nevertheless apodeictic propositions1 since no other intuition than the ,euclidean is possible to us#hg4 ,i have already criticized this line of argument in the beginning of the present chapter4 ,helmholtz's criticism1 however1 was different3 admitting the internal consistency of the argument1 he denied one of its premisses4 ,we can imagine non-,euclidean spaces1 he said1 though their unfamiliarity makes this difficult4 ,from this view it followed1 of course1 that ,kant's argument1 even if it were formally valid1 could not prove the apriority of ,euclidean space in particular1 but only of that general space which included ,euclid and non-,euclid alike#hh4 ,although ,i agree with ,helmholtz in thinking the distinction between ,euclidean and non-,euclidean spaces empirical1 ,i cannot think his argument on the imaginability of the latter a very happy one4 ,the validity of any proof must turn1 obviously1 on the definition of imaginability4 ,the definition which ,helmholtz gives in his answer to ,land is as follows3 ,imaginability requires die vollstndige ,vorstellbarkeit derjenigen ,sinneseindrcke1 welche das betreffende ,object in uns nach den bekannten ,gesetzen unserer ,sinnesorgane unter allen denkbaren ,bedingungen der ,beobachtung erregen1 und wodurch es sich von anderen hnlichen ,objecten unterscheiden wrde 7,wiss4 ,abh4 ,,ii4 p4 #fdd74 ,this definition is not very clear1 owing to the ambiguity of the word ,vorstellbarkeit4 ,the following definition seems less ambiguous3 ,wenn die ,reihe der ,sinneseindrcke vollstndig und eindeutig angegeben werden kann1 muss man m4 ,e4 die ,sache fr anschaulich vorstellbar erklren 7,vortrge und ,reden1 ,,ii4 p4 #bcd74 ,this makes clear1 what also appears from his manner of proof1 that he regards things as imaginable which can be described in conceptual terms4 ,such1 as ,land remarks 7,mind1 ,vol4 ,,ii4 p4 #de71 is not the sense required for argumentation in this case4 ,that ,land's criticism is just1 is shown by ,helmholtz's proof for non-,euclidean spaces1 for it consists only in an analogy to the volume inside a sphere1 which is mathematically obtained thus3 ,we take the symbols representing magnitudes in pseudo-spherical 7hyperbolic7 space1 and give them a new ,euclidean meaning2 thus all our symbolic propositions become capable of two interpretations1 one for pseudo-spherical space1 and one for the volume inside a sphere4 ,it is1 however1 sufficiently obvious that this procedure1 though it enables us to describe our new space1 does not enable us to imagine it1 in the sense of calling up images of the way things would look in it4 ,we really derive1 from this analogy1 no more knowledge than a man born blind may derive1 as to light1 from an analogy with heat4 ,the dictum ,nihil est in intellectu quod non fuerit ante in sensu1 would unquestionably be true1 if for intellect we were to substitute imagination2 it is vain1 therefore1 if our actual space be ,euclidean1 to hope for a power of imagining a non-,euclidean space4 ,what ,helmholtz might1 ,i believe with perfect truth1 have urged against ,land1 is that the image we actually have of space is not sufficiently accurate to exclude1 in the actual space we know1 all possibility of a slight departure from the ,euclidean type4 ,but in maintaining that we cannot imagine1 though we can conceive and describe1 a space different from that we actually have1 ,land is1 in my opinion1 unquestionably in the right4 ,for a pure ,kantian1 who maintains1 with ,land1 that none of the axioms can be proved1 this question is of great importance4 ,but if1 as ,i have maintained1 some of the axioms are susceptible of a transcendental proof1 while the others can be verified empirically1 the question is freed from psychological implications1 and the imaginability or non-imaginability of metageometrical spaces becomes unimportant4 #fi4 ,we come now to the third and most important question1 the relation of ,geometry to ,mechanics4 ,there are three senses in which ,helmholtz's appeal to rigid bodies may be taken3 the first1 ,i think1 is the sense in which he originally intended it2 the second seems to be the sense which he adopted in his defence against ,land2 while the third is admitted by ,land1 and will be admitted in the following argument4 ,these three senses are as follows3 7#a7 ,it may be asserted that the actual meaning of the axiom of ,free ,mobility lies in the assertion of empirical rigid bodies1 and that the two propositions are equivalent to one another4 ,this is certainly false4 7#b7 ,the axiom of ,free ,mobility1 it may be said1 is logically distinguishable from the assertion of rigid bodies1 and may even be not empirical2 but it is barren1 even for pure ,geometry1 without the aid of measures1 which must themselves be empirical rigid bodies4 ,this sense is more plausible than the first1 but ,i believe we can show that1 in this sense also1 the proposition is false4 7#c7 ,for pure ,geometry and the abstract study of space1 it may be said1 ,free ,mobility1 as applied to an abstract geometrical matter1 gives a sufficient possibility of quantitative comparison2 but the moment we extend our results to mixed mathematics1 and apply them to empirically given matter1 we require also1 as measures1 empirically given rigid bodies1 or bodies1 at least1 whose departures from rigidity are empirically known4 ,in this sense1 ,i admit1 the proposition is correct#hi4 ,in discussing these three meanings1 ,i shall not confine myself strictly to the text of ,helmholtz or ,land3 if ,i endeavoured to do so1 ,i should be met by the difficulty that neither of them defines the priori1 and that each is too much inclined1 in my opinion1 to test it by psychological criteria4 ,i shall1 therefore1 take the three meanings in turn1 without laying stress on their historical adequacy to the views of ,land or ,helmholtz4 #gj4 7#a7 ,congruence may be taken to mean--as ,helmholtz would certainly seem to desire--that we find actual bodies1 in our mechanical experience1 to preserve their shapes with approximate constancy1 and that we infer1 from this experience1 the homogeneity of space4 ,this view1 in my opinion1 radically misconceives the nature of measurement1 and of the axioms involved in it4 ,for what is meant by the non-rigidity of a body8 ,we mean1 simply1 that it has changed its shape4 ,but this involves the possibility of comparison with its former shape1 in other words1 of measurement4 ,in order1 therefore1 that there may be any question of rigidity or non-rigidity1 the measurement of spatial magnitudes must be already possible4 ,it follows that measurement cannot1 without a vicious circle1 be itself derived from experience of rigid bodies4 ,geometrical measurement1 in fact1 is the comparison of spatial magnitudes1 and such comparison involves1 as will be proved at length in ,chapter ,,iii41 the homogeneity of space4 ,this is1 therefore1 the logical prerequisite of all experience of rigid bodies1 and cannot be the result of such experience4 ,without the homogeneity of space1 the very notion of rigidity or non-rigidity could not exist1 since these mean1 respectively1 the constancy or inconstancy of spatial magnitude in pieces of matter1 and both alike1 therefore1 presuppose the possibility of spatial measurement4 ,from the homogeneity of space1 we learn that a body1 when it moves1 will not change its shape without some physical cause2 that it actually does not change its shape1 is never asserted1 and is indeed known to be false4 ,as soon as measurement is possible1 actual changes of shape can be estimated1 and their empirical causes can be sought4 ,but if space were not homogeneous1 measurement would be impossible1 constant shape would be a meaningless phrase1 and rigidity could never be experienced4 ,congruence asserts1 in short1 that a body can1 so far as mere space is concerned1 move without change of shape2 rigidity asserts that it actually does so move--a very different proposition1 involving obviously1 as its logical prius1 the former geometrical proposition4 ,this argument may be summed up by the following disjunction3 ,if bodies change their shapes in motion--and to some extent1 since no body is perfectly rigid1 they must all do so--then one of two cases must occur4 ,either the changes of shape1 as bodies move from place to place1 follow no geometrical law1 are not1 for instance1 functions of the amount or direction of motion2 in which case the law of causation requires that they should not be effects of the change of place1 but of some simultaneous non-geometrical change1 such as temperature4 ,or the changes are regular1 and the shape ,s becomes1 in a new position p1 ,sf7p74 ,in this case1 the law of concomitant variations leads us to attribute the change of shape to the mere motion1 and shape thus becomes a function of absolute position4 ,but this is absurd1 for position means merely a relation or set of relations2 it is impossible1 therefore1 that mere position should be able to effect changes in a body4 ,position is one term in a relation1 not a thing per se2 it cannot1 therefore1 act on a thing1 nor exist by itself1 apart from the other terms of the relation4 ,thus ,helmholtz's view1 that ,congruence depends on the existence of rigid bodies1 must1 since it involves absolute position1 be condemned as a logical fallacy4 ,congruence1 in fact1 as ,i shall prove more fully in ,chapter ,,iii41 is an priori deduction from the relativity of position4 #ga4 7#b7 ,the above argument seems to me to answer satisfactorily ,helmholtz's contention in the precise form which he first gave it4 ,the axiom of ,congruence1 we must agree1 is logically distinguishable from the existence of rigid bodies4 ,nevertheless some reference to matter is logically involved in ,geometry#ij1 but whether this reference makes ,geometry empirical1 or does not1 rather1 show an priori element in dynamics1 is a further question4 ,the reference to matter is necessitated by the homogeneity of empty space4 ,for so long as we leave matter out of account1 one position is perfectly indistinguishable from another1 and a science of the relations of positions is impossible4 ,indeed1 before spatial relations can arise at all1 the homogeneity of empty space must be destroyed1 and this destruction must be effected by matter4 ,the blank page is useless to the geometer until he defaces its homogeneity by lines in ink or pencil4 ,no spatial figures1 in short1 are conceivable1 without a reference to a not purely spatial matter4 ,again1 if ,congruence is ever to be used1 there must be motion3 but a purely geometrical point1 being defined solely by its spatial attributes1 cannot be supposed to move without a contradiction in terms4 ,what moves1 therefore1 must be matter4 ,hence1 in order that motion may afford a test of equality1 we must have some matter which is known to be unaffected throughout the motion1 that is1 we must have some rigid bodies4 ,and the difficulty is1 that these bodies must not only undergo no change due solely to the nature of space1 but must1 further1 be unchanged by their changing relation to other bodies4 ,and here we have a requisite which can no longer be fulfilled priori3 which1 indeed1 we know to be1 in strictness1 untrue4 ,for the forces acting on a body depend upon its spatial relations to other bodies1 and changing forces are liable to produce changing configuration4 ,hence1 it would seem1 actual measurement must be purely empirical1 and must depend on the degree of rigidity to be obtained1 during the process of measurement1 in the bodies with which we are conversant4 ,this conclusion1 ,i believe1 is valid of all actual measurement4 ,but the possibility of such empirical and approximate rigidity1 ,i must insist1 depends on the priori law that mere motion1 apart from the action of other matter1 cannot effect a change of shape4 ,for without this law1 the effect of other matter would not be discoverable2 the laws of motion would be absurd1 and ,physics would be impossible4 ,consider the second law1 for example3 ,how could we measure the change of motion1 if motion itself produced a change in our measures8 ,or consider the law of gravitation3 ,how could we establish the inverse square1 unless we were able1 independently of ,dynamics1 to measure distances8 ,the whole science of ,dynamics1 in short1 is fundamentally dependent on ,geometry1 and but for the independent possibility of measuring spatial magnitudes1 none of the magnitudes of ,dynamics could be measured4 ,time1 force1 and mass are alike measured by spatial correlates3 these correlates are given1 for time1 by the first law1 for force and mass1 by the second and third4 ,it is true1 then1 that an empirical element appears unavoidably in all actual measurement1 inasmuch as we can only know empirically that a given piece of matter preserves its shape throughout the necessary change of dynamical relations to other matter involved in motion2 but it is further true that1 for ,geometry--which regards matter simply as supplying the necessary breach in the homogeneity of space1 and the necessary term for spatial relations1 not as the bearer of forces which change the configuration of other material systems--for ,geometry1 which deals with this abstract and merely kinematical matter1 rigidity is priori1 in so far as the only changes with which it is cognizant--changes of mere position1 namely--are incapable of affecting the shapes of the imaginary and abstract bodies with which it deals4 ,to use a scholastic distinction1 we may say that matter is the causa essendi of space1 but ,geometry is the causa cognoscendi of ,physics4 ,without a ,geometry independent of ,physics1 ,physics itself1 which necessarily assumes the results of ,geometry1 could never arise2 but when ,geometry is used in ,physics1 it loses some of its priori certainty1 and acquires the empirical and approximate character which belongs to all accounts of actual phenomena4 #gb4 7#c7 ,this argument leads us to ,land's distinction of physical and geometrical rigidity4 ,the distinction may be expressed--and ,i think it is better expressed--by distinguishing between the conceptions of matter proper to ,dynamics and to ,geometry respectively4 ,in ,dynamics1 we are concerned with matter as subject to and as causing motion1 as affected by and as exerting force4 ,we are therefore concerned with the changes of spatial configuration to which material systems are liable3 the description and explanation of these changes is the proper subject-matter of all ,dynamics4 ,but in order that such a science may exist1 it is obviously necessary that spatial configuration should be already measurable4 ,if this were not the case1 motion1 acceleration and force would remain perfectly indeterminate4 ,geometry1 therefore1 must already exist before ,dynamics becomes possible3 to make ,geometry dependent for its possibility on the laws of motion or any of their consequences1 is a gross 4 ,nevertheless1 as we have seen1 some sort of matter is essential to ,geometry4 ,but this geometrical matter is a more abstract and wholly different matter from that of ,dynamics4 ,in order to study space by itself1 we reduce the properties of matter to a bare minimum3 we avoid entirely the category of causation1 so essential to ,dynamics1 and retain nothing1 in our matter1 but its spatial adjectives#ia4 ,the kind of rigidity affirmed of this abstract matter--a kind which suffices for the theory of our science1 though not for its application to the objects of daily life--is purely geometrical1 and asserts no more than this3 ,that since our matter is devoid1 ex hypothesi1 of causal properties1 there remains nothing1 in mere empty space1 which is capable of changing the configuration of any geometrical system4 ,a change of absolute position1 it asserts1 is nothing2 therefore the only real change involved in motion is a change of relation to other matter2 but such other matter1 for the purposes of our science1 is regarded as destitute of causal powers2 hence no change can occur1 in the configuration of our system1 by the mere effect of motion through empty space4 ,the necessity of such a principle may be shown by a simple reductio ad absurdum1 as follows4 ,a motion of translation of the universe as a whole1 with constant direction and velocity1 is dynamically negligeable2 indeed it is1 philosophically1 no motion at all1 for it involves no change in the condition or mutual relations of the things in the universe4 ,but if our geometrical rigidity were denied1 the change in the parameter of space might cause all bodies to change their shapes owing to the mere change of absolute position1 which is obviously absurd4 ,to make quite plain the function of rigid bodies in ,geometry1 let us suppose a liquid geometer in a liquid world4 ,we cannot suppose the liquid perfectly homogeneous and undifferentiated1 in the first place because such a liquid would be indistinguishable from empty space1 in the second place because our geometer's body--unless he be a disembodied spirit--will itself constitute a differentiation for him4 ,we may therefore assume dim beams1 ,which amid the streams ,weave a network of coloured light1 and we may suppose this network to form the occasion for our geometer's reflections4 ,then he will be able to imagine a network in which the lines are straight1 or circular1 or parabolic1 or any other shape1 and he will be able to infer that such a network1 if it can be woven in one part of the fluid1 can be woven in another4 ,this will form sufficient basis for his deductions4 ,the superposition he is concerned with--since not actual equality1 but only the formal conditions of equality1 are the subject-matter of ,geometry--is purely ideal1 and is unaffected by the impossibility of congealing any actual network4 ,but in order to apply his ,geometry to the exigencies of life1 he would need some standard of comparison between actual networks1 and here1 it is true1 he would need either a rigid body1 or a knowledge of the conditions under which similar networks arose4 ,moreover these conditions1 being necessarily empirical1 could hardly be known apart from previous measurement4 ,hence for applied1 though not for pure ,geometry1 one rigid body at least seems essential4 #gc4 ,the utility1 for ,dynamics1 of our abstract geometrical matter1 is sufficiently evident4 ,for having1 by its means1 a power of determining the configurations of material systems in whatever part of space1 and knowing that changes of configuration are not due to mere change of place1 we are able to attribute these changes to the action of other matter1 and thus to establish the notion of force1 which would be impossible if change of shape might be due to empty space4 ,thus1 to conclude3 ,geometry requires1 if it is to be practically possible1 some body or bodies which are either rigid 7in the dynamical sense71 or known to undergo some definite changes of shape according to some definite law4 7,these changes1 we may suppose1 are known by the laws of ,physics1 which have been experimentally established1 and which throughout assume the truth of ,geometry47 ,one or more such bodies are necessary to applied ,geometry--but only in the sense in which rulers and compasses are necessary4 ,they are necessary as1 in making the ,ordnance ,survey1 an elaborate apparatus was necessary for measuring the base line on ,salisbury ,plain4 ,but for the theory of ,geometry1 geometrical rigidity suffices1 and geometrical rigidity means only that a shape1 which is possible in one part of space1 is possible in any other4 ,the empirical element in practice1 arising from the purely empirical nature of physical rigidity1 is comparable to the empirical inaccuracies arising from the failure to find straight lines or circles in the world--which no one but ,mill has regarded as rendering ,geometry itself empirical or inaccurate4 ,but to make ,geometry await the perfection of ,physics1 is to make ,physics1 which depends throughout on ,geometry1 forever impossible4 ,as well might we leave the formation of numbers until we had counted the houses in ,piccadilly4 ,erdmann4 #gd4 ,in connection with ,riemann and ,helmholtz1 it is natural to consider ,erdmann's philosophical work on their theories#ib4 ,this is certainly the most important book on the subject which has appeared from the philosophical side1 and in spite of the fact that1 like the whole theory of ,riemann and ,helmholtz1 it is inapplicable to projective ,geometry1 it still deserves a very full discussion4 ,erdmann agrees throughout with the conclusions of ,riemann and ,helmholtz1 except on a few points of minor importance2 and his views1 as this agreement would lead one to expect1 are ultra-empirical4 ,indeed his logic seems--though ,i say this with hesitation--to be incompatible with any system but that of ,mill3 there is apparently no distinction1 to him1 between the general and the universal1 and consequently no concept not embodied in a series of instances4 ,such a theory of logic1 to my mind1 vitiates most of his work1 as it vitiated ,riemann's philosophy#ic4 ,this general criticism will find abundant illustration in the course of our account of ,erdmann's views4 #ge4 ,after a general introduction1 and a short history of the development of ,metageometry1 ,erdmann proceeds1 in his second chapter1 to discuss what are the axioms of ,euclidean ,geometry4 ,the arithmetical axioms1 as they are called1 he leaves aside1 as applying to magnitude in general2 what we want here1 he says1 is a definition of space1 for which the geometrical axioms are alone relevant4 ,but a definition of space1 he says--following ,riemann--demands a genus of which space shall be a species1 and this1 since our space is psychologically unique1 can only be furnished by analytical mathematics 7p4 #cf74 ,now the space-forms dealt with by ,geometry are magnitudes1 and conceptions of magnitude are everywhere applied in ,geometry4 ,but before ,riemann1 only particular determinations of space could be exhibited as magnitudes1 and thus the desired definition was impossible to obtain4 ,now1 however1 we can subsume space as a whole under a general conception of magnitude1 and thus obtain1 besides the space-intuition and the space-conception1 a third form1 namely1 the conception of space as a magnitude 7,grssenbegriff vom ,raum1 pp4 #ch-ci74 ,the definition of this will give us the complete1 but not redundant1 system of axioms1 which could not be obtained by transforming the general intuition of space into the space-conception1 for want of a plurality of instances 7p4 #dj74 #gf4 ,before considering the subsequent method of definition1 let us reflect on the theories involved in the above account of the conception of space as a magnitude4 ,in the first place1 it is assumed that conceptions cannot be formed unless we have a series of separate objects from which to abstract a common property--in other words1 that the universal is always the general4 ,in the second place1 it is assumed that all definition is classification under a genus4 ,in the third place1 the conception of magnitude1 if ,i am not mistaken1 is fundamentally misunderstood when it is supposed applicable to space as a whole4 ,but in the fourth place1 even if such a conception existed1 it could give none of the essential properties of space4 ,let us consider these four points successively4 #gg4 ,as regards the first point1 it is to be observed that people certainly had some conception of space before ,riemann invented the notion of a manifold1 and that this conception was certainly something other than the common qualities of all the points1 lines or figures in space4 ,in the second place1 ,erdmann's view would make it impossible to conceive ,god1 unless one were a polytheist1 or the universe--unless1 like ,leibnitz1 one imagined a series of possible worlds1 set over against ,god1 and none of them1 therefore1 a true ,universe--or1 to take an instance more likely to appeal to an empiricist1 the necessarily unique centre of mass of the material universe4 ,any universal1 in short1 which is a bond or unity between things1 and not merely a common property among independent objects1 becomes impossible on ,erdmann's view4 ,we cannot1 therefore1 unless we adopt ,mill's philosophy intact1 regard the conception of space as demanding a series of instances from which to abstract4 ,but even if we did so regard it1 ,riemann's manifolds would leave us without resources4 ,for ,euclidean space still appears as unique1 at the end of his series of determinations4 ,we have instances of manifolds1 but not instances of ,euclidean space4 ,thus if ,erdmann's theory of conceptions were correct1 he would still be left searching in vain for the conception of ,euclidean space4 #gh4 ,the second point1 the view that all definition is classification1 is closely allied to the first1 and the two together plunge us into the depths of scholastic formal logic4 ,the same instances of things which could not1 on ,erdmann's view1 be conceived1 may now be adduced as things which cannot be defined4 ,whatever was said above applies here also1 and the point need not1 therefore1 be further discussed#id4 #gi4 ,as regards the third point1 the impossibility of applying conceptions of magnitude to space as a whole1 a longer argument will be necessary1 for we are concerned1 here1 with the whole question of the logical nature of judgments of magnitude4 ,as we had before too much comparison for our needs1 so we have now too little4 ,i will endeavour to explain this point1 which is of great importance1 and underlies1 ,i think1 most of the philosophical fallacies of ,riemann's school4 ,a judgment of magnitude is always a judgment of comparison1 and what is more1 the comparison is never concerned with quality1 but only with quantity4 ,quality1 in the judgment of magnitude1 is supposed identical1 in the object whose magnitude is stated1 and in the unit with which it is compared4 ,but quality1 except in pure number1 and in pure quantity as dealt with by the ,calculus1 is always present1 and is partly absorbed into quantity1 partly untouched by the judgment of magnitude4 ,as ,bosanquet says 7,logic1 ,vol4 ,i4 p4 #abd72 ,quantitative comparison is not prima facie coordinate with qualitative1 but rather stands in its place as the effect of comparison on quality1 which so far as comparable becomes quantity1 and so far as incomparable furnishes the distinction of parts essential to the quantitative whole 7italics in the original74 ,thus1 if we are to regard space as a magnitude1 we must be able to adduce all those series of instances of which ,erdmann speaks1 and which1 for the conception of space1 seemed irrelevant4 ,but it remains to be proved that the comparison1 which we can institute between various spaces1 is capable of expression in a quantitative form4 ,rather it would seem that the difference of quality is such as to preclude quantitative comparison between different spaces1 and therefore also to preclude all judgments of magnitude about space as a whole4 ,here an exception might seem to be demanded by non-,euclidean spaces1 whose space-constants give a definite magnitude1 inherent in space as a whole1 and therefore1 one might think1 characterizing space as a magnitude4 ,but this is a mistake4 ,for the space-constant1 in such spaces1 is the ultimate unit1 the fixed term in all quantitative comparison2 it is itself1 therefore1 destitute of quantity1 since there is no independently given magnitude with which to compare it4 ,a non-,euclidean world1 in which the space-constant and all lines and figures were suddenly multiplied in a constant ratio1 would be wholly unchanged2 the lines1 as measured against the space-constant1 would have the same magnitude as before1 and the space-constant itself1 having no outside standard of comparison1 would be destitute of quantity1 and therefore not subject to change of quantity4 ,such an enlargement of a non-,euclidean world1 in other words1 is unmeaning2 and this proves how inapplicable is the notion of quantity to space as a whole4 ,it might be objected that this only proves the absence of quantitative difference between different spaces of positive space-constant1 or between those of negative space-constant3 the quantitative difference persists1 it might be said1 between those of positive curvature in general and those of negative curvature in general1 or between both together and ,euclidean space4 ,this ,i entirely deny4 ,there is no qualitatively similar unit1 in the three kinds of space1 by which quantitative comparison could be effected4 ,the straight lines of one space cannot be put into the other3 the two straight lines1 in one space1 whose product is the reciprocal of the measure of curvature1 have no corresponding curves in the other space1 and the measures of curvature cannot1 therefore1 be quantitatively compared with each other4 ,that the one may be regarded as positive1 the other negative1 ,i admit1 but their values are indeterminate1 and the units in the two cases are qualitatively different4 ,a debt of #cjj may be represented as the asset of -#cjj1 and the height of the ,eiffel ,tower is #cjj metres2 but it does not follow that the two are quantitatively comparable4 ,so with space-constants3 the space-constant is itself the unit for magnitudes in its own space1 and differs qualitatively from the space-constant of another kind of space4 ,again1 to proceed to a more philosophical argument1 two different spaces cannot co-exist in the same world3 we may be unable to decide between the alternatives of the disjunction1 but they remain1 none the less1 absolutely incompatible alternatives4 ,hence we cannot get that coexistence of two spaces which is essential to comparison4 ,the fact seems to be that ,erdmann1 in his admiration for ,riemann and ,helmholtz1 has fallen in with their mathematical bias1 and assumed1 as mathematicians naturally tend to assume1 that quantity is everywhere and always applicable and adequate1 and can deal with more than the mere comparison of things whose qualities are already known as similar#ie4 #hj4 ,this suggests the fourth and last of the above points1 that the qualities of space1 even if space could be successfully regarded as a magnitude1 would have to be entirely omitted in such a manner of regarding it1 and that1 therefore1 none of its important or essential properties would emerge from such treatment4 ,for to regard space as a magnitude involves1 as we saw1 a comparison with something qualitatively similar1 and an abstraction from the similar qualities4 ,to some extent and by the help of certain doubtful arguments1 such a comparison is instituted by ,riemann and ,erdmann2 but when they have instituted it1 they forget all about the common qualities on which its possibility depends4 ,but these are precisely the fundamental properties of space1 and those from which1 as ,i shall endeavour to prove in ,chapter ,,iii41 the axioms common to ,euclid and ,metageometry follow priori4 ,such are the dangers of the quantitative bias4 #ha4 ,after this protest against the initial assumptions in ,erdmann's deduction of space1 let us return to consider the manner1 in which this deduction is carried out4 ,here there will be less ground for criticism1 as the deduction1 given its presuppositions1 is1 ,i think1 as good as such a deduction can be4 ,to define space as a magnitude1 he says1 let us start with two of its most obvious properties1 continuity and the three dimensions4 ,tones and colours afford other instances of a manifold with these two properties1 but differ from space in that their dimensions are not homogeneous and interchangeable4 ,to designate this difference1 ,erdmann introduces a useful pair of terms3 in the general case1 he calls a manifold n-determined 7n-bestimmt72 in the case where1 as in space1 the dimensions are homogeneous1 he calls the manifold n-extended 7n-ausgedehnt74 ,manifolds of the latter sort he calls extents 7,ausgedehntheiten74 ,that the difference between the two kinds is one of quality1 not of quantity1 he seems not to perceive2 he also overlooks the fact that1 in the second kind1 from its very definition1 the axiom of ,congruence must hold1 on account of the qualitative similarity of different parts4 ,in spite of this fact1 he defines space as an extent1 and then regards ,congruence as empirical1 and as possibly false in the infinitesimal4 ,this is the more strange1 as he actually proves 7p4 #ej7 that measurement is impossible1 in an extent1 unless the parts are independent of their place1 and can be carried about unaltered as measures4 ,in spite of this1 he proceeds immediately to discuss whether the measure of curvature is constant or variable1 without investigating how1 in the latter case1 ,geometry could exist4 ,we cannot know1 he says1 from geometrical superposition1 that geometrical bodies are independent of place1 for if their dimensions altered in motion according to any fixed law1 two bodies which could be superposed in one place could be superposed in any other4 ,that such a hypothesis involves absolute position1 and denies the qualitative similarity of the parts of space1 which he declares 7p4 #aga7 to be the principle of his theory of ,geometry1 is nowhere perceived4 ,but what is more1 his notion that magnitude is something absolute1 independent of comparison1 has prevented him from seeing that such a hypothesis is unmeaning4 ,he says himself that1 even on this hypothesis1 a geometrical body can be defined as one whose points retain constant distances from each other1 for1 since we have no absolute measure1 measurement could not reveal to us the change of absolute magnitude 7p4 #fj74 ,but is not this a reductio ad absurdum8 ,for magnitude is nothing apart from comparison1 and the comparison here can only be effected by superposition2 if1 then1 as on the above hypothesis1 superposition always gives the same result1 by whatever motion it is effected1 there is no sense in speaking of magnitudes as no longer equal when separated3 absolute magnitude is an absurdity1 and the magnitude resulting from comparison does not differ from that which would result if the dimensions of bodies were unchanged in motion4 ,therefore1 since magnitude is only intelligible as the result of comparison1 the dimensions of bodies are unchanged in motion1 and the suggested hypothesis is unmeaning4 ,on this subject ,i shall have more to say in ,chapter ,,iii4#if #hb4 ,this hypothesis1 however1 is not introduced for its own sake1 but only to usher in the ,helmholtzian deus ex machina1 ,mechanics4 ,for ,mechanics proves--so ,erdmann confidently continues--that rigidity must hold1 not merely as to ratios1 in the above restricted geometrical sense1 but as to absolute magnitudes 7p4 #fb74 ,hence we get at last true ,congruence1 empirical as ,mechanics is empirical1 and impossible to prove apart from ,mechanics4 ,i have already criticized ,helmholtz's view of the dependence of ,geometry on ,mechanics1 and need not here speak of it at length4 ,it is a pity that ,erdmann has in no way specified the procedure by which ,mechanics decides the geometrical alternatives--indeed he seems to rely on the ipse dixit of ,helmholtz4 ,how1 if ,geometry would be totally unable to discover a change in dimensions of the kind suggested1 the ,laws of ,motion1 which throughout depend on ,geometry1 should be able to discover it if it existed1 ,i am wholly at a loss to understand4 ,uniform motion in a straight line1 for example1 presupposes geometrical measurement2 if this measurement is mistaken1 what ,mechanics imagines to be uniform motion is not really such1 but ,mechanics can never discover the discrepancy4 ,if the ,laws of ,motion had been regarded as priori1 ,geometry might possibly have been reinforced by them2 but so long as they are empirical1 they presuppose geometrical measurement1 and cannot therefore condition or affect it4 ,erdmann's conclusion1 in the second chapter1 is that ,congruence is probable1 but cannot be verified in the infinitesimal2 that its truth involves the actual existence of rigid bodies 7though1 by the way1 we know these to be1 strictly speaking1 non-existent71 that rigid bodies are freely moveable1 and do not alter their size in rotation 7,helmholtz's ,monodromy72 that the axiom of three dimensions is certain1 since small errors are impossible2 and that the remaining axioms of ,euclid--those of the straight line and of parallels--are approximately1 if not accurately1 true of our actual space 7pp4 #gh1 #hc74 ,he does not discuss how ,congruence1 on the above view1 is compatible with the atomic theory1 or even with the observed deformations of approximately rigid bodies2 nor how1 if space1 as he assumes1 is homogeneous1 rigid bodies can fail to be freely moveable through space4 ,the axioms are all lumped together as empirical1 and it appears1 in the following chapters1 that ,erdmann regards their empirical nature as sufficiently proved by their applicability to empirical material 7cf4 pp4 #aei1 #afe7--;a strange criterion1 which would prove the same conclusion1 with equal facility1 of ,arithmetic and of the laws of thought4 #hc4 ,the third chapter1 on the philosophical consequences of ,metageometry1 need not be discussed at length1 since it deals rather with space than with ,geometry4 ,at the same time1 it will be worth while to treat briefly of ,erdmann's criterion of apriority4 ,on this subject it is very difficult to discover his meaning1 since it seems to vary with the topic he is discussing4 ,thus at one time 7p4 #adg7 he rejects most emphatically the ,kantian connection of the priori and the subjective#ig1 and yet at another time 7p4 #if7 he regards every presentation of external things as partly priori1 partly empirical1 merely because such a presentation is due to an interaction between ourselves and things1 and is therefore partly due to subjective activity1 partly due to outside objects4 ,hence1 he says1 the distinction is not between different presentations1 but between different aspects of one and the same presentation4 ,this seems to return wholly to the ,kantian psychological criterion of subjectivity1 with the added disadvantage that it makes the distinction1 like that of analytic and synthetic1 epistemologically worthless4 ,and yet he never hesitates to pronounce every piece of knowledge in turn empirical4 ,the fact seems to be1 that where he wants a more logical criterion1 he adopts a modification of ,helmholtz's criterion for sensations4 ,if space be an priori form1 he says1 no experience could possibly change it 7p4 #ajh72 but this ,metageometry has proved not to be the case1 since we can intuit the perceptions which non-,euclidean space would give us 7p4 #aae74 ,i have criticised this argument in discussing ,helmholtz2 at present we are concerned with ,erdmann's criterion of apriority4 ,the subjectivity-criterion--though he certainly uses it in discussing the apriority of space1 and solemnly decides1 by its means1 that space is both priori and empirical since a change either in us or in the outer world could change it 7p4 #ig7--would seem1 like several of his other tests1 to be a lapse on his part3 the criterion which he means to use is ,helmholtz's4 ,this criterion1 ,i think1 with a slight change of wording1 might be accepted2 it seems to me a necessary1 but not a sufficient condition4 ,the priori1 we may say1 is not only that which no experience can change1 but that without which experience would become impossible4 ,it is the omission to discuss the conditions which render geometrical 7and mechanical7 experience possible1 to my mind1 which vitiates the empirical conclusions of ,helmholtz and ,erdmann4 ,why certain conditions should be necessary for experience--whether on account of the constitution of the mind1 or for some other reason--is a further question1 which introduces the relation of the priori to the subjective4 ,but in discussing the question as to what knowledge is priori1 as opposed to the question concerning the further consequences of apriority1 it is well to keep to the purely logical criterion1 and so preserve our independence of psychological controversies4 ,the fact1 if it be a fact1 that the world might be such as to defy our attempts to know it1 will not1 with the above criterion1 invalidate the conclusion that certain elements in knowledge are priori2 for whether fulfilled or not1 they remain necessary conditions for the existence of any knowledge at all4 #hd4 ,with this caution as to the meaning of apriority1 we shall find1 ,i think1 that the conclusions of ,erdmann's final chapter1 on the principles of a theory of ,geometry1 are largely invalidated by the diversity and inadequacy of his tests of the priori4 ,he begins by asserting1 in conformity with the quantitative bias noticed above1 that the question as to the nature of geometrical axioms is completely analogous to the corresponding question of the foundations of pure mathematics 7p4 #ach74 ,this is1 ,i think1 a radical error3 for the function of the axioms seems to be1 to establish that qualitative basis on which1 as we saw1 all qualitative comparison must rest4 ,but in pure mathematics1 this qualitative basis is irrelevant1 for we deal there with pure quantity1 i4e4 with the merely quantitative result of quantitative comparison1 wherever it is possible1 independently of the qualities underlying the comparison4 ,geometry1 as ,grassmann insists#ih1 ought not to be classed with pure mathematics1 for it deals with a matter which is given to the intellect1 not created by it4 ,the axioms give the means by which this matter is made amenable to quantity1 and cannot1 therefore1 be themselves deduced from purely quantitative considerations4 ,leaving this point aside1 however1 let us return to ,erdmann4 ,he distinguishes1 within space1 a form and a matter3 the form is to contain the properties common to all extents1 the matter the properties which distinguish space from other extents4 ,this distinction1 he says1 is purely logical1 and does not correspond with ,kant's3 matter and form1 for ,erdmann1 are alike empirical4 ,the axioms and definitions of ,geometry1 he says1 deal exclusively with the matter of space4 ,it seems a pity1 having made this distinction1 to put it to so little use3 after a few pages1 it is dropped1 and no epistemological consequences are drawn from it4 ,the reason is1 ,i think1 that ,erdmann has not perceived how much can be deduced from his definition of an extent1 as a manifold in which the dimensions are homogeneous and interchangeable4 ,for this property suffices to prove the complete homogeneity of an extent1 and hence--from the absence of qualitative differences among elements--the relativity of position and the axiom of ,congruence4 ,this deduction will be made at length in the sequel#ii2 at present1 ,i have only to observe that every extent1 on this view1 possesses all the properties 7except the three dimensions7 common to ,euclidean and non-,euclidean spaces4 ,the axioms which express these properties1 therefore1 apply to the form of space1 and follow from homogeneity alone1 which ,erdmann allows 7p4 #aga7 as the principle of any theory of space4 ,the above distinction of form and matter1 therefore1 corresponds1 when its full consequences are deduced1 to the distinction between the axioms which follow from the homogeneity of space and those which do not4 ,since1 then1 homogeneity is equivalent to the relativity of position1 and the relativity of position is of the very essence of a form of externality1 it would seem that his distinction of form and matter can also be made coextensive with the distinction of the priori and empirical in ,geometry4 ,on this subject1 ,i shall have more to say in ,chapter ,,iii4 ,in the remainder of the chapter1 ,erdmann insists that the straight line1 etc41 though not abstracted from experience1 which nowhere presents straight lines1 must yet1 as applicable to admittedly empirical sciences1 be empirical 7p4 #aei7--;a criterion which he appears to employ only when all other grounds for an empirical opinion fail1 and one which1 obviously1 can never refuse to do its work1 since all elements of knowledge are susceptible of employment on some empirical material4 ,he also defines the straight line 7p4 #aee7 as a line of constant curvature zero1 as though curvature could be measured independently of the straight line4 ,even the arithmetical axioms are declared empirical 7p4 #afe71 since in a world where things were all hopelessly different from one another1 these axioms could not be applied4 ,after this reminder of ,mill1 we are not surprised1 a few pages later 7p4 #agb71 at a vague appeal to ,english logicians as having proved ,geometry to be an inductive science4 ,nevertheless1 ,erdmann declares1 almost on the last page of his book 7p4 #agc71 that ,geometry is distinguished from all other sciences by the homogeneity of its material3 a principle of which no single application occurs throughout his book1 and which1 as we shall see in ,chapter ,,iii41 flatly contradicts the philosophical theories advocated throughout his preceding pages4 ,on the whole1 then1 it cannot be said that ,erdmann has done much to strengthen the philosophical position of ,riemann and ,helmholtz4 ,i have criticized him at length1 because his book has the appearance of great thoroughness1 and because it is undoubtedly the best defence extant of the position which it takes up4 ,we shall now have the opposite task to perform1 in defending ,metageometry1 on its mathematical side1 from the attacks of ,lotze and others1 and in vindicating for it that measure of philosophical importance--far inferior1 indeed1 to the hopes of ,erdmann--which it seems really to possess4 ,lotze4 #he4 ,lotze's argument as regards ,geometry#ajj--which follows a metaphysical argument as to the ontological nature of space1 and assumes the results of this argument--consists of two parts3 the first discusses the various meanings logically assignable 7pp4 #bcc-bdg7 to the proposition that other spaces than ,euclid's are possible1 and the second criticizes1 in detail1 the procedure of ,metageometry4 ,the first of these questions is very important1 and demands considerable care as to the logical import of a judgment of possibility4 ,although ,lotze's discussion is excellent in many respects1 ,i cannot persuade myself that he has hit on the only true sense in which non-,euclidean spaces are possible4 ,i shall endeavour to make good this statement in the following pages4 #hf4 ,lotze opens with a somewhat startling statement1 which1 though philosophically worthy to be true1 does not appear to be historically borne out4 ,euclidean ,geometry has been chiefly shaken1 he says1 by the ,kantian notion of the exclusive subjectivity of space--if space is only our private form of intuition1 to which there exists no analogue in the objective world1 then other beings may have other spaces1 without supposing any difference in the world which they arrange in these spaces 7p4 #bcc74 ,this certainly seems a legitimate deduction from the subjectivity of space1 which1 so far from establishing the universal validity of ,euclid1 establishes his validity only after an empirical investigation of the nature of space as intuited by ,tom1 ,dick or ,harry4 ,but as a matter of fact1 those who have done most to further non-,euclidean ,geometry--with the exception of ,riemann1 who was a disciple of ,herbart--have usually inherited from ,newton a nave realism as regards absolute space4 ,i might instance the passage quoted from ,bolyai in ,chapter ,i41 or ,clifford1 who seems to have thought that we actually see the images of things on the retina#aja1 or again ,helmholtz's belief in the dependence of ,geometry on the behaviour of rigid bodies4 ,this belief led to the view that ,geometry1 like ,physics1 is an experimental science1 in which objective truth can be attained1 it is true1 but only by empirical methods4 ,however1 ,lotze's ground for uncertainty about ,euclid is a philosophically tenable ground1 and it will be instructive to observe the various possibilities which arise from it4 ,if space is only a subjective form--so ,lotze opens his argument--other beings may have a different form4 ,if this corresponds to a different world1 the difference1 he says1 is uninteresting3 for our world alone is relevant to any metaphysical discussion4 ,but if this different space corresponds to the same world which we know under the ,euclidean form1 then1 in his opinion1 we get a question of genuine philosophic interest4 ,and here he distinguishes two cases3 either the relations between things1 which are presented to these hypothetical beings under the form of some different space1 are relations which do not appear to us1 or at any rate do not appear spatial2 or they are the same relations which appear to us as figures in ,euclidean space 7p4 #bce74 ,the first possibility would be illustrated1 he says1 by beings to whom the tone or colour-manifolds appeared extended2 but we cannot1 in his opinion1 imagine a manifold1 such as is required for this case1 to have its dimensions homogeneous and comparable inter se1 and therefore the contents of the various presentations constituting such a manifold could not be combined into a single content containing them all4 ,but the possibility of such a combination is of the essence of anything worth calling a space3 therefore the first of the above possibilities is unmotived and uninteresting4 ,lotze's conclusion on this point1 ,i think1 is undeniable1 but ,i doubt whether his argument is very cogent4 ,however1 as this possibility has no connection with that contemplated by non-,euclideans1 it is not worth while to discuss it further4 ,the second possibility also1 ,lotze thinks1 is not that of ,metageometry1 but in truth it comes nearer to it than any of the other possibilities discussed4 ,if a non-,euclidean were at the same time a believer in the subjectivity of space1 he would have to be an adherent of this view4 ,let us see more precisely what the view is4 ,in ,book ,,ii41 ,chapter ,i41 ,lotze has accepted the argument of the ,transcendental ,aesthetic1 but rejected that of the mathematical antinomies3 he has decided that space is1 as ,kant believed1 subjective1 but possesses nevertheless1 what ,kant denied it1 an objective counterpart4 ,the relation of presented space to its objective counterpart1 as conceived by ,lotze1 is rather hard to understand4 ,it seems scarcely to resemble the relation of sensation to its object--e4g4 of light to ether-vibrations--for if it did1 space would not be in any peculiar sense subjective4 ,it seems rather to resemble the relation of a perceived bodily motion to the state of mind of the person willing the motion4 ,however this may be1 the objective counterpart of space is supposed to consist of certain immediate interactions of monads1 who experience the interactions as modifications of their internal states4 ,such interactions1 it is plain1 do not form the subject-matter of ,geometry1 which deals only with our resulting perceptions of spatial figures4 ,now if ,lotze's construction of space be correct1 there seems certainly no reason why these resulting perceptions should not1 for one and the same interaction between monads1 be very different in beings differently constituted from ourselves4 ,but if they were different1 says ,lotze1 they would have to be utterly different--as different1 for example1 as the interval between two notes is from a straight line4 ,the possibility is1 therefore1 in his opinion1 one about which we can know nothing1 and one which must remain always a mere empty idea4 ,this seems to me to go too far3 for whatever the objective counterpart may be1 any argument which gives us information about it must1 when reversed1 give us information about any possible form of intuition in which this counterpart is presented4 ,the argument which ,lotze has used in his former chapter1 for example1 deducing1 from the relativity of position1 the merely relational nature of the objective counterpart1 allows us1 conversely1 to infer1 from this relational nature1 the complete relativity of position in any possible space-intuition--unless1 indeed1 it bore a wholly deceitful relation to those interactions of monads which form its objective counterpart4 ,but the complete relativity of position1 as ,i shall endeavour to establish in ,chapter ,,iii41 suffices to prove that our ,geometry must be ,euclidean1 elliptic1 spherical or pseudo-spherical4 ,we have1 therefore1 it would seem1 very considerable knowledge1 on ,lotze's theory of space1 of the manner in which what appears to us as space must appear to any beings with our laws of thought4 ,we cannot know1 it is true1 what psychological theory of space-perception would apply to such beings3 they might have a sense different from any of ours1 and they might have no sense in any way resembling ours1 but yet their ,geometry would have points of resemblance to ours1 as that of the blind coincides with that of the seeing4 ,if space has any objective counterpart whatever1 in short1 and if any inference is possible1 as ,lotze holds it to be1 from space to its counterpart1 then a converse argument is also possible1 though it may give some only of the qualities of ,euclidean space1 since some only of these qualities may be found to have a necessary analogue in the counterpart4 #hg4 ,admitting1 then1 in ,lotze's sense1 the subjectivity of space1 the above possibility does not seem so empty as he imagines4 ,he discusses it briefly1 however1 in order to pass on to what he regards as the real meaning of ,metageometry4 ,in this he is guilty of a mathematical mistake1 which causes much irrelevant reasoning4 ,for he believes that ,metageometry constructs its spaces out of straight lines and angles in all respects similar to ,euclid's1 whence he derives an easy victory in proving that these elements can lead only to the one space4 ,in this he has been misled by the phraseology of non-,euclideans1 as well as by ,euclid's separation of definitions and axioms4 ,for the fact is1 of course1 that straight lines are only fully defined when we add to the formal definition the axioms of the straight line and of parallels4 ,within ,euclidean space1 ,euclid's definition suffices to distinguish the straight line from all other curves2 the two axioms referred to are then absorbed into the definition of space4 ,but apart from the restriction to ,euclidean space1 the definition has to be supplemented by the two axioms1 in order to define completely the ,euclidean straight line4 ,thus ,lotze has misconceived the bearing of non-,euclidean constructions1 and has simply missed the point in arguing as he does4 ,the possibility contemplated by a non-,euclidean1 if it fell under any of ,lotze's cases1 would fall under the second case discussed above4 #hh4 ,but the bearing of ,metageometry is really1 ,i think1 different from anything imagined by ,lotze2 and as few writers seem clear on this point1 ,i will enter somewhat fully into what ,i conceive to be its purpose4 ,in the first place1 there are some writers--notably ,clifford--who1 being nave realists as regards space1 hold that our evidence is wholly insufficient1 as yet1 to decide as to its nature in the infinite or in the infinitesimal 7cf4 ,essays1 ,vol4 ,i4 p4 #cbj73 these writers are not concerned with any possibility of beings different from ourselves1 but simply with the everyday space we know1 which they investigate in the spirit of a chemist discussing whether hydrogen is a metal1 or an astronomer discussing the nebular hypothesis4 ,but these are a minority3 most1 more cautious1 admit that our space1 so far as observation extends1 is ,euclidean1 and if not accurately ,euclidean1 must be only slightly spherical or pseudo-spherical4 ,here again1 it is the space of daily life which is under discussion1 and here further the discussion is1 ,i think1 independent of any philosophical assumption as to the nature of our space-intuition4 ,for even if this be purely subjective1 the translation of an intuition into a conception can only be accomplished approximately1 within the errors of observation incident to self-analysis2 and until the intuition of space has become a conception1 we get no scientific ,geometry4 ,the apodeictic certainty of the axiom of parallels shrinks to an unmotived subjective conviction1 and vanishes altogether in those who entertain non-,euclidean doubts4 ,to reinforce the ,euclidean faith1 reason must now be brought to the aid of intuition2 but reason1 unfortunately1 abandons us1 and we are left to the mercy of approximate observations of stellar triangles--a meagre support1 indeed1 for the cherished religion of our childhood4 #hi4 ,but the possibility of an inaccuracy so slight1 that our finest instruments and our most distant parallaxes show no trace of it1 would trouble men's minds no more than the analogous chance of inaccuracy in the law of gravitation1 were it not for the philosophical import of even the slenderest possibility in this sphere4 ,and it is the philosophical bearing of ,metageometry alone1 ,i think1 which constitutes its real importance4 ,even if1 as we will suppose for the moment1 observation had established1 beyond the possibility of doubt1 that our space might be safely regarded as ,euclidean1 still ,metageometry would have shown a philosophical possibility1 and on this ground alone it could claim1 ,i think1 very nearly all the attention which it at present deserves4 ,but what is this possibility8 ,a thing is possible1 according to ,bradley 7,logic1 p4 #ahg71 when it would follow from a certain number of conditions1 some of which are known to be realized4 ,now the conditions to which a form of externality must conform1 in order to be affirmed1 are3 first1 of course1 that it should be experienced1 or legitimately inferred from something experienced2 but secondly1 that it should conform to certain logical conditions1 detailed in ,chapter ,,iii41 which may be summed up in the relativity of position4 ,now what ,metageometry has done1 in any case1 is to suggest the proof that the second of these conditions is fulfilled by non-,euclidean spaces4 ,euclid is affirmed1 therefore1 on the ground of immediate experience alone1 and his truth1 as unmediated by logical necessity1 is merely assertorical1 or1 if we prefer it1 empirical4 ,this is the most important sense1 it seems to me1 in which non-,euclidean spaces are possible4 ,they are1 in short1 a step in a philosophical argument1 rather than in the investigation of fact3 they throw light on the nature of the grounds for ,euclid1 rather than on the actual conformation of space#ajb4 ,this import of ,metageometry is denied by ,lotze1 on the ground that non-,euclidean logic is faulty1 a ground which he endeavours1 by much detail and through many pages1 to make good--with what success1 we will now proceed to examine4 #ij4 ,lotze's attack on ,metageometry--although it remains1 so far as ,i know1 the best hostile criticism extant1 and although its arguments have become part of the regular stock-in-trade of ,euclidean philosophers--contains1 if ,i am not mistaken1 several misunderstandings due to insufficient mathematical knowledge of the subject4 ,as these misunderstandings have been widely spread among philosophers1 and cannot be easily removed except by a critic who has gone into non-,euclidean ,geometry with some care1 it seems desirable to discuss ,lotze's strictures point by point4 ,illustration #ia4 ,the mathematical criticism begins 7 #aca7 with a somewhat question-begging definition of parallel straight lines4 ,two straight lines a1 b1 according to this definition1 are parallel when--a and b being arbitrary points on the two lines--if a b1 then ab 1 where 1 are two other points on the two straight lines respectively4 ,this definition--which contains ,euclid's axiom and definition combined in a very convenient and enticing form--is of course thoroughly suitable to ,euclidean ,geometry1 and leads immediately to all the ,euclidean propositions about parallels4 ,but it is perhaps more honest to follow ,euclid's course2 when an axiom is thus buried in a definition1 it is apt to seem1 since definitions are supposed to be arbitrary1 as though the difficulty had been overcome1 while in reality1 the possibility of parallels1 as above defined1 involves the very point in question1 namely1 the disputed axiom of parallels4 ,for what this axiom asserts is simply the existence of lines conforming to ,lotze's definition4 ,the deduction of the principal propositions on parallels1 with which ,lotze follows up his definition1 is of course a very simple proceeding--a proceeding1 however1 in which the first step begs the question4 #ib4 ,the next argument for the apriority of ,euclidean ,geometry has1 oddly enough1 an exactly opposite bearing1 although it is a great favourite with opponents of ,metageometry4 ,measurements of stellar triangles1 and all similar attempts at an empirical determination of the space-constant are1 according to ,lotze1 beside the mark2 for any observed departure from two right angles1 or any finite annual parallax for distant stars1 would be attributed to some new kind of refraction1 or1 as in the case of aberration1 to some other physical cause1 and never to the geometrical nature of space4 ,this is a strong argument for the empirical validity of ,euclid1 but as an argument for the apodeictic certainty of the orthodox system1 it has an opposite tendency4 ,for observations of the kind contemplated would have to be due to departures from ,euclidean straightness1 hitherto unknown1 on the part of stellar light-rays4 ,such departure could1 in certain cases1 be accounted for by a finite space-constant1 but it could also1 probably1 be accounted for by a change in ,optics1 for example1 by attributing refractive properties to the ether4 ,such properties could only exist if ether were of varying density1 if 7say7 it were denser in the neighbourhood of any of the heavenly bodies4 ,but such an assumption would1 ,i believe1 destroy the utility of ether for ,physics2 a slight alteration in our ,geometry1 so slight as not appreciably to affect distances within the ,solar ,system1 would probably be in the end1 therefore1 should such errors ever be discovered1 a simpler explanation than any that ,physics could offer4 ,but this is not the point of my contention4 ,the point is that1 if the physical explanation1 as ,lotze holds1 be possible in the above case1 the converse must also hold3 it must be possible to explain the present phenomena by supposing ether refractive and space non-,euclidean4 ,from this conclusion there is no escape4 ,if every conceivable behaviour of light-rays can be explained1 within ,euclid1 by physical causes1 it must also be possible1 by a suitable choice of hypothetical physical causes1 to explain the actual phenomena as belonging to a non-,euclidean space4 ,such a hypothesis would be rightly rejected by ,science1 for the present1 on account of its unnecessary complexity4 ,nevertheless it would remain1 for philosophy1 a possibility to be reckoned with1 and the choice could only be decided upon empirical grounds of simplicity4 ,it may well be doubted whether1 in the world we know1 the phenomena could be attributed to a distinctly non-,euclidean space1 but this conclusion follows inevitably from the contention that no phenomena could force us to assume such a space4 ,lotze's argument1 therefore1 if pushed home1 disproves his own view1 and puts ,euclidean space1 as an empirical explanation of phenomena1 on a level with luminiferous ether#ajc4 #ic4 ,lotze now proceeds 7 #acb7 to a detailed criticism of ,helmholtz1 whom he regards as a typical exponent of ,metageometry4 ,it is possible that1 at the time when he wrote1 ,helmholtz really did occupy this position2 but it is unfortunate that1 in the minds of philosophers1 he should still continue to do so1 after the very material advances brought about by the projective treatment of the subject4 ,it is also unfortunate that his somewhat careless attempts to popularise mathematical results have so often been disposed of1 without due attention to his more technical and solid contributions4 ,thus his romances about ,flatland and ,sphereland--at best only fairy-tale analogies of doubtful value--have been attacked as if they formed an essential feature of ,metageometry4 ,but to proceed to particulars3 ,lotze readily allows that the ,flatlanders would set up ,plane ,geometry1 as we know it1 but refuses to admit that the ,spherelanders could1 without inferring the third dimension1 set up a two-dimensional spherical ,geometry which should be free from contradictions4 ,i will endeavour to give a free rendering of ,lotze's argument on this point4 ,illustration ,suppose1 he says1 a north and south pole1 ,n and ,s1 arbitrarily fixed1 and an equator ,,ew4 ,suppose a being1 ,b1 capable of impressions only from things on the surface of the sphere1 to move in a meridian ,,nbs4 ,let ,b start from some point a1 and finally1 after describing a great circle1 return to the same point a4 ,if a is known only by the quality of the impression it makes on ,b1 ,b may imagine he has not reached the same point a1 but another similar point a1 bearing a relation to a similar to that of the octave in singing3 he might even not arrange his impressions spatially at all4 ,in order that this may occur1 we require the further assumption1 that every difference in the above-mentioned feelings 7as he describes the meridian7 may be presented as a spatial distance between two places4 ,even now1 ,b may think he is describing a ,euclidean straight line1 containing similar points at certain intervals4 ,allowing1 however1 that he realizes the identity of a with his initial position1 he will now seem1 by motion in a straight line1 to have returned to the point from which he started1 for his motion cannot1 without the third dimension1 seem to him other than rectilinear4 ,up to this point1 there seems little ground for objection1 except1 perhaps1 to the idea of a straight line with periodical similar points--if ,b were as philosophical as1 in these discussions1 we usually suppose him to be1 he would probably object to this interpretation of his experiences1 on the ground that it regards empty space as something independent of the objects in it4 ,it is worth pointing out1 also1 that ,b would not need to describe the whole circle1 in order suddenly to find himself home again with his old friends4 ,accurate measurements of small triangles would suffice to determine his space-constant1 and show him the length of a great circle 7or straight line1 as he would call it74 ,we must admit1 also1 that so hypothetical a being as ,b might form no space-intuition at all1 but as he is introduced solely for the purposes of the analogy1 it is convenient to allow him all possible qualifications for his post4 ,but these points do not touch the kernel of the argument1 which lies in the statement that such a straight line1 returning into itself after a finite time1 would appear to ,b as an unendurable contradiction1 and thus force him1 for logical though not for sensational purposes1 into the assumption of a third dimension4 ,this assertion seems to me quite unwarranted3 the whole of ,metageometry is a solid array in disproof of it4 ,helmholtz's argument is1 it must be remembered1 only an analogy1 and the contradiction would exist only for a ,euclidean4 ,a complete three-dimensional ,geometry has1 we have seen in ,chapter ,i41 been developed on the assumption that straight lines are of finite length4 ,a constant value for the measure of curvature1 as our discussion of ,riemann showed1 involves neither reference to the fourth dimension1 nor any kind of internal contradiction4 ,this fact disproves ,lotze's contention1 which arises solely from inability to divest his imagination of ,euclidean ideas4 ,lotze next attacks ,helmholtz for the assertion that ,b would know nothing of parallel lines--parallel straight lines1 as the context shows1 he meant to say#ajd4 ,lotze1 however1 takes him as meaning1 apparently1 mere curves of constant distance from a given straight line1 which are part of the regular stock-in-trade of ,metageometry4 ,parallels of latitude1 in the geographical sense1 would not--with the exception of the equator--appear to ,b as straight lines1 but as circles4 ,great circles he would call straight1 and this fact seems to have misled ,lotze into thinking all circles were to be treated as straight lines4 ,parallels of latitude1 therefore1 though ,b might call them parallels1 would not invalidate ,helmholtz's contention1 which applies only to straight lines4 ,the argument that such small circles would be parallel1 which we have just disposed of1 is only the preface to another proof that ,b would need a third dimension4 ,let us call two of these parallels of latitude ln and ls1 and let them be equidistant from the equator1 one in the northern1 one in the southern hemisphere4 ,consecutive tangent planes1 along these parallels1 converge1 in the one case northwards1 in the other southwards4 ,either ,b could become aware of their difference1 says ,lotze1 or he could not4 ,in the former case1 which he regards as the more probable1 he easily proves that ,b would infer a third dimension4 ,but this alternative is1 ,i think1 wholly inadmissible4 ,tangent planes1 like ,euclidean planes in general1 would have no meaning to ,b2 unless1 indeed1 he were a metageometrician1 which1 with all his metaphysical and mathematical subtlety1 the argument supposes him not to be--and to such a supposition ,lotze1 surely1 is the last person who has a right to object4 ,lotze's attempted proof that this is the right alternative rests1 if ,i understand him aright1 on a sheer error in ordinary spherical ,geometry4 ,b would observe1 he says1 that the meridians made smaller angles with his path towards the nearer than towards the further pole--as a matter of fact1 they would be simply perpendicular to his path in both directions4 ,what ,lotze means is1 perhaps1 that all the meridians would meet sooner in one direction than in the other1 and this1 of course1 is true4 ,but the poles1 in which the meridians meet1 would appear to ,b as the centres of the respective parallels1 while the parallels themselves would appear to be circles4 ,now ,i am at a loss to see what difficulty would arise1 to ,b1 in supposing two different circles to have different centres#aje4 ,we must1 therefore1 take the first alternative1 that ,b would have no sort of knowledge as to the direction in which the tangent planes converged4 ,here ,lotze attempts1 if ,i have not misunderstood him1 to prove a reductio ad absurdum3 ,b would think1 he says1 that he was describing two paths wholly the same in direction1 and then he might regard both paths as circles in a plane4 ,it may be observed that direction1 when applied to a circle as a whole1 is meaningless2 indeed direction1 in all ,metageometry1 can only mean1 even when applied to straight lines1 direction towards a point4 ,to speak of two lines1 which do not meet1 as having the same direction1 is a surreptitious introduction of the axiom of parallels4 ,apart from this1 ,i cannot conceive any objection1 on ,b's part1 to such a view--one should say must1 not might4 ,the whole argumentation1 therefore1 unless its obscurity has led me astray1 must be pronounced fruitless and inconclusive4 #id4 ,after this preliminary discussion of ,sphereland1 ,lotze proceeds to the question of a fourth dimension1 and thence to spherical and pseudo-spherical space4 ,as before1 he appears to know only the more careless and popular utterances of ,helmholtz and ,riemann1 and to have taken no trouble to understand even the foundations of mathematical ,metageometry4 ,by this neglect1 much of what he says is rendered wholly worthless4 ,to begin with1 he regards1 as the purpose of ,helmholtz's fairy tale1 the suggestion of a possible fourth dimension1 whereas the real purpose was quite the opposite--to make intelligible a purely three-dimensional non-,euclidean space4 ,helmholtz introduced ,flatland only because its relation to ,sphereland is analogous to the relation of ours to spherical space#ajf4 ,but ,lotze says3 ,the ,flatlanders would find no difficulty in a third dimension1 since it would in no way contradict their own ,geometry1 while the people in ,sphereland1 from the contradictions in their two-dimensional system1 would already have been led to it4 ,the latter contention ,i have already tried to answer2 the former has an odd sound1 in view of the attempt1 a few pages later1 to prove priori that all forms of intuition1 in any way analogous to space1 must have three dimensions4 ,one cannot help suspecting that the ,flatlanders1 with two instead of three dimensions1 would make a similar attempt4 ,but to return to ,lotze's argument3 ,neither analogy can be used1 he says1 to prove that we ought perhaps to set up a fourth dimension1 since1 for us1 no contradictions or otherwise inexplicable phenomena exist4 ,the only people1 so far as ,i know1 who have used this analogy1 are ,dr ,abbot and a few ,spiritualists--the former in joke1 the latter to explain certain phenomena more simply explained1 perhaps1 by ,maskelyne and ,cooke4 ,but although ,lotze's conclusion in this matter is sound1 and one with which ,helmholtz might have agreed1 his arguments1 to my mind1 are irrelevant and unconvincing4 ,there is this difference1 he says1 between us and the ,spherelanders3 the latter were logically forced to a new dimension1 and found it possible2 we are not forced to it1 and find it1 in our space1 impossible4 ,i have contended that1 on the contrary1 nothing would force the ,spherelanders to assume a third dimension1 while they would find it impossible exactly as we find a fourth impossible--not logically1 that is to say1 but only as a presentable construction in given space4 ,after a somewhat elephantine piece of humour1 about socialistic whales in a four-dimensional sea of ,fourrier's eau sucre1 ,lotze proceeds to a proof1 by logic1 that every form of intuition1 which embraces the whole system of ordered relations of a coexisting manifold1 must have three dimensions4 ,one might object1 on priori grounds1 to any such attempt3 what belongs to pure intuition could hardly1 one would have thought1 be determined by priori reasoning#ajg4 ,i will not1 however1 develop this argument here1 but endeavour to point out1 as far as its obscurity will allow1 the particular fallacy of the proof in question4 ,illustration ,lotze's argument is as follows4 ,in this discussion1 though our terminology is necessarily taken from space1 we are really concerned with a much more general conception4 ,we assume1 in order to preserve the homogeneity of dimensions1 that the difference 7distance7 between any two elements 7points7 of our manifold--to borrow ,riemann's word--is of the same kind as1 and commensurable with1 the difference between any other two elements4 ,let us take a series of elements at successive distances x such that the distance between any two is the sum of the distances between intermediate elements4 ,such a series corresponds to a straight line1 which is taken as the x-axis4 ,then a series ,,oy is called perpendicular to the x-axis ,,ox1 when the distances of any element y1 on ,,oy1 from mx and -mx are equal4 ,by our hypothesis1 these distances are comparable with1 and qualitatively similar to1 x and y4 ,so long as ,,oy is defined only by relation to ,,ox1 it is conceptually unique4 ,but now let us suppose the same relation as that between ,,ox and ,,oy1 to be possible between ,,oy and a new series ,,oz2 we then get a third series ,,oz perpendicular to ,,oy1 and again conceptually unique1 so long as it is defined by relation to ,,oy alone4 ,we might proceed1 in the same way1 to a fourth line ,,ou perpendicular to ,,oz4 ,but it is necessary1 for our purposes1 that ,,oz should be perpendicular to ,,ox as well as ,,oy4 ,without this condition1 ,,oz might extend into another world1 and have no corresponding relation to ,,ox--this is a possibility only excluded by our unavoidable spatial images4 ,at this point comes the crux of the argument4 ,that ,,oz1 says ,lotze1 which1 besides being perpendicular to ,,oy1 is also perpendicular to ,,ox1 must be among the series of ,,oy's1 for these were defined only by perpendicularity to ,,ox4 ,hence1 he concludes1 there can only be even a third dimension if ,,oz coincides with one1 and--as soon as ,,ox is considered fixed--with only one1 of the many members of the ,,oy series4 ,in this argument it is difficult--to me at any rate--to see any force at all4 ,the only way ,i can account for it is1 to suppose that ,lotze has neglected the possibility of any but single infinities4 ,on this interpretation1 the argument might be stated thus3 ,there is an infinite series of continuously varying ,,oy's2 to the common property of these1 we add another property1 which will divide their total number by infinity4 ,the remaining ,,oz1 therefore1 must be uniquely determined4 ,the same form of argument1 however1 would prove that two surfaces can only cut one another in a single point1 and numberless other absurdities4 ,the fact is1 that infinities may be of different orders4 ,for example1 the number of points in a line may be taken as a single infinity1 and so may the number of lines in a plane through any point2 hence1 by multiplication1 the number of points in a plane is a double infinity1 #b1 and if we divide this number by a single infinity1 we get still an infinite number left4 ,thus ,lotze's argument assumes what he has to prove1 that the number of lines perpendicular to a given line1 through any point1 is a single infinity1 which is equivalent to the axiom of three dimensions4 ,the whole passage is so obscure1 that its meaning may have escaped me4 ,it is obvious priori1 however1 as ,i pointed out in the beginning1 that any proof of the axiom must be fallacious somewhere1 and the above interpretation of the argument is the only one ,i have been able to find4 #ie4 ,the rest of the ,chapter is devoted to an attack on spherical and pseudo-spherical space1 on the ground that they interfere with the homogeneity of the three dimensions1 and with the similarity of all parts of space4 ,this is simply false4 ,such spaces1 like the surface of a sphere1 are exactly alike throughout4 ,lotze shows1 here and elsewhere1 that he has not taken the pains to find out what ,metageometry really is4 ,i hold myself1 and have tried to prove in this ,essay1 that ,congruence is an priori axiom1 without which ,geometry would be impossible2 but the wish to uphold this axiom is1 as ,lotze ought to have known1 the precise motive which led ,metageometry to limit itself to spaces of constant measure of curvature4 ,we see here the importance of distinguishing between ,helmholtz the philosopher and ,helmholtz the mathematician4 ,though the philosopher wished to dispense with ,congruence1 the mathematician1 as we saw in ,chapter ,i41 retained and strongly emphasized it4 ,a little later ,lotze shows1 again1 how he has been misled by the unfortunate analogy of ,sphereland4 ,a spherical surface1 he says1 he can understand2 but how are we to pass from this to a spherical space8 ,either this surface is the whole of our space1 as in ,sphereland1 or it generates space by a gradually growing radius4 ,such concentric spheres1 as ,lotze triumphantly points out1 of course generate ,euclidean space4 ,his disjunction1 however1 is utterly and entirely false1 and could never have been suggested by any one with even a superficial knowledge of ,metageometry4 ,this point is less laboured than the former1 which1 in all its nakedness1 is thus re-stated in the last sentence of the ,chapter3 ,i cannot persuade myself that one could1 without the elements of homogeneous space1 even form or define the presentation of heterogeneous spaces1 or of such as had variable measures of curvature4 ,as though such spaces were ever set up by non-,euclidean mathematics6 ,in conclusion1 ,lotze expresses a hope that ,philosophy1 on this point1 will not allow itself to be imposed upon by ,mathematics4 ,i must1 instead1 rejoice that ,mathematics has not been imposed upon by ,philosophy1 but has developed freely an important and self-consistent system1 which deserves1 for its subtle analysis into logical and factual elements1 the gratitude of all who seek for a philosophy of space4 #if4 ,the objections to non-,euclidean ,geometry which have just been discussed fall under four heads3 ,i4 ,non-,euclidean spaces are not homogeneous2 ,metageometry therefore unduly reifies space4 ,,ii4 ,they involve a reference to a fourth dimension4 ,,iii4 ,they cannot be set up without an implicit reference to ,euclidean space1 or to the ,euclidean straight line1 on which they are therefore dependent4 ,,iv4 ,they are self-contradictory in one or more ways4 ,the reader who has followed me in regarding these four objections as fallacious1 will have no difficulty in disposing of any other critic of ,metageometry1 as these are the only mathematical arguments1 so far as ,i know1 ever urged against non-,euclideans#ajh4 ,the logical validity of ,metageometry1 and the mathematical possibility of three-dimensional non-,euclidean spaces1 will therefore be regarded1 throughout the remainder of the work1 as sufficiently established4 #ig4 ,two other objections may1 indeed1 be urged against ,metageometry1 but these are rather of a philosophical than of a strictly mathematical import4 ,the first of these1 which has been made the base of operations by ,delbuf1 applies equally to all non-,euclidean spaces4 ,the second1 which has not1 so far as ,i know1 been much employed1 but yet seems to me deserving of notice1 bears directly against spaces of positive curvature alone2 but if it could discredit these1 it might throw doubt on the method by which all alike are obtained4 ,the two objections are3 ,i4 ,space must be such as to allow of similarity1 i4e4 of the increase or diminution1 in a constant ratio1 of all the lines in a figure1 without change of angles2 whereas in non-,euclid1 lines1 like angles1 have absolute magnitude4 ,,ii4 ,space must be infinite1 whereas spherical and elliptic spaces are finite4 ,i will discuss the first objection in connection with ,delbuf's articles referred to above4 ,the second1 which has not1 to my knowledge1 been widely used in criticism1 will be better deferred to ,chapter ,,iii4 ,delbuf4 #ih4 ,m4 ,delbuf's four articles in the ,revue ,philosophique contain much matter that has already been dealt with in the criticism of ,lotze1 and much that is irrelevant for our present purpose4 ,the only point1 which ,i wish to discuss here1 is the question of absolute magnitude1 as it is called--the question1 that is1 whether the possibility of similar but unequal geometrical figures can be known priori#aji4 ,in discussing this question1 it is important1 to begin with1 to distinguish clearly the sense in which absolute magnitude is required in non-,euclidean ,geometry1 from another sense1 in which it would be absurd to regard any magnitude as absolute4 ,judgments of magnitude can only result from comparison1 and if ,metageometry required magnitudes which could be determined without comparison1 it would certainly deserve condemnation4 ,but this is not required4 ,all we require is1 that it shall be impossible1 while the rest of space is unaffected1 to alter the magnitude of any figure1 as compared with other figures1 while leaving the relative internal magnitudes of its parts unchanged4 ,this construction1 which is possible in ,euclid1 is impossible in ,metageometry4 ,we have to discuss whether such an impossibility renders non-,euclidean spaces logically faulty4 ,m4 ,delbuf's position on this axiom--which he calls the postulate of homogeneity#aaj--;is1 that all ,geometry must presuppose it1 and that ,metageometry1 consequently1 though logically sound1 is logically subsequent to ,euclid1 and can only make its constructions within a ,euclidean homogeneous space 7,rev4 ,phil4 ,vol4 ,,xxxvii41 pp4 #chj-a74 ,he would appear to think1 nevertheless1 that homogeneity 7in his sense7 is learnt from experience1 though on this point he is not very explicit4 7,see ,vol4 ,,xxxviii41 p4 #abi47 ,no priori proof1 at any rate1 is offered in his articles4 ,as a result of experience1 every one would admit1 similarity is known to be possible within the limits of observation2 but the fact that this possibility extends to ,ordnance maps1 which deal with a spherical surface1 should make us chary of inferring1 from such a datum1 the certainty of ,euclid for large spaces4 ,moreover if homogeneity be empirical1 ,metageometry1 which dispenses with it1 is not necessarily in logical dependence upon ,euclid1 since homogeneity and isogeneity are logically separable4 ,i shall assume1 therefore1 as the only contention which can be interesting to our argument1 that homogeneity is regarded as priori1 and as logically essential to ,geometry4 #ii4 ,now we saw1 in discussing ,erdmann's views of the judgment of quantity1 that in non-,euclidean space1 as in ,euclidean1 a change of all spatial magnitudes1 in the same ratio1 would be no change at all2 the ratios of all magnitudes to the space-constant would be unchanged1 and the space-constant1 as the ultimate standard of comparison1 cannot1 in any intelligible sense1 be said to have any particular magnitude4 ,the absolute magnitudes of ,metageometry1 therefore1 are absolute only as against any other particular magnitude1 not as against other magnitudes in general4 ,if this were not the case1 the comparative nature of the judgment of magnitude would be contradicted1 and metrical ,metageometry would become absurd4 ,but as it is1 the difference from ,euclid consists only in this3 that in ,metageometry we have1 while in ,euclid we have not1 a standard of comparison involved in the nature of our space as a whole1 which we call the space-constant4 ,we have to discuss whether the assertion of such a standard involves an undue reification of space4 ,i do not believe that this is the case4 ,for an undue reification of space would only arise1 if we were no longer able to regard position as wholly relative1 and as geometrically definable only by departure from other positions4 ,but the relativity of position1 as we have abundantly seen1 is preserved by all spaces of constant curvature--in all of these1 positions can only be defined1 geometrically1 by relations to fresh positions#aaa4 ,this series of definitions may lead to an infinite regress1 but it may also1 as in spherical space1 form a vicious circle1 and return again to the position from which it started4 ,no reification of space1 no independent existence of mere relations1 seems involved in such a procedure4 ,the whole of ,metageometry1 in short1 is a proof that the relativity of position is compatible with absolute magnitude1 in the only sense required by non-,euclidean spaces4 ,we must conclude1 therefore1 that there is nothing incompatible1 in a denial of homogeneity 7in ,delbuf's sense71 either with the relational nature of space1 or with the comparative nature of magnitude4 ,this last priori objection to ,metageometry1 therefore1 cannot be maintained1 and the issue must be decided on empirical grounds alone4 #ajj4 ,the foundations of ,geometry have been the subject of much recent speculation in ,france1 and this seems to demand some notice4 ,but in spite of the splendid work which the ,french have done on the allied question of number and continuous quantity1 ,i cannot persuade myself that they have succeeded in greatly advancing the subject of geometrical philosophy4 ,the chief writers have been1 from the mathematical side1 ,calinon and ,poincar1 from the philosophical1 ,renouvier and ,delbuf2 as a mediator between mathematics and philosophy1 ,lechalas4 ,calinon1 in an interesting article on the geometrical indeterminateness of the universe1 maintains that any ,geometry may be applied to the actual world by a suitable hypothesis as to the course of light-rays4 ,for the earth only is known to us otherwise than by ,optics1 and the earth is an infinitesimal part of the universe4 ,this line of argument has been already discussed in connection with ,lotze1 but ,calinon adds a new suggestion1 that the space-constant may perhaps vary with the time4 ,this would involve a causal connection between space and other things1 which seems hardly conceivable1 and which1 if regarded as possible1 must surely destroy ,geometry1 since ,geometry depends throughout on the irrelevance of ,causation#aab4 ,moreover1 in all operations of measurement1 some time is spent2 unless we knew that space was unchanging throughout the operation1 it is hard to see how our results could be trustworthy1 and how1 consequently1 a change in the parameter could be discovered4 ,the same difficulties would arise1 in fact1 as those which result from supposing space not homogeneous4 ,poincar maintains that the question1 whether ,euclid or ,metageometry should be accepted1 is one of convenience and convention1 not of truth2 axioms are definitions in disguise1 and the choice between definitions is arbitrary4 ,this view has been discussed in ,chapter ,i41 in connection with ,cayley's theory of distance1 on which it depends4 ,lechalas is a philosophical disciple of ,calinon4 ,he is a rationalist of the pre-,kantian type1 but a believer in the validity of ,metageometry4 ,he holds that ,geometry can dispense with all purely spatial postulates1 and work with axioms of magnitude alone#aac1 which1 in his opinion1 are purely analytic4 ,the principle of contradiction1 to him1 is the sole and only test of truth2 we make long chains of reasoning from our premisses to see if contradictions will emerge4 ,it might be objected that this view1 though it saves general ,geometry from being logically empirical1 leaves it only empirically logical2 this must1 in fact1 be the fate of every piece of priori knowledge1 if ,m4 ,lechalas's were the only test of truth4 ,however1 he concludes that general ,geometry is apodeictic1 while the space of our actual world1 like all other phenomena1 is contingent4 ,delbuf criticizes non-,euclidean space from an ultra-realist standpoint3 he holds that real space is neither homogeneous nor isogeneous1 but that conceived space1 as abstracted from real space1 has both these properties4 ,he offers no justification for his real space1 which seems to be maintained in the spirit of nave realism1 nor does he show how he has acquired his intimate knowledge of its constitution#aad4 ,his arguments against ,metageometry1 in so far as they are not repetitions of ,lotze1 have been discussed above4 ,renouvier1 finally1 is a pure ,kantian1 of the most orthodox type4 ,his views as to the importance1 for ,geometry1 of the distinction between synthetic and analytic judgments1 have been discussed1 in connection with ,kant1 at the beginning of the present ,chapter#aae4 #aja4 ,before beginning the constructive argument of the next ,chapter1 let us endeavour briefly to sum up the theories which have been polemically advocated throughout the criticisms we have just concluded4 ,we agreed to accept1 with ,kant1 necessity for any possible experience as the test of the priori1 but we refused1 for the present1 to discuss the connection of the priori with the subjective1 regarding the purely logical test as sufficient for our immediate purpose4 ,we also refused to attach importance to the distinction of analytic and synthetic1 since it seemed to apply1 not to different judgments1 but only to different aspects of any judgment4 ,we then discussed ,riemann's attempt to identify the empirical element in ,geometry with the element not deducible from ideas of magnitude1 and we decided that this identification was due to a confusion as to the nature of magnitude4 ,for judgments of magnitude1 we said1 require always some qualitative basis1 which is not quantitatively expressible4 ,in criticizing ,helmholtz1 we decided that ,mechanics logically presupposes ,geometry1 though space presupposes matter2 but that the matter which space presupposes1 and to which ,geometry indirectly refers1 is a more abstract matter than that of ,mechanics1 a matter destitute of force and of causal attributes1 and possessed only of the purely spatial attributes required for the possibility of spatial figures4 ,but we conceded that ,geometry1 when applied to mixed mathematics or to daily life1 demands more than this1 demands1 in fact1 some means of discovering1 in the more concrete matter of ,mechanics1 either a rigid body1 or a body whose departure from rigidity follows some empirically discoverable law4 ,actual measurement1 therefore1 we agreed to regard as empirical4 ,our conclusions1 as regards the empiricism of ,riemann and ,helmholtz1 were reinforced by a criticism of ,erdmann4 ,we then had an opposite task to perform1 in defending ,metageometry against ,lotze4 ,here we saw that there are two senses in which ,metageometry is possible4 ,the first concerns our actual space1 and asserts that it may have a very small space-constant2 the second concerns philosophical theories of space1 and asserts a purely logical possibility1 which leaves the decision to experience4 ,we saw also that ,lotze's mathematical strictures arose from insufficient knowledge of the subject1 and could all be refuted by a better acquaintance with ,metageometry4 ,finally1 we discussed the question of absolute magnitude1 and found in it no logical obstacle to non-,euclidean spaces4 ,our conclusion1 then1 in so far as we are as yet entitled to a conclusion1 is that all spaces with a space-constant are priori justifiable1 and that the decision between them must be the work of experience4 ,spaces without a space-constant1 on the other hand1 spaces1 that is1 which are not homogeneous throughout1 we found logically unsound and impossible to know1 and therefore to be condemned priori4 ,the constructive proof of this thesis will form the argument of the following chapter4 ,,footnotes3 #fg ,the ,critical ,philosophy of ,kant1 ,vol4 ,i4 p4 #bhg4 #fh ,for a discussion of ,kant from a less purely mathematical standpoint1 see ,chap4 ,,iv4 #fi ,cf4 ,vaihinger's ,commentar1 ,,ii4 pp4 #bjb1 #bfe4 ,also p4 #ccf ff4 #gj ,e4g4 second edition1 p4 #ci3 ,so werden auch alle geometrischen ,grundstze1 z4 ,b4 dass in einem ,triangel zwei ,seiten zusammen grsser sind als die dritte1 niemals aus allgemeinen ,begriffen von ,linie und ,triangel1 sondern aus der ,anschauung1 und zwar priori mit apodiktischer ,gewissheit abgeleitet4 #ga ,cf4 ,bradley's ,logic1 ,bk4 ,,iii4 ,pt4 ,i4 ,chap4 ,,vi42 ,bosanquet's ,logic1 ,bk4 ,i4 ,chap4 ,i4 pp4 #ig-ajc4 #gb ,philosophie de la ,rgle et du ,compas1 ,anne ,philosophique1 ,,ii4 pp4 #a-ff4 #gc ,i have stated this doctrine dogmatically1 as a proof would require a whole treatise on ,logic4 ,i accept the proofs offered by ,bradley and ,bosanquet1 to which the reader is referred4 #gd ,for a further discussion of this point1 see ,chaps4 ,,iii4 and ,,iv4 #ge ,see ,chap4 ,,iv4 for a discussion of this argument4 #gf ,see ,chap4 ,,iv4 #ahe4 #gg ,an ,otherness of substance1 rather than of attribute1 is here intended2 an ,otherness which may perhaps be called real as opposed to logical diversity4 #gh ,this proposition will be argued at length in ,chap4 ,,iv4 #gi ,see ,psychologie als ,wissenschaft1 ,i4 ,section ,,iii4 ,chap4 ,,vii42 ,,ii4 ,section ,i4 ,chap4 ,,iii4 and ,section ,,ii4 ,chap4 ,,iii4 ,compare also ,synechologie1 ,section ,i4 ,chaps4 ,,ii4 and ,,iii4 #hj ,on the influence of ,herbart on ,riemann1 compare ,erdmann1 ,die ,axiome der ,geometrie1 p4 #cj4 #ha ,i do not mean that measurement of colours is effected without reference to their relations1 since all measurement is essentially comparison4 ,but in colours1 it is the elements which are compared1 while in space1 it is the relations between elements4 #hb ,for a discussion of this point1 see ,chap4 ,,iii4 ,sec4 ,b1 #agf4 #hc ,the works of ,helmholtz on geometrical philosophy comprise1 in addition to the articles quoted in ,chap4 ,i41 the following articles3 ,ursprung und ,sinn der geometrischen ,axiome1 gegen ,land1 ,wiss4 ,abh4 ,vol4 ,,ii4 p4 #fdj1 #ahgh4 7,also ,mind1 ,vol4 ,,iii43 an answer to ,land in ,mind1 ,vol4 ,,ii47 ,ursprung und ,bedeutung der geometrischen ,axiome1 #ahgj1 ,vortrge und ,reden1 ,vol4 ,,ii4 p4 #a4 7,also ,mind1 ,vol4 ,i47 ,two ,appendices to ,die ,thatsachen in der ,wahrnehmung1 entitled3 ,,ii4 ,der ,raum kann transcendental sein1 ohne dass es die ,axiome sind2 and ,,iii4 ,die ,anwendbarkeit der ,axiome auf die physische ,welt1 #ahgh1 ,vortrge und ,reden1 ,vol4 ,,ii4 p4 #bef ff4 ,the two ,appendices last mentioned are popularizings and expansions of the article in ,mind1 ,vol4 ,,iii4 ,the most widely read1 though also1 to my mind1 the least valuable1 of all ,helmholtz's writings on ,geometry1 is the article in ,mind1 ,vol4 ,i4 ,this contains the famous and much misunderstood analogies of ,flatland and ,sphereland1 which will be discussed1 and as far as possible defended1 in answering ,lotze's attack on ,metageometry--an attack based1 apparently1 almost entirely on this one popular article4 ,the present discussion1 therefore1 may be confined almost entirely to ,mind1 ,vol4 ,,iii41 and the philosophical portions of the two papers quoted in ,chap4 ,i41 i4e4 to the articles in ,wiss4 ,abh4 ,vol4 ,,ii4 pp4 #faj-ffj4 ,his other works are popular1 and important only because of the large public to which they appeal4 #hd ,in the answer to ,land1 ,mind1 ,vol4 ,,iii4 and ,wiss4 ,abh4 ,,ii4 p4 #fdj4 #he ,see also ,die ,thatsachen in der ,wahrnehmung1 ,zusatz ,,ii41 ,der ,raum kann transcendental sein1 ohne dass es die ,axiome sind4 ,vortrge und ,reden1 ,vol4 ,,ii4 #hf ,see below1 criticism of ,erdmann1 #hd4 #hg ,see ,prof4 ,land1 in ,mind1 ,vol4 ,,ii4 #hh ,see concluding paragraph of ,helmholtz's article in ,mind1 ,vol4 ,,iii4 #hi ,cf4 ,veronese1 ,grundzge der ,geometrie 7,german translation71 p4 ix4 ,also pp4 xxxiv1 #cjd1 and ,note ,,ii4 pp4 #fib-d4 #ij ,see ,chap4 ,,iv4 #aig ff4 #ia ,cf4 the opinion of ,bolyai1 quoted by ,erdmann1 ,axiome1 p4 #bf2 cf4 also ib4 p4 #fj4 #ib ,die ,axiome der ,geometrie3 ,eine philosophische ,untersuchung der ,riemann-,helmholtz'schen ,raumtheorie1 ,leipzig1 #ahgg4 #ic ,on the influence of ,mill1 cf4 ,stallo1 ,concepts of ,modern ,physics1 p4 #baf4 #id ,this view seems to be derived1 through ,riemann1 from ,herbart4 ,see ,psych4 als ,wiss4 ed4 ,hart4 ,vol4 ,v4 p4 #bfb4 #ie ,the same irreducibility of space to mere magnitude is proved by ,kant's hands and spherical triangles1 in which a difference persists in spite of complete quantitative equality4 #if ,see #adf-g4 #ig ,jeder ,versuch1 ,kant's ,lehre von der ,aprioritt als des subjectiven1 von aller ,erfahrung absolut unabhngigen ,erkenntnissfactors1 trotzdem zu halten1 ist deshalb von voruherein aussichtslos4 #ih ,ausdehnungslehre von #ahdd1 #bnd edition1 pp4 xxii4 xxiii4 #ii ,see #abi ff4 #ajj ,metaphysik1 ,book ,,ii4 ,chap4 ,,ii4 ,my references are to the original4 #aja ,see ,lectures and ,essays1 ,vol4 ,i4 p4 #bfa4 #ajb ,on the meaning of geometrical possibility1 cf4 ,veronese1 ,grundzge der ,geometrie 7,german translation71 pp4 xi4-xiii4 #ajc ,compare ,calinon1 ,sur l',indtermination gomtrique de l',univers1 ,revue ,philosophique1 #ahic1 ,vol4 ,,xxxvi4 pp4 #eie-fjg4 #ajd ,vortrge und ,reden1 ,vol4 ,,ii4 p4 #i3 ,parallele ,linien wrden die ,bewohner der ,kugel gar nicht kennen4 ,sie wrden behaupten1 dass jede beliebige zwei geradeste ,linien1 gehrig verlangert1 sich schliesslich nicht nur in einem1 sondern in zwei ,punkten schneiden mssten4 7,the italics are mine47 ,the omission of straight in such phrases is a frequent laxity of mathematicians4 #aje ,it has been suggested to me that ,lotze regards the meridians as projected on to a plane1 as in a map4 ,if this be so1 there is an obviously illegitimate introduction of the third dimension4 #ajf ,this is proved by ,helmholtz's remark at the end of a detailed attempt to make spherical and pseudo-spherical spaces imaginable 7l4c4 p4 #bh73 ,anders ist es mit den drei ,dimensionen des ,raumes4 ,da alle unsere ,mittel sinnlicher ,anschauung sich nur auf einen ,raum von drei ,dimensionen erstrecken1 und die vierte ,dimension nicht bloss eine ,abnderung von ,vorhandenem1 sondern etwas vollkommen ,neues wre1 so befinden wir uns schon wegen unserer krperlichen ,organisation in der absoluten ,unmglichkeit1 uns eine ,anschauungsweise einer vierten ,dimension vorzustellen4 #ajg ,cf4 ,grassmann1 ,ausdehnungslehre von #ahdd1 #bnd ,edition1 p4 xxiii4 #ajh ,see especially ,stallo1 ,concepts of ,modern ,physics1 ,international ,science ,series1 ,vol4 ,,xlii4 ,chaps4 ,,xiii4 and ,,xiv42 ,renouvier1 ,philosophie de la rgle et du compas1 ,anne ,philosophique1 ,,ii42 ,delbuf1 ,l'ancienne et les nouvelles gomtries1 ,revue ,philosophique1 ,vols4 ,,xxxvi4-,,xxxix4 #aji ,m4 ,delbuf deserves credit for having based ,euclid1 already in #ahfj1 in his ,prolgomnes ,philosophiques de la ,gomtrie1 on this axiom--certainly a better basis1 at first sight1 than the axiom of parallels4 #aaj ,this meaning of homogeneity must not be confounded with the sense in which ,i have used the word4 ,in ,delbuf's sense1 it means that figures may be similar though of different sizes2 in my sense it means that figures may be similar though in different places4 ,this property of space is called by ,delbuf isogeneity4 #aaa ,for a full proof of this proposition1 see ,chap4 ,,iii4 #aab ,see ,chap4 ,,iii41 especially #acc4 #aac ,for a criticism of this view1 see the above discussions on ,riemann and ,erdmann4 #aad ,cf4 ,couturat1 ,de l',infini ,mathmatique1 ,paris1 ,flix ,alcan1 #ahif1 p4 #edd4 #aae ,the following is a list of the most important recent ,french philosophical writings on ,geometry1 so far as ,i am acquainted with them4 ,andrade3 ,les bases exprimentales de la gomtrie euclidienne2 ,rev4 ,phil4 #ahij1 ,,ii41 and #ahia1 ,i4 ,bonnel3 ,les hypothses dans la gomtrie2 ,gauthier-,villars1 #ahig4 ,l',abb de ,broglie3 ,la gomtrie non-euclidienne1 two articles2 ,annales de ,phil4 ,chrt4 #ahij4 ,calinon3 ,les espaces gomtriques2 ,rev4 ,phil4 #ahhi1 ,i41 and #ahia1 ,,ii4 ,sur l'indtermination gomtrique de l'univers2 ib4 #ahic1 ,,ii4 ,couturat3 ,l',anne ,philosophique de ,f4 ,pillon1 ,rev4 de ,mt4 et de ,morale1 ,jan4 #ahic4 ,note sur la gomtrie non-euclidienne et la relativit de l'espace2 ib41 ,may1 #ahic4 tudes sur l'espace et le temps1 ib4 ,sep4 #ahif4 ,delbuf3 ,l'ancienne et les nouvelles gomtries1 four articles2 ,rev4 ,phil4 #ahic-e4 ,lechalas3 ,la gomtrie gnrale2 ,crit4 ,phil4 #ahhi4 ,la gomtrie gnrale et les jugements synthtiques priori and ,les bases exprimentales de la gomtrie2 ,rev4 ,phil4 #ahij1 ,,ii4 ,m4 ,delbuf et ,le problme des mondes semblables2 ib4 #ahid1 ,i4 ,note sur la gomtrie non-euclidienne et le principe de similitude2 ,rev4 de ,mt4 et de ,morale1 ,march1 #ahic4 ,la courbure et la distance en gomtrie gnrale2 ib41 ,march1 #ahif4 ,la gomtrie gnrale et l'intuition2 ,annales de ,phil4 ,chrt41 #ahij4 ,etude sur l'espace et le temps2 ,paris1 ,alcan1 #ahif4 ,liard3 ,des dfinitions gomtrie et des dfinitions empiriques1 #bnd ed42 ,paris1 ,alcan1 #ahhh4 ,mansion3 ,premiers principes de la mtagomtrie2 two articles in ,rev4 ,no-,scholastique1 #ahif4 ,separately published1 ,gauthier-,villars1 #ahif4 ,milhaud3 ,la gomtrie non-euclidienne et la thorie de la connaissance2 ,rev4 ,phil4 #ahhh1 ,i4 ,poincar3 ,non-,euclidian ,geometry2 ,nature1 ,vol4 ,,xlv41 #ahia-b4 ,l'espace et la gomtrie2 ,rev4 de ,mt4 et de ,morale1 ,nov4 #ahie4 ,rsponse quelques critiques1 ib4 ,jan4 #ahig4 ,renouvier3 ,philosophie de la rgle et du compas2 ,crit4 ,phil41 #ahhi1 and ,l',anne ,phil41 ,,iime anne1 #ahia4 ,sorel3 ,sur la gomtrie non-euclidienne2 ,rev4 ,phil41 #ahia1 ,i4 ,tannery3 ,thorie de la connaissance mathmatique2 ,rev4 ,phil41 #ahid1 ,,ii4 ,,chapter ,,iii4 ,section ,a4 ,,the ,,axioms ,,of ,,projective ,,geometry4 #ajb4 ,projective ,geometry proper1 as we saw in ,chapter ,i41 does not employ the conception of magnitude1 and does not1 therefore1 require those axioms which1 in the systems of the second or metrical period1 were required solely to render possible the application of magnitude to space4 ,but we saw1 also1 that ,cayley's reduction of metrical to projective properties was purely technical and philosophically irrelevant4 ,now it is in metrical properties alone--apart from the exception to the axiom of the straight line1 which itself1 however1 presupposes metrical properties#aaf--that non-,euclidean and ,euclidean spaces differ4 ,the properties dealt with by projective ,geometry1 therefore1 in so far as these are obtained without the use of imaginaries1 are properties common to all spaces4 ,finally1 the differences which appear between the ,geometries of different spaces of the same curvature--e4g4 between the ,geometries of the plane and the cylinder--are differences in projective properties#aag4 ,thus the necessity which arises1 in metrical ,geometry1 for further qualifications besides those of constant curvature1 disappears when our general space is defined by purely projective properties4 #ajc4 ,we have good ground for expecting1 therefore1 that the axioms of projective ,geometry will be the simplest and most complete expression of the indispensable requisites of any geometrical reasoning3 and this expectation1 ,i hope1 will not be disappointed4 ,projective ,geometry1 in so far as it deals only with the properties common to all spaces1 will be found1 if ,i am not mistaken1 to be wholly priori1 to take nothing from experience1 and to have1 like ,arithmetic1 a creature of the pure intellect for its object4 ,if this be so1 it is that branch of pure mathematics which ,grassmann1 in his ,ausdehnungslehre of #ahdd1 felt to be possible1 and endeavoured1 in a brilliant failure1 to construct without any appeal to the space of intuition4 #ajd4 ,but unfortunately1 the task of discovering the axioms of projective ,geometry is far from easy4 ,they have1 as yet1 found no ,riemann or ,helmholtz to formulate them philosophically4 ,many geometers have constructed systems1 which they intended to be1 and which1 with sufficient care in interpretation1 really are1 free from metrical presuppositions4 ,but these presuppositions are so rooted in all the very elements of ,geometry1 that the task of eliminating them demands a reconstruction of the whole geometrical edifice4 ,thus ,euclid1 for example1 deals1 from the start1 with spatial equality--he employs the circle1 which is necessarily defined by means of equality1 and he bases all his later propositions on the congruence of triangles as discussed in ,book ,i4#aah ,before we can use any elementary proposition of ,euclid1 therefore1 even if this expresses a projective property1 we have to prove that the property in question can be deduced by projective methods4 ,this has not1 in general1 been done by projective geometers1 who have too often assumed1 for example1 that the quadrilateral construction--by which1 as we saw in ,chap4 ,i41 they introduce projective coordinates--or anharmonic ratio1 which is prim facie metrical1 could be satisfactorily established on their principles4 ,both these assumptions1 however1 can be justified1 and we may admit1 therefore1 that the claims of projective ,geometry to logical independence of measurement or congruence are valid4 ,let us see1 then1 how it proceeds4 #aje4 ,in the first place1 it is important to realize that when coordinates are used1 in projective ,geometry1 they are not coordinates in the ordinary metrical sense1 i4e4 the numerical measures of certain spatial magnitudes4 ,on the contrary1 they are a set of numbers1 arbitrarily but systematically assigned to different points1 like the numbers of houses in a street1 and serving only1 from a philosophical standpoint1 as convenient designations for points which the investigation wishes to distinguish4 ,but for the brevity of the alphabet1 in fact1 they might1 as in ,euclid1 be replaced by letters4 ,how they are introduced1 and what they mean1 has been discussed in ,chapter ,i4 ,here we have only to repeat a caution1 whose neglect has led to much misunderstanding4 #ajf4 ,the distinction between various points1 then1 is not a result1 but a condition1 of the projective coordinate system4 ,the coordinate system is a wholly extraneous1 and merely convenient1 set of marks1 which in no way touches the essence of projective ,geometry4 ,what we must begin with1 in this domain1 is the possibility of distinguishing various points from one another4 ,this may be designated1 with ,veronese1 as the first axiom of ,geometry#aai4 ,how we are to define a point1 and how we distinguish it from other points1 is for the moment irrelevant2 for here we only wish to discover the nature of projective ,geometry1 and the kind of properties which it uses and demonstrates4 ,how1 and with what justification1 it uses and demonstrates them1 we will discuss later4 #ajg4 ,now it is obvious that a mere collection of points1 distinguished one from another1 cannot found a ,geometry3 we must have some idea of the manner in which the points are interrelated1 in order to have an adequate subject-matter for discussion4 ,but since all ideas of quantity are excluded1 the relations of points cannot be relations of distance in the ordinary sense1 nor even1 in the sense of ordinary ,geometry1 anharmonic ratios1 for anharmonic ratios are usually defined as the ratios of four distances1 or of four sines1 and are thus quantitative4 ,but since all quantitative comparison presupposes an identity of quality1 we may expect to find1 in projective ,geometry1 the qualitative substrata of the metrical superstructure4 ,and this1 we shall see1 is actually the case4 ,we have not distance1 but we have the straight line2 we have not quantitative anharmonic ratio1 but we have the property1 in any four points on a line1 of being the intersections with the rays of a given pencil4 ,and from this basis1 we can build up a qualitative science of abstract externality1 which is projective ,geometry4 ,how this happens1 ,i shall now proceed to show4 #ajh4 ,all geometrical reasoning is1 in the last resort1 circular3 if we start by assuming points1 they can only be defined by the lines or planes which relate them2 and if we start by assuming lines or planes1 they can only be defined by the points through which they pass4 ,this is an inevitable circle1 whose ground of necessity will appear as we proceed4 ,it is1 therefore1 somewhat arbitrary to start either with points or with lines1 as the eminently projective principle of duality mathematically illustrates2 nevertheless we will elect1 with most geometers1 to start with points#abj4 ,we suppose1 therefore1 as our datum1 a set of discrete points1 for the moment without regard to their interconnections4 ,but since connections are essential to any reasoning about them as a system1 we introduce1 to begin with1 the axiom of the straight line4 ,any two of our points1 we say1 lie on a line which those two points completely define4 ,this line1 being determined by the two points1 may be regarded as a relation of the two points1 or an adjective of the system formed by both together4 ,this is the only purely qualitative adjective--as will be proved later--of a system of two points4 ,now projective ,geometry can only take account of qualitative adjectives1 and can distinguish between different points only by their relations to other points1 since all points1 per se1 are qualitatively similar4 ,hence it comes that1 for projective ,geometry1 when two points only are given1 they are qualitatively indistinguishable from any two other points on the same straight line1 since any two such other points have the same qualitative relation4 ,reciprocally1 since one straight line is a figure determined by any two of its points1 and all points are qualitatively similar1 it follows that all straight lines are qualitatively similar4 ,we may regard a point1 therefore1 as determined by two straight lines which meet in it1 and the point1 on this view1 becomes the only qualitative relation between the two straight lines4 ,hence1 if the point only be regarded as given1 the two straight lines are qualitatively indistinguishable from any other pair through the point4 #aji4 ,the extension of these two reciprocal principles is the essence of all projective transformations1 and indeed of all projective ,geometry4 ,the fundamental operations1 by which figures are projectively transformed1 are called projection and section4 ,the various forms of projection and section are defined in ,cremona's ,projective ,geometry1 ,chapter ,i41 from which ,i quote the following account4 ,to project from a fixed point ,s 7the centre of projection7 a figure 7,,abcd 444 abcd 4447 composed of points and straight lines1 is to construct the straight lines or projecting rays ,,sa1 ,,sb1 ,,sc1 ,,sd1 444 and the planes 7projecting planes7 ,sa1 ,sb1 ,sc1 ,sd1 444 ,we thus obtain a new figure composed of straight lines and planes which all pass through the centre ,s4 ,to cut by a fixed plane 7transversal plane7 a figure 7 444 abcd 4447 made up of planes and straight lines1 is to construct the straight lines or traces 1 1 444 and the points or traces a1 b1 c4444#aba ,by this means we obtain a new figure composed of straight lines and points lying in the plane 4 ,to project from a fixed straight line s 7the axis7 a figure ,,abcd composed of points1 is to construct the planes s,a1 s,b1 s,c4444 ,the figure thus obtained is composed of planes which all pass through the axis s4 ,to cut by a fixed straight line s 7a transversal7 a figure 444 composed of planes1 is to construct the points s1 s1 s4444 ,in this way a new figure is obtained1 composed of points all lying on the fixed transversal s4 ,if a figure is composed of straight lines a1 b1 c 444 which all pass through a fixed point or centre ,s1 it can be projected from a straight line or axis s passing through ,s2 the result is a figure composed of planes sa1 sb1 sc4444 ,if a figure is composed of straight lines a1 b1 c 444 all lying in a fixed plane1 it may be cut by a straight line 7transversal7 s lying in the same plane2 the figure which results is formed by the points sa1 sb1 sc4444 #aaj4 ,the successive application1 to any figure1 of two reciprocal operations of projection and section1 is regarded as producing a figure protectively indistinguishable from the first1 provided only that the dimensions of the original figure were the same as those of the resulting figure1 that1 for example1 if the second operation be section by a plane1 the original figure shall have been a plane figure4 ,the figures obtained from a given figure1 by projection or section alone1 are related to that figure by the principle of duality1 of which we shall have to speak later on4 ,i shall endeavour to show1 in what follows1 first1 in what sense figures obtained from each other by projective transformation are qualitatively alike2 secondly1 what axioms1 or adjectives of space1 are involved in the principle of projective transformation2 and thirdly1 that these adjectives must belong to any form of externality with more than one dimension1 and are1 therefore1 priori properties of any possible space4 ,for the sake of simplicity1 ,i shall in general confine myself to two dimensions4 ,in so doing1 ,i shall introduce no important difference of principle1 and shall greatly simplify the mathematics involved4 #aaa4 ,the two mathematically fundamental things in projective ,geometry are anharmonic ratio1 and the quadrilateral construction4 ,everything else follows mathematically from these two4 ,now what is meant1 in projective ,geometry1 by anharmonic ratio8 ,illustration ,if we start from anharmonic ratio as ordinarily defined1 we are met by the difficulty of its quantitative nature#abb4 ,but among the properties deduced from this definition1 many1 if not most1 are purely qualitative4 ,the most fundamental of these is that1 if through any four points in a straight line we draw four straight lines which meet in a point1 and if we then draw a new straight line meeting these four1 the four new points of intersection have the same anharmonic ratio as the four points we started with4 ,thus1 in the figure1 abcd1 abcd1 abcd1 all have the same anharmonic ratio4 ,the reciprocal relation holds for the anharmonic ratio of four straight lines4 ,here we have1 plainly1 the required basis for a qualitative definition4 ,the definition must be as follows3 ,two sets of four points each are defined as having the same anharmonic ratio1 when 7#a7 each set of four lies in one straight line1 and 7#b7 corresponding points of different sets lie two by two on four straight lines through a single point1 or when both sets have this relation to any third set#abc4 ,and reciprocally3 ,two sets of four straight lines are defined as having the same anharmonic ratio when 7#a7 each set of four passes through a single point1 and 7#b7 corresponding lines of different sets pass1 two by two1 through four points in one straight line1 or when both sets have this relation to any third set4 ,two sets of points or of lines1 which have the same anharmonic ratio1 are treated by projective ,geometry as equivalent3 this qualitative equivalence replaces the quantitative equality of metrical ,geometry1 and is obviously included1 by its definition1 in the above account of projective transformations in general4 #aab4 ,we have next to consider the quadrilateral construction#abd4 ,this has a double purpose3 first1 to define the important special case known as a harmonic range2 and secondly1 to afford an unambiguous and exhaustive method of assigning different numbers to different points4 ,this last method has1 again1 a double purpose3 first1 the purpose of giving a convenient symbolism for describing and distinguishing different points1 and of thus affording a means for the introduction of analysis2 and secondly1 of so assigning these numbers that1 if they had the ordinary metrical significance1 as distances from some point on the numbered straight line1 they would yield -#a as the anharmonic ratio of a harmonic range1 and that1 if four points have the same anharmonic ratio as four others1 so have the corresponding numbers4 ,this last purpose is due to purely technical motives3 it avoids the confusion with our preconceptions which would result from any other value for a harmonic range2 it allows us1 when metrical interpretations of projective results are desired1 to make these interpretations without tedious numerical transformations1 and it enables us to perform projective transformations by algebraical methods4 ,at the same time1 from the strictly projective point of view1 as observed above1 the numbers introduced have a purely conventional meaning2 and until we pass to metrical ,geometry1 no reason can be shown for assigning the value -#a to a harmonic range4 ,with this preliminary1 let us see in what the quadrilateral construction consists4 ,illustration #aac4 ,a harmonic range1 in elementary ,geometry1 is one whose anharmonic ratio is -#a1 or one in which the three segments formed by the four points are in harmonic progression1 or again1 one in which the ratio of the two internal segments is equal to the ratio of the two external segments4 ,if a1 b1 c1 d be the four points1 it is easily seen that these definitions are equivalent to one another3 they give respectively3 7ab_/bc7_/7ad_/dc7 -#a1 7#a_/;ab7 - 7#a_/;ac7 7#a_/;ac7 - 7#a_/;ad71 and 7ab_/bc7 7ad_/cd74 ,but as they are all quantitative1 they cannot be used for our present purpose4 ,nor are any definitions which involve bisection of lines or angles available4 ,we must have a definition which proceeds entirely by the help of straight lines and points1 without measurement of distances or angles4 ,now from the above definitions of a harmonic range1 we see that a1 b1 c1 d have the same anharmonic ratio as c1 b1 a1 d4 ,this gives us the property we require for our definition4 ,for it shows that1 in a harmonic range1 we can find a projective transformation which will interchange a and c4 ,this is a necessary and sufficient condition for a harmonic range1 and the quadrilateral construction is the general method for giving effect to it4 ,illustration ,given any three points ,a1 ,b1 ,d in one straight line1 the quadrilateral construction finds the point ,c harmonic to ,a with respect to ,b1 ,d by the following method3 ,take any point ,o outside the straight line ,,abd1 and join it to ,b and ,d4 ,through ,a draw any straight line cutting ,,od1 ,,ob in ,p and ,q4 ,join ,,dq1 ,,bp1 and let them intersect in ,r4 ,join ,,or1 and let ,,or meet ,,abd in ,c4 ,then ,c is the point required4 ,to prove this1 let ,,drq meet ,,oa in ,t1 and draw ,,ar1 meeting ,,od in ,s4 ,then a projective transformation of ,a1 ,b1 ,c1 ,d from ,r on to ,,od gives the points ,s1 ,p1 ,o1 ,d1 which1 projected from ,a on to ,,dq1 give ,r1 ,q1 ,t1 ,d4 ,but these again1 projected from ,o on to ,,abd1 give ,c1 ,b1 ,a1 ,d4 ,hence ,a1 ,b1 ,c1 ,d can be projectively transformed into ,c1 ,b1 ,a1 ,d1 and therefore form a harmonic range4 ,from this point1 the proof that the construction is unique and general follows simply#abe4 ,the introduction of numbers1 by this construction1 offers no difficulties of principle--except1 indeed1 those which always attend the application of number to continua--and may be studied satisfactorily in ,klein's ,nicht-,euklid 7,i4 p4 #ccg ff474 ,the principle of it is1 to assign the numbers #j1 #a1 to ,a1 ,b1 ,d and therefore the number #b to ,c1 in order that the differences ,,ab1 ,,ac1 ,,ad may be in harmonic progression4 ,by taking ,b1 ,c1 ,d as a new triad corresponding to ,a1 ,b1 ,d1 we find a point harmonic to ,b with respect to ,c1 ,d and assign to it the number #c1 and so on4 ,in this way1 we can obtain any number of points1 and we are sure of having no number and no point twice over1 so that our coordinates have the essential property of a unique correspondence with the points they denote1 and vice versa4 #aad4 ,the point of importance in the above construction1 however1 and the reason why ,i have reproduced it in detail1 is that it proceeds entirely by means of the general principles of transformation enunciated above4 ,from this stage onwards1 everything is effected by means of the two fundamental ideas we have just discussed1 and everything1 therefore1 depends on our general principle of projective equivalence4 ,this principle1 as regards two dimensions1 may be stated more simply than in the passage quoted from ,cremona4 ,it starts1 in two dimensions1 from the following definitions3 ,to project the points ,a1 ,b1 ,c1 ,d 444 from a centre ,o1 is to construct the straight lines ,,oa1 ,,ob1 ,,oc1 ,,od4444 ,to cut a number of straight lines a1 b1 c1 d 444 by a transversal s1 is to construct the points sa1 sb1 sc1 sd4444#abf ,the successive application of these two operations1 provided the original figure consisted of points on one straight line or of straight lines through one point1 gives a figure projectively indistinguishable from the former figure2 and hence1 by extension1 if any points in one straight line in the original figure lie in one straight line in the derived figure1 and reciprocally for straight lines through points1 the two operations have given projectively similar figures4 ,this general principle may be regarded as consisting of two parts1 according to the order of the operations3 if we begin with projection and end with section1 we transform a figure of points into another figure of points2 by the converse order1 we transform a figure of lines into another figure of lines4 #aae4 ,before we can be clear as to the meaning of our principle1 we must have some notion as to our definition of points and straight lines4 ,but this definition1 in projective ,geometry1 cannot be given without some discussion of the principle of duality1 the mathematical form of the philosophical circle involved in geometrical definitions4 ,confining ourselves for the moment to two dimensions1 the principle asserts1 roughly speaking1 that any theorem1 dealing with lines through a point and points on a line1 remains true if these two terms1 wherever they occur1 are interchanged4 ,thus3 two points lie on one straight line which they completely determine2 and two straight lines meet in one point1 which they completely determine4 ,the four points of intersection of a transversal with four lines through a point have an anharmonic ratio independent of the particular transversal2 and the four lines joining four points on one straight line to a fifth point have an anharmonic ratio independent of that fifth point4 ,so also our general principle of projective transformation has two sides3 one in which points move along fixed lines1 and one in which lines turn about fixed points4 ,this duality suggests that any definition of points must be effected by means of the straight line1 and any definition of the straight line must be effected by means of points4 ,when we take the third dimension into account1 it is true1 the duality is no longer so simple2 we have now to take account also of the plane1 but this only introduces a circle of three terms1 which is scarcely preferable to a circle of two terms4 ,we now say3 ,three points1 or a line and a point1 determine a plane3 but conversely1 three planes1 or a line and plane1 determine a point4 ,we may regard the straight line as a relation between two of its points1 but we may also regard the point as a relation between two straight lines through it4 ,we may regard the plane as a relation between three points1 or between a point and a line1 but we may also regard the point as a relation between three planes1 or between a line and a plane1 which meet in it4 #aaf4 ,how are we to get outside this circle8 ,the fact is that1 in pure ,geometry1 we cannot get outside it4 ,for space1 as we shall see more fully hereafter1 is nothing but relations2 if1 therefore1 we take any spatial figure1 and seek for the terms between which it is a relation1 we are compelled1 in ,geometry1 to seek these terms within space1 since we have nowhere else to seek them1 but we are doomed1 since anything purely spatial is a mere relation1 to find our terms melting away as we grasp them4 ,thus the relativity of space1 while it is the essence of the principle of duality1 at the same time renders impossible the expression of that principle1 or of any other principle of pure ,geometry1 in a manner which shall be free from contradictions4 ,nevertheless1 if we are to advance at all with our analysis of geometrical reasoning and with our definitions of lines and points1 we must1 for a while1 ignore this contradiction2 we must argue as though it did not exist1 so as to free our science from any contradictions which are not inevitable4 #aag4 ,in accordance with this procedure1 then1 let us define our points as the terms of spatial relations1 regarding whatever is not a point as a relation between points4 ,what1 on this view1 must our points be taken to be8 ,obviously1 if extension is mere relativity1 they must be taken to contain no extension2 but if they are to supply the terms for spatial relations1 e4g4 for straight lines1 these relations must exhibit them as the terms of the figures they relate4 ,in other words1 since what can really be taken1 without contradiction1 as the term of a spatial relation1 is unextended1 we must take1 as the term to be used in ,geometry1 where we cannot go outside space1 the least spatial thing which ,geometry can deal with1 the thing which1 though in space1 contains no space2 and this thing we define as the point#abg4 ,neglecting1 then1 the fundamental contradiction in this definition1 the rest of our definitions follow without difficulty4 ,the straight line is the relation between two points1 and the plane is the relation between three4 ,these definitions will be argued and defended at length in section ,b of this ,chapter#abh1 where we can discuss at the same time the alternative metrical definitions2 for our present purpose1 it is sufficient to observe that projective ,geometry1 from the first1 regards the straight line as determined by two points1 and the plane as determined by three1 from which it follows1 if we take points as possible terms for spatial relations1 that the straight line and the plane may be regarded as relations between two and three points respectively4 ,if we agree on these definitions1 we can proceed to discuss the fundamental principle of projective ,geometry1 and to analyse the axioms implicated in its truth4 #aah4 ,projective ,geometry1 we have seen1 does not deal with quantity1 and therefore recognizes no difference where the difference is purely quantitative4 ,now quantitative comparison depends on a recognized identity of quality2 the recognition of qualitative identity1 therefore1 is logically prior to quantity1 and presupposed by every judgment of quantity4 ,hence all figures1 whose differences can be exhaustively described by quantity1 i4e4 by pure measurement1 must have an identity of quality1 and this must be recognizable without appeal to quantity4 ,it follows that1 by defining the word quality in geometrical matters1 we shall discover what sets of figures are projectively indiscernible4 ,if our definition is correct1 it ought to yield the general projective principle with which we set out4 #aai4 ,we agreed to regard points as the terms of spatial relations1 and we agreed that different points could be distinguished4 ,but we postponed the discussion of the conditions under which this distinction could be effected4 ,this discussion will yield us the definition of quality and the proof of our general projective principle4 ,points1 to begin with1 have been defined as nothing but the terms for spatial relations4 ,they have1 therefore1 no intrinsic properties2 but are distinguished solely by means of their relations4 ,now the relation between two points1 we said1 is the straight line on which they lie4 ,this gives that identity of quality for all pairs of points on the same straight line1 which is required both by our projective principle and by metrical ,geometry4 7,for only where there is identity of quality can quantity be properly applied47 ,if only two points are given1 they cannot1 without the use of quantity1 be distinguished from any two other points on the same straight line2 for the qualitative relation between any two such points is the same as for the original pair1 and only by a difference of relation can points be distinguished from one another4 ,but conversely1 one straight line is nothing but the relation between two of its points1 and all points are qualitatively alike4 ,hence there can be nothing to distinguish one straight line from another except the points through which it passes1 and these are distinguished from other points only by the fact that it passes through them4 ,thus we get the reciprocal transformation3 if we are given only one point1 any pair of straight lines through that point is qualitatively indistinguishable from any other4 ,this again is1 on the one hand1 the basis of the second part of our general projective principle1 and on the other hand the condition of applying quantity1 in the measurement of angles1 to the departure of two intersecting straight lines4 #abj4 ,we can now see the reason for what may have hitherto seemed a somewhat arbitrary fact1 namely1 the necessity of four collinear points for anharmonic ratio4 ,recurring to the quadrilateral construction and the consequent introduction of number1 we see that anharmonic ratio is an intrinsic projective relation of four collinear points or concurrent straight lines1 such that given three terms and the relation1 the fourth term can be uniquely determined by projective methods4 ,now consider first a pair of points4 ,since all straight lines are projectively equivalent1 the relation between one pair of points is precisely equivalent to that between another pair4 ,given one point only1 therefore1 no projective relation1 to any second point1 can be assigned1 which shall in any way limit our choice of the second point4 ,given two points1 however1 there is such a relation--the third point may be given collinear with the first two4 ,this limits its position to one straight line1 but since two points determine nothing but one straight line1 the third point cannot be further limited4 ,thus we see why no intrinsic projective relation can be found between three points1 which shall enable us1 from two1 uniquely to determine the third4 ,with three given collinear points1 however1 we have more given than a mere straight line1 and the quadrilateral construction enables us uniquely to determine any number of fresh collinear points4 ,this shows why anharmonic ratio must be a relation between four points1 rather than between three4 #aba4 ,we can now prove1 ,i think1 that two figures1 which are projectively related1 are qualitatively similar4 ,let us begin with a collection of points on a straight line4 ,so long as these are considered without reference to other points or figures1 they are all qualitatively similar4 ,they can be distinguished by immediate intuition1 but when we endeavour1 without quantity1 to distinguish them conceptually1 we find the task impossible1 since the only qualitative relation of any two of them1 the straight line1 is the same for any other two4 ,but now let us choose1 at hap-hazard1 some point outside the straight line4 ,the points of our line now acquire new adjectives1 namely their relations to the new point1 i4e4 the straight lines joining them to this new point4 ,but these straight lines1 reciprocally1 alone define our external point1 and all straight lines are qualitatively similar4 ,if we take some other external point1 therefore1 and join it to the same points of our original straight line1 we obtain a figure in which1 so long as quantity is excluded1 there is no conceptual difference from the former figure4 ,immediate intuition can distinguish the two figures1 but qualitative discrimination cannot do so4 ,thus we obtain a projective transformation of four lines into four other lines1 as giving a figure qualitatively indistinguishable from the original figure4 ,a similar argument applies to the other projective transformations4 ,thus the only reason1 within projective ,geometry1 for not regarding projective figures as actually identical1 is the intuitive perception of difference of position4 ,this is fundamental1 and must be accepted as a datum4 ,it is presupposed in the distinction of various points1 and forms the very life of ,geometry4 ,it is1 in fact1 the essence of the notion of a form of externality1 which notion forms the subject-matter of projective ,geometry4 #abb4 ,we may now sum up the results of our analysis of projective ,geometry1 and state the axioms on which its reasoning is based4 ,we shall then have to prove that these axioms are necessary to any form of externality1 with which we shall pass1 from mere analysis1 to a transcendental argument4 ,the axioms which have been assumed in the above analysis1 and which1 it would seem1 suffice to found projective ,geometry1 may be roughly stated as follows3 ,i4 ,we can distinguish different parts of space1 but all parts are qualitatively similar1 and are distinguished only by the immediate fact that they lie outside one another4 ,,ii4 ,space is continuous and infinitely divisible2 the result of infinite division1 the zero of extension1 is called a point#abi4 ,,iii4 ,any two points determine a unique figure1 called a straight line1 any three in general determine a unique figure1 the plane4 ,any four determine a corresponding figure of three dimensions1 and for aught that appears to the contrary1 the same may be true of any number of points4 ,but this process comes to an end1 sooner or later1 with some number of points which determine the whole of space4 ,for if this were not the case1 no number of relations of a point to a collection of given points could ever determine its relation to fresh points1 and ,geometry would become impossible#acj4 ,this statement of the axioms is not intended to have any exclusive precision3 other statements equally valid could easily be made4 ,for all these axioms1 as we shall see hereafter1 are philosophically interdependent1 and may1 therefore1 be enunciated in many ways4 ,the above statement1 however1 includes1 if ,i am not mistaken1 everything essential to projective ,geometry1 and everything required to prove the principle of projective transformation4 ,before discussing the apriority of these axioms1 let us once more briefly recapitulate the ends which they are intended to attain4 #abc4 ,from the exclusively mathematical standpoint1 as we have seen1 projective ,geometry discusses only what figures can be obtained from each other by projective transformations1 i4e4 by the operations of projection and section4 ,these operations1 in all their forms1 presuppose the point1 straight line1 and plane#aca1 whose necessity for projective ,geometry1 from the purely mathematical point of view1 is thus self-evident from the start4 ,but philosophically1 projective ,geometry has1 as we saw1 a wider aim4 ,this wider aim1 which gives1 to the investigation of projectively equivalent figures1 its chief importance1 consists in the determination of qualitative spatial similarity1 in the determination1 that is1 of all the figures which1 when any one figure is given1 can be distinguished from the given figure1 so long as quantity is excluded1 only by the mere fact that they are external to it4 #abd4 ,now when we consider what is involved in such absolute qualitative equivalence1 we find at once1 as its most obvious prerequisite1 the perfect homogeneity of space4 ,for it is assumed that a figure can be completely defined by its internal relations1 and that the external relations1 which constitute its position1 though they suffice to distinguish it from other figures1 in no way affect its internal properties1 which are regarded as qualitatively identical with those of figures with quite different external relations4 ,if this were not the case1 anything analogous to projective transformation would be impossible4 ,for such transformation always alters the position1 i4e4 the external relations1 of a figure1 and could not1 therefore1 if figures were dependent on their relations to other figures or to empty space1 be studied without reference to other figures1 or to the absolute position of the original figure4 ,we require for our principle1 in short1 what may be called the mutual passivity and reciprocal independence of two parts or figures of space4 ,this passivity and this independence involve the homogeneity of space1 or its equivalent1 the relativity of position4 ,for if the internal properties of a figure are the same1 whatever its external relations may be1 it follows that all parts of space are qualitatively similar1 since a change of external relation is a change in the part of space occupied4 ,it follows1 also1 that all position is relative and extrinsic1 i4e41 that the position of a point1 or the part of space occupied by a figure1 is not1 and has no effect upon1 any intrinsic property of the point or figure1 but is exclusively a relation to other points or figures in space1 and remains without effect except where such relations are considered4 #abe4 ,the homogeneity of space and the relativity of position1 therefore1 are presupposed in the qualitative spatial comparison with which projective ,geometry deals4 ,the latter1 as we saw1 is also the basis of the principle of duality4 ,but these properties1 as ,i shall now endeavour to prove1 belong of necessity to any form of externality1 and are thus priori properties of all possible spaces4 ,to prove this1 however1 we must first define the notion of a form of externality in general4 ,let us observe1 to begin with1 that the distinction between ,euclidean and non-,euclidean ,geometries1 so important in metrical investigations1 disappears in projective ,geometry proper4 ,this suggests that projective ,geometry1 though originally invented as the science of ,euclidean space1 and subsequently of non-,euclidean spaces also1 deals really with a wider conception1 a conception which includes both1 and neglects the attributes in which they differ4 ,this conception ,i shall speak of as a form of externality4 #abf4 ,in ,grassmann's profound philosophical introduction to his ,ausdehnungslehre of #ahdd1 he suggested that ,geometry1 though improperly regarded as pure1 was really a branch of applied mathematics1 since it dealt with a subject-matter not created1 like number1 by the intellect1 but given to it1 and therefore not wholly subject to its laws alone4 ,but it must be possible--so he contended--to construct a branch of pure mathematics1 a science1 that is1 in which our object should be wholly a creature of the intellect1 which should yet deal1 as ,geometry does1 with extension--extension as conceived1 however1 not as empirically perceived in sensation or intuition4 ,from this point of view1 the controversy between ,kantians and anti-,kantians becomes wholly irrelevant1 since the distinction between pure and mixed mathematics does not lie in the distinction between the subjective and the objective1 but between the purely intellectual on the one hand1 and everything else on the other4 ,now ,kant had contended1 with great emphasis1 that space was not an intellectual construction1 but a subjective intuition4 ,geometry1 therefore1 with ,grassmann's distinction1 belongs to mixed mathematics as much on ,kant's view as on that of his opponents4 ,and ,grassmann's distinction1 ,i contend1 is the more important for ,epistemology1 and the one to be adopted in distinguishing the priori from the empirical4 ,for what is merely intuitional can change1 without upsetting the laws of thought1 without making knowledge formally impossible3 but what is purely intellectual cannot change1 unless the laws of thought should change1 and all our knowledge simultaneously collapse4 ,i shall therefore follow ,grassmann's distinction in constructing an priori and purely conceptual form of externality4 #abg4 ,the pure doctrine of extension1 as constructed by ,grassmann1 need not be discussed--it included much empirical material1 and was philosophically a failure4 ,but his principles1 ,i think1 will enable us to prove that projective ,geometry1 abstractly interpreted1 is the science which he foresaw1 and deals with a matter which can be constructed by the pure intellect alone4 ,if this be so1 however1 it must be observed that projective ,geometry1 for the moment1 is rendered purely hypothetical#acb4 ,all necessary truth1 as ,bradley has shown1 is hypothetical#acc1 and asserts1 prim facie1 only the ground on which rests the necessary connection of premisses and conclusion4 ,if we construct a mere conception of externality1 and thus abandon our actually given space1 the result of our construction1 until we return to something actually given1 remains without existential import--if there be experienced externality1 it asserts1 then there must be a form of externality with such and such properties4 ,that there must be experienced externality1 ,kant's first argument about space proves1 ,i think1 to those who admit experience of a world of diverse but interrelated things4 ,but this is a question which belongs to the next ,chapter4 ,what we have to do here is1 not to discuss whether there is a form of externality1 but whether1 if there be such a form1 it must possess the properties embodied in the axioms of projective ,geometry4 ,now first of all1 what do we mean by such a form8 #abh4 ,in any world in which perception presents us with various things1 with discriminated and differentiated contents1 there must be1 in perception1 at least one principle of differentiation#acd1 an element1 that is1 by which the things presented are distinguished as various4 ,this element1 taken in isolation1 and abstracted from the content which it differentiates1 we may call a form of externality4 ,that it must1 when taken in isolation1 appear as a form1 and not as a mere diversity of material content1 is1 ,i think1 fairly obvious4 ,for a diversity of material content cannot be studied apart from that material content2 what we wish to study here1 on the contrary1 is the bare possibility of such diversity1 which forms the residuum1 as ,i shall try to prove hereafter#ace1 when we abstract from any sense-perception all that is distinctive of its particular matter4 ,this possibility1 then1 this principle of bare diversity1 is our form of externality4 ,how far it is necessary to assume such a form1 as distinct from interrelated things1 ,i shall consider later on#acf4 ,for the present1 since space1 as dealt with by ,geometry1 is certainly a form of this kind1 we have only to ask3 ,what properties must such a form1 when studied in abstraction1 necessarily possess8 #abi4 ,in the first place1 externality is an essentially relative conception--nothing can be external to itself4 ,to be external to something is to be another with some relation to that thing4 ,hence1 when we abstract a form of externality from all material content1 and study it in isolation1 position will appear1 of necessity1 as purely relative--a position can have no intrinsic quality1 for our form consists of pure externality1 and externality contains no shadow or trace of an intrinsic quality4 ,thus we obtain our fundamental postulate1 the relativity of position1 or1 as we may put it1 the complete absence1 on the part of our form1 of any vestige of thinghood4 ,the same argument may also be stated as follows3 ,if we abstract the conception of externality1 and endeavour to deal with it per se1 it is evident that we must obtain an object alike destitute of elements and of totality4 ,for we have abstracted from the diverse matter which filled our form1 while any element1 or any whole1 would retain some of the qualities of a matter4 ,either an element or a whole1 in fact1 would have to be a thing not external to itself1 and would thus contain something not pure externality4 ,hence arise infinite divisibility1 with the self-contradictory notion of the point1 in the search for elements1 and unbounded extension1 with the contradiction of an infinite regress or a vicious circle1 in the search for a completed whole4 ,thus again1 our form contains neither elements nor totality1 but only endless relations--the terms of these relations being excluded by our abstraction from the matter which fills our form4 #acj4 ,in like manner we can deduce the homogeneity of our form4 ,the diversity of content1 which was possible only within the form of externality1 has been abstracted from1 leaving nothing but the bare possibility of diversity1 the bare principle of differentiation1 itself uniform and undifferentiated4 ,for if diversity presupposes such a form1 the form cannot1 unless it were contained in a fresh form1 be itself diverse or differentiated4 ,or we may deduce the same property from the relativity of position4 ,for any quality in one position1 by which it was marked out from another1 would be necessarily more or less intrinsic1 and would contradict the pure relativity4 ,hence all positions are qualitatively alike1 i4e4 the form is homogeneous throughout4 #aca4 ,from what has been said of homogeneity and relativity1 follows one of the strangest properties of a form of externality4 ,this property is1 that the relation of externality between any two things is infinitely divisible1 and may be regarded1 consequently1 as made up of an infinite number of the would-be elements of our form1 or again as the sum of two relations of externality#acg4 ,to speak of dividing or adding relations may well sound absurd--indeed it reveals the impropriety of the word relation in this connexion4 ,it is difficult1 however1 to find an expression which shall be less improper4 ,the fact seems to be1 that externality is not so much a relation as bare relativity1 or the bare possibility of a relation4 ,on this subject1 ,i shall enlarge in ,chapter ,,iv4#ach ,at this point it is only important to realize1 what the subsequent argument will assume1 that the relation--if we may so call it--of externality between two or more things must1 since our form is homogeneous1 be capable of continuous alteration1 and must1 since our infinitely divisible form is constituted by such relations1 be capable of infinite division4 ,but the result of infinite division is defined as the element of our form4 7,our form has no elements1 but we have to imagine elements in order to reason about it1 as will be shown more fully in ,chapter ,,iv47 ,hence it follows1 that every relation of externality may be regarded1 for scientific purposes1 as an infinite congeries of elements1 though philosophically1 the relations alone are valid1 and the elements are a self-contradictory result of hypostatizing the form of externality4 ,this way of regarding relations of externality is important in understanding the meaning of such ideas as three or four collinear points4 ,as this point is difficult and important1 ,i will repeat1 in somewhat greater detail1 the explanation of the manner in which straight lines and planes come to be regarded as congeries of points4 ,from the strictly projective standpoint1 though all other figures are merely a collection of any required number of points1 lines or planes1 given by some projective construction1 straight lines and planes themselves are given integrally1 and are not to be considered as divisible or composed of parts4 ,to say that a point lies on a straight line means1 for projective ,geometry proper1 that the straight line is a relation between this and some other point4 ,here the points concerned1 if our statement is to be freed from contradictions1 must be regarded1 if ,i may use such an expression1 as real points--i4e4 as unextended material centres#aci4 ,straight lines and planes are then relations between these material atoms4 ,they are relations1 however1 which may undergo a metrical alteration while remaining projectively unchanged4 ,when the projective relation between the two points ,a1 ,b is the same as that between the two points ,a1 ,c1 while the metrical relation 7distance7 is different1 the three points ,a1 ,b1 ,c are said to be collinear4 ,now the metrical manner of regarding spatial figures demands that they should be hypostatized1 and no longer regarded as mere relations4 ,for when we regard a quantity as extensive1 i4e4 as divisible into parts1 we necessarily regard it as more than a mere relation or adjective1 since no mere relation or adjective can be divided4 ,for quantitative treatment1 therefore1 spatial relations must be hypostatized#adj4 ,when this is done1 we obtain1 as we saw above1 a homogeneous and infinitely divisible form of externality4 ,we find now that distance1 for example1 may be continuously altered without changing the straight line on which it is measured4 ,we thus obtain1 on the straight line in question1 a continuous series of points1 which1 since it is continuous1 we regard as constituting our straight line4 ,it is thus solely from the hypostatizing of relations1 which metrical ,geometry requires1 that the view of straight lines and planes as composed of points arises1 and it is from this hypostatizing that the difficulties of metrical ,geometry spring4 #acb4 ,the next step1 in defining a form of externality1 is obtained from the idea of dimensions4 ,positions1 we have seen1 are defined solely by their relations to other positions4 ,but in order that such definition may be possible1 a finite number of relations must suffice1 since infinite numbers are philosophically inadmissible4 ,a position must be definable1 therefore1 if knowledge of our form is to be possible at all1 by some finite integral number of relations to other positions4 ,every relation thus necessary for definition we call a dimension4 ,hence we obtain the proposition3 ,any form of externality must have a finite integral number of dimensions4 #acc4 ,the above argument1 it may be urged1 has overlooked a possibility4 ,it has used a transcendental argument1 so an opponent may contend1 without sufficiently proving that knowledge about externality must be possible without reference to the matters external to each other4 ,the definition of a position may be impossible1 so long as we neglect the matter which fills the form1 but may become possible when this matter is taken into account4 ,such an objection can1 ,i think1 be successfully met1 by a reference to the passivity and homogeneity of our form4 ,for any dependence of the definition of a position on the particular matter filling that position1 would involve some kind of interaction between the matter and its position1 some effect of the diverse content on the homogeneous form4 ,but since the form is totally destitute of thinghood1 perfectly impassive1 and perfectly void of differences between its parts1 any such effect is inconceivable4 ,an effect on a position would have to alter it in some way1 but how could it be altered8 ,it has no qualities except those which make it the position it is1 as opposed to other positions2 it cannot change1 therefore1 without becoming a different position4 ,but such a change contradicts the law of identity4 ,hence it is not the position which has changed1 but the content which has moved in the form4 ,thus it must be possible1 if knowledge of our form can be obtained at all1 to obtain this knowledge in logical independence of the particular matter which fills it4 ,the above argument1 therefore1 granted the possibility of knowledge in the department in question1 shows the necessity of a finite integral number of dimensions4 #acd4 ,let us repeat our original argument in the light of this elucidation4 ,a position is completely defined when1 and only when1 enough relations are known to enable us to determine its relation to any fresh known position4 ,only by relations within the form of externality1 as we have just seen1 and never by relations which involve a reference to the particular matter filling the form1 can such a definition be effected4 ,but the possibility of such a definition follows from the ,law of ,excluded ,middle1 when this law is interpreted to mean1 as ,bosanquet makes it mean1 that ,reality 444 is a system of reciprocally determinate parts#ada4 ,for this implies that1 given the relations of a part ,a to other parts ,b1 ,c 4441 a sufficient wealth of such relations throws light on the relations of ,b to ,c1 etc4 ,if this were not the case1 the parts ,a1 ,b1 ,c 444 could not be said to form such a system2 for in such a system1 to define ,a is to define1 at the same time1 all the other members1 and to give an adjective to ,a1 is to give an adjective to ,b and ,c4 ,but the relations between positions are1 when we restore the matter from which the positions were abstracted1 relations between the things occupying those positions1 and these relations1 we have seen1 can be studied without reference to the particular nature1 in other respects1 of the related things4 ,it follows that1 when we apply the general principle of systematic unity to these relations in particular1 we find these relations to be dependent on each other1 since they are not dependent1 for their definition1 on anything else4 ,this gives the axiom of dimensions1 in the above general form1 as the result1 on our abstract geometrical level1 of the relativity of position and the law of excluded middle4 #ace4 ,before proceeding further1 it is necessary to discuss the important special case where a form of externality has only one dimension4 ,of the two such forms1 given in experience1 one1 namely time1 presents an instance of this special case4 ,but it may be shown1 ,i think1 that the function1 in constituting the possibility of experience1 which we demand of such forms1 could not be accomplished by a one-dimensional form alone4 ,for in a one-dimensional form1 the various contents may be arranged in a series1 and cannot1 without interpenetration1 change the order of contents in the series4 ,but interpenetration is impossible1 since a form of externality is the mere expression of diversity among things1 from which it follows that things cannot occupy the same position in a form1 unless there is another form by which to differentiate them4 ,for without externality1 there is no diversity#adb4 ,thus two bodies may occupy the same space1 but only at different times3 two things may exist simultaneously1 but only at different places4 ,a form of one dimension1 therefore1 could not1 by itself1 allow that change of the relations of externality1 by which alone a varied world of interrelated things can be brought into consciousness4 ,in a one-dimensional space1 for example1 only a single object1 which must appear as a point1 or two objects at most1 one in front and one behind1 could ever be perceived4 ,thus two or more dimensions seem an essential condition of anything worth calling an experience of interrelated things4 #acf4 ,it may be objected1 to this argument1 that its validity depends upon the assumption that the change of a relation of externality must be continuous4 ,both to make and to meet this objection1 in a manner which shall not imply time1 seems almost impossible4 ,for we cannot speak of change1 whether continuous or discrete1 without imagining time4 ,let us1 therefore1 allow time to be known1 and discuss whether the temporal change1 in any other form of externality1 is necessarily continuous#adc4 ,we must reply1 ,i think1 that continuity is necessary4 ,the change of relation1 in our non-temporal form1 may be safely described as motion1 and the law of ,causality--since we have already assumed time--may be applied to this motion4 ,it then follows that discrete motion would involve a finite effect from an infinitesimal cause1 for a cause acting only for a moment of time would be infinitesimal4 ,it involves1 also1 a validity in the point of time1 whereas what is valid in any form of externality is not1 as we have already seen1 the infinitesimal and self-contradictory element resulting from infinite division1 but the finite relation which mathematics analyzes into vanishing elements4 ,hence change must be continuous1 and the possibility of serial arrangement holds good4 ,in a one-dimensional form other than time1 the same argument must hold4 ,for something analogous to ,causality would be necessary to experience1 and the relativity of the form would still necessarily hold4 ,hence1 since only these two properties of time have been assumed1 the above contention would remain valid of any second form whose relations were correlated with those of the first1 as the analogue of ,causality would require them to be4 #acg4 ,the next step in the argument1 which assumes two or more dimensions1 is concerned with the general analogues of straight lines and planes1 i4e4 with figures--which may be regarded either as relations between positions or as series of positions--uniquely determined by two or by three positions4 ,if this step can be successfully taken1 our deduction of the above projective axioms will be complete1 and descriptive ,geometry will be established as the abstract priori doctrine of forms of externality4 ,to prove this contention1 consider of what nature the relations can be by which positions are defined4 ,we have seen already that our form is purely relational and infinitely divisible1 and that positions 7points7 are the self-contradictory outcome of the search for something other than relations4 ,what we really mean1 therefore1 by the relations defining a position1 is1 when we undo our previous abstraction1 the relations of externality by which some thing is related to other things4 ,but how1 when we remain in the abstract form1 must such relations appear8 #ach4 ,we have to prove that two positions must have a relation independent of any reference to other positions4 ,to prove this1 let us recur to what was said1 in connection with dimensions1 as to the passivity and homogeneity of our form4 ,since positions are defined only by relations1 there must be relations1 within the form1 between positions4 ,but if there are such relations1 there must be a relation which is intrinsic to two positions4 ,for to suppose the contrary1 is to attribute an interaction or causal connection1 of some kind1 between those two positions and other positions--a supposition which the perfect homogeneity of our form renders absurd1 since all positions are qualitatively similar1 and cannot be changed without losing their identity4 ,we may put this argument thus3 since positions are only defined by their relations1 such definition could never begin1 unless it began with a relation between only two positions4 ,for suppose three positions ,a1 ,b1 ,c were necessary1 and gave rise to the relation abc between the three4 ,then there would remain no means of defining the different pairs ,,bc1 ,,ca1 ,,ab1 since the only relation defining them would be one common to all three pairs4 ,nothing would be gained1 in this case1 by reference to fresh points1 for it follows1 from the homogeneity and passivity of the form1 that these fresh points could not affect the internal relations of our triad1 which relations1 if they can give definiteness at all1 must give it without the aid of external reference4 ,two positions must1 therefore1 if definition is to be possible1 have some relation which they by themselves suffice to define4 ,precisely the same argument applies to three positions1 or to four2 the argument loses its scope only when we have exhausted the dimensions of the form considered4 ,thus1 in three dimensions1 five positions have no fresh relation1 not deducible from those already known1 for by the definition of dimensions1 all the relations involved can be deduced from those of the fourth point to the first three1 together with those of the fifth to the first three4 ,we may give the argument a more concrete1 and perhaps a more convincing shape1 by considering the matter arranged in our form4 ,if two things are mutually external1 they must since they belong to the same world1 have some relation of externality2 there is1 therefore1 a relation of externality between two things4 ,but since our form is homogeneous1 the same relation of externality may subsist in other parts of the form1 i4e4 while the two things considered alter their relations of externality to other things4 ,the relation of externality between two things is1 therefore1 independent of other things4 ,hence1 when we return to the abstract language of the form1 two positions have a relation determined by those two positions alone1 and independent of other positions4 ,precisely the same argument applies to the relations of three positions1 and in each case the relation must appear in the form as not a mere inference from the positions it relates4 ,for relations1 as we have seen1 actually constitute a form of externality1 and are not mere inferences from terms1 which are nowhere to be found in the form#add4 ,to sum up3 ,since position is relative1 two positions must have some relation to each other2 and since our form of externality is homogeneous1 this relation can be kept unchanged while the two positions change their relations to other positions4 ,hence their relation is intrinsic1 and independent of other positions4 ,since the form is a mere complex of relations1 the relation in question must1 if the form is sensuous or intuitive1 be itself sensuous or intuitive1 and not a mere inference4 ,in this case1 a unique relation must be a unique figure--in spatial terms1 the straight line joining the two points4 #aci4 ,with this1 our deduction of projective ,geometry from the priori conceptual properties of a form of externality is completed4 ,that such a form1 when regarded as an independent thing1 is self-contradictory1 has been abundantly evident throughout the discussion4 ,but the science of the form has been founded on the opposite way of regarding it3 we have held it throughout to be a mere complex of relations1 and have deduced its properties exclusively from this view of it4 ,the many difficulties1 in applying such an priori deduction to intuitive space1 and in explaining1 as logical necessities1 properties which appear as sensuous or intuitional data1 must be postponed to ,chapter ,,iv4 ,for the present1 ,i wish to point out that projective ,geometry is wholly priori2 that it deals with an object whose properties are logically deduced from its definition1 not empirically discovered from data2 that its definition1 again1 is founded on the possibility of experiencing diversity in relation1 or multiplicity in unity2 and that our whole science1 therefore1 is logically implied in1 and deducible from1 the possibility of such experience4 #adj4 ,in metrical ,geometry1 on the contrary1 we shall find a very different result4 ,although the geometrical conditions which render spatial measurement possible1 will be found identical1 except for slight differences in the form of statement1 with the priori axioms discussed above1 yet the actual measurement--which deals with actually given space1 not the mere intellectual construction we have been just discussing--gives results which can only be known empirically and approximately1 and can be deduced by no necessity of thought4 ,the ,euclidean and non-,euclidean spaces give the various results which are priori possible2 the axioms peculiar to ,euclid--which are properly not axioms1 but empirical results of measurement--determine1 within the errors of observation1 which of these priori possibilities is realized in our actual space4 ,thus measurement deals throughout with an empirically given matter1 not with a creature of the intellect1 and its priori elements are only the conditions presupposed in the possibility of measurement4 ,what these conditions are1 we shall see in the second section of this chapter4 ,section ,b4 ,,the ,,axioms ,,of ,,metrical ,,geometry4 #ada4 ,we have now reviewed the axioms of projective ,geometry1 and have seen that they are priori deductions from the fact that we can experience externality1 i4e4 a coexistent multiplicity of different but interrelated things4 ,but projective ,geometry1 in spite of its claims1 is not the whole science of space1 as is sufficiently proved by the fact that it cannot discriminate between ,euclidean and non-,euclidean spaces#ade4 ,for this purpose1 spatial measurement is required3 metrical ,geometry1 with its quantitative tests1 can alone effect the discrimination4 ,for all application of ,geometry to physics1 also1 measurement is required2 the law of gravitation1 for example1 requires the determination of actual distances4 ,for many purposes1 in short1 projective ,geometry is wholly insufficient3 thus it is unable to distinguish between different kinds of conics1 though their distinction is of fundamental importance in many departments of knowledge4 ,metrical ,geometry is1 then1 a necessary part of the science of space1 and a part not included in descriptive ,geometry4 ,its priori element1 nevertheless1 so far as this is spatial and not arithmetical1 is the same as the postulate of projective ,geometry1 namely1 the homogeneity of space1 or its equivalent1 the relativity of position4 ,we can see1 in fact1 that the priori element in both is likely to be the same4 ,for the priori in metrical ,geometry will be whatever is presupposed in the possibility of spatial measurement1 i4e4 of quantitative spatial comparison4 ,but such comparison presupposes simply a known identity of quality1 the determination of which is precisely the problem of projective ,geometry4 ,hence the conditions for the possibility of measurement1 in so far as they are not arithmetical1 will be precisely the same as those for projective ,geometry4 #adb4 ,metrical ,geometry1 therefore1 though distinct from projective ,geometry1 is not independent of it1 but presupposes it1 and arises from its combination with the extraneous idea of quantity4 ,nevertheless the mathematical form of the axioms1 in metrical ,geometry1 is slightly different from their form in projective ,geometry4 ,the homogeneity of space is replaced by its equivalent1 the axiom of ,free ,mobility4 ,the axiom of the straight line is replaced by the axiom of distance3 ,two points determine a unique quantity1 distance1 which is unaltered in any motion of the two points as a single figure4 ,this axiom1 indeed1 will be found to involve the axiom of the straight line--such a quantity could not exist unless the two points determined a unique curve--but its mathematical form is changed4 ,another important change is the collapse of the principle of duality3 quantity can be applied to the straight line1 because it is divisible into similar parts1 but cannot be applied to the indivisible point4 ,we thus obtain a reason1 which was wanting in descriptive ,geometry1 for preferring points1 as spatial elements1 to straight lines or planes#adf4 ,finally1 an entirely new idea is introduced with quantity1 namely1 the idea of ,motion4 ,not that we study motion1 or that any of our results have reference to motion1 but that they cannot1 though in projective ,geometry they could1 be obtained without at least an ideal motion of our figures through space4 ,let us now examine in detail the prerequisites of spatial measurement4 ,we shall find three axioms1 without which such measurement would be impossible1 but with which it is adequate to decide1 empirically and approximately1 the ,euclidean or non-,euclidean nature of our actual space4 ,we shall find1 further1 that these three axioms can be deduced from the conception of a form of externality1 and owe nothing to the evidence of intuition4 ,they are1 therefore1 like their equivalents the axioms of projective ,geometry1 priori1 and deducible from the conditions of spatial experience4 ,this experience1 accordingly1 can never disprove them1 since its very existence presupposes them4 ,i4 ,the ,axiom of ,free ,mobility4 #adc4 ,metrical ,geometry1 to begin with1 may be defined as the science which deals with the comparison and relations of spatial magnitudes4 ,the conception of magnitude1 therefore1 is necessary from the start4 ,some of ,euclid's axioms1 accordingly1 have been classed as arithmetical1 and have been supposed to have nothing particular to do with space4 ,such are the axioms that equals added to or subtracted from equals give equals1 and that things which are equal to the same thing are equal to one another4 ,these axioms1 it is said1 are purely arithmetical1 and do not1 like the others1 ascribe an adjective to space4 ,as regards their use in arithmetic1 this is of course true4 ,but if an arithmetical axiom is to be applied to spatial magnitudes1 it must have some spatial import#adg1 and thus even this class is not1 in ,geometry1 merely arithmetical4 ,fortunately1 the geometrical element is the same in all the axioms of this class--we can see at once1 in fact1 that it can amount to no more than a definition of spatial magnitude#adh4 ,again1 since the space with which ,geometry deals is infinitely divisible1 a definition of spatial magnitude reduces itself to a definition of spatial equality1 for1 as soon as we have this last1 we can compare two spatial magnitudes by dividing each into a number of equal units1 and counting the number of such units in each#adi4 ,the ratio of the number of units is1 of course1 the ratio of the two magnitudes4 #add4 ,we require1 then1 at the very outset1 some criterion of spatial equality3 without such a criterion metrical ,geometry would become wholly impossible4 ,it might appear1 at first sight1 as though this need not be an axiom1 but might be a mere definition4 ,in part this is true1 but not wholly4 ,the part which is merely a definition is given in ,euclid's eighth axiom3 ,magnitudes which exactly coincide are equal4 ,but this gives a sufficient criterion only when the magnitudes to be compared already occupy the same position4 ,when1 as will normally be the case1 the two spatial magnitudes are external to one another--as1 indeed1 must be the case1 if they are distinct1 and not whole and part--the two magnitudes can only be made to coincide by a motion of one or both of them4 ,in order1 therefore1 that our definition of spatial magnitude may give unambiguous results1 coincidence when superposed1 if it can ever occur1 must occur always1 whatever path be pursued in bringing it about4 ,hence1 if mere motion could alter shapes1 our criterion of equality would break down4 ,it follows that the application of the conception of magnitude to figures in space involves the following axiom#aej3 ,spatial magnitudes can be moved from place to place without distortion2 or1 as it may be put1 ,shapes do not in any way depend upon absolute position in space4 ,the above axiom is the axiom of ,free ,mobility#aea4 ,i propose to prove 7#a7 that the denial of this axiom would involve logical and philosophical absurdities1 so that it must be classed as wholly priori2 7#b7 that metrical ,geometry1 if it refused this axiom1 would be unable1 without a logical absurdity1 to establish the notion of spatial magnitude at all4 ,the conclusion will be1 that the axiom cannot be proved or disproved by experience1 but is an priori condition of metrical ,geometry4 ,as ,i shall thus be maintaining a position which has been much controverted1 especially by ,helmholtz and ,erdmann1 ,i shall have to enter into the arguments at some length4 #ade4 ,a4 ,philosophical ,argument4 ,the denial of the axiom involves absolute position1 and an action of mere space1 per se1 on things4 ,for the axiom does not assert that real bodies1 as a matter of empirical fact1 never change their shape in any way during their passage from place to place3 on the contrary1 we know that such changes do occur1 sometimes in a very noticeable degree1 and always to some extent4 ,but such changes are attributed1 not to the change of place as such1 but to physical causes3 changes of temperature1 pressure1 etc4 ,what our axiom has to deal with is not actual material bodies1 but geometrical figures#aeb1 and it asserts that a figure which is possible in any one position in space is possible in every other4 ,its meaning will become clearer by reference to a case where it does not hold1 say the space formed by the surface of an egg4 ,here1 a triangle drawn near the equator cannot be moved without distortion to the point1 as it would no longer fit the greater curvature of the new position3 a triangle drawn near the point cannot be fitted on to the flatter end1 and so on4 ,thus the method of superposition1 such as ,euclid employs in ,book ,i4 ,prop4 ,,iv41 becomes impossible2 figures cannot be freely moved about1 indeed1 given any figure1 we can determine a certain series of possible positions for it on the egg1 outside which it becomes impossible4 ,what ,i assert is1 then1 that there is a philosophic absurdity in supposing space in general to be of this nature4 ,on the egg we have marked points1 such as the two ends2 the space formed by its surface is not homogeneous1 and if things are moved about in it1 it must of itself exercise a distorting effect upon them1 quite independently of physical causes2 if it did not exercise such an effect1 the things could not be moved4 ,thus such a space would not be homogeneous1 but would have marked points1 by reference to which bodies would have absolute position1 quite independently of any other bodies4 ,space would no longer be passive1 but would exercise a definite effect upon things1 and we should have to accommodate ourselves to the notion of marked points in empty space2 these points being marked1 not by the bodies which occupied them1 but by their effects on any bodies which might from time to time occupy them4 ,this want of homogeneity and passivity is1 however1 absurd2 space must1 since it is a form of externality1 allow only of relative1 not of absolute1 position1 and must be completely homogeneous throughout4 ,to suppose it otherwise1 is to give it a thinghood which no form of externality can possibly possess4 ,we must1 then1 on purely philosophical grounds1 admit that a geometrical figure which is possible anywhere is possible everywhere1 which is the axiom of ,free ,mobility4 #adf4 ,b4 ,geometrical ,argument4 ,let us see next what sort of ,geometry we could construct without this axiom4 ,the ultimate standard of comparison of spatial magnitudes must1 as we saw in introducing the axiom1 be equality when superposed2 but need we1 from this equality1 infer equality when separated8 ,it has been urged by ,erdmann that1 for the more immediate purposes of ,geometry1 this would be unnecessary#aec4 ,we might construct a new ,geometry1 he thinks1 in which sizes varied with motion on any definite law4 ,such a view1 as ,i shall show below1 involves a logical error as to the nature of magnitude4 ,but before pointing this out1 let us discuss the geometrical consequences of assuming its truth4 ,suppose the length of an infinitesimal arc in some standard position were ds2 then in any other position p its length would be ds4f7p71 where the form of the function f7p7 must be supposed known4 ,but how are we to determine the position p8 ,for this purpose1 we require p's coordinates1 i4e41 some measurement of distance from the origin4 ,but the distance from the origin could only be measured if we assumed our law f7p7 to measure it by4 ,for suppose the origin to be ,o1 and ,op to be a straight line whose length is required4 ,if we have a measuring rod with which we travel along the line and measure successive infinitesimal arcs1 the measuring rod will change its size as we move1 so that an arc which appears by the measure to be ds will really be f7s74ds1 where s is the previously traversed distance4 ,if1 on the other hand1 we move our line ,op slowly through the origin1 and measure each piece as it passes through1 our measure1 it is true1 will not alter1 but now we have no means of discovering the law by which any element has changed its length in coming to the origin4 ,hence1 until we assume our function f7p71 we have no means of determining p1 for we have just seen that distances from the origin can only be estimated by means of the law f7p74 ,it follows that experience can neither prove nor disprove the constancy of shapes throughout motion1 since1 if shapes were not constant1 we should have to assume a law of their variation before measurement became possible1 and therefore measurement could not itself reveal that variation to us#aed4 ,nevertheless1 such an arbitrarily assumed law does1 at first sight1 give a mathematically possible ,geometry4 ,the fundamental proposition1 that two magnitudes which can be superposed in any one position can be superposed in any other1 still holds4 ,for two infinitesimal arcs1 whose lengths in the standard position are ds#a and ds#b1 would1 in any other position p1 have lengths f7p74ds#a and f7p74ds#b1 so that their ratio would be unaltered4 ,from this constancy of ratio1 as we know through ,riemann and ,helmholtz1 the above proposition follows4 ,hence all that ,geometry requires1 it would seem1 as a basis for measurement1 is an axiom that the alteration of shapes during motion follows a definite known law1 such as that assumed above4 #adg4 ,there is1 however1 in such a view1 as ,i remarked above1 a logical error as to the nature of magnitude4 ,this error has been already pointed out in dealing with ,erdmann#aee1 and need only be briefly repeated here4 ,a judgment of magnitude is essentially a judgment of comparison3 in unmeasured quantity1 comparison as to the mere more or less1 but in measured magnitude1 comparison as to the precise how many times4 ,to speak of differences of magnitude1 therefore1 in a case where comparison cannot reveal them1 is logically absurd4 ,now in the case contemplated above1 two magnitudes1 which appear equal in one position1 appear equal also when compared in another position4 ,there is no sense1 therefore1 in supposing the two magnitudes unequal when separated1 nor in supposing1 consequently1 that they have changed their magnitudes in motion4 ,this senselessness of our hypothesis is the logical ground of the mathematical indeterminateness as to the law of variation4 ,since1 then1 there is no means of comparing two spatial figures1 as regards magnitude1 except superposition1 the only logically possible axiom1 if spatial magnitude is to be self-consistent1 is the axiom of ,free ,mobility in the form first given above4 #adh4 ,although this axiom is priori1 its application to the measurement of actual bodies1 as we found in discussing ,helmholtz's views1 always involves an empirical element#aef4 ,our axiom1 then1 only supplies the priori condition for carrying out an operation which1 in the concrete1 is empirical--just as arithmetic supplies the priori condition for a census4 ,as this topic has been discussed at length in ,chapter ,,ii41 ,i shall say no more about it here4 #adi4 ,there remain1 however1 a few objections and difficulties to be discussed4 ,first1 how do we obtain equality in solids1 and in ,kant's cases of right and left hands1 or of right and left-handed screws1 where actual superposition is impossible8 ,secondly1 how can we take congruence as the only possible basis of spatial measurement1 when we have before us the case of time1 where no such thing as congruence is conceivable8 ,thirdly1 it might be urged that we can immediately estimate spatial equality by the eye1 with more or less accuracy1 and thus have a measure independent of congruence4 ,fourthly1 how is metrical ,geometry possible on non-congruent surfaces1 if congruence be the basis of spatial measurement8 ,i will discuss these objections successively4 #aej4 7#a7 ,how do we measure the equality of solids8 ,these could only be brought into actual congruence if we had a fourth dimension to operate in#aeg1 and from what ,i have said before of the absolute necessity of this test1 it might seem as though we should be left here in utter ignorance4 ,euclid is silent on the subject1 and in all works on ,geometry it is assumed as self-evident that two cubes of equal side are equal4 ,this assumption suggests that we are not so badly off as we should have been without congruence1 as a test of equality in one or two dimensions2 for now we can at least be sure that two cubes have all their sides and all their faces equal4 ,two such cubes differ1 then1 in no sensible spatial quality save position1 for volume1 in this case at any rate1 is not a sensible quality4 ,they are1 therefore1 as far as such qualities are concerned1 indiscernible4 ,if their places were interchanged1 we might know the change by their colour1 or by some other non-geometrical property2 but so far as any property of which ,geometry can take cognisance is concerned1 everything would seem as before4 ,to suppose a difference of volume1 then1 would be to ascribe an effect to mere position1 which we saw to be inadmissible while discussing ,free ,mobility4 ,except as regards position1 they are geometrically indiscernible1 and we may call to our aid the ,identity of ,indiscernibles to establish their agreement in the one remaining geometrical property of volume4 ,this may seem rather a strange principle to use in ,mathematics1 and for ,geometry their equality is1 perhaps1 best regarded as a definition2 but if we demand a philosophical ground for this definition1 it is1 ,i believe1 only to be found in the ,identity of ,indiscernibles4 ,we can1 without error1 make our definition of three-dimensional equality rest on two-dimensional congruence4 ,for since direct comparison as to volume is impossible1 we are at liberty to define two volumes as equal1 when all their various lines1 surfaces1 angles and solid angles are congruent1 since there remains1 in such a case1 no measurable difference between the figures composing the two volumes4 ,of course1 as soon as we have established this one case of equality of volumes1 the rest of the theory follows2 as appears from the ordinary method of integrating volumes1 by dividing them into small cubes4 ,thus congruence helps to establish three-dimensional equality1 though it cannot directly prove such equality2 and the same philosophical principle1 of the homogeneity of space1 by which congruence was proved1 comes to our rescue here4 ,but how about right-handed and left-handed screws8 ,here we can no longer apply the ,identity of ,indiscernibles1 for the two are very well discernible4 ,but as with solids1 so here1 ,free ,mobility can help us much4 ,it can enable us1 by ordinary measurement1 to show that the internal relations of both screws are the same1 and that the difference lies only in their relation to other things in space4 ,knowing these internal relations1 we can calculate1 by the ,geometry which ,free ,mobility has rendered possible1 all the geometrical properties of these screws--radius1 pitch1 etc4--and can show them to be severally equal in both4 ,but this is all we require4 ,mediate comparison is possible1 though immediate comparison is not4 ,both can1 for instance1 be compared with the cylinder on which both would fit1 and thus their equality can be proved4 ,a precisely similar proof holds1 of course1 for the other cases1 right and left hands1 spherical triangles1 etc4 ,on the whole1 these cases confirm my argument2 for they show1 as ,kant intended them to show#aeh1 the essential relativity of space4 #aea4 7#b7 ,as regards time1 no congruence is here conceivable1 for to effect congruence requires always--as we saw in the case of solids--one more dimension than belongs to the magnitudes compared4 ,no day can be brought into temporal coincidence with any other day1 to show that the two exactly cover each other2 we are therefore reduced to the arbitrary assumption that some motion or set of motions1 given us in experience1 is uniform4 ,fortunately1 we have a large set of motions which all roughly agree2 the swing of the pendulum1 the rotation and revolution of the earth and the planets1 etc4 ,these do not exactly agree1 but they lead us to the laws of motion1 by which we are able1 on our arbitrary hypothesis1 to estimate their small departures from uniformity2 just as the assumption of ,free ,mobility enabled us to measure the departures of actual bodies from rigidity4 ,but here1 as there1 another possibility is mathematically open to us1 and can only be excluded by its philosophic absurdity2 we might have assumed that the above set of approximately agreeing motions all had velocities which varied approximately as some arbitrarily assumed function of the time1 f7t7 say1 measured from some arbitrary origin4 ,such an assumption would still keep them as nearly synchronous as before1 and would give an equally possible1 though more complex1 system of ,mechanics2 instead of the first law of motion1 we should have the following3 ,a particle perseveres in its state of rest1 or of rectilinear motion with velocity varying as f7t71 except in so far as it is compelled to alter that state by the action of external forces4 ,such a hypothesis is mathematically possible1 but1 like the similar one for space1 it is excluded logically by the comparative nature of the judgment of quantity1 and philosophically by the fact that it involves absolute time1 as a determining agent in change1 whereas time can never1 philosophically1 be anything but a passive form1 abstracted from change4 ,i have introduced this parallel from time1 not as directly bearing on the argument1 but as a simpler case which may serve to illustrate my reasoning in the more complex case of space4 ,for since time1 in mathematics1 is one-dimensional1 the mathematical difficulties are simpler than in ,geometry2 and although nothing accurately corresponds to congruence1 there is a very similar mixture of mathematical and philosophical necessity1 giving1 finally1 a thoroughly definite axiom as the basis of time-measurement1 corresponding to congruence as the basis of space-measurement#aei4 #aeb4 7#c7 ,the case of time-measurement suggests the third of the above objections to the absolute necessity of the axiom of ,free ,mobility4 ,psycho-physics has shown that we have an approximate power1 by means of what may be called the sense of duration1 of immediately estimating equal short times4 ,this establishes a rough measure independent of any assumed uniform motion1 and in space also1 it may be said1 we have a similar power of immediate comparison4 ,we can see1 by immediate inspection1 that the sub-divisions on a foot rule are not grossly inaccurate2 and so1 it may be said1 we both have a measure independent of congruence1 and also could discover1 by experience1 any gross departure from ,free ,mobility4 ,against this view1 however1 there is at the outset a very fundamental psychological objection4 ,it has been urged that all our comparison of spatial magnitudes proceeds by ideal superposition4 ,thus ,james says 7,psychology1 ,vol4 ,,ii4 p4 #aeb73 ,even where we only feel one sub-division to be vaguely larger or less1 the mind must pass rapidly between it and the other sub-division1 and receive the immediate sensible shock of the more1 and so far as the sub-divisions of a sense-space are to be measured exactly against each other1 objective forms occupying one sub-division must be directly or indirectly superposed upon the other#afj4 ,even if we waive this fundamental objection1 however1 others remain4 ,to begin with1 such judgments of equality are only very rough approximations1 and cannot be applied to lines of more than a certain length1 if only for the reason that such lines cannot well be seen together4 ,thus this method can only give us any security in our own immediate neighbourhood1 and could in no wise warrant such operations as would be required for the construction of maps c41 much less the measurement of astronomical distances4 ,they might just enable us to say that some lines were longer than others1 but they would leave ,geometry in a position no better than that of the ,hedonical ,calculus1 in which we depend on a purely subjective measure4 ,so inaccurate1 in fact1 is such a method acknowledged to be1 that the foot-rule is as much a need of daily life as of science4 ,besides1 no one would trust such immediate judgments1 but for the fact that the stricter test of congruence to some extent confirms them2 if we could not apply this test1 we should have no ground for trusting them even as much as we do4 ,thus we should have1 here1 no real escape from our absolute dependence upon the axiom of ,free ,mobility4 #aec4 7#d7 ,one last elucidatory remark is necessary before our proof of this axiom can be considered complete4 ,we spoke above of the ,geometry on an egg1 where ,free ,mobility does not hold4 ,what1 ,i may be asked1 is there about a thoroughly non-congruent ,geometry1 more impossible than this ,geometry on the egg8 ,the answer is obvious4 ,the ,geometry of non-congruent surfaces is only possible by the use of infinitesimals1 and in the infinitesimal all surfaces become plane4 ,the fundamental formula1 that for the length of an infinitesimal arc1 is only obtained on the assumption that such an arc may be treated as a straight line1 and that ,euclidean ,plane ,geometry may be applied in the immediate neighbourhood of any point4 ,if we had not our ,euclidean measure1 which could be moved without distortion1 we should have no method of comparing small arcs in different places1 and the ,geometry of non-congruent surfaces would break down4 ,thus the axiom of ,free ,mobility1 as regards three-dimensional space1 is necessarily implied and presupposed in the ,geometry of non-congruent surfaces2 the possibility of the latter1 therefore1 is a dependent and derivative possibility1 and can form no argument against the priori necessity of congruence as the test of equality4 #aed4 ,it is to be observed that the axiom of ,free ,mobility1 as ,i have enunciated it1 includes also the axiom to which ,helmholtz gives the name of ,monodromy4 ,this asserts that a body does not alter its dimensions in consequence of a complete revolution through four right angles1 but occupies at the end the same position as at the beginning4 ,the supposed mathematical necessity of making a separate axiom of this property of space has been disproved by ,sophus ,lie 7v4 ,chap4 ,i4 #de72 philosophically1 it is plainly a particular case of ,free ,mobility#afa1 and indeed a particularly obvious case1 for a translation really does make some change in a body1 namely1 a change in position1 but a rotation through four right angles may be supposed to have been performed any number of times without appearing in the result1 and the absurdity of ascribing to space the power of making bodies grow in the process is palpable2 everything that was said above on congruence in general applies with even greater evidence to this special case4 #aee4 ,the axiom of ,free ,mobility involves1 if it is to be true1 the homogeneity of space1 or the complete relativity of position4 ,for if any shape1 which is possible in one part of space1 be always possible in another1 it follows that all parts of space are qualitatively similar1 and cannot1 therefore1 be distinguished by any intrinsic property4 ,hence positions in space1 if our axiom be true1 must be wholly defined by external relations1 i4e4 ,position is not an intrinsic1 but a purely relative1 property of things in space4 ,if there could be such a thing as absolute position1 in short1 metrical ,geometry would be impossible4 ,this relativity of position is the fundamental postulate of all ,geometry1 to which each of the necessary metrical axioms leads1 and from which1 conversely1 each of these axioms can be deduced4 #aef4 ,this converse deduction1 as regards ,free ,mobility1 is not very difficult1 and follows from the argument of ,section ,a#afb1 which ,i will briefly recapitulate4 ,in the first place1 externality is an essentially relative conception--nothing can be external to itself4 ,to be external to something is to be an other with some relation to that thing4 ,hence1 when we abstract a form of externality from all material content1 and study it in isolation1 position will appear of necessity as purely relative--it can have no intrinsic quality1 for our form consists of pure externality1 and externality contains no shadow or trace of an intrinsic quality4 ,hence we derive our fundamental postulate1 the relativity of position4 ,from this follows the homogeneity of our form1 for any quality in one position1 which marked out that position from another1 would be necessarily more or less intrinsic1 and would contradict the pure relativity4 ,finally ,free ,mobility follows from homogeneity1 for our form would not be homogeneous unless it allowed1 in every part1 shapes or systems of relations1 which it allowed in any other part4 ,free ,mobility1 therefore1 is a necessary property of every possible form of externality4 #aeg4 ,in summing up the argument we have just concluded1 we may exhibit it1 in consequence of the two preceding paragraphs1 in the form of a completed circle4 ,starting from the conditions of spatial measurement1 we found that the comparison1 required for measurement1 could only be effected by superposition4 ,but we found1 further1 that the result of such comparison will only be unambiguous1 if spatial magnitudes and shapes are unaltered by motion in space1 if1 in other words1 shapes do not depend upon absolute position in space4 ,but this axiom can only be true if space is homogeneous and position merely relative4 ,conversely1 if position is assumed to be merely relative1 a change of magnitude in motion--involving as it does1 the assertion of absolute position--is impossible1 and our test of spatial equality is therefore adequate4 ,but position in any form of externality must be purely relative1 since externality cannot be an intrinsic property of anything4 ,our axiom1 therefore1 is priori in a double sense4 ,it is presupposed in all spatial measurement1 and it is a necessary property of any form of externality4 ,a similar double apriority1 we shall see1 appears in our other necessary axioms4 ,,ii4 ,the ,axiom of ,dimensions#afc4 #aeh4 ,we have seen1 in discussing the axiom of ,free ,mobility1 that all position is relative1 that is1 a position exists only by virtue of relations#afd4 ,it follows that1 if positions can be defined at all1 they must be uniquely and exhaustively defined by some finite number of such relations4 ,if ,geometry is to be possible1 it must happen that1 after enough relations have been given to determine a point uniquely1 its relations to any fresh known point are deducible from the relations already given4 ,hence we obtain1 as an priori condition of ,geometry1 logically indispensable to its existence1 the axiom that ,space must have a finite integral number of ,dimensions4 ,for every relation required in the definition of a point constitutes a dimension1 and a fraction of a relation is meaningless4 ,the number of relations required must be finite1 since an infinite number of dimensions would be practically impossible to determine4 ,if we remember our axiom of ,free ,mobility1 and remember also that space is a continuum1 we may state our axiom1 for metrical ,geometry1 in the form given by ,helmholtz 7v4 ,chap4 ,i4 #be73 ,in a space of n dimensions1 the position of every point is uniquely determined by the measurement of n continuous independent variables 7coordinates74#afe #aei4 ,so much1 then1 is priori necessary to metrical ,geometry4 ,the restriction of the dimensions to three seems1 on the contrary1 to be wholly the work of experience#aff4 ,this restriction cannot be logically necessary1 for as soon as we have formulated any analytical system1 it appears wholly arbitrary4 ,why1 we are driven to ask1 cannot we add a fourth coordinate to our x1 y1 z1 or give a geometrical meaning to x#d8 ,in this more special form1 we are tempted to regard the axiom of dimensions1 like the number of inhabitants of a town1 as a purely statistical fact1 with no greater necessity than such facts have4 ,geometry affords intrinsic evidence of the truth of my division of the axiom of dimensions into an priori and empirical portion4 ,for while the extension of the number of dimensions to four1 or to n1 alters nothing in plane and solid ,geometry1 but only adds a new branch which interferes in no way with the old1 some definite number of dimensions is assumed in all ,geometries1 nor is it possible to conceive of a ,geometry which should be free from this assumption#afg4 #afj4 ,let us1 since the point seems of some interest1 repeat our proof of the apriority of this axiom from a slightly different point of view4 ,we will begin1 this time1 from the most abstract conception of space1 such as we find in ,riemann's dissertation1 or in ,erdmann's extents4 ,we have here1 an ordered manifold1 infinitely divisible and allowing of ,free ,mobility#afh4 ,free ,mobility involves1 as we saw1 the power of passing continuously from any one point to any other1 by any course which may seem pleasant to us2 it involves1 also1 that1 in such a course1 no changes occur except changes of mere position1 i4e41 positions do not differ from one another in any qualitative way4 7,this absence of qualitative difference is the distinguishing mark of space as opposed to other manifolds1 such as the colour- and tone-systems3 in these1 every element has a definite qualitative sensational value1 whereas in space1 the sensational value of a position depends wholly on its spatial relation to our own body1 and is thus not intrinsic1 but relative47 ,from the absence of qualitative differences among positions1 it follows logically that positions exist only by virtue of other positions2 one position differs from another just because they are two1 not because of anything intrinsic in either4 ,position is thus defined simply and solely by relation to other positions4 ,any position1 therefore1 is completely defined when1 and only when1 enough such relations have been given to enable us to determine its relation to any new position1 this new position being defined by the same number of relations4 ,now1 in order that such definition may be at all possible1 a finite number of relations must suffice4 ,but every such relation constitutes a dimension4 ,therefore1 if ,geometry is to be possible1 it is priori necessary that space should have a finite integral number of dimensions4 #afa4 ,the limitation of the dimensions to three is1 as we have seen1 empirical2 nevertheless1 it is not liable to the inaccuracy and uncertainty which usually belong to empirical knowledge4 ,for the alternatives which logic leaves to sense are discrete--if the dimensions are not three1 they must be two or four or some other number--so that small errors are out of the question#afi4 ,hence the final certainty of the axiom of three dimensions1 though in part due to experience1 is of quite a different order from that of 7say7 the law of ,gravitation4 ,in the latter1 a small inaccuracy might exist and remain undetected2 in the former1 an error would have to be so considerable as to be utterly impossible to overlook4 ,it follows that the certainty of our whole axiom1 that the number of dimensions is three1 is almost as great as that of the priori element1 since this element leaves to sense a definite disjunction of discrete possibilities4 ,,iii4 ,the ,axiom of ,distance4 #afb4 ,we have already seen1 in discussing projective ,geometry1 that two points must determine a unique curve1 the straight line4 ,in metrical ,geometry1 the corresponding axiom is1 that two points must determine a unique spatial quantity1 distance4 ,i propose to prove1 in what follows1 7#a7 that if distance1 as a quantity completely determined by two points1 did not exist1 spatial magnitude would not be measurable2 7#b7 that distance can only be determined by two points1 if there is an actual curve in space determined by those two points2 7#c7 that the existence of such a curve can be deduced from the conception of a form of externality1 and 7#d7 that the application of quantity to such a curve necessarily leads to a certain magnitude1 namely distance1 uniquely determined by any two points which determine the curve4 ,the conclusion will be1 if these propositions can be successfully maintained1 that the axiom of distance is priori in the same double sense as the axiom of ,free ,mobility1 i4e4 it is presupposed in the possibility of measurement1 and it is necessarily true of any possible form of externality4 #afc4 7#a7 ,the possibility of spatial measurement allows us to infer the existence of a magnitude uniquely determined by any two points4 ,the proof of this depends on the axiom of ,free ,mobility1 or its equivalent1 the homogeneity of space4 ,we have seen that these are involved in the possibility of spatial measurement2 we may employ them1 therefore1 in any argument as to the conditions of this possibility4 ,now to begin with1 two points must1 if ,geometry is to be possible1 have some relation to each other1 for we have seen that such relations alone constitute position or localization4 ,but if two points have a relation to each other1 this must be an intrinsic relation4 ,for it follows1 from the axiom of ,free ,mobility1 that two points1 forming a figure congruent with the given pair1 can be constructed in any part of space4 ,if this were not possible1 we have seen that metrical ,geometry could not exist4 ,but both the figures may be regarded as composed of two points and their relation2 if the two figures are congruent1 therefore1 it follows that the relation is quantitatively the same for both figures1 since congruence is the test of spatial equality4 ,hence the two points have a quantitative relation1 which is such that they can traverse all space in a combined motion without in any way altering that relation4 ,but in such a general motion1 any external relation of the two points1 any relation involving other points or figures in space1 must be altered#agj4 ,hence the relation between the two points1 being unaltered1 must be an intrinsic relation1 a relation involving no other point or figure in space2 and this intrinsic relation we call distance#aga4 #afd4 ,it might be objected1 to the above argument1 that it involves a petitio principii4 ,for it has been assumed that the two points and their relation form a figure1 to which other figures can be congruent4 ,now if two points have no intrinsic relation1 it would seem that they cannot form such a figure4 ,the argument1 therefore1 apparently assumes what it had to prove4 ,why1 it may be asked1 should not three points be required1 before we obtain any relation1 which ,free ,mobility allows us to construct afresh in other parts of space8 ,the answer to this1 as to the corresponding question in the first section of this chapter1 lies1 ,i think1 in the passivity of space1 or the mutual independence of its parts4 ,for it follows1 from this independence1 that any figure1 or any assemblage of points1 may be discussed without reference to other figures or points4 ,this principle is the basis of infinite divisibility1 of the use of quantity in ,geometry1 and of all possibility of isolating particular figures for discussion4 ,it follows that two points cannot be dependent1 as to their relation1 on any other points or figures1 for if they were so dependent1 we should have to suppose some action of such points or figures on the two points considered1 which would contradict the mutual independence of different positions4 ,to illustrate by an example3 the relation of two given points does not depend on the other points of the straight line on which the given points lie4 ,for only through their relation1 i4e4 through the straight line which they determine1 can the other points of the straight line be known to have any peculiar connection with the given pair4 #afe4 ,but why1 it may be asked1 should there be only one such relation between two points8 ,why not several8 ,the answer to this lies in the fact that points are wholly constituted by relations1 and have no intrinsic nature of their own#agb4 ,a point is defined by its relations to other points1 and when once the relations necessary for definition have been given1 no fresh relations to the points used in definition are possible1 since the point defined has no qualities from which such relations could flow4 ,now one relation to any one other point is as good for definition as more would be1 since however many we had1 they would all remain unaltered in a combined motion of both points4 ,hence there can only be one relation determined by any two points4 #aff4 7#b7 ,we have thus established our first proposition--two points have one and only one relation uniquely determined by those two points4 ,this relation we call their distance apart4 ,it remains to consider the conditions of the measurement of distance1 i4e41 how far a unique value for distance involves a curve uniquely determined by the two points4 ,in the first place1 some curve joining the two points is involved in the above notion of a combined motion of the two points1 or of two other points forming a figure congruent with the first two4 ,for without some such curve1 the two point-pairs cannot be known as congruent1 nor can we have any test by which to discover when a point-pair is moving as a single figure#agc4 ,distance must be measured1 therefore1 by some line which joins the two points4 ,but need this be a line which the two points completely determine8 #afg4 ,we are accustomed to the definition of the straight line as the shortest distance between two points1 which implies that distance might equally well be measured by curved lines4 ,this implication ,i believe to be false1 for the following reasons4 ,when we speak of the length of a curve1 we can give a meaning to our words only by supposing the curve divided into infinitesimal rectilinear arcs1 whose sum gives the length of an equivalent straight line2 thus unless we presuppose the straight line1 we have no means of comparing the lengths of different curves1 and can therefore never discover the applicability of our definition4 ,it might be thought1 perhaps1 that some other line1 say a circle1 might be used as the basis of measurement4 ,but in order to estimate in this way the length of any curve other than a circle1 we should have to divide the curve into infinitesimal circular arcs4 ,now two successive points do not determine a circle1 so that an arc of two points would have an indeterminate length4 ,it is true that1 if we exclude infinitesimal radii for the measuring circles1 the lengths of the infinitesimal arcs would be determinate1 even if the circles varied1 but that is only because all the small circular arcs through two consecutive points coincide with the straight line through those two points4 ,thus1 even with the help of the arbitrary restriction to a finite radius1 all that happens is that we are brought back to the straight line4 ,if1 to mend matters1 we take three consecutive points of our curve1 and reckon distance by the arc of the circle of curvature1 the notion of distance loses its fundamental property of being a relation between two points4 ,for two consecutive points of the arc could not then be said to have any corresponding distance apart--three points would be necessary before the notion of distance became applicable4 ,thus the circle is not a possible basis for measurement1 and similar objections apply1 of course1 with increased force1 to any other curve4 ,all this argument is designed to show1 in detail1 the logical impossibility of measuring distance by any curve not completely defined by the two points whose distance apart is required4 ,if in the above we had taken distance as measured by circles of given radius1 we should have introduced into its definition a relation to other points besides the two whose distance was to be measured1 which we saw to be a logical fallacy4 ,moreover1 how are we to know that all the circles have equal radii1 until we have an independent measure of distance8 #afh4 ,a straight line1 then1 is not the shortest distance1 but is simply the distance between two points--so far1 this conclusion has stood firm4 ,but suppose we had two or more curves through two points1 and that all these curves were congruent inter se4 ,we should then say1 in accordance with the definition of spatial equality1 that the lengths of all these curves were equal4 ,now it might happen that1 although no one of the curves was uniquely determined by the two end-points1 yet the common length of all the curves was so determined4 ,in this case1 what would hinder us from calling this common length the distance apart1 although no unique figure in space corresponded to it8 ,this is the case contemplated by spherical ,geometry1 where1 as on a sphere1 antipodes can be joined by an infinite number of geodesics1 all of which are of equal length4 ,the difficulty supposed is1 therefore1 not a purely imaginary one1 but one which modern ,geometry forces us to face4 ,i shall consequently discuss it at some length4 #afi4 ,to begin with1 ,i must point out that my axiom is not quite equivalent to ,euclid's4 ,euclid's axiom states that two straight lines cannot enclose a space1 i4e41 cannot have more than one common point4 ,now if every two points1 without exception1 determine a unique straight line1 it follows1 of course1 that two different straight lines can have only one point in common--so far1 the two axioms are equivalent4 ,but it may happen1 as in spherical space1 that two points in general determine a unique straight line1 but fail to do so when they have to each other the special relation of being antipodes4 ,in such a system every pair of straight lines in the same plane meet in two points1 which are each other's antipodes2 but two points1 in general1 still determine a unique straight line4 ,we are still able1 therefore1 to obtain distances from unique straight lines1 except in limiting cases2 and in such cases1 we can take any point intermediate between the two antipodes1 join it by the same straight line to both antipodes1 and measure its distance from those antipodes in the usual way4 ,the sum of these distances then gives a unique value for the distance between the antipodes4 ,thus even in spherical space1 we are greatly assisted by the axiom of the straight line2 all linear measurement is effected by it1 and exceptional cases can be treated1 through its help1 by the usual methods for limits4 ,spherical space1 therefore1 is not so adverse as it at first appeared to be to the priori necessity of the axiom4 ,nevertheless we have1 so far1 not attacked the kernel of the objection which spherical space suggested4 ,to this attack it is now our duty to proceed4 #agj4 ,it will be remembered that1 in our priori proof that two points must have one definite relation1 we held it impossible for those two points to have1 to the rest of space1 any relation which would be unaltered by motion4 ,now in spherical space1 in the particular case where the two points are antipodes1 they have a relation1 unaltered by motion1 to the rest of space--the relation1 namely1 that their distance is half the circumference of the universe4 ,in our former discussion1 we assumed that any relation to outside space must be a relation of position--and a relation of position must be altered by motion4 ,but with a finite space1 in which we have absolute magnitude1 another relation becomes possible1 namely1 a relation of magnitude4 ,antipodal points1 accordingly1 like coincident points1 no longer determine a unique straight line4 ,and it is instructive to observe that there is1 in consequence1 an ambiguity in the expression for distance1 like the ordinary ambiguity in angular measurement4 ,if #a_/k#b be the space constant1 and d be one value for the distance between two points1 #bkn d1 where n is any integer1 is an equally good value4 ,distance is1 in short1 a periodic function like angle4 ,thus such a state of things rather confirms than destroys my contention1 that distance depends on a curve uniquely determined by two points4 ,for as soon as we drop this unique determination1 we see ambiguities creeping into our expression for distance4 ,distance still has a set of discrete values1 corresponding to the fact that1 given one point1 the straight line is uniquely determined for all other points but one1 the antipodal point4 ,it is tempting to go on1 and say3 ,if through every pair of points there were an infinite number of the curves used in measuring distance1 distance would be able1 for the same pair of points1 to take1 not only a discrete series1 but an infinite continuous series of values4 #aga4 ,this1 however1 is mere speculation4 ,i come now to the pice de rsistance of my argument4 ,the ambiguity in spherical space arose1 as we saw1 from a relation of magnitude to the rest of space--such a relation being unaltered by a motion of the two points1 and therefore falling outside our introductory reasoning4 ,but what is this relation of magnitude8 ,simply a relation of the distance between the two points to a distance given in the nature of the space in question4 ,it follows that such a relation presupposes a measure of distance1 and need not1 therefore1 be contemplated in any argument which deals with the priori requisites for the possibility of definite distances#agd4 #agb4 ,i have now shown1 ,i hope conclusively1 that spherical space affords no objection to the apriority of my axiom4 ,any two points have one relation1 their distance1 which is independent of the rest of space1 and this relation requires1 as its measure1 a curve uniquely determined by those two points4 ,i might have taken the bull by the horns1 and said3 ,two points can have no relation but what is given by lines which join them1 and therefore1 if they have a relation independent of the rest of space1 there must be one line joining them which they completely determine4 ,thus ,james says#age3 ,just as1 in the field of quantity1 the relation between two numbers is another number1 so in the field of space the relations are facts of the same order with the facts they relate4444 ,when we speak of the relation of direction of two points towards each other1 we mean simply the sensation of the line that joins the two points together4 ,the line is the relation4444 ,the relation of position between the top and bottom points of a vertical line is that line1 and nothing else4 ,if ,i had been willing to use this doctrine at the beginning1 ,i might have avoided all discussion4 ,a unique relation between two points must in this case1 involve a unique line between them4 ,but it seemed better to avoid a doctrine not universally accepted1 the more so as ,i was approaching the question from the logical1 not the psychological1 side4 ,after disposing of the objections1 however1 it is interesting to find this confirmation of the above theory from so different a standpoint4 ,indeed1 ,i believe ,james's doctrine could be proved to be a logical necessity1 as well as a psychological fact4 ,for what sort of thing can a spatial relation between two distinct points be8 ,it must be something spatial1 and it must1 since points are wholly constituted by their relations1 be something at least as real and tangible as the points it relates4 ,there seems nothing which can satisfy these requirements1 except a line joining them4 ,hence1 once more1 a unique relation must involve a unique line4 ,that is1 linear magnitude is logically impossible1 unless space allows of curves uniquely determined by any two of their points4 #agc4 7#c7 ,but farther1 the existence of curves uniquely determined by two points can be deduced from the nature of any form of externality#agf4 ,for we saw1 in discussing ,free ,mobility1 that this axiom1 together with homogeneity and the relativity of position1 can be so deduced1 and we saw in the beginning of our discussion on distance1 that the existence of a unique relation between two points could be deduced from the homogeneity of space4 ,since position is relative1 we may say1 any two points must have some relation to each other3 since our form of externality is homogeneous1 this relation can be kept unchanged while the two points move in the form1 i4e41 change their relations to other points2 hence their relation to each other is an intrinsic relation1 independent of their relations to other points4 ,but since our form is merely a complex of relations1 a relation of externality must appear in the form1 with the same evidence as anything else in the form2 thus if the form be intuitive or sensational1 the relation must be immediately presented1 and not a mere inference4 ,hence the intrinsic relation between two points must be a unique figure in our form1 i4e4 in spatial terms1 the straight line joining the two points4 #agd4 7#d7 ,finally1 we have to prove that the existence of such a curve necessarily leads1 when quantity is applied to the relation between two points1 to a unique magnitude1 which those two points completely determine4 ,with this1 we shall be brought back to distance1 from which we started1 and shall complete the circle of our argument4 ,we saw1 in section ,a #aai1 that the figure formed by two points is projectively indistinguishable from that formed by any two other points in the same straight line2 the figure1 in both cases1 is1 from the projective standpoint1 simply the straight line on which the two points lie4 ,the difference of relation1 in the two cases1 is not qualitative1 since projective ,geometry cannot deal with it2 nevertheless1 there is some difference of relation4 ,for instance1 if one point be kept fixed1 while the other moves1 there is obviously some change of relation4 ,this change1 since all parts of the straight line are qualitatively alike1 must be a change of quantity4 ,if two points1 therefore1 determine a unique figure1 there must exist1 for the distinction between the various other points of this figure1 a unique quantitative relation between the two determining points1 and therefore1 since these points are arbitrary1 between only two points4 ,this relation is distance1 with which our argument began1 and to which it at least returns4 #age4 ,to sum up3 ,if points are defined simply by relations to other points1 i4e41 if all position is relative1 every point must have to every other point one1 and only one1 relation independent of the rest of space4 ,this relation is the distance between the two points4 ,now a relation between two points can only be defined by a line joining them--nay further1 it may be contended that a relation can only be a line joining them4 ,hence a unique relation involves a unique line1 i4e41 a line determined by any two of its points4 ,only in a space which admits of such a line is linear magnitude a logically possible conception4 ,but when once we have established the possibility1 in general1 of drawing such lines1 and therefore of measuring linear magnitudes1 we may find that a certain magnitude has a peculiar relation to the constitution of space4 ,the straight line may turn out to be of finite length1 and in this case its length will give a certain peculiar magnitude1 the space-constant4 ,two antipodal points1 that is1 points which bisect the entire straight line1 will then have a relation of magnitude which1 though unaltered by motion1 is rendered peculiar by a certain constant relation to the rest of space4 ,this peculiarity presupposes a measure of linear magnitude in general1 and cannot1 therefore1 upset the apriority of the axiom of the straight line4 ,but it destroys1 for points having the peculiar antipodal relation to each other1 the argument which proved that the relation between two points could not1 since it was unchanged by motion1 have reference to the rest of space4 ,thus it is intelligible that1 for such special points1 the axiom breaks down1 and an infinite number of straight lines are possible between them2 but unless we had started with assuming the general validity of the axiom1 we could never have reached a position in which antipodal points could have been known to be peculiar1 or1 indeed1 a position which would have enabled us to give any quantitative definition whatever of particular points4 ,distance and the straight line1 as relations uniquely determined by two points1 are thus priori necessary to metrical ,geometry4 ,but further1 they are properties which must belong to any form of externality4 ,since their necessity for ,geometry was deduced from homogeneity and the relativity of position1 and since these are necessary properties of any form of externality1 the same argument proves both conclusions4 ,we thus obtain1 as in the case of ,free ,mobility1 a double apriority3 ,the axiom of ,distance1 and its implication1 the axiom of the ,straight ,line1 are1 on the one hand1 presupposed in the possibility of spatial magnitude1 and cannot1 therefore1 be contradicted by any experience resulting from the measurement of space2 while they are consequences1 on the other hand1 of the necessary properties of any form of externality which is to render possible experience of an external world4 #agf4 ,in connection with the straight line1 it will be convenient to discuss the conditions of a metrical coordinate system4 ,the projective coordinate system1 as we have seen1 aims only at a convenient nomenclature for different points1 and can be set up without introducing the notion of spatial quantity4 ,but a metrical coordinate system does much more than this4 ,it defines every point quantitatively1 by its quantitative spatial relations to a certain coordinate figure4 ,only when the system of coordinates is thus metrical1 i4e41 when every coordinate represents some spatial magnitude1 which is itself a relation of the point defined to some other point or figure--can operations with coordinates lead to a metrical result4 ,when1 as in projective ,geometry1 the coordinates are not spatial magnitudes1 no amount of transformation can give a metrical result4 ,i wish to prove1 here1 that a metrical coordinate system necessarily involves the straight line1 and cannot1 without a logical fallacy1 be set up on any other basis4 ,the projective system of coordinates1 as we saw1 is entirely based on the straight line2 but the metrical system is more important1 since its quantities embody actual information as to spatial magnitudes1 which1 in projective ,geometry1 is not the case4 ,in the first place1 a point's metrical coordinates constitute a complete quantitative definition of it2 now a point can only be defined1 as we have seen1 by its relations to other points1 and these relations can only be defined by means of the straight line4 ,consequently1 any metrical system of coordinates must involve the straight line1 as the basis of its definitions of points4 ,this priori argument1 however1 though ,i believe it to be quite sound1 is not likely to carry conviction to any one persuaded of the opposite4 ,let us1 therefore1 examine metrical coordinate systems in detail1 and show1 in each case1 their dependence on the straight line4 ,we have already seen that the notion of distance is impossible without the straight line4 ,we cannot1 therefore1 define our coordinates in any of the ordinary ways1 as the distances from three planes1 lines1 points1 spheres1 or what not4 ,polar coordinates are impossible1 since1--waiving the straightness of the radius vector--the length of the radius vector becomes unmeaning4 ,triangular coordinates involve not only angles1 which must in the limit be rectilinear1 but straight lines1 or at any rate some well-defined curves4 ,now curves can only be metrically defined in two ways3 ,either by relation to the straight line1 as1 e4g41 by the curvature at any point1 or by purely analytical equations1 which presuppose an intelligible system of metrical coordinates4 ,what methods remain for assigning these arbitrary values to different points8 ,nay1 how are we to get any estimate of the difference--to avoid the more special notion of distance--between two points8 ,the very notion of a point has become illusory4 ,when we have a coordinate system1 we may define a point by its three coordinates2 in the absence of such a system1 we may define the notion of point in general as the intersection of three surfaces or of two curves4 ,here we take surfaces and curves as notions which intuition makes plain1 but if we wish them to give us a precise numerical definition of particular points1 we must specify the kind of surface or curve to be used4 ,now this1 as we have seen1 is only possible when we presuppose either the straight line1 or a coordinate system4 ,it follows that every coordinate system presupposes the straight line1 and is logically impossible without it4 #agg4 ,the above three axioms1 we have seen1 are priori necessary to metrical ,geometry4 ,no others can be necessary1 since metrical systems1 logically as unassailable as ,euclid's1 and dealing with spaces equally homogeneous and equally relational1 have been constructed by the metageometers1 without the help of any other axioms4 ,the remaining axioms of ,euclidean ,geometry--the axiom of parallels1 the axiom that the number of dimensions is three1 and ,euclid's form of the axiom of the straight line 7two straight lines cannot enclose a space7--are not essential to the possibility of metrical ,geometry1 i4e41 are not deducible from the fact that a science of spatial magnitudes is possible4 ,they are rather to be regarded as empirical laws1 obtained1 like the empirical laws of other sciences1 by actual investigation of the given subject-matter--in this instance1 experienced space4 #agh4 ,in summing up the distinctive argument of this ,section1 we may give it a more general form1 and discuss the conditions of measurement in any continuous manifold1 i4e41 the qualities necessary to the manifold1 in order that quantities in it may be determinable1 not only as to the more or less1 but as to the precise how much4 ,measurement1 we may say1 is the application of number to continua1 or1 if we prefer it1 the transformation of mere quantity into number of units4 ,using quantity to denote the vague more or less1 and magnitude to denote the precise number of units1 the problem of measurement may be defined as the transformation of quantity into magnitude4 ,now a number1 to begin with1 is a whole consisting of smaller units1 all of these units being qualitatively alike4 ,in order1 therefore1 that a continuous quantity may be expressible as a number1 it must1 on the one hand1 be itself a whole1 and must1 on the other hand1 be divisible into qualitatively similar parts4 ,in the aspect of a whole1 the quantity is intensive2 in the aspect of an aggregate of parts1 it is extensive4 ,a purely intensive quantity1 therefore1 is not numerable--a purely extensive quantity1 if any such could be imagined1 would not be a single quantity at all1 since it would have to consist of wholly unsynthesized particulars4 ,a measurable quantity1 therefore1 is a whole divisible into similar parts4 ,but a continuous quantity1 if divisible at all1 must be infinitely divisible4 ,for otherwise the points at which it could be divided would form natural barriers1 and so destroy its continuity4 ,but further1 it is not sufficient that there should be a possibility of division into mutually external parts2 while the parts1 to be perceptible as parts1 must be mutually external1 they must also1 to be knowable as equal parts1 be capable of overcoming their mutual externality4 ,for this1 as we have seen1 we require superposition1 which involves ,free ,mobility and homogeneity--the absence of ,free ,mobility in time1 where all other requisites of measurement are fulfilled1 renders direct measurement of time impossible4 ,hence infinite divisibility1 free mobility1 and homogeneity are necessary for the possibility of measurement in any continuous manifold1 and these1 as we have seen1 are equivalent to our three axioms4 ,these axioms are necessary1 therefore1 not only for spatial measurement1 but for all measurement4 ,the only manifold given in experience1 in which these conditions are satisfied1 is space4 ,all other exact measurement--as could be proved1 ,i believe1 for every separate case--is effected1 as we saw in the case of time1 by reduction to a spatial correlative4 ,this explains the paramount importance1 to exact science1 of the mechanical view of nature1 which reduces all phenomena to motions in time and space4 ,for number is1 of all conceptions1 the easiest to operate with1 and science seeks everywhere for an opportunity to apply it1 but finds this opportunity only by means of spatial equivalents to phenomena#agg4 #agi4 ,we have now seen in what the priori element of ,geometry consists4 ,this priori element may be defined as the axioms common to ,euclidean and non-,euclidean spaces1 as the axioms deducible from the conception of a form of externality1 or--in metrical ,geometry--as the axioms required for the possibility of measurement4 ,it remains to discuss1 in a final chapter1 some questions of a more general philosophic nature1 in which we shall have to desert the firm ground of mathematics and enter on speculations which ,i put forward very tentatively1 and with little faith in their ultimate validity4 ,the chief questions for this final chapter will be two3 7#a7 ,how is such priori and purely logical necessity possible1 as applied to an actually given subject-matter like space8 7#b7 ,how can we remove the contradictions which have haunted us in this chapter1 arising out of the relativity1 infinite divisibility1 and unbounded extension of space8 ,these two questions are forced upon us by the present chapter1 but as they open some of the fundamental problems of philosophy1 it would be rash to expect a conclusive or wholly satisfactory answer4 ,a few hints and suggestions may be hoped for1 but a complete solution could only be obtained from a complete philosophy1 of which the prospects are far too slender to encourage a confident frame of mind4 ,,footnotes3 #aaf ,see infra1 ,axiom of ,distance1 in ,sec4 ,b4 of this ,chapter4 #aag ,thus on a cylinder1 two geodesics1 e4g4 a generator and a helix1 may have any number of intersections--a very important difference from the plane4 #aah ,cf4 ,cremona1 ,projective ,geometry 7,clarendon ,press1 #bnd ed4 #ahic7 p4 #ej3 ,most of the propositions in ,euclid's ,elements are metrical1 and it is not easy to find among them an example of a purely descriptive theorem4 #aai ,op4 cit4 p4 #bbf4 #abj ,some ground for this choice will appear when we come to metrical ,geometry4 #aba ,the straight line a denotes the straight line common to the planes and a1 the point a denotes the point common to the plane and the straight line a1 and similarly for the rest of the notation4 #abb ,cremona 7op4 cit4 ,chap4 ,,ix4 p4 #ej7 defines anharmonic ratio as a metrical property which is unaltered by projection4 ,this1 however1 destroys the logical independence of projective ,geometry1 which can only be maintained by a purely descriptive definition4 #abc ,there is no corresponding property of three points on a line1 because they can be projectively transformed into any other three points on the same line4 ,see #abj4 #abd ,due to v4 ,staudt's ,geometrie der ,lage4 #abe ,see ,cremona1 op4 cit4 ,chapter ,,viii4 #abf ,the corresponding definitions1 for the two-dimensional manifold of lines through a point1 follow by the principle of duality4 #abg ,it is important to observe that this definition of the ,point introduces metrical ideas4 ,without metrical ideas1 we saw1 nothing appears to give the ,point precedence of the straight line1 or indeed to distinguish it conceptually from the straight line4 ,a reference to quantity is therefore inevitable in defining the ,point1 if the definition is to be geometrical4 ,a non-metrical definition would have to be also non-geometrical4 ,see ,chap4 ,,iv4 #aif-aii4 #abh #afc-age4 #abi ,on this axiom1 however1 compare #aca4 #acj ,for the proof of this proposition1 see ,chap4 ,,iii4 ,sec4 ,b1 ,axiom of ,dimensions4 #aca ,the straight line and plane1 in all discussions of general ,geometry1 are not necessarily ,euclidean4 ,they are simply figures determined1 in general1 by two and by three points respectively2 whether they conform to the axiom of parallels and to ,euclid's form of the axiom of the straight line1 is not to be considered in the general definition4 #acb ,that projective ,geometry must have existential import1 ,i shall attempt to prove in ,chapter ,,iv4 #acc ,logic1 ,book ,i4 ,chapter ,,ii4 #acd ,cf4 ,bradley's ,logic1 p4 #fc4 ,it will be seen that the sense in which ,i have spoken of space as a principle of differentiation is not the sense of a principle of individuation which ,bradley objects to4 #ace ,chap4 ,,iv4 #ahf-aia4 #acf ,chap4 ,,iv4 #bja ff4 #acg ,it is important to observe1 however1 that this way of regarding spatial relations is metrical2 from the projective standpoint1 the relation between two points is the whole unbounded straight line on which they lie1 and need not be regarded as divisible into parts or as built up of points4 #ach #bjg1 #bjh4 ,cf4 ,hegel1 ,naturphilosophie1 #bed4 #aci ,see ,chap4 ,,iv4 #aif-aii4 #adj ,see a forthcoming article on ,the relations of number and quantity by the present writer in ,mind1 ,july1 #ahig4 #ada ,logic1 ,vol4 ,,ii4 ,chap4 ,,vii4 p4 #baa4 #adb ,real1 as opposed to logical1 diversity is throughout intended4 ,diverse aspects may coexist in a thing at one time and place1 but two diverse real things cannot so coexist4 #adc ,on the insufficiency of time alone1 see ,chapter ,,iv4 #aia4 #add ,geometrically1 the axiom of the plane is1 not that three points determine a figure at all1 which follows from the axiom of the straight line1 but that the straight line joining two casual points of the plane lies wholly in the plane4 ,this axiom requires a projective method of constructing the plane1 i4e4 of finding all the triads of points which determine the same projective figure as the given triad4 ,the required construction will be obtained if we can find any projective figure determined by three points1 and any projective method of reaching other points which determine the same figure4 ,illustration ,let ,o1 ,p1 ,q be the three points whose projective relation is required4 ,then we have given us the three straight lines ,,pq1 ,,qo1 ,,op4 ,metrically1 the relation between these points is made up of the area1 and the magnitude of the sides and angles1 of the triangle ,,opq1 just as the relation between two points is distance4 ,but projectively1 the figure is unchanged when ,p and ,q travel along ,,op and ,,oq1 or when ,,op and ,,oq turn about ,o in such a way as still to meet ,,pq4 ,this is a result of the general principle of projective equivalence enunciated above 7 #ajh1 #aji74 ,hence the projective relation between ,o1 ,p1 ,q is the same as that between ,o1 p1 q or ,o1 ,p1 ,q2 that is1 p1 q and ,p1 ,q lie in the plane ,,opq4 ,in this way1 any number of points on the plane may be obtained1 and by repeating the construction with fresh triads1 every point of the plane can be reached4 ,we have to prove that1 when the plane is so constructed1 the straight line joining any two points of the plane lies wholly in the plane4 ,it is evident1 from the manner of construction1 that any point of ,,pq1 ,,op1 ,,oq1 ,,op or ,,oq lies in the plane4 ,if we can prove that any point of pq lies in the plane1 we shall have proved all that is required1 since pq may be transformed1 by successive repetitions of the same construction1 into any straight line joining two points of the plane4 ,but we have seen that the same plane is determined by ,o1 p1 q and by ,o1 ,p1 ,q4 ,the straight lines ,,pq1 pq have1 therefore1 the same relation to the plane4 ,but ,,pq lies wholly in the plane2 therefore pq also lies wholly in the plane4 ,hence our axiom is proved4 #ade ,a detailed proof has been given above1 ,chap4 ,i4 #crd period4 ,it is to be observed that any reference to infinitely distant elements involves metrical ideas4 #adf ,cf4 ,section ,a1 #aae-aag4 #adg ,contrast ,erdmann1 op4 cit4 p4 #ach4 #adh ,cf4 ,erdmann1 op4 cit4 p4 #afd4 #adi ,strictly speaking1 this method is only applicable where the two magnitudes are commensurable4 ,but if we take infinite divisibility rigidly1 the units can theoretically be taken so small as to obtain any required degree of approximation4 ,the difficulty is the universal one of applying to continua the essentially discrete conception of number4 #aej ,cf4 ,erdmann1 op4 cit4 p4 #ej4 #aea ,also called the axiom of congruence4 ,i have taken congruence to be the definition of spatial equality by superposition1 and shall therefore generally speak of the axiom as ,free ,mobility4 #aeb ,for the sense in which these figures are to be regarded as material1 see criticism of ,helmholtz1 ,chapter ,,ii4 #fi ff4 #aec ,op4 cit4 p4 #fj4 #aed ,the view of ,helmholtz and ,erdmann1 that mechanical experience suffices here1 though geometrical experience fails us1 has been discussed above1 ,chapter ,,ii4 #gc1 #hb4 #aee ,chapter ,,ii4 #ha4 #aef ,chapter ,,ii4 #gb4 #aeg ,contrast ,delbuf1 ,l'ancienne et les nouvelles gomtries1 ,,ii4 ,rev4 ,phil4 #ahid1 ,vol4 xxxvii4 p4 #ced4 #aeh ,prolegomena1 #ac4 ,see ,vaihinger's ,commentar1 ,,ii4 pp4 #eah-ecb esp4 pp4 #eba-b4 ,the above was ,kant's whole purpose in #agfh1 but only part of his purpose in the ,prolegomena1 where the intuitive nature of space was also to be proved4 #aei ,on the subject of time measurement1 cf4 ,bosanquet's ,logic1 ,vol4 i4 pp4 #agh-ahc4 ,since time1 in the above account1 is measured by motion1 its measurement presupposes that of spatial magnitudes4 #afj ,cf4 ,stumpf4 ,ursprung der ,raumvorstellung1 p4 #fh4 #afa ,as is ,helmholtz's other axiom1 that the possibility of superposition is independent of the course pursued in bringing it about4 #afb ,cf4 #abi1 #acj4 #afc ,this deduction is practically the same as that in ,sec4 ,a1 but ,i have stated it here with more special reference to space and to metrical ,geometry4 #afd ,the question3 ,relations to what8 is a question involving many difficulties4 ,it will be touched on later in this chapter1 and answered1 as far as possible1 in the fourth chapter4 ,for the present1 in spite of the glaring circle involved1 ,i shall take the relations as relations to other positions4 #afe ,wiss4 ,abh4 ,vol4 ,,ii4 p4 #fad4 #aff ,cp4 ,grassmann1 ,ausdehnungslehre von #ahdd1 #bnd ed4 p4 ,,xxiii4 #afg ,delbuf1 it is true1 speaks of ,geometries with m_/n dimensions1 but gives no reference 7,rev4 ,phil4 ,t4 xxxvi4 p4 #dej74 #afh ,in criticizing ,erdmann1 it will be remembered1 we saw that ,free ,mobility is a necessary property of his extents1 though he does not regard it as such4 #afi ,cf4 ,riemann1 ,hypothesen welche der ,geometrie zu ,grunde liegen1 ,gesammelte ,werke1 p4 #bff2 also ,erdmann1 op4 cit4 p4 #aed4 #agj ,this is subject1 in spherical space1 to the modification pointed out below1 in dealing with the exception to the axiom of the straight line4 ,see #afh-aga4 #aga ,in speaking of distance at once as a quantity and as an intrinsic relation1 ,i am anxious to guard against an apparent inconsistency4 ,i have spoken of the judgment of quantity1 throughout1 as one of comparison2 how1 then1 can a quantity be intrinsic8 ,the reply is that1 although measurement and the judgment of quantity express the result of comparison1 yet the terms compared must exist before the comparison2 in this case1 the terms compared in measuring distances1 i4e4 in comparing them inter se1 are intrinsic relations between points4 ,thus1 although the measurement of distance involves a reference to other distances1 and its expression as a magnitude requires such a reference1 yet its existence does not depend on any external reference1 but exclusively on the two points whose distance it is4 #agb ,see the end of the argument on ,free ,mobility1 #aee ff4 #agc ,in ,frischauf's ,absolute ,geometrie nach ,johann ,bolyai1 ,anhang1 there is a series of definitions1 starting from the sphere1 as the locus of congruent point-pairs when one point of the pair is fixed1 and hence obtaining the circle and the straight line4 ,from the above it follows1 that the sphere so defined already involves a curve between the points of the point-pair1 by which various point-pairs can be known as congruent2 and it will appear1 as we proceed1 that this curve must be a straight line4 ,frischauf's definition by means of the sphere involves1 therefore1 a vicious circle1 since the sphere presupposes the straight line1 as the test of congruent point-pairs4 #agd ,nor in any argument which1 like those of projective ,geometry1 avoids the notion of magnitude or distance altogether4 ,it follows that the propositions of projective ,geometry apply1 without reserve1 to spherical space1 since the exception to the axiom of the straight line arises only on metrical ground4 #age ,psychology1 ,vol4 ,,ii4 pp4 #adi-aej4 #agf ,this step in the argument has been put very briefly1 since it is a mere repetition of the corresponding argument in ,section ,a1 and is inserted here only for the sake of logical completeness4 ,see #acg ff4 #agg ,cf4 ,hannequin1 ,essai critique sur l'hypothse des atomes1 ,paris1 #ahie1 passim4 ,,chapter ,,iv4 ,,philosophical ,,consequences4 #ahj4 ,in the present chapter1 we have to discuss two questions which1 though scarcely geometrical1 are of fundamental importance to the theory of ,geometry propounded above4 ,the first of these questions is this3 ,what relation can a purely logical and deductive proof1 like that from the nature of a form of externality1 bear to an experienced subject-matter such as space8 ,you have merely framed a general conception1 ,i may be told1 containing space as a particular species1 and you have then shown1 what should have been obvious from the beginning1 that this general conception contained some of the attributes of space4 ,but what ground does this give for regarding these attributes as priori8 ,the conception ,mammal has some of the attributes of a horse2 but are these attributes therefore priori adjectives of the horse8 ,the answer to this obvious objection is so difficult1 and involves so much general philosophy1 that ,i have kept it for a final chapter1 in order not to interrupt the argument on specially geometrical topics4 #aha4 ,i have already indicated1 in general terms1 the ground for regarding as priori the properties of any form of externality4 ,this ground is transcendental1 i4e4 it is to be found in the conditions required for the possibility of experience4 ,the form of externality1 like ,riemann's manifolds1 is a general class-conception1 including time as well as ,euclidean and non-,euclidean spaces4 ,it is not motived1 however1 like the manifolds1 by a quantitative resemblance to space1 but by the fact that it fulfils1 if it has more than one dimension1 all those functions which1 in our actual world1 are fulfilled by space4 ,but a form of externality1 in order to accomplish this1 must be1 not a mere conception1 but an actually experienced intuition4 ,hence the conception of such a form is the general conception1 containing under it every logically possible intuition which can fulfil the function actually fulfilled by space4 ,and this function is1 to render possible experience of diverse but interrelated things4 ,some form in sense-perception1 then1 whose conception is included under our form of externality1 is priori necessary to experience of diversity in relation1 and without experience of this1 we should1 as modern logic shows1 have no experience at all4 ,this still leaves untouched the relation of the priori to the subjective3 the form of externality is necessary to experience1 but is not1 on that account1 to be declared purely subjective4 ,of course1 necessity for experience can only arise from the nature of the mind which experiences2 but it does not follow that the necessary conditions could be fulfilled1 unless the objective world had certain properties4 ,the ground of necessity1 we may safely say1 arises from the mind2 but it by no means follows that the truth of what is necessary depends only on the constitution of the mind4 ,where this is not the case1 our conclusion1 when a piece of knowledge has been declared priori1 can only be3 ,owing to the constitution of the mind1 experience will be impossible unless the world accepts certain adjectives4 ,such1 in outline1 will be the argument of the first half of this chapter1 and such will be the justification for regarding as priori those axioms of ,geometry1 which were deduced above from the conception of a form of externality4 ,for these axioms1 and these only1 are necessarily true of any world in which experience is possible4 #ahbagh4 ,the view suggested has1 obviously1 much in common with that of the ,transcendental ,aesthetic4 ,indeed the whole of it1 ,i believe1 can be obtained by a certain limitation and interpretation of ,kant's classic arguments4 ,but as it differs1 in many important points1 from the conclusions aimed at by ,kant1 and as the agreement may easily seem greater than it is1 ,i will begin by a brief comparison1 and endeavour1 by reference to authoritative criticisms1 to establish the legitimacy of my divergence from him4 #ahc4 ,in the first place1 the psychological element is much larger in ,kant's thesis than in mine4 ,i shall contend1 it is true1 that a form of externality1 if it is to do its work1 must not be a mere conception or a mere inference1 but must be a given element in sense-perception--not1 of course1 originally given in isolation1 but discoverable1 through analysis1 by attention to the object of sense-perception#agi4 ,but ,kant contended1 not only that this element is given1 but also that it is subjective4 ,space1 for him1 is1 on the one hand1 not conceptual1 but on the other hand1 not sensational4 ,it forms1 for him1 no part of the data of sense1 but is added by a subjective intuition1 which he regards as not only logically1 but psychologically1 prior to objects in space#ahj4 ,this part of ,kant's argument is wholly irrelevant for us4 ,whether a form of externality be given in sense1 or in a pure intuition1 is for us unimportant1 since we neglect the question as to the connection of the priori and the subjective2 while the temporal priority of space to objects in it has been generally recognized as irrelevant to ,epistemology1 and has often been regarded as forming no part of ,kant's thesis#aha4 ,if we call intuitional whatever is given in sense-perception1 then we may contend that a form of externality must be intuitional2 but whether it is a pure intuition1 in ,kant's sense1 or not1 is irrelevant to us1 as is its priority to the objects in it4 ,that the non-sensational nature of space is no essential part of ,kant's logical teaching1 appears from an examination of his argument4 ,he has made1 in the introduction1 the purely logical distinction of matter and form1 but has given to this distinction1 in the very moment of suggesting it1 a psychological implication4 ,this he does by the assertion that the form1 in which the matter of sensations is ordered1 cannot itself be sensational4 ,from this assumption it follows1 of course1 that space cannot be sensational4 ,but the assumption is totally unsupported by argument1 being set forth1 apparently1 as a self-evident axiom2 it has been severely criticized by ,stumpf#ahb and others#ahc1 and has been described by ,vaihinger as a fatal petitio principii#ahd2 it is irrelevant to the logical argument1 when this argument is separated1 as we have separated it1 from all connection with psychological subjectivity2 and finally1 it leaves us a prey to psychological theories of space1 which have seemed1 of late1 but little favourable to the pure ,kantian doctrine4 #ahd4 ,we have a right1 therefore1 in an epistemological inquiry1 to neglect ,kant's psychological teaching--in so far1 at any rate1 as it distinguishes spatial intuition from sensation--and attend rather to the logical aspect alone4 ,that part of his psychological teaching1 which maintains that space is not a mere conception1 is1 with certain limitations1 sufficiently evident as applied to actual space2 but for us1 it must be transformed into a much more difficult thesis1 namely1 that no form of externality1 which renders experience of diversity in relation possible1 can be merely conceptual4 ,this question1 to which we must return later1 is no longer psychological1 but belongs wholly to ,epistemology4 #ahe4 ,what1 then1 remains the kernel1 for our purposes1 of ,kant's first argument for the apriority of space8 ,his argument1 in the form in which he gave it1 is concerned with the eccentric projection of sensations4 ,in order that ,i may refer sensations1 he says1 to something outside myself1 ,i must already have the subjective space-form in the mind4 ,in this shape1 as ,vaihinger points out 7,commentar1 ,,ii4 pp4 #fi1 #afe71 the argument rests on a petitio principii1 for only if sensations are necessarily non-spatial does their projection demand a subjective space-form4 ,but1 further1 is the logical apriority of space concerned with the externality of things to ourselves8 ,space seems to perform two functions3 on the one hand1 it reveals things1 by the eccentric projection of sensations1 as external to the self1 while1 on the other hand1 it reveals simultaneously presented things as mutually external4 ,these two functions1 though often treated as coordinate and almost equivalent#ahe1 seem to me widely different4 ,before we discuss the apriority of space1 we must carefully distinguish1 ,i think1 between these two functions1 and decide which of them we are to argue about4 ,now externality to the ,self1 it would seem1 must necessarily raise the whole question of the nature and limits of the ,ego1 and what is more1 it cannot be derived from spatial presentation1 unless we give the ,self a definite position in space4 ,but things acquire a position in space only when they can appear in sense-perception2 we are forced1 therefore1 if we adopt this view of the function of space1 to regard the ,self as a phenomenon presented to sense-perception4 ,but this reduces externality to the ,self to externality to the body4 ,the body1 however1 is a presented object like any other1 and externality of objects to it is1 therefore1 a special case of the mutual externality of presented things4 ,hence we cannot regard space as giving1 primarily at any rate1 externality to the ,self1 but only the mutual externality of the things presented to sense-perception#ahf4 #ahf4 ,this1 then1 is the kind of externality we are to expect from space1 and our question must be3 ,would the existence of diverse but interrelated things be unknowable1 if there were not1 in sense-perception1 some form of externality8 ,this is the crucial question1 on which turns the apriority of our form1 and hence of the necessary axioms of ,geometry4 #ahg4 ,the converse argument to mine1 the argument from the spatio-temporal element in perception to a world of interrelated but diverse things1 is developed at length in ,bradley's ,logic4 ,it is put briefly in the following sentence 7p4 #dd1 note73 ,if space and time are continuous1 and if all appearance must occupy some time or space--and it is not hard to support both these theses--we can at once proceed to the conclusion1 no mere particular exists4 ,every phenomenon will exist in more times or spaces than one2 and against that diversity will be itself an universal#ahg4 ,the importance of this fact appears1 when we consider that1 if any mere particular existed1 all judgment and inference as to that particular would be impossible1 since all judgment and inference necessarily operate by means of universal4 ,but all reality is constructed from the ,this of immediate presentation1 from which judgment and inference necessarily spring4 ,owing1 however1 to the continuity and relativity of space and time1 no ,this can be regarded either as simple or as self-subsistent4 ,every ,this1 on the one hand1 can be analyzed into ,thises1 and on the other hand1 is found to be necessarily related to other things1 outside the limits of the given object of sense-perception4 ,this function of space and time is presupposed in the following statement from ,bosanquet's ,logic 7,vol4 ,i4 pp4 #gg-gh73 ,reality is given for me in present sensuous perception1 and in the immediate feeling of my own sentient existence that goes with it4 ,the real world1 as a definite organized system1 is for me an extension of this present sensation and self feeling by means of judgment1 and it is the essence of judgment to effect and sustain such an extension4444 ,the subject in every judgment of ,perception is some given spot or point in sensuous contact with the percipient self4 ,but1 as all reality is continuous1 the subject is not merely this given spot or point4 #ahh4 ,this doctrine of ,bradley and ,bosanquet is the converse of the epistemological doctrine ,i have to advocate4 ,owing to the continuity and relativity of space and time1 they say1 we are able to construct a systematic world1 by judgment and inference1 out of that fragmentary and yet necessarily complex existence which is given in sense-perception4 ,my contention is1 conversely1 that since all knowledge is necessarily derived by an extension of the ,this of sense-perception1 and since such extension is only possible if the ,this has that fragmentary and yet complex character conferred by a form of externality1 therefore some form of externality1 given with the ,this1 is essential to all knowledge1 and is thus logically priori4 ,bradley's argument1 if sound1 already proves this contention2 for while1 on the one hand1 he uses no properties of space and time but those which belong to every form of externality1 he proves1 on the other hand1 that judgment and inference require the ,this to be neither single nor self-subsistent4 ,but ,i will endeavour1 since the point is of fundamental importance1 to reproduce the proof1 in a form more suited than ,bradley's to the epistemological question4 #ahi4 ,the essence of my contention is that1 if experience is to be possible1 every sensational ,this must1 when attended to1 be found1 on the one hand1 resolvable into ,thises1 and on the other hand dependent1 for some of its adjectives1 on external reference4 ,the second of these theses follows from the first1 for if we take one of the ,thises contained in the first ,this1 we get a new ,this necessarily related to the other ,thises which make up the original ,this4 ,i may1 therefore1 confine myself to the first proposition1 which affirms that the object of perception must contain a diversity1 not only of conceptual content1 but of existence1 and that this can only be known if sense-perception contains1 as an element1 some form of externality4 ,my premiss1 in this argument1 is that all knowledge involves a recognition of diversity in relation1 or1 if we prefer it1 of identity in difference4 ,this premiss ,i accept from ,logic1 as resulting from the analysis of judgment and inference4 ,to prove such a premiss1 would require a treatise on ,logic2 ,i must refer the reader1 therefore1 to the works of ,bradley and ,bosanquet on the subject4 ,it follows at once1 from my premiss1 that knowledge would be impossible1 unless the object of attention could be complex1 i4e4 not a mere particular4 ,now could the mental object--i4e41 in this connection1 the object of a cognition--be complex1 if the object of immediate perception were always simple8 #aij4 ,we might be inclined1 at first sight1 to answer this question affirmatively4 ,but several difficulties1 ,i think1 would prevent such an answer4 ,in the first place1 knowledge must start from perception4 ,hence1 either we could have no knowledge except of our present perception1 or else we must be able to contrast and compare it with some other perception4 ,now in the first case1 since the present perception1 by hypothesis1 is a mere particular1 knowledge of it is impossible1 according to our premiss4 ,but in the second case1 the other perception1 with which we compare our first1 must have occurred at some other time1 and with time1 we have at once a form of externality4 ,but what is more1 our present perception is no longer a mere particular4 ,for the power of comparing it with another perception involves a point of identity between the two1 and thus renders both complex4 ,moreover1 time must be continuous1 and the present1 as ,bradley points out1 is no mere point of time#ahh4 ,thus our present perception contains the complexity involved in duration throughout the specious present3 its mere particularity and its simplicity are lost4 ,its self-subsistence is also lost1 for beyond the specious present1 lie the past and the future1 to which our present perception thus unavoidably refers us4 ,time at least1 therefore1 is essential to that identity in difference1 which all knowledge postulates4 #aia4 ,but we have derived1 from all this1 no ground for affirming a multiplicity of real things1 or a form of externality of more than one dimension1 which1 we saw1 was necessary for the truth of two out of our three axioms4 ,this brings us to the question3 ,have we enough1 with time alone as a form of externality1 for the possibility of knowledge8 ,this question we must1 ,i think1 answer in the negative4 ,with time alone1 we have seen1 our presented object must be complex1 but its complexity must1 if ,i may use such a phrase1 be merely adjectival4 ,without a second form of externality1 only one thing can be given at one moment#ahi1 and this one thing1 therefore1 must constitute the whole of our world4 ,the object of past perception must--since our one thing has nothing external to it1 by which it could be created or destroyed--be regarded as the same thing in a different state4 ,the complexity1 therefore1 will lie only in the changing states of our one thing--it will be adjectival1 not substantival4 ,moreover we have the following dilemma3 ,either the one thing must be ourselves1 or else self-consciousness could never arise4 ,but the chief difficulty of such a world would lie in the changes of the thing4 ,what could cause these changes1 since we should know of nothing external to our thing8 ,it would be like a ,leibnitzian monad1 without any ,god outside it to prearrange its changes4 ,causality1 in such a world1 could not be applied1 and change would be wholly inexplicable4 ,hence we require also the possibility of a diversity of simultaneously existing things1 not merely of successive adjectives2 and this1 we have seen1 cannot be given by time alone1 but only by a form of externality for simultaneous parts of one presentation4 ,we could never1 in other words1 infer the existence of diverse but interrelated things1 unless the object of sense-perception could have substantival complexity1 and for such complexity we require a form of externality other than time4 ,such a form1 moreover1 as was shown in ,chapter ,,iii41 ,section ,a 7 #ace71 can only fulfil its functions if it has more than one dimension4 ,in our actual world1 this form is given by space2 in any world1 knowable to beings with our laws of thought1 some such form1 as we have now seen1 must be given in sense-perception4 ,this argument may be briefly summed up1 by assuming the doctrine of ,bradley1 that all knowledge is obtained by inference from the ,this of sense-perception4 ,for1 if this be so1 the ,this--in order that inference1 which depends on identity in difference1 may be possible at all--must itself be complex1 and must1 on analysis1 reveal adjectives having a reference beyond itself4 ,but this1 as was shown above1 can only happen by means of a form of externality4 ,this establishes the priori axioms of ,geometry1 as necessarily having existential import and validity in any intelligible world4 #aib4 ,the above argument1 ,i hope1 has explained why ,i hold it possible to deduce1 from a mere conception like that of a form of externality1 the logical apriority of certain axioms as to experienced space4 ,the ,kantian argument--which was correct1 if our reasoning has been sound1 in asserting that real diversity1 in our actual world1 could only be known by the help of space--was only mistaken1 so far as its purely logical scope extends1 in overlooking the possibility of other forms of externality1 which could1 if they existed1 perform the same task with equal efficiency4 ,in so far as space differs1 therefore1 from these other conceptions of possible intuitional forms1 it is a mere experienced fact1 while in so far as its properties are those which all such forms must have1 it is priori necessary to the possibility of experience4 ,i cannot hope1 however1 that no difficulty will remain1 for the reader1 in such a deduction1 from abstract conceptions1 of the properties of an actual datum in sense-perception4 ,let us consider1 for example1 such a property as impenetrability4 ,to suppose two things simultaneously in the same position in a form of externality1 is a logical contradiction2 but can we say as much of actual space and time8 ,is not the impossibility1 here1 a matter of experience rather than of logic8 ,not if the above argument has been sound1 ,i reply4 ,for in that case1 we infer real diversity1 i4e4 the existence of different things1 only from difference of position in space or time4 ,it follows1 that to suppose two things in the same point of space and time1 is still a logical contradiction3 not because we have constructed the data of sense out of logic1 but because logic is dependent1 as regards its application1 on the nature of these data4 ,this instance illustrates1 what ,i am anxious to make plain1 that my argument has not attempted to construct the living wealth of sense-perception out of bloodless categories1 but only to point out that1 unless sense-perception contained a certain element1 these categories would be powerless to grapple with it4 #aic4 ,how we are to account for the fortunate realization of these requirements--whether by a pre-established harmony1 by ,darwinian adaptation to our environment1 by the subjectivity of the necessary element in sense-perception1 or by a fundamental identity and unity between ourselves and the rest of reality--is a further question1 belonging rather to metaphysics than to our present line of argument4 ,the priori1 we have said throughout1 is that which is necessary for the possibility of experience1 and in this we have a purely logical criterion1 giving results which only ,logic and ,epistemology can prove or disprove4 ,what is subjective in experience1 on the contrary1 is primarily a question for psychology1 and should be decided on psychological grounds alone4 ,when these two questions have been separately answered1 but not till then1 we may frame theories as to the connection of the priori and the subjective2 to allow such theories to influence our decision1 on either of the two previous questions1 is liable1 surely1 to confuse the issue1 and prevent a clear discrimination between fundamentally different points of view4 #aid4 ,i come now to the second question with which this chapter has to deal1 the question1 namely3 ,what are we to do with the contradictions which obtruded themselves in ,chapter ,,iii41 whenever we came to a point which seemed fundamental8 ,i shall treat this question briefly1 as ,i have little to add to answers with which we are all familiar4 ,i have only to prove1 first1 that the contradictions are inevitable1 and therefore form no objection to my argument2 secondly1 that the first step in removing them is to restore the notion of matter1 as that which1 in the data of sense-perception1 is localized and interrelated in space4 #aie4 ,the contradictions in space are an ancient theme--as ancient1 in fact1 as ,zeno's refutation of motion4 ,they are1 roughly1 of two kinds1 though the two kinds cannot be sharply divided4 ,there are the contradictions inherent in the notion of the continuum1 and the contradictions which spring from the fact that space1 while it must1 to be knowable1 be pure relativity1 must also1 it would seem1 since it is immediately experienced1 be something more than mere relations4 ,the first class of contradictions has been encountered more frequently in this essay1 and is also1 ,i think1 the more definite1 and the more important for our present purpose4 ,i doubt1 however1 whether the two classes are really distinct2 for any continuum1 ,i believe1 in which the elements are not data1 but intellectual constructions resulting from analysis1 can be shown to have the same relational and yet not wholly relational character as belongs to space4 ,the three following contradictions1 which ,i shall discuss successively1 seem to me the most prominent in a theory of ,geometry4 7#a7 ,though the parts of space are intuitively distinguished1 no conception is adequate to differentiate them4 ,hence arises a vain search for elements1 by which the differentiation could be accomplished1 and for a whole1 of which the parts of space are to be components4 ,thus we get the point1 or zero extension1 as the spatial element1 and an infinite regress or a vicious circle in the search for a whole4 7#b7 ,all positions being relative1 positions can only be defined by their relations1 i4e4 by the straight lines or planes through them2 but straight lines and planes1 being all qualitatively similar1 can only be defined by the positions they relate4 ,hence1 again1 we get a vicious circle4 7#c7 ,spatial figures must be regarded as relations4 ,but a relation is necessarily indivisible1 while spatial figures are necessarily divisible ad infinitum4 #aif4 7#a7 ,points4 ,the antinomy of the point--which arises wherever a continuum is given1 and elements have to be sought in it--is fundamental to ,geometry4 ,it has been given1 perhaps unintentionally1 by ,veronese as the first axiom1 in the form3 ,there are different points4 ,all points are identical 7op4 cit4 p4 #bbf74 ,we saw1 in discussing projective ,geometry1 that straight lines and planes must be regarded1 on the one hand as relations between points1 and on the other hand as made up of points#aij4 ,we saw again1 in dealing with measurement1 how space must be regarded as infinitely divisible1 and yet as mere relativity4 ,but what is divisible and consists of parts1 as space does1 must lead at last1 by continued analysis1 to a simple and unanalyzable part1 as the unit of differentiation4 ,for whatever can be divided1 and has parts1 possesses some thinghood1 and must1 therefore1 contain two ultimate units1 the whole namely1 and the smallest element possessing thinghood4 ,but in space this is notoriously not the case4 ,after hypostatizing space1 as ,geometry is compelled to do1 the mind imperatively demands elements1 and insists on having them1 whether possible or not4 ,of this demand1 all the geometrical applications of the infinitesimal calculus are evidence#aia4 ,but what sort of elements do we thus obtain8 ,analysis1 being unable to find any earlier halting-place1 finds its elements in points1 that is1 in zero quanta of space4 ,such a conception is a palpable contradiction1 only rendered tolerable by its necessity and familiarity4 ,a point must be spatial1 otherwise it would not fulfil the function of a spatial element2 but again it must contain no space1 for any finite extension is capable of further analysis4 ,points can never be given in intuition1 which has no concern with the infinitesimal3 they are a purely conceptual construction1 arising out of the need of terms between which spatial relations can hold4 ,if space be more than relativity1 spatial relations must involve spatial relata2 but no relata appear1 until we have analyzed our spatial data down to nothing4 ,the contradictory notion of the point1 as a thing in space without spatial magnitude1 is the only outcome of our search for spatial relata4 ,this reductio ad absurdum surely suffices1 by itself1 to prove the essential relativity of space4 #aig4 ,thus ,geometry is forced1 since it wishes to regard space as independent1 to hypostatize its abstractions1 and therefore to invent a self-contradictory notion as the spatial element4 ,a similar absurdity appears1 even more obviously1 in the notion of a whole of space4 ,the antinomy may1 therefore1 be stated thus3 ,space1 as we have seen throughout1 must1 if knowledge of it is to be possible1 be mere relativity2 but it must also1 if independent knowledge of it1 such as ,geometry seeks1 is to be possible1 be something more than mere relativity1 since it is divisible and has parts4 ,but we saw1 in ,chap4 ,,iii41 ,section ,a 7 #acc7 that knowledge of a form of externality must be logically independent of the particular matter filling the form4 ,how then are we to extricate ourselves from this dilemma8 ,the only way1 ,i think1 is1 not to make ,geometry dependent on ,physics1 which we have seen to be erroneous#aib1 but to give every geometrical proposition a certain reference to matter in general4 ,and at this point an important distinction must be made4 ,we have hitherto spoken of space as relational1 and of spatial figures as relations4 ,but space1 it would seem1 is rather relativity than relations--itself not a relation1 it gives the bare possibility of relations between diverse things#aic4 ,as applied to a spatial figure1 which can only arise by a differentiation of space1 and hence by the introduction of some differentiating matter1 the word relation is1 perhaps1 less misleading than any other2 as applied to empty undifferentiated space1 it seems by no means an accurate description4 ,but a bare possibility cannot exist1 or be given in sense-perception6 ,what becomes1 then1 of the arguments of the first part of this chapter8 ,i reply1 it is not empty space1 but spatial figures1 which sense-perception reveals1 and spatial figures1 as we have just seen1 involve a differentiation of space1 and therefore a reference to the matter which is in space4 ,it is spatial figures1 also1 and not empty space1 with which ,geometry has to deal4 ,the antinomy discussed above arises then--so it would seem--from the attempt to deal with empty space1 rather than with spatial figures and the matter to which they necessarily refer4 #aih4 ,let us see whether1 by this change1 we can overcome the antinomy of the point4 ,spatial figures1 we shall now say1 are relations between the matter which differentiates empty space4 ,their divisibility1 which seemed to contradict their relational character1 may be explained in two ways3 first1 as holding of the figures considered as parts of empty space1 which is itself not a relation2 second1 as denoting the possibility of continuous change in the relation expressed by the spatial figure4 ,these two ways are1 at bottom1 the same2 for empty space is a possibility of relations1 and the figure1 when viewed in connection with empty space1 thus becomes a possible relation1 with which other possible relations may be contrasted or compared4 ,but the second way of regarding divisibility is the better way1 since it introduces a reference to the matter which differentiates empty space1 without which1 spatial figures1 and therefore ,geometry1 could not exist4 ,it is empty space1 then--so we must conclude--which gives rise to the antinomy in question2 for empty space is a bare possibility of relations1 undifferentiated and homogeneous1 and thus wholly destitute of parts or of thinghood4 ,to speak of parts of a possibility is nonsense2 the parts and differentiations arise only through a reference to the matter which is differentiated in space4 #aii4 ,but what nature must we ascribe to this matter1 which is to be involved in all geometrical propositions8 ,in criticizing ,helmholtz 7,chap4 ,,ii4 #gc71 it may be remembered1 we decided that ,geometry refers to a peculiar and abstract kind of matter1 which is not regarded as possessing any causal qualities1 as exerting or as subject to the action of forces4 ,and this is the matter1 ,i think1 which we require for the needs of the moment4 ,not that we affirm1 of course1 that actual matter can be destitute of the properties with which ,physics is cognizant1 but that we abstract from these properties1 as being irrelevant to ,geometry4 ,all that we require1 for our immediate purpose1 is a subject of that diversity which space renders possible1 or terms for those relations by which empty space1 if space is to be studied at all1 must be differentiated4 ,but how must a matter1 which is to fulfil this function1 be regarded8 ,empty space1 we have said1 is a possibility of diversity in relation1 but spatial figures1 with which ,geometry necessarily deals1 are the actual relations rendered possible by empty space4 ,our matter1 therefore1 must supply the terms for these relations4 ,it must be differentiated1 since such differentiation1 as we have seen1 is the special work of space4 ,we must find1 therefore1 in our matter1 that unit of differentiation1 or atom#aid1 which in space we could not find4 ,this atom must be simple1 i4e4 it must contain no real diversity2 it must be a ,this not resolvable into ,thises4 ,being simple1 it can contain no relations within itself1 and consequently1 since spatial figures are mere relations1 it cannot appear as a spatial figure2 for every spatial figure involves some diversity of matter4 ,but our atom must have spatial relations with other atoms1 since to supply terms for these relations is its only function4 ,it is also capable of having these relations1 since it is differentiated from other atoms4 ,hence we obtain an unextended term for spatial relations1 precisely of the kind we require4 ,so long as we sought this term without reference to anything more than space1 the self-contradictory notion of the point was the only outcome of our search2 but now that we allow a reference to the matter differentiated by space1 we find at once the term which was needed1 namely1 a non-spatial simple element1 with spatial relations to other elements4 ,to ,geometry such a term will appear1 owing to its spatial relations1 as a point2 but the contradiction of the point1 as we now see1 is a result only of the undue abstraction with which ,geometry deals4 #bjj4 7#b7 ,the circle in the definition of straight lines and planes4 ,this difficulty need not long detain us1 since we have already1 with the material atom1 broken through the relativity which caused our circle4 ,straight lines1 in the purely geometrical procedure1 are defined only by points1 and points only by straight lines4 ,but points1 now1 are replaced by material atoms3 the duality of points and lines1 therefore1 has disappeared1 and the straight line may be defined as the spatial relation between two unextended atoms4 ,these atoms have spatial adjectives1 derived from their relations to other atoms2 but they have no intrinsic spatial adjectives1 such as could belong to them if they had extension or figure4 ,thus straight lines and planes are the true spatial units1 and points result only from the attempt to find1 within space1 those terms for spatial relations which exist only in a more than spatial matter4 ,straight lines1 planes and volumes are the spatial relations between two1 three or four unextended atoms1 and points are a merely convenient geometrical fiction1 by which possible atoms are replaced4 ,for1 since space1 as we saw1 is a possibility1 ,geometry deals not with actually realized spatial relations1 but with the whole scheme of possible relations4 #bja4 7#c7 ,space is at once relational and more than relational4 ,we have already touched on the question how far space is other than relations1 but as this question is quite fundamental1 as relation is an ambiguous and dangerous word1 as ,i have made constant use of the relativity of space without attempting to define a relation1 it will be necessary to discuss this antinomy at length4 #bjb4 ,now for this discussion it is essential to distinguish clearly between empty space and spatial figures4 ,empty space1 as a form of externality1 is not actual relations1 but the possibility of relations3 if we ascribe existential import to it1 as the ground1 in reality1 of all diversity in relation1 we at once have space as something not itself relations1 though giving the possibility of all relations4 ,in this sense1 space is to be distinguished from spatial order4 ,spatial order1 it may be said1 presupposes space1 as that in which this order is possible4 ,thus ,stumpf says#aie3 ,there is no order or relation without a positive absolute content1 underlying it1 and making it possible to order anything in this manner4 ,why and how should we otherwise distinguish one order from another8444 ,to distinguish different orders from one another1 we must everywhere recognize a particular absolute content1 in relation to which the order takes place4 ,and so space1 too1 is not a mere order1 but just that by which the spatial order1 side-by-sideness 7,nebeneinander7 distinguishes itself from the rest4 ,may we not1 then1 resolve the antinomy very simply1 by a reference to this ambiguity of space8 ,bradley contends 7,appearance and ,reality1 pp4 #cf-g7 that1 on the one hand1 space has parts1 and is therefore not mere relations1 while on the other hand1 when we try to say what these parts are1 we find them after all to be mere relations4 ,but cannot the space which has parts be regarded as empty space1 ,stumpf's absolute underlying content1 which is not mere relations1 while the parts1 in so far as they turn out to be mere relations1 are those relations which constitute spatial order1 not empty space8 ,if this can be maintained1 the antinomy no longer exists4 ,but such an explanation1 though ,i believe it to be a first step towards a solution1 will1 ,i fear1 itself demand almost as much explanation as the original difficulty4 ,for the connection of empty space with spatial order is itself a question full of difficulty1 to be answered only after much labour4 #bjc4 ,let us consider what this empty space is4 7,i speak of empty space without necessarily implying the absence of matter1 but only to denote a space which is not a mere order of material things47 ,stumpf regards it as given in sense2 ,kant1 in the last two arguments of his metaphysical deduction1 argues that it is an intuition1 not a concept1 and must be known before spatial order becomes possible4 ,i wish to maintain1 on the contrary1 that it is wholly conceptual2 that space is given only as spatial order2 that spatial relations1 being given1 appear as more than mere relations1 and so become hypostatized2 that when hypostatized1 the whole collection of them is regarded as contained in empty space2 but that this empty space itself1 if it means more than the logical possibility of space-relations1 is an unnecessary and self-contradictory assumption4 ,let us begin by considering ,kant's arguments on this point4 ,leibnitz had affirmed that space was only relations1 while ,newton had maintained the objective reality of absolute space4 ,kant adopted a middle course3 he asserted absolute space1 but regarded it as purely subjective4 ,the assertion of absolute space is the object of his second argument2 for if space were mere relations between things1 it would necessarily disappear with the disappearance of the things in it2 but this the second argument denies#aif4 ,now spatial order obviously does disappear with matter1 but absolute or empty space may be supposed to remain4 ,it is this1 then1 which ,kant is arguing about1 and it is this which he affirms to be a pure intuition1 necessarily presupposed by spatial order#aig4 #bjd4 ,but can we agree in regarding empty space1 the infinite given whole1 as really given8 ,must we not1 in spite of ,kant's argument1 regard it as wholly conceptual8 ,it is not required1 in the first place1 by the argument of the first half of this chapter1 which required only that every ,this of sense-perception should be resolvable into ,thises1 and thus involved only an order among ,thises1 not anything given originally without reference to them at all4 ,in the second place1 ,kant's two arguments#aih designed to prove that empty space is not conceptual1 are inadequate to their purpose4 ,the argument that the parts of space are not contained under it1 but in it1 proves certainly that space is not a general conception1 of which spatial figures are the instances2 but it by no means follows that empty space is not a conception4 ,empty space is undifferentiated and homogeneous2 parts of space1 or spatial figures1 arise only by reference to some differentiating matter1 and thus belong rather to spatial order than to empty space4 ,if empty space be the pre-condition of spatial order1 we cannot expect it to be connected with spatial relations as genus with species4 ,but empty space may nevertheless be a universal conception2 it may be related to spatial order as the state to the citizens4 ,these are not instances of the state1 but are contained in it2 they also1 in a sense1 presuppose it1 for a man can only become a citizen by being related to other citizens in a state#aii4 ,the uniqueness of space1 again1 seems hardly a valid argument for its intuitional nature2 to regard it as an argument implies1 indeed1 that all conceptions are abstracted from a series of instances--a view which has been criticized in ,chapter ,,ii4 7 #gg71 and need not be further discussed here#bjj4 ,there is no ground1 therefore1 in ,kant's two arguments for the intuitional nature of empty space1 which can be maintained against criticism4 #bje4 ,another ground for condemning empty space is to be found in the mathematical antinomies4 ,for it is no solution1 as ,lotze points out 7,metaphysik1 ,bk4 ,,ii4 ,chap4 ,i41 #ajf71 to regard empty space as purely subjective3 contradictions in a necessary subjective intuition form as great a difficulty as in anything else4 ,but these antinomies arise only in connection with empty space1 not with spatial order as an aggregate of relations4 ,for only when space is regarded as possessed of some thinghood1 can a whole or a true element be demanded4 ,this we have seen already in connection with the ,point4 ,when space is regarded1 so far as it is valid1 as only spatial order1 unbounded extension and infinite divisibility both disappear4 ,what is divided is not spatial relations1 but matter2 and if matter1 as we have seen that ,geometry requires1 consists of unextended atoms with spatial relations1 there is no reason to regard matter either as infinitely divisible1 or as consisting of atoms of finite extension4 #bjf4 ,but whence arises1 on this view1 the paradox that we cannot but regard space as having more or less thinghood1 and as divisible ad infinitum8 ,this must be explained1 ,i think1 as a psychological illusion1 unavoidably arising from the fact that spatial relations are immediately presented4 ,they thus have a peculiar psychical quality1 as immediate experiences1 by which quality they can be distinguished from time-relations or any other order in which things may be arranged4 ,to ,stumpf1 whose problem is psychological1 such a psychical quality would constitute an absolute underlying content1 and would fully justify his thesis2 to us1 however1 whose problem is epistemological1 it would not do so1 but would leave the meaning of the spatial element in sense-perception free from any implication of an absolute or empty space#bja4 ,may we not1 then1 abandon empty space1 and say3 ,spatial order consists of felt relations1 and qu felt has1 for ,psychology1 an existence not wholly resolvable into relations1 and unavoidably seeming to be more than mere relations4 ,but when we examine the information1 as to space1 which we derive from sense-perception1 we find ourselves plunged in contradictions1 as soon as we allow this information to consist of more than relations4 ,this leaves spatial order alone in the field1 and reduces empty space to a mere name for the logical possibility of spatial relations4 #bjg4 ,the apparent divisibility of the relations which constitute spatial order1 then1 may be explained in two ways1 though these are at bottom equivalent4 ,we may take the relation as considered in connection with empty space1 in which case it becomes more than a relation2 but being falsely hypostatized1 it appears as a complex thing1 necessarily composed of elements1 which elements1 however1 nowhere emerge until we analyze the pseudo-thing down to nothing1 and arrive at the point4 ,in this sense1 the divisibility of spatial relations is an unavoidable illusion4 ,or again1 we may take the relation in connection with the material atoms it relates4 ,in this case1 other atoms may be imagined1 differently localized by different spatial relations4 ,if they are localized on the straight line joining two of the original atoms1 this straight line appears as divided by them4 ,but the original relation is not really divided3 all that has happened is1 that two or more equivalent relations have replaced it1 as two compounded relations of father and son may replace the equivalent relation of grandfather and grandson4 ,these two ways of viewing the apparent divisibility are equivalent3 for empty space1 in so far as it is not illusion1 is a name for the aggregate of possible space-relations4 ,to regard a figure in empty space as divided1 therefore1 means1 if it means anything1 to regard two or more other possible relations as substituted for it1 which gives the second way of viewing the question4 ,the same reference to matter1 then1 by which the antinomy of the ,point was solved1 solves also the antinomy as to the relational nature of space4 ,space1 if it is to be freed from contradictions1 must be regarded exclusively as spatial order1 as relations between unextended material atoms4 ,empty space1 which arises1 by an inevitable illusion1 out of the spatial element in sense-perception1 may be regarded1 if we wish to retain it1 as the bare principle of relativity1 the bare logical possibility of relations between diverse things4 ,in this sense1 empty space is wholly conceptual2 spatial order alone is immediately experienced4 #bjh4 ,but in what sense does spatial order consist of relations8 ,we have hitherto spoken of externality as a relation1 and in a sense such a manner of speaking is justified4 ,externality1 when predicated of anything1 is an adjective of that thing1 and implies a reference to some other thing4 ,to this extent1 then1 externality is analogous to other relations2 and only to this extent1 in our previous arguments1 has it been regarded as a relation4 ,but when we take account of further qualities of relations1 externality begins to appear1 not so much as a relation1 but rather as a necessary aspect or element in every relation4 ,and this is borne out by the necessity1 for the existence of relations1 of some given form of externality4 ,every relation1 we may say1 involves a diversity between the related terms1 but also some unity4 ,mere diversity does not give a ground for that interaction1 and that interdependence1 which a relation requires4 ,mere unity leaves the terms identical1 and thus destroys the reference of one to another required for a relation4 ,mere externality1 taken in abstraction1 gives only the element of diversity required for a relation1 and is thus more abstract than any actual relation4 ,but mere diversity does not give that indivisible whole of which any actual relation must consist1 and is thus1 when regarded abstractly1 not subject to the restrictions of ordinary relations4 ,but with mere diversity1 we seem to have returned to empty space1 and abandoned spatial order4 ,mere diversity1 surely1 is either complete or non-existent2 degrees of diversity1 or a quantitative measure of it1 are nonsense4 ,we cannot1 therefore1 reduce spatial order to mere diversity4 ,two things1 if they occupy different positions in space1 are necessarily diverse1 but are as necessarily something more2 otherwise spatial order becomes unmeaning4 ,empty space1 then1 in the above sense of the possibility of spatial relations1 contains only one aspect of a relation1 namely the aspect of diversity2 but spatial order1 by its reference to matter1 becomes more concrete1 and contains also the element of unity1 arising out of the connection of the different material atoms4 ,spatial order1 then1 consists of relations in the ordinary sense2 its merely spatial element1 however--if one may make such a distinction--the element1 that is1 which can be abstracted from matter and regarded as constituting empty space1 is only one aspect of a relation1 but an aspect which1 in the concrete1 must be inseparably bound up with the other aspect4 ,here1 once more1 we see the ground of the contradictions in empty space1 and the reason why spatial order is free from these contradictions4 ,conclusion4 #bji4 ,we have now completed our review of the foundations of ,geometry4 ,it will be well1 before we take leave of the subject1 briefly to review and recapitulate the results we have won4 ,in the first chapter1 we watched the development of a branch of ,mathematics designed1 at first1 only to establish the logical independence of ,euclid's axiom of parallels1 and the possibility of a self-consistent ,geometry which dispensed with it4 ,we found the further development of the subject entangled1 for a while1 in philosophical controversy2 having shown one axiom to be superfluous1 the geometers of the second period hoped to prove the same conclusion of all the others1 but failed to construct any system free from three fundamental axioms4 ,being concerned with analytical and metrical ,geometry1 they tended to regard ,algebra as priori1 but held that those properties of spatial magnitudes1 which were not deducible from the laws of ,algebra1 must be empirical4 ,in all this1 they aimed as much at discrediting ,kant as at advancing ,mathematics4 ,but with the third period1 the interest in ,philosophy diminishes1 the opposition to ,euclid becomes less marked1 and most important of all1 measurement is no longer regarded as fundamental1 and space is dealt with by descriptive rather than quantitative methods4 ,but nevertheless1 three axioms1 substantially the same as those retained in the second period1 are still retained by all geometers4 ,in the second chapter1 we endeavoured1 by a criticism of some geometrical philosophies1 to prepare the ground for a constructive theory of ,geometry4 ,we saw that ,kant1 in applying the argument of the ,transcendental ,aesthetic to space1 had gone too far1 since its logical scope extended only to some form of externality in general4 ,we saw that ,riemann1 ,helmholtz and ,erdmann1 misled by the quantitative bias1 overlooked the qualitative substratum required by all judgments of quantity1 and thus mistook the direction in which the necessary axioms of ,geometry are to be found4 ,we rejected1 also1 ,helmholtz's view that ,geometry depends on ,physics1 because we found that ,physics must assume a knowledge of ,geometry before it can become possible4 ,but we admitted1 in ,geometry1 a reference to matter--not1 however1 to matter as empirically known in ,physics1 but to a more abstract matter1 whose sole function is to appear in space1 and supply the terms for spatial relations4 ,we admitted1 however1 besides this1 that all actual measurement must be effected by means of actual matter1 and is only empirically possible1 through the empirical knowledge of approximately rigid bodies4 ,in criticizing ,lotze1 we saw that the most important sense1 in which non-,euclidean spaces are possible1 is a philosophical sense1 namely1 that they are not condemned by any priori argument as to the necessity of space for experience1 and that consequently1 if they are not affirmed1 this must be on empirical grounds alone4 ,lotze's strictures on the mathematical procedure of ,metageometry we found to be wholly due to ignorance of the subject4 ,proceeding1 in the third chapter1 to a constructive theory of ,geometry1 we saw that projective ,geometry1 which has no reference to quantity1 is necessarily true of any form of externality4 ,its three axioms--homogeneity1 dimensions1 and the straight line--were all deduced from the conception of a form of externality1 and1 since some such form is necessary to experience1 were all declared priori4 ,in metrical ,geometry1 on the contrary1 we found an empirical element1 arising out of the alternatives of ,euclidean and non-,euclidean space4 ,three priori axioms1 common to these spaces1 and necessary conditions of the possibility of measurement1 still remained2 these were the axiom of ,free ,mobility1 the axiom that space has a finite integral number of dimensions1 and the axiom of distance4 ,except for the new idea of motion1 these were found equivalent to the projective triad1 and thus necessarily true of any form of externality4 ,but the remaining axioms of ,euclid--the axiom of three dimensions1 the axiom that two straight lines can never enclose a space1 and the axiom of parallels--were regarded as empirical laws1 derived from the investigation and measurement of our actual space1 and true only1 as far as the last two are concerned1 within the limits set by errors of observation4 ,in the present chapter1 we completed our proof of the apriority of the projective and equivalent metrical axioms1 by showing the necessity1 for experience1 of some form of externality1 given by sensation or intuition1 and not merely inferred from other data4 ,without this1 we said1 a knowledge of diverse but interrelated things1 the corner-stone of all experience1 would be impossible4 ,finally1 we discussed the contradictions arising out of the relativity and continuity of space1 and endeavoured to overcome them by a reference to matter4 ,this matter1 we found1 must consist of unextended atoms1 localized by their spatial relations1 and appearing1 in ,geometry1 as points4 ,but the non-spatial adjectives of matter1 we contended1 are irrelevant to ,geometry1 and its causal properties may be left out of account4 ,to deal with the new contradictions1 involved in such a notion of matter1 would demand a fresh treatise1 leading us1 through ,kinematics1 into the domains of ,dynamics and ,physics4 ,but to discuss the special difficulties of space is all that is possible in an essay on the ,foundations of ,geometry4 ,,footnotes3 #agh ,compare1 with the following paragraphs1 the admirable discussion in ,mr ,hobhouse's ,theory of ,knowledge 7,methuen #ahif71 ,part ,i4 ,chapter ,,ii4 #agi ,i speak of sense-perception instead of sensation1 so as not to prejudge the issue as to the sensational nature of space4 #ahj ,see ,vaihinger's ,commentar1 ,,ii4 pp4 #hf-g1 #afh-aga4 #aha ,see ,caird1 ,critical ,philosophy of ,kant1 ,vol4 ,i4 p4 #bhg4 #ahb ,ursprung der ,raumvorstellung1 pp4 #ab-cj4 #ahc ,see the references in ,vaihinger's ,commentar1 ,,ii4 p4 #gf ff4 #ahd ,commentar1 ,,ii4 p4 #ga ff4 #ahe ,e4g4 by ,caird1 op4 cit4 ,vol4 ,i4 p4 #bhf4 #ahf ,i have no wish to deny1 however1 that space is essential in the subsequent distinction of ,self and not-,self4 #ahg ,see also ,book ,i4 ,chap ,,ii4 passim2 especially p4 #ea ff4 and pp4 #gj-a4 #ahh ,logic1 p4 #ea ff4 #ahi ,for the ,this1 on such a hypothesis1 has a purely temporal complexity1 and is not resolvable into coexisting ,thises4 #aij ,chapter ,,iii4 ,section ,a1 7 #aca74 #aia ,cf4 ,hannequin1 ,essai critique sur l'hypothse des atomes1 ,paris #ahie1 ,chap4 ,i4 ,section ,,iii42 especially p4 #dc4 #aib ,see ,chapter ,,ii4 #fi ff4 #aic ,see third antinomy below1 #bja ff4 #aid ,this atom1 of course1 must not be confounded with the atom of ,chemistry4 #aie ,ursprung der ,raumvorstellung1 p4 #ae4 #aif ,see ,vaihinger's ,commentar1 ,,ii4 pp4 #ahi-aij4 #aig ,see ibid4 p4 #bbd ff4 for ,kant's inconsistencies on this point4 #aih ,the fourth and fifth in the first edition1 the third and fourth in the second4 #aii ,cf4 ,vaihinger's ,commentar1 ,,ii4 p4 #bah4 #bjj ,cf4 ,vaihinger's ,commentar1 ,,ii4 p4 #bjg4 #bja ,cf4 ,james1 ,psychology1 ,vol4 ,,ii41 p4 #adh ff4 ,cambridge3 ,,printed ,,by ,j4 ,,and ,c4 ,f4 ,,clay1 ,,at ,,the ,,university ,,press4 ,,transcriber',s ,,note ,italic text is denoted by underscores4 ,bold text is denoted by equal signs4 ,a superscript is denoted by x4 ,for example1 y#d indicates y to the power #d4 ,a subscript is denoted by x4 ,for example1 aik indicates the ik-th element of array a4 ,algebraic expressions in the original text usually had the variables in italic4 ,the italic underscore markup has been removed for readability1 so for example3 dxi dxi dxi has been changed to3 dxi dxi dxi ,obvious typographical errors and punctuation errors have been corrected after careful comparison with other occurrences within the text and consultation of external sources4 ,except for those changes noted below1 all misspellings in the text1 and inconsistent or archaic usage1 have been retained4 ,for example1 co-exist1 coexist2 every-day1 everyday2 connexion2 assertorial2 apodeictic2 premisses4 #hb1 'so ,erdmannn confidently' replaced by 'so ,erdmann confidently'4 #aej ,footnote #aeg1 ',delboeuf' replaced by ',delbuf' for consistency4 #aeb1 'one subdivison must' replaced by 'one sub-division must'4 #aei ,footnote #afg1 ',delboeuf' replaced by ',delbuf' for consistency4 #bjd1 'and homogenous2' replaced by 'and homogeneous2'4