{"collection":"philosophy-ethics","full_text_available":true,"graph_url":"https://unchartedknowledge.5gfusion.net/v2/api/page/gutenberg_52091/graph","html_url":"https://unchartedknowledge.5gfusion.net/v2/page/gutenberg_52091","id":"gutenberg_52091","license":"Public Domain (US)","rights_basis":"gutenberg_metadata_copyright_false","source":"gutenberg","source_url":"https://www.gutenberg.org/ebooks/52091","summary":null,"tags":["Category: Mathematics","Category: Philosophy & Ethics","Geometry -- Foundations","Nobel Prizes in Literature"],"text":"TRANSCRIBER'S NOTE\n\n Italic text is denoted by _underscores_.\n\n Bold text is denoted by =equal signs=.\n\n A superscript is denoted by ^{x}. For example, y^{4} indicates y to\n the power 4.\n\n A subscript is denoted by {x}. For example, a{ik} indicates the ik-th\n element of array a.\n\n Algebraic expressions in the original text usually had the variables in\n italic. The italic underscore markup has been removed for readability,\n so for example: _dx_{i}_ = \u03bb\u2032_d_\u2032_x_{i}_ + \u03bb\u2033_d\u2033x_{i}_\n has been changed to: dx{i} = \u03bb\u2032d\u2032x{i} + \u03bb\u2033d\u2033x{i}\n\n Obvious typographical errors and punctuation errors have been\n corrected after careful comparison with other occurrences within\n the text and consultation of external sources.\n\n More detail can be found at the end of the book.\n\n\n\n\n THE FOUNDATIONS OF GEOMETRY.\n\n\n\n\n London: C. J. CLAY AND SONS,\n\n CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,\n AVE MARIA LANE.\n\n Glasgow: 263, ARGYLE STREET.\n\n [Illustration]\n\n Leipzig: F. A. BROCKHAUS.\n New York: THE MACMILLAN COMPANY.\n Bombay: GEORGE BELL AND SONS.\n\n\n\n\n AN ESSAY\n\n ON THE\n\n FOUNDATIONS OF GEOMETRY\n\n BY\n\n BERTRAND A. W. RUSSELL. M.A.\n\n FELLOW OF TRINITY COLLEGE, CAMBRIDGE.\n\n CAMBRIDGE:\n AT THE UNIVERSITY PRESS.\n\n 1897\n\n [_All Rights reserved._]\n\n\n Cambridge:\n\n PRINTED BY J. AND C. F. CLAY,\n AT THE UNIVERSITY PRESS.\n\n\n\n\nPREFACE.\n\n\nThe present work is based on a dissertation submitted at the\nFellowship Examination of Trinity College, Cambridge, in the year\n1895. Section B of the third chapter is in the main a reprint, with\nsome serious alterations, of an article in _Mind_ (New Series, No.\n17). The substance of the book has been given in the form of lectures\nat the Johns Hopkins University, Baltimore, and at Bryn Mawr College,\nPennsylvania.\n\nMy chief obligation is to Professor Klein. Throughout the first\nchapter, I have found his \"Lectures on non-Euclidean Geometry\"\nan invaluable guide; I have accepted from him the division of\nMetageometry into three periods, and have found my historical work\nmuch lightened by his references to previous writers. In Logic, I\nhave learnt most from Mr Bradley, and next to him, from Sigwart and\nDr Bosanquet. On several important points, I have derived useful\nsuggestions from Professor James's \"Principles of Psychology.\"\n\nMy thanks are due to Mr G. F. Stout and Mr A. N. Whitehead for\nkindly reading my proofs, and helping me by many useful criticisms.\nTo Mr Whitehead I owe, also, the inestimable assistance of constant\ncriticism and suggestion throughout the course of construction,\nespecially as regards the philosophical importance of projective\nGeometry.\n\n HASLEMERE.\n\n _May, 1897._\n\n\n\n\n TO\n\n JOHN McTAGGART ELLIS McTAGGART\n\n TO WHOSE DISCOURSE AND FRIENDSHIP IS OWING\n THE EXISTENCE OF THIS BOOK.\n\n\n\n\nTABLE OF CONTENTS.\n\n\n INTRODUCTION.\n\n OUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC,\n PSYCHOLOGY AND MATHEMATICS.\n PAGE\n\n 1. The problem first received a modern form through Kant, who\n connected the _\u00e0 priori_ with the subjective 1\n\n 2. A mental state is subjective, for Psychology, when its immediate\n cause does not lie in the outer world 2\n\n 3. A piece of knowledge is _\u00e0 priori_, for Epistemology, when\n without it knowledge would be impossible 2\n\n 4. The subjective and the _\u00e0 priori_ belong respectively to\n Psychology and to Epistemology. The latter alone will be\n investigated in this essay 3\n\n 5. My test of the _\u00e0 priori_ will be purely logical: what knowledge\n is necessary for experience? 3\n\n 6. But since the necessary is hypothetical, we must include, in\n the _\u00e0 priori_, the ground of necessity 4\n\n 7. This may be the essential postulate of our science, or the\n element, in the subject-matter, which is necessary to\n experience; 4\n\n 8. Which, however, are both at bottom the same ground 5\n\n 9. Forecast of the work 5\n\n\n CHAPTER I.\n\n A SHORT HISTORY OF METAGEOMETRY.\n\n 10. Metageometry began by rejecting the axiom of parallels 7\n\n 11. Its history may be divided into three periods: the synthetic,\n the metrical and the projective 7\n\n 12. The first period was inaugurated by Gauss, 10\n\n 13. Whose suggestions were developed independently by\n Lobatchewsky 10\n\n 14. And Bolyai 11\n\n 15. The purpose of all three was to show that the axiom of parallels\n could not be deduced from the others, since its denial did\n not lead to contradictions 12\n\n 16. The second period had a more philosophical aim, and was\n inspired chiefly by Gauss and Herbart 13\n\n 17. The first work of this period, that of Riemann, invented two\n new conceptions: 14\n\n 18. The first, that of a manifold, is a class-conception, containing\n space as a species, 14\n\n 19. And defined as such that its determinations form a collection\n of magnitudes 15\n\n 20. The second, the measure of curvature of a manifold, grew out\n of curvature in curves and surfaces 16\n\n 21. By means of Gauss's analytical formula for the curvature of\n surfaces, 19\n\n 22. Which enables us to define a _constant_ measure of curvature\n of a three-dimensional space without reference to a fourth\n dimension 20\n\n 23. The main result of Riemann's mathematical work was to\n show that, if magnitudes are independent of place, the\n measure of curvature of space must be constant 21\n\n 24. Helmholtz, who was more of a philosopher than a mathematician, 22\n\n 25. Gave a new but incorrect formulation of the essential axioms, 23\n\n 26. And deduced the quadratic formula for the infinitesimal arc,\n which Riemann had assumed 24\n\n 27. Beltrami gave Lobatchewsky's planimetry a Euclidean\n interpretation, 25\n\n 28. Which is analogous to Cayley's theory of distance; 26\n\n 29. And dealt with _n_-dimensional spaces of constant negative\n curvature 27\n\n 30. The third period abandons the metrical methods of the second,\n and extrudes the notion of spatial quantity 27\n\n 31. Cayley reduced metrical properties to projective properties,\n relative to a certain conic or quadric, the Absolute; 28\n\n 32. And Klein showed that the Euclidean or non-Euclidean systems\n result, according to the nature of the Absolute; 29\n\n 33. Hence Euclidean _space_ appeared to give rise to all the kinds\n of Geometry, and the question, which is true, appeared\n reduced to one of convention 30\n\n 34. But this view is due to a confusion as to the nature of the\n coordinates employed 30\n\n 35. Projective coordinates have been regarded as dependent on\n distance, and thus really metrical 31\n\n 36. But this is not the case, since anharmonic ratio can be\n projectively defined 32\n\n 37. Projective coordinates, being purely descriptive, can give no\n information as to metrical properties, and the reduction of\n metrical to projective properties is purely technical 33\n\n 38. The true connection of Cayley's measure of distance with\n non-Euclidean Geometry is that suggested by Beltrami's\n Saggio, and worked out by Sir R. Ball, 36\n\n 39. Which provides a Euclidean equivalent for every non-Euclidean\n proposition, and so removes the possibility of contradictions\n in Metageometry 38\n\n 40. Klein's elliptic Geometry has not been proved to have a\n corresponding variety of space 39\n\n 41. The geometrical use of imaginaries, of which Cayley demanded\n a philosophical discussion, 41\n\n 42. Has a merely technical validity, 42\n\n 43. And is capable of giving geometrical results only when it\n begins and ends with real points and figures 45\n\n 44. We have now seen that projective Geometry is logically prior\n to metrical Geometry, but cannot supersede it 46\n\n 45. Sophus Lie has applied projective methods to Helmholtz's\n formulation of the axioms, and has shown the axiom of\n Monodromy to be superfluous 46\n\n 46. Metageometry has gradually grown independent of philosophy,\n but has grown continually more interesting to philosophy 50\n\n 47. Metrical Geometry has three indispensable axioms, 50\n\n 48. Which we shall find to be not results, but conditions, of\n measurement, 51\n\n 49. And which are nearly equivalent to the three axioms of\n projective Geometry 52\n\n 50. Both sets of axioms are necessitated, not by facts, but by\n logic 52\n\n\n CHAPTER II.\n\n CRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL\n THEORIES OF GEOMETRY.\n\n 51. A criticism of representative modern theories need not begin\n before Kant 54\n\n 52. Kant's doctrine must be taken, in an argument about Geometry,\n on its purely logical side 55\n\n 53. Kant contends that since Geometry is apodeictic, space must\n be _\u00e0 priori_ and subjective, while since space is _\u00e0 priori_\n and subjective, Geometry must be apodeictic 55\n\n 54. Metageometry has upset the first line of argument, not the\n second 56\n\n 55. The second may be attacked by criticizing either the distinction\n of synthetic and analytic judgments, or the first two arguments\n of the metaphysical deduction of space 57\n\n 56. Modern Logic regards every judgment as both synthetic and\n analytic, 57\n\n 57. But leaves the _\u00e0 priori_, as that which is presupposed in the\n possibility of experience 59\n\n 58. Kant's first two arguments as to space suffice to prove _some_\n form of externality, but not necessarily Euclidean space, a\n necessary condition of experience 60\n\n 59. Among the successors of Kant, Herbart alone advanced the\n theory of Geometry, by influencing Riemann 62\n\n 60. Riemann regarded space as a particular kind of manifold, i.e.\n wholly quantitatively 63\n\n 61. He therefore unduly neglected the qualitative adjectives of\n space 64\n\n 62. His philosophy rests on a vicious disjunction 65\n\n 63. His definition of a manifold is obscure, 66\n\n 64. And his definition of measurement applies only to space 67\n\n 65. Though mathematically invaluable, his view of space as a\n manifold is philosophically misleading 69\n\n 66. Helmholtz attacked Kant both on the mathematical and on\n the psychological side; 70\n\n 67. But his criterion of apriority is changeable and often\n invalid; 71\n\n 68. His proof that non-Euclidean spaces are imaginable is\n inconclusive; 72\n\n 69. And his assertion of the dependence of measurement on rigid\n bodies, which may be taken in three senses, 74\n\n 70. Is wholly false if it means that the axiom of Congruence\n actually asserts the existence of rigid bodies, 75\n\n 71. Is untrue if it means that the necessary reference of geometrical\n propositions to matter renders pure Geometry empirical, 76\n\n 72. And is inadequate to his conclusion if it means, what is\n true, that _actual_ measurement involves approximately rigid\n bodies 78\n\n 73. Geometry deals with an abstract matter, whose physical\n properties are disregarded; and Physics must presuppose\n Geometry 80\n\n 74. Erdmann accepted the conclusions of Riemann and Helmholtz, 81\n\n 75. And regarded the axioms as necessarily successive steps in\n classifying space as a species of manifold 82\n\n 76. His deduction involves four fallacious assumptions, namely: 82\n\n 77. That conceptions must be abstracted from a series of\n instances; 83\n\n 78. That all definition is classification; 83\n\n 79. That conceptions of magnitude can be applied to space as\n a whole; 84\n\n 80. And that if conceptions of magnitude could be so applied, all\n the adjectives of space would result from their application 86\n\n 81. Erdmann regards Geometry alone as incapable of deciding on\n the truth of the axiom of Congruence, 86\n\n 82. Which he affirms to be empirically proved by Mechanics. 88\n\n 83. The variety and inadequacy of Erdmann's tests of apriority 89\n\n 84. Invalidate his final conclusions on the theory of Geometry 90\n\n 85. Lotze has discussed two questions in the theory of Geometry: 93\n\n 86. (1) He regards the possibility of non-Euclidean spaces as\n suggested by the subjectivity of space, 93\n\n 87. And rejects it owing to a mathematical misunderstanding, 96\n\n 88. Having missed the most important sense of their possibility, 96\n\n 89. Which is that they fulfil the logical conditions to which any\n form of externality must conform 97\n\n 90. (2) He attacks the mathematical procedure of Metageometry 98\n\n 91. The attack begins with a question-begging definition of\n parallels 99\n\n 92. Lotze maintains that all apparent departures from Euclid\n could be physically explained, a view which really makes\n Euclid empirical 99\n\n 93. His criticism of Helmholtz's analogies rests wholly on\n mathematical mistakes 101\n\n 94. His proof that space must have three dimensions rests on\n neglect of different orders of infinity 104\n\n 95. He attacks non-Euclidean spaces on the mistaken ground\n that they are not homogeneous 107\n\n 96. Lotze's objections fall under four heads 108\n\n 97. Two other semi-philosophical objections may be urged, 109\n\n 98. One of which, the absence of similarity, has been made the\n basis of attack by Delb\u0153uf, 110\n\n 99. But does not form a valid ground of objection 111\n\n 100. Recent French speculation on the foundations of Geometry\n has suggested few new views 112\n\n 101. All homogeneous spaces are _\u00e0 priori_ possible, and the\n decision between them is empirical 114\n\n\n CHAPTER III.\n\n SECTION A. THE AXIOMS OF PROJECTIVE GEOMETRY.\n\n 102. Projective Geometry does not deal with magnitude, and\n applies to all spaces alike 117\n\n 103. It will be found wholly _\u00e0 priori_ 117\n\n 104. Its axioms have not yet been formulated philosophically 118\n\n 105. Coordinates, in projective Geometry, are not spatial\n magnitudes, but convenient names for points 118\n\n 106. The possibility of distinguishing various points is an axiom 119\n\n 107. The qualitative relations between points, dealt with by\n projective Geometry, are presupposed by the quantitative\n treatment 119\n\n 108. The only qualitative relation between two points is the\n straight line, and all straight lines are qualitatively\n similar 120\n\n 109. Hence follows, by extension, the principle of projective\n transformation 121\n\n 110. By which figures qualitatively indistinguishable from a\n given figure are obtained 122\n\n 111. Anharmonic ratio may and must be descriptively defined 122\n\n 112. The quadrilateral construction is essential to the\n projective definition of points, 123\n\n 113. And can be projectively defined, 124\n\n 114. By the general principle of projective transformation 126\n\n 115. The principle of duality is the mathematical form of a\n philosophical circle, 127\n\n 116. Which is an inevitable consequence of the relativity of\n space, and makes any definition of the point contradictory 128\n\n 117. We define the point as that which is spatial, but contains\n no space, whence other definitions follow 128\n\n 118. What is meant by qualitative equivalence in Geometry? 129\n\n 119. Two pairs of points on one straight line, or two pairs of\n straight lines through one point, are qualitatively equivalent 129\n\n 120. This explains why _four_ collinear points are needed, to give\n an intrinsic relation by which the fourth can be descriptively\n defined when the first three are given 130\n\n 121. Any two projectively related figures are qualitatively\n equivalent, i.e. differ in no non-quantitative conceptual\n property 131\n\n 122. Three axioms are used by projective Geometry, 132\n\n 123. And are required for qualitative spatial comparison, 132\n\n 124. Which involves the homogeneity, relativity and passivity\n of space 133\n\n 125. The conception of a form of externality, 134\n\n 126. Being a creature of the intellect, can be dealt with by\n pure mathematics 134\n\n 127. The resulting doctrine of extension will be, for the moment,\n hypothetical 135\n\n 128. But is rendered assertorical by the necessity, for\n experience, of some form of externality 136\n\n 129. Any such form must be relational 136\n\n 130. And homogeneous 137\n\n 131. And the relations constituting it must appear infinitely\n divisible 137\n\n 132. It must have a finite integral number of dimensions, 139\n\n 133. Owing to its passivity and homogeneity 140\n\n 134. And to the systematic unity of the world 140\n\n 135. A one-dimensional form alone would not suffice for\n experience 141\n\n 136. Since its elements would be immovably fixed in a series 142\n\n 137. Two positions have a relation independent of other\n positions, 143\n\n 138. Since positions are wholly defined by mutually independent\n relations 143\n\n 139. Hence projective Geometry is wholly _\u00e0 priori_, 146\n\n 140. Though metrical Geometry contains an empirical element 146\n\n\n SECTION B. THE AXIOMS OF METRICAL GEOMETRY.\n\n 141. Metrical Geometry is distinct from projective, but has the\n same fundamental postulate 147\n\n 142. It introduces the new idea of motion, and has three\n _\u00e0 priori_ axioms 148\n\n\n I. _The Axiom of Free Mobility._\n\n 143. Measurement requires a criterion of spatial equality 149\n\n 144. Which is given by superposition, and involves the axiom\n of Free Mobility 150\n\n 145. The denial of this axiom involves an action of empty\n space on things 151\n\n 146. There is a mathematically possible alternative to the axiom, 152\n\n 147. Which, however, is logically and philosophically untenable 153\n\n 148. Though Free Mobility is _\u00e0 priori_, actual measurement is\n empirical 154\n\n 149. Some objections remain to be answered, concerning-- 154\n\n 150. (1) The comparison of volumes and of Kant's symmetrical\n objects 154\n\n 151. (2) The measurement of time, where congruence is impossible 156\n\n 152. (3) The immediate perception of spatial magnitude; and 157\n\n 153. (4) The Geometry of non-congruent surfaces 158\n\n 154. Free Mobility includes Helmholtz's Monodromy 159\n\n 155. Free Mobility involves the relativity of space 159\n\n 156. From which, reciprocally, it can be deduced 160\n\n 157. Our axiom is therefore _\u00e0 priori_ in a double sense 160\n\n\n II. _The Axiom of Dimensions._\n\n 158. Space must have a finite integral number of dimensions 161\n\n 159. But the restriction to three is empirical 162\n\n 160. The general axiom follows from the relativity of position 162\n\n 161. The limitation to three dimensions, unlike most empirical\n knowledge, is accurate and certain 163\n\n\n III. _The Axiom of Distance._\n\n 162. The axiom of distance corresponds, here, to that of the\n straight line in projective Geometry 164\n\n 163. The possibility of spatial measurement involves a magnitude\n uniquely determined by two points, 164\n\n 164. Since two points must have some relation, and the passivity\n of space proves this to be independent of external reference 165\n\n 165. There can be only one such relation 166\n\n 166. This must be measured by a curve joining the two points, 166\n\n 167. And the curve must be uniquely determined by the two\n points 167\n\n 168. Spherical Geometry contains an exception to this axiom, 168\n\n 169. Which, however, is not quite equivalent to Euclid's 168\n\n 170. The exception is due to the fact that two points, in\n spherical space, may have an external relation unaltered by\n motion, 169\n\n 171. Which, however, being a relation of linear magnitude,\n presupposes the possibility of linear magnitude 170\n\n 172. A relation between two points must be a line joining them 170\n\n 173. Conversely, the existence of a unique line between two\n points can be deduced from the nature of a form of\n externality, 171\n\n 174. And necessarily leads to distance, when quantity is applied\n to it 172\n\n 175. Hence the axiom of distance, also, is _\u00e0 priori_ in a double\n sense 172\n\n 176. No metrical coordinate system can be set up without the\n straight line 174\n\n 177. No axioms besides the above three are necessary to metrical\n Geometry 175\n\n 178. But these three are necessary to the direct measurement\n of any continuum 176\n\n 179. Two philosophical questions remain for a final chapter 177\n\n\n CHAPTER IV.\n\n PHILOSOPHICAL CONSEQUENCES.\n\n 180. What is the relation to experience of a form of externality\n in general? 178\n\n 181. This form is the class-conception, containing every possible\n intuition of externality; and some such intuition is\n necessary to experience 178\n\n 182. What relation does this view bear to Kant's? 179\n\n 183. It is less psychological, since it does not discuss whether\n space is given in sensation, 180\n\n 184. And maintains that not only space, but any form of\n externality which renders experience possible, must be given in\n sense-perception 181\n\n 185. Externality should mean, not externality to the Self, but\n the mutual externality of presented things 181\n\n 186. Would this be unknowable without a given form of\n externality? 182\n\n 187. Bradley has proved that space and time preclude the\n existence of mere particulars, 182\n\n 188. And that knowledge requires the _This_ to be neither simple\n nor self-subsistent 183\n\n 189. To prove that experience requires a form of externality, I\n assume that all knowledge requires the recognition of identity in\n difference 184\n\n 190. Such recognition involves time 184\n\n 191. And some other form giving simultaneous diversity 185\n\n 192. The above argument has not deduced sense-perception from\n the categories, but has shown the former, unless it contains\n a certain element, to be unintelligible to the latter 186\n\n 193. How to account for the realization of this element, is a\n question for metaphysics 187\n\n 194. What are we to do with the contradictions in space? 188\n\n 195. Three contradictions will be discussed in what follows 188\n\n 196. (1) The antinomy of the Point proves the relativity of\n space, 189\n\n 197. And shows that Geometry must have some reference to\n matter, 190\n\n 198. By which means it is made to refer to spatial order, not\n to empty space 191\n\n 199. The causal properties of matter are irrelevant to Geometry,\n which must regard it as composed of unextended atoms,\n by which points are replaced 191\n\n 200. (2) The circle in defining straight lines and planes is\n overcome by the same reference to matter 192\n\n 201. (3) The antinomy that space is relational and yet more\n than relational, 193\n\n 202. Seems to depend on the confusion of empty space with\n spatial order 193\n\n 203. Kant regarded empty space as the subject-matter of Geometry, 194\n\n 204. But the arguments of the Aesthetic are inconclusive on this\n point, 195\n\n 205. And are upset by the mathematical antinomies, which prove\n that spatial order should be the subject-matter of Geometry 196\n\n 206. The apparent thinghood of space is a psychological illusion,\n due to the fact that spatial relations are immediately given 196\n\n 207. The apparent divisibility of spatial relations is either an\n illusion, arising out of empty space, or the expression of the\n possibility of quantitatively different spatial relations 197\n\n 208. Externality is not a relation, but an aspect of relations.\n Spatial order, owing to its reference to matter, is a real\n relation 198\n\n 209. Conclusion 199\n\n\n\n\nINTRODUCTION.\n\nOUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC, PSYCHOLOGY AND\nMATHEMATICS.\n\n\n=1.= Geometry, throughout the 17th and 18th centuries, remained, in the\nwar against empiricism, an impregnable fortress of the idealists.\nThose who held--as was generally held on the Continent--that certain\nknowledge, independent of experience, was possible about the real\nworld, had only to point to Geometry: none but a madman, they said,\nwould throw doubt on its validity, and none but a fool would deny its\nobjective reference. The English Empiricists, in this matter, had,\ntherefore, a somewhat difficult task; either they had to ignore the\nproblem, or if, like Hume and Mill, they ventured on the assault,\nthey were driven into the apparently paradoxical assertion that\nGeometry, at bottom, had no certainty of a different _kind_ from that\nof Mechanics--only the perpetual presence of spatial impressions,\nthey said, made our experience of the truth of the axioms so wide as\nto seem absolute certainty.\n\nHere, however, as in many other instances, merciless logic drove\nthese philosophers, whether they would or no, into glaring opposition\nto the common sense of their day. It was only through Kant, the\ncreator of modern Epistemology, that the geometrical problem\nreceived a modern form. He reduced the question to the following\nhypotheticals: If Geometry has apodeictic certainty, its matter,\n_i.e._ space, must be _\u00e0 priori_, and as such must be purely\nsubjective; and conversely, if space is purely subjective, Geometry\nmust have apodeictic certainty. The latter hypothetical has more\nweight with Kant, indeed it is ineradicably bound up with his whole\nEpistemology; nevertheless it has, I think, much less force than\nthe former. Let us accept, however, for the moment, the Kantian\nformulation, and endeavour to give precision to the terms _\u00e0 priori_\nand _subjective_.\n\n=2.= One of the great difficulties, throughout this controversy,\nis the extremely variable use to which these words, as well as\nthe word _empirical_, are put by different authors. To Kant, who\nwas nothing of a psychologist, _\u00e0 priori_ and _subjective_ were\nalmost interchangeable terms[1]; in modern usage there is, on the\nwhole, a tendency to confine the word _subjective_ to Psychology,\nleaving _\u00e0 priori_ to do duty for Epistemology. If we accept this\ndifferentiation, we may set up, corresponding to the problems of\nthese two sciences, the following provisional definitions: _\u00e0 priori_\napplies to any piece of knowledge which, though perhaps elicited by\nexperience, is _logically_ presupposed in experience: _subjective_\napplies to any mental state whose immediate cause lies, not in the\nexternal world, but within the limits of the subject. The latter\ndefinition, of course, is framed exclusively for Psychology: from the\npoint of view of physical Science all mental states are subjective.\nBut for a Science whose matter, strictly speaking, is _only_ mental\nstates, we require, if we are to use the word to any purpose,\nsome differentia among mental states, as a mark of a more special\nsubjectivity on the part of those to which this term is applied.\n\nNow the only mental states whose immediate causes lie in the external\nworld are _sensations_. A pure sensation is, of course, an impossible\nabstraction--we are never wholly passive under the action of an\nexternal stimulus--but for the purposes of Psychology the abstraction\nis a useful one. Whatever, then, is not sensation, we shall, in\nPsychology, call subjective. It is in sensation alone that we are\ndirectly affected by the external world, and only here does it give\nus direct information about itself.\n\n=3.= Let us now consider the epistemological question, as to the sort\nof knowledge which can be called _\u00e0 priori_. Here we have nothing to\ndo--in the first instance, at any rate--with the cause or genesis of\na piece of knowledge; we accept knowledge as a datum to be analysed\nand classified. Such analysis will reveal a formal and a material\nelement in knowledge. The formal element will consist of postulates\nwhich are required to make knowledge possible at all, and of all\nthat can be deduced from these postulates; the material element, on\nthe other hand, will consist of all that comes to fill in the form\ngiven by the formal postulates--all that is contingent or dependent\non experience, all that might have been otherwise without rendering\nknowledge impossible. We shall then call the formal element _\u00e0\npriori_, the material element empirical.\n\n=4.= Now what is the connection between the subjective and the _\u00e0\npriori_? It is a connection, obviously--if it exists at all--from the\noutside, _i.e._ not deducible directly from the nature of either,\nbut provable--if it can be proved--only by a general view of the\nconditions of both. The question, what knowledge is _\u00e0 priori_, must,\non the above definition, depend on a logical analysis of knowledge,\nby which the conditions of possible experience may be revealed; but\nthe question, what elements of a cognitive state are subjective, is\nto be investigated by pure Psychology, which has to determine what,\nin our perceptions, belongs to sensation, and what is the work of\nthought or of association. Since, then, these two questions belong\nto different sciences, and can be settled independently, will it not\nbe wise to conduct the two investigations separately? To decree that\nthe _\u00e0 priori_ shall always be subjective, seems dangerous, when we\nreflect that such a view places our results, as to the _\u00e0 priori_, at\nthe mercy of empirical psychology. How serious this danger is, the\ncontroversy as to Kant's pure intuition sufficiently shows.\n\n=5.= I shall, therefore, throughout the present Essay, use the\nword _\u00e0 priori_ without any psychological implication. My test of\napriority will be purely logical: Would experience be impossible, if\na certain axiom or postulate were denied? Or, in a more restricted\nsense, which gives apriority only within a particular science: Would\nexperience as to the subject-matter of that science be impossible,\nwithout a certain axiom or postulate? My results also, therefore,\nwill be purely logical. If Psychology declares that some things,\nwhich I have declared _\u00e0 priori_, are not subjective, then, failing\nan error of detail in my proofs, the connection of the _\u00e0 priori_ and\nthe subjective, so far as those things are concerned, must be given\nup. There will be no discussion, accordingly, throughout this Essay,\nof the relation of the _\u00e0 priori_ to the subjective--a relation which\ncannot determine what pieces of knowledge are _\u00e0 priori_, but rather\ndepends on that determination, and belongs, in any case, rather to\nMetaphysics than to Epistemology.\n\n=6.= As I have ventured to use the word _\u00e0 priori_ in a slightly\nunconventional sense, I will give a few elucidatory remarks of a\ngeneral nature.\n\nThe _\u00e0 priori_, since Kant at any rate, has generally stood for the\nnecessary or apodeictic element in knowledge. But modern logic has\nshown that necessary propositions are always, in one aspect at least,\nhypothetical. There may be, and usually is, an implication that the\nconnection, of which necessity is predicated, has some existence,\nbut still, necessity always points beyond itself to a _ground_ of\nnecessity, and asserts this ground rather than the actual connection.\nAs Bradley points out, \"arsenic poisons\" remains true, even if it\nis poisoning no one. If, therefore, the _\u00e0 priori_ in knowledge be\nprimarily the necessary, it must be the necessary on some hypothesis,\nand the _ground_ of necessity must be included as _\u00e0 priori_. But\nthe ground of necessity is, so far as the necessary connection in\nquestion can show, a mere fact, a merely categorical judgment. Hence\nnecessity alone is an insufficient criterion of apriority.\n\nTo supplement this criterion, we must supply the hypothesis or\nground, on which alone the necessity holds, and this ground will\nvary from one science to another, and even, with the progress of\nknowledge, in the same science at different times. For as knowledge\nbecomes more developed and articulate, more and more necessary\nconnections are perceived, and the merely categorical truths, though\nthey remain the foundation of apodeictic judgments, diminish in\nrelative number. Nevertheless, in a fairly advanced science such as\nGeometry, we can, I think, pretty completely supply the appropriate\nground, and establish, within the limits of the isolated science, the\ndistinction between the necessary and the merely assertorical.\n\n=7.= There are two grounds, I think, on which necessity may be sought\nwithin any science. These may be (very roughly) distinguished as\nthe ground which Kant seeks in the _Prolegomena_, and that which\nhe seeks in the _Pure Reason_. We may start from the existence of\nour science as a fact, and analyse the reasoning employed with a\nview to discovering the fundamental postulate on which its logical\npossibility depends; in this case, the postulate, and all which\nfollows from it alone, will be _\u00e0 priori_. Or we may accept the\nexistence of the subject-matter of our science as our basis of fact,\nand deduce dogmatically whatever principles we can from the essential\nnature of this subject-matter. In this latter case, however, it is\nnot the whole empirical nature of the subject-matter, as revealed by\nthe subsequent researches of our science, which forms our ground; for\nif it were, the whole science would, of course, be _\u00e0 priori_. Rather\nit is that element, in the subject-matter, which makes _possible_\nthe branch of experience dealt with by the science in question[2].\nThe importance of this distinction will appear more clearly as we\nproceed[3].\n\n=8.= These two grounds of necessity, in ultimate analysis, fall\ntogether. The _methods_ of investigation in the two cases differ\nwidely, but the _results_ cannot differ. For in the first case, by\nanalysis of the science, we discover the postulate on which alone\nits reasonings are possible. Now if reasoning in the science is\nimpossible without some postulate, this postulate must be essential\nto experience of the subject-matter of the science, and thus we\nget the second ground. Nevertheless, the two methods are useful as\nsupplementing one another, and the first, as starting from the actual\nscience, is the safest and easiest method of investigation, though\nthe second seems the more convincing for exposition.\n\n=9.= The course of my argument, therefore, will be as follows: In\nthe first chapter, as a preliminary to the logical analysis of\nGeometry, I shall give a brief history of the rise and development\nof non-Euclidean systems. The second chapter will prepare the ground\nfor a constructive theory of Geometry, by a criticism of some\nprevious philosophical views; in this chapter, I shall endeavour\nto exhibit such views as partly true, partly false, and so to\nestablish, by preliminary polemics, the truth of such parts of my own\ntheory as are to be found in former writers. A large part of this\ntheory, however, cannot be so introduced, since the whole field of\nprojective Geometry, so far as I am aware, has been hitherto unknown\nto philosophers. Passing, in the third chapter, from criticism to\nconstruction, I shall deal first with projective Geometry. This, I\nshall maintain, is necessarily true of any form of externality, and\nis, since some such form is necessary to experience, completely _\u00e0\npriori_. In metrical Geometry, however, which I shall next consider,\nthe axioms will fall into two classes: (1) Those common to Euclidean\nand non-Euclidean spaces. These will be found, on the one hand,\nessential to the possibility of measurement in any continuum, and\non the other hand, necessary properties of any form of externality\nwith more than one dimension. They will, therefore, be declared\n_\u00e0 priori_. (2) Those axioms which distinguish Euclidean from\nnon-Euclidean spaces. These will be regarded as wholly empirical.\nThe axiom that the number of dimensions is three, however, though\nempirical, will be declared, since small errors are here impossible,\nexactly and certainly true of our actual world; while the two\nremaining axioms, which determine the value of the space-constant,\nwill be regarded as only approximately known, and certain only\nwithin the errors of observation[4]. The fourth chapter, finally,\nwill endeavour to prove, what was assumed in Chapter III., that some\nform of externality is necessary to experience, and will conclude by\nexhibiting the logical impossibility, if knowledge of such a form is\nto be freed from contradictions, of wholly abstracting this knowledge\nfrom all reference to the matter contained in the form.\n\nI shall hope to have touched, with this discussion, on all the main\npoints relating to the Foundations of Geometry.\n\nFOOTNOTES:\n\n[1] Cf. Erdmann, Axiome der Geometrie, p. 111: \"F\u00fcr Kant sind\nApriorit\u00e4t und ausschliessliche Subjectivit\u00e4t allerdings\nWechselbegriffe.\"\n\n[2] I use \"experience\" here in the widest possible sense, the sense\nin which the word is used by Bradley.\n\n[3] Where the branch of experience in question is essential to all\nexperience, the resulting apriority may be regarded as absolute;\nwhere it is necessary only to some special science, as relative to\nthat science.\n\n[4] I have given no account of these empirical proofs, as they seem\nto be constituted by the whole body of physical science. Everything\nin physical science, from the law of gravitation to the building of\nbridges, from the spectroscope to the art of navigation, would be\nprofoundly modified by any considerable inaccuracy in the hypothesis\nthat our actual space is Euclidean. The observed truth of physical\nscience, therefore, constitutes overwhelming empirical evidence that\nthis hypothesis is very approximately correct, even if not rigidly\ntrue.\n\n\n\n\nCHAPTER I.\n\nA SHORT HISTORY OF METAGEOMETRY.\n\n\n=10.= When a long established system is attacked, it usually happens\nthat the attack begins only at a single point, where the weakness of\nthe established doctrine is peculiarly evident. But criticism, when\nonce invited, is apt to extend much further than the most daring, at\nfirst, would have wished.\n\n \"First cut the liquefaction, what comes last,\n But Fichte's clever cut at God himself?\"\n\nSo it has been with Geometry. The liquefaction of Euclidean orthodoxy\nis the axiom of parallels, and it was by the refusal to admit this\naxiom without proof that Metageometry began. The first effort in\nthis direction, that of Legendre[5], was inspired by the hope of\ndeducing this axiom from the others--a hope which, as we now know,\nwas doomed to inevitable failure. Parallels are defined by Legendre\nas lines in the same plane, such that, if a third line cut them, it\nmakes the sum of the interior and opposite angles equal to two right\nangles. He proves without difficulty that such lines would not meet,\nbut is unable to prove that non-parallel lines in a plane must meet.\nSimilarly he can prove that the sum of the angles of a triangle\ncannot exceed two right angles, and that if any one triangle has a\nsum equal to two right angles, all triangles have the same sum; but\nhe is unable to prove the existence of this one triangle.\n\n=11.= Thus Legendre's attempt broke down; but mere failure could\nprove nothing. A bolder method, suggested by Gauss, was carried out\nby Lobatchewsky and Bolyai[6]. If the axiom of parallels is logically\ndeducible from the others, we shall, by denying it and maintaining\nthe rest, be led to contradictions. These three mathematicians,\naccordingly, attacked the problem indirectly: they denied the axiom\nof parallels, and yet obtained a logically consistent Geometry. They\ninferred that the axiom was logically independent of the others, and\nessential to the Euclidean system. Their works, being all inspired by\nthis motive, may be distinguished as forming the first period in the\ndevelopment of Metageometry.\n\nThe second period, inaugurated by Riemann, had a much deeper import:\nit was largely philosophical in its aims and constructive in its\nmethods. It aimed at no less than a logical analysis of all the\nessential axioms of Geometry, and regarded space as a particular\ncase of the more general conception of a _manifold_. Taking its\nstand on the methods of analytical metrical Geometry, it established\ntwo non-Euclidean systems, the first that of Lobatchewsky, the\nsecond--in which the axiom of the straight line, in Euclid's form,\nwas also denied--a new variety, by analogy called spherical. The\nleading conception in this period is the _measure of curvature_, a\nterm invented by Gauss, but applied by him only to surfaces. Gauss\nhad shown that free mobility on surfaces was only possible when the\nmeasure of curvature was constant; Riemann and Helmholtz extended\nthis proposition to _n_ dimensions, and made it the fundamental\nproperty of space.\n\nIn the third period, which begins with Cayley, the philosophical\nmotive, which had moved the first pioneers, is less apparent, and\nis replaced by a more technical and mathematical spirit. This\nperiod is chiefly distinguished from the second, in a mathematical\npoint of view, by its method, which is projective instead of\nmetrical. The leading mathematical conception here is the Absolute\n(_Grundgebild_), a figure by relation to which all metrical\nproperties become projective. Cayley's work, which was very brief,\nand attracted little attention, has been perfected and elaborated by\nF. Klein, and through him has found general acceptance. Klein has\nadded to the two kinds of non-Euclidean Geometry already known, a\nthird, which he calls elliptic; this third kind closely resembles\nHelmholtz's spherical Geometry, but is distinguished by the important\ndifference that, in it, two straight lines meet in only one point[7].\nThe distinctive mark of the spaces represented by both is that,\nlike the surface of a sphere, they are finite but unbounded. The\nreduction of metrical to projective properties, as will be proved\nhereafter, has only a technical importance; at the same time,\nprojective Geometry is able to deal directly with those purely\ndescriptive or qualitative properties of space which are common to\nEuclid and Metageometry alike. The third period has, therefore, great\nphilosophical importance, while its method has, mathematically, much\ngreater beauty and unity than that of the second; it is able to\ntreat all kinds of space at once, so that every symbolic proposition\nis, according to the meaning given to the symbols, a proposition\nin whichever Geometry we choose. This has the advantage of proving\nthat further research cannot lead to contradictions in non-Euclidean\nsystems, unless it at the same moment reveals contradictions in\nEuclid. These systems, therefore, are logically as sound as that of\nEuclid himself.\n\nAfter this brief sketch of the characteristics of the three periods,\nI will proceed to a more detailed account. It will be my aim to\navoid, as far as possible, all technical mathematics, and bring\ninto relief only those fundamental points in the mathematical\ndevelopment, which seem of logical or philosophical importance.\n\n\nFirst Period.\n\n=12.= The originator of the whole system, _Gauss_, does not appear,\nas regards strictly non-Euclidean Geometry, in any of his hitherto\npublished papers, to have given more than results; his proofs remain\nunknown to us. Nevertheless he was the first to investigate the\nconsequences of denying the axiom of parallels[8], and in his letters\nhe communicated these consequences to some of his friends, among whom\nwas Wolfgang Bolyai. The first mention of the subject in his letters\noccurs when he was only 18; four years later, in 1799, writing to\nW. Bolyai, he enunciates the important theorem that, in hyperbolic\nGeometry, there is a maximum to the area of a triangle. From later\nwritings it appears that he had worked out a system nearly, if not\nquite, as complete as those of Lobatchewsky and Bolyai[9].\n\nIt is important to remember, however, that Gauss's work on curvature,\nwhich _was_ published, laid the foundation for the whole method of\nthe second period, and was undertaken, according to Riemann and\nHelmholtz[10], with a view to an (unpublished) investigation of the\nfoundations of Geometry. His work in this direction will, owing to\nits method, be better treated of under the second period, but it is\ninteresting to observe that he stood, like many pioneers, at the head\nof two tendencies which afterwards diverged.\n\n=13.= _Lobatchewsky_, a professor in the University of Kasan, first\npublished his results, in their native Russian, in the proceedings\nof that learned body for the years 1829-1830. Owing to this double\nobscurity of language and place, they attracted little attention,\nuntil he translated them into French[11] and German[12]: even then,\nthey do not appear to have obtained the notice they deserved, until,\nin 1868, Beltrami unearthed the article in Crelle, and made it the\ntheme of a brilliant interpretation.\n\nIn the introduction to his little German book, Lobatchewsky laments\nthe slight interest shown in his writings by his compatriots, and the\ninattention of mathematicians, since Legendre's abortive attempt, to\nthe difficulties in the theory of parallels. The body of the work\nbegins with the enunciation of several important propositions which\nhold good in the system proposed as well as in Euclid: of these, some\nare in any case independent of the axiom of parallels, while others\nare rendered so by substituting, for the word \"parallel,\" the phrase\n\"not intersecting, however far produced.\" Then follows a definition,\nintentionally framed so as to contradict Euclid's: With respect to\na given straight line, all others in the same plane may be divided\ninto two classes, those which cut the given straight line, and those\nwhich do not cut it; a line which is the limit between the two\nclasses is called _parallel_ to the given straight line. It follows\nthat, from any external point, two parallels can be drawn, one in\neach direction. From this starting-point, by the Euclidean synthetic\nmethod, a series of propositions are deduced; the most important of\nthese is, that in a triangle the sum of the angles is always less\nthan, or always equal to two right angles, while in the latter case\nthe whole system becomes orthodox. A certain analogy with spherical\nGeometry--whose meaning and extent will appear later--is also proved,\nconsisting roughly in the substitution of hyperbolic for circular\nfunctions.\n\n=14.= Very similar is the system of _Johann Bolyai_, so similar,\nindeed, as to make the independence of the two works, though a\nwell-authenticated fact, seem all but incredible. Johann Bolyai\nfirst published his results in 1832, in an appendix to a work by his\nfather Wolfgang, entitled; \"Appendix, scientiam spatii absolute veram\nexhibens: a veritate aut falsitate Axiomatis XI. Euclidei (a priori\nhaud unquam decidenda) independentem; adjecta ad casum falsitatis,\nquadratura circuli geometrica.\" Gauss, whose bosom friend he became\nat college and remained through life, was, as we have seen, the\ninspirer of Wolfgang Bolyai, and used to say that the latter was the\nonly man who appreciated his philosophical speculations on the axioms\nof Geometry; nevertheless, Wolfgang appears to have left to his son\nJohann the detailed working out of the hyperbolic system. The works\nof both the Bolyai are very rare, and their method and results are\nknown to me only through the renderings of Frischauf and Halsted[13].\nBoth as to method and as to results, the system is very similar to\nLobatchewsky's, so that neither need detain us here. Only the initial\npostulates, which are more explicit than Lobatchewsky's, demand a\nbrief attention. Frischauf's introduction, which has a philosophical\nand Newtonian air, begins by setting forth that Geometry deals with\nabsolute (empty) space, obtained by abstracting from the bodies in\nit, that two figures are called congruent when they differ only in\nposition, and that the axiom of Congruence is indispensable in all\ndetermination of spatial magnitudes. Congruence was to refer to\ngeometrical bodies, with none of the properties of ordinary bodies\nexcept impenetrability (Erdmann, Axiome der Geometrie, p. 26). A\nstraight line is defined as determined by two of its points[14],\nand a plane as determined by three. These premisses, with a slight\nexception as to the straight line, we shall hereafter find essential\nto every Geometry. I have drawn attention to them, as it is often\nsupposed that non-Euclideans deny the axiom of Congruence, which,\nhere and elsewhere, is never the case. The stress laid on this axiom\nby Bolyai is probably due to the influence of Gauss, whose work on\nthe curvature of surfaces laid the foundation for the use made of\ncongruence by Helmholtz.\n\n=15.= It is important to remember that, throughout the period we have\njust reviewed, the purpose of hyperbolic Geometry is indirect: not\nthe truth of the latter, but the logical independence of the axiom\nof parallels from the rest, is the guiding motive of the work. If,\nby denying the axiom of parallels while retaining the rest, we can\nobtain a system free from logical contradictions, it follows that\nthe axiom of parallels cannot be implicitly contained in the others.\nIf this be so, attempts to dispense with the axiom, like Legendre's,\ncannot be successful; Euclid must stand or fall with the suspected\naxiom. Of course, it remained possible that, by further development,\nlatent contradictions might have been revealed in these systems. This\npossibility, however, was removed by the more direct and constructive\nwork of the second period, to which we must now turn our attention.\n\n\nSecond Period.\n\n=16.= The work of Lobatchewsky and Bolyai remained, for nearly a\nquarter of a century, without issue--indeed, the investigations\nof Riemann and Helmholtz, when they came, appear to have been\ninspired, not by these men, but rather by Gauss[15] and Herbart. We\nfind, accordingly, very great difference, both of aim and method,\nbetween the first period and the second. The former, beginning\nwith a criticism of one point in Euclid's system, preserved his\nsynthetic method, while it threw over one of his axioms. The latter,\non the contrary, being guided by a philosophical rather than a\nmathematical spirit, endeavoured to classify the conception of\nspace as a species of a more general conception: it treated space\nalgebraically, and the properties it gave to space were expressed\nin terms, not of intuition, but of algebra. The aim of Riemann and\nHelmholtz was to show, by the exhibition of logically possible\nalternatives, the empirical nature of the received axioms. For this\npurpose, they conceived space as a particular case of a manifold,\nand showed that various relations of magnitude (_Massverh\u00e4ltnisse_)\nwere mathematically possible in an extended manifold. Their\nphilosophy, which seems to me not always irreproachable, will be\ndiscussed in Chapter II.; here, while it is important to remember\nthe philosophical motive of Riemann and Helmholtz, we shall confine\nour attention to the mathematical side of their work. In so doing,\nwhile we shall, I fear, somewhat maim the system of their thoughts,\nwe shall secure a closer unity of subject, and a more compact\naccount of the purely mathematical development. But there is, in my\nopinion, a further reason for separating their philosophy from their\nmathematics. While their philosophical purpose was, to prove that all\nthe axioms of Geometry are empirical, and that a different content of\nour experience might have changed them all, the unintended result of\ntheir mathematical work was, if I am not mistaken, to afford material\nfor an _\u00e0 priori_ proof of certain axioms. These axioms, though they\nbelieved them to be unnecessary, were always introduced in their\nmathematical works, before laying the foundations of non-Euclidean\nsystems. I shall contend, in Chapter III., that this retention was\nlogically inevitable, and was not merely due, as they supposed, to a\ndesire for conformity with experience. If I am right in this, there\nis a divergence between Riemann and Helmholtz the philosophers, and\nRiemann and Helmholtz the mathematicians. This divergence makes it\nthe more desirable to trace the mathematical development apart from\nthe accompanying philosophy.\n\n=17.= _Riemann's_ epoch-making work, \"_Ueber die Hypothesen, welche\nder Geometrie zu Grande liegen_[16]\", was written, and read to a\nsmall circle, in 1854; owing, however, to some changes which he\ndesired to make in it, it remained unpublished till 1867, when it\nwas published by his executors. The two fundamental conceptions, on\nwhose invention rests the historic importance of this dissertation,\nare that of a _manifold_, and that of the _measure of curvature_\nof a manifold. The former conception serves a mainly philosophical\npurpose, and is designed, principally, to exhibit space as an\ninstance of a more general conception. On this aspect of the\nmanifold, I shall have much to say in Chapter II.; its mathematical\naspect, which alone concerns us here, is less complicated and less\nfruitful of controversy. The latter conception also serves a double\npurpose, but its mathematical use is the more prominent. We will\nconsider these two conceptions successively.\n\n=18.= (1) _Conception of a manifold[17]._ The general purpose of\nRiemann's dissertation is, to exhibit the axioms as successive steps\nin the classification of the species space. The axioms of Geometry,\nlike the marks of a scholastic definition, appear as successive\ndeterminations of class-conceptions, ending with Euclidean space.\nWe have thus, from the analytical point of view, about as logical\nand precise a formulation as can be desired--a formulation in which,\nfrom its classificatory character, we seem certain of having nothing\nsuperfluous or redundant, and obtain the axioms explicitly in the\nmost desirable form, namely as adjectives of the conception of\nspace. At the same time, it is a pity that Riemann, in accordance\nwith the metrical bias of his time, regarded space as primarily a\nmagnitude[18], or assemblage of magnitudes, in which the main problem\nconsists in assigning quantities to the different elements or points,\nwithout regard to the qualitative nature of the quantities assigned.\nConsiderable obscurity thus arises as to the whole nature of\nmagnitude[19]. This view of Geometry underlies the definition of the\nmanifold, as the general conception of which space forms a special\ncase. This definition, which is not very clear, may be rendered as\nfollows.\n\n=19.= Conceptions of magnitude, according to Riemann, are possible\nthere only, where we have a general conception, capable of various\ndeterminations (_Bestimmungsweisen_). The various determinations of\nsuch a conception together form a _manifold_, which is continuous\nor discrete, according as the passage from one determination to\nanother is continuous or discrete. Particular bits of a manifold, or\nquanta, can be compared by counting when discrete, and by measurement\nwhen continuous. \"Measurement consists in a superposition of the\nmagnitudes to be compared. If this be absent, magnitudes can only be\ncompared when one is part of another, and then only the more or less,\nnot the how much, can be decided\" (p. 256). We thus reach the general\nconception of a manifold of several dimensions, of which space and\ncolours are mentioned as special cases.\n\nTo the absence of this conception Riemann attributes the \"obscurity\"\nwhich, on the subject of the axioms, \"lasted from Euclid to\nLegendre\" (p. 254). And Riemann certainly has succeeded, from an\nalgebraic point of view, in exhibiting, far more clearly than any\nof his predecessors, the axioms which distinguish spatial quantity\nfrom other quantities with which mathematics is conversant. But\nby the assumption, from the start, that space can be regarded as\na quantity, he has been led to state the problem as: What sort of\nmagnitude is space? rather than: What must space be in order that\nwe may be able to regard it as a magnitude at all? He does not\nrealise, either--indeed in his day there were few who realized--that\nan elaborate Geometry is possible which does not deal with space\nas a quantity at all. His definition of space as a species of\nmanifold, therefore, though for analytical purposes it defines, most\nsatisfactorily, the nature of spatial magnitudes, leaves obscure the\ntrue ground for this nature, which lies in the nature of space as a\nsystem of relations, and is anterior to the possibility of regarding\nit as a system of magnitudes at all.\n\nBut to proceed with the mathematical development of Riemann's\nideas. We have seen that he declared measurement to consist in\na superposition of the magnitudes to be compared. But in order\nthat this may be a possible means of determining magnitudes, he\ncontinues, these magnitudes must be independent of their position\nin the manifold (p. 259). This can occur, he says, in several ways,\nas the simplest of which, he assumes that the lengths of lines are\nindependent of their position. One would be glad to know what other\nways are possible: for my part, I am unable to imagine any other\nhypothesis on which magnitude would be independent of place. Setting\nthis aside, however, the problem, owing to the fact that measurement\nconsists in superposition, becomes identical with the determination\nof the most general manifold in which magnitudes are independent of\nplace. This brings us to Riemann's other fundamental conception,\nwhich seems to me even more fruitful than that of a manifold.\n\n=20.= (2) _Measure of curvature._ This conception is due to Gauss,\nbut was applied by him only to surfaces; the novelty in Riemann's\ndissertation was its extension to a manifold of _n_ dimensions.\nThis extension, however, is rather briefly and obscurely expressed,\nand has been further obscured by Helmholtz's attempts at popular\nexposition. The term _curvature_, also, is misleading, so that\nthe phrase has been the source of more misunderstanding, even\namong mathematicians, than any other in Pangeometry. It is often\nforgotten, in spite of Helmholtz's explicit statement[20], that the\n\"measure of curvature\" of an _n_-dimensional manifold is a purely\nanalytical expression, which has only a symbolic affinity to ordinary\ncurvature. As applied to three-dimensional space, the implication\nof a four-dimensional \"plane\" space is wholly misleading; I shall,\ntherefore, generally use the term space-constant instead[21].\nNevertheless, as the conception grew, historically, out of that of\ncurvature, I will give a very brief exposition of the historical\ndevelopment of theories of curvature.\n\nJust as the notion of _length_ was originally derived from the\nstraight line, and extended to other curves by dividing them into\ninfinitesimal straight lines, so the notion of _curvature_ was\nderived from the circle, and extended to other curves by dividing\nthem into infinitesimal circular arcs. Curvature may be regarded,\noriginally, as a measure of the amount by which a curve departs from\na straight line; in a circle, which is similar throughout, this\namount is evidently constant, and is measured by the reciprocal\nof the radius. But in all other curves, the amount of curvature\nvaries from point to point, so that it cannot be measured without\ninfinitesimals. The measure which at once suggests itself is, the\ncurvature of the circle most nearly coinciding with the curve at the\npoint considered. Since a circle is determined by three points, this\ncircle will pass through three consecutive points of the curve. We\nhave thus defined the curvature of any curve, plane or tortuous; for,\nsince any three points lie in a plane, such a circle can always be\ndescribed.\n\nIf we now pass to a surface, what we want is, by analogy, a measure\nof its departure from a plane. The curvature, as above defined, has\nbecome indeterminate, for through any point of the surface we can\ndraw an infinite number of arcs, which will not, in general, all\nhave the same curvature. Let us, then, draw all the geodesics joining\nthe point in question to neighbouring points of the surface in all\ndirections. Since these arcs form a singly infinite manifold, there\nwill be among them, if they have not all the same curvature, one arc\nof maximum, and one of minimum curvature[22]. The product of these\nmaximum and minimum curvatures is called the _measure of curvature_\nof the surface at the point under consideration. To illustrate by a\nfew simple examples: on a sphere, the curvatures of all such lines\nare equal to the reciprocal of the radius of the sphere, hence the\nmeasure of curvature everywhere is the square of the reciprocal\nof the radius of the sphere. On any surface, such as a cone or\na cylinder, on which straight lines can be drawn, these have no\ncurvature, so that the measure of curvature is everywhere zero--this\nis the case, in particular, with the plane. In general, however, the\nmeasure of curvature of a surface varies from point to point.\n\nGauss, the inventor of this conception[23], proved that, in order\nthat two surfaces may be developable upon each other--_i.e._ may\nbe such that one can be bent into the shape of the other without\nstretching or tearing--it is necessary that the two surfaces should\nhave equal measures of curvature at corresponding points. When\nthis is the case, every figure which is possible on the one is, in\ngeneral, possible on the other, and the two have practically the same\nGeometry[24]. As a corollary, it follows that a necessary condition,\nfor the free mobility of figures on any surface, is the constancy\nof the measure of curvature[25]. This condition was proved to be\nsufficient, as well as necessary, by Minding[26].\n\n=21.= So far, all has been plain sailing--we have been dealing with\npurely geometrical ideas in a purely geometrical manner--but we have\nnot, as yet, found any sense of the measure of curvature, in which it\ncan be extended to space, still less to an _n_-dimensional manifold.\nFor this purpose, we must examine Gauss's method, which enables us to\ndetermine the measure of curvature of a surface at any point as an\ninherent property, quite independent of any reference to the third\ndimension.\n\nThe method of determining the measure of curvature from within is,\nbriefly, as follows: If any point on the surface be determined by two\ncoordinates, _u_, _v_, then small arcs of the surface are given by\nthe formula\n\n ds^{2} = Edu^{2} + 2Fdu dv + Gdv^{2},\n\nwhere _E_, _F_, _G_ are, in general, functions of _u_, _v_.[27]\nFrom this formula alone, without reference to any space outside the\nsurface, we can determine the measure of curvature at the point _u_,\n_v_, as a function of _E_, _F_, _G_ and their differentials with\nrespect to _u_ and _v_. Thus we may regard the measure of curvature\nof a surface as an inherent property, and the above geometrical\ndefinition, which involved a reference to the third dimension, may\nbe dropped. But at this point a caution is necessary. It will appear\nin Chap. III. (\u00a7 176), that it is logically impossible to set up a\nprecise coordinate system, in which the coordinates represent spatial\nmagnitudes, without the axiom of Free Mobility, and this axiom, as we\nhave just seen, holds on surfaces only when the measure of curvature\nis constant. Hence our definition of the measure of curvature will\nonly be _really_ free from reference to the third dimension, when we\nare dealing with a surface of constant measure of curvature--a point\nwhich Riemann entirely overlooks. This caution, however, applies\nonly in space, and if we take the coordinate system as presupposed\nin the conception of a manifold, we may neglect the caution\naltogether--while remembering that the possibility of a coordinate\nsystem in space involves axioms to be investigated later. We can thus\nsee how a meaning might be found, without reference to any higher\ndimension, for a constant measure of curvature of three-dimensional\nspace, or for any measure of curvature of an _n_-dimensional manifold\nin general.\n\n=22.= Such a meaning is supplied by Riemann's dissertation, to which,\nafter this long digression, we can now return. We may define a\ncontinuous manifold as any continuum of elements, such that a single\nelement is defined by _n_ continuously variable magnitudes. This\ndefinition does not really include space, for coordinates in space do\nnot define a point, but its relations to the origin, which is itself\narbitrary. It includes, however, the analytical conception of space\nwith which Riemann deals, and may, therefore, be allowed to stand for\nthe moment. Riemann then assumes that the difference--or distance,\nas it may be loosely called--between any two elements is comparable,\nas regards magnitude, to the difference between any other two. He\nassumes further, what it is Helmholtz's merit to have proved, that\nthe difference _ds_ between two consecutive elements can be expressed\nas the square root of a quadratic function of the differences of the\ncoordinates: _i.e._\n\n ds^{2} = \u03a3{1}^{n} \u03a3{1}^{n} a{ik} dx{i}.dx{k},\n\nwhere the coefficients _a{ik}_ are, in general, functions of the\ncoordinates _x{1} x{2} ... x{n}_.[28] The question is: How are\nwe to obtain a definition of the measure of curvature out of this\nformula? It is noticeable, in the first place, that, just as in a\nsurface we found an infinite number of _radii_ of curvature at a\npoint, so in a manifold of three or more dimensions we must find an\ninfinite number of _measures_ of curvature at a point, one for every\ntwo-dimensional manifold passing through the point, and contained\nin the higher manifold. What we have first to do, therefore, is to\ndefine such two-dimensional manifolds. They must consist, as we saw\non the surface, of a singly infinite series of geodesics through the\npoint. Now a geodesic is completely determined by one point and its\ndirection at that point, or by one point and the next consecutive\npoint. Hence a geodesic through the point considered is determined\nby the ratios of the increments of coordinates, _dx{1} dx{2} ...\ndx{n}_. Suppose we have two such geodesics, in which the _i_\u2032th\nincrements are respectively _d\u2032x{i}_ and _d\u2033x{i}_. Then all the\ngeodesics given by\n\n dx{i} = \u03bb\u2032d\u2032x{i} + \u03bb\u2033d\u2033x{i}\n\n[Illustration]\n\nform a singly infinite series, since they contain one parameter,\nnamely \u03bb\u2032: \u03bb\u2033. Such a series of geodesics, therefore, must form a\ntwo-dimensional manifold, with a measure of curvature in the ordinary\nGaussian sense. This measure of curvature can be determined from\nthe above formula for the elementary arc, by the help of Gauss's\ngeneral formula alluded to above. We thus obtain an infinite number\nof measures of curvature at a point, but from n.(n - 1)/2 of these,\nthe rest can be deduced (Riemann, Gesammelte Werke, p. 262). When all\nthe measures of curvature at a point are constant, and equal to all\nthe measures of curvature at any other point, we get what Riemann\ncalls a manifold of constant curvature. In such a manifold free\nmobility is possible, and positions do not differ intrinsically from\none another. If _a_ be the measure of curvature, the formula for the\narc becomes, in this case,\n\n ds^{2} = \u03a3dx^{2}/(1 + a/4 \u03a3x^{2})^{2}.\n\nIn this case only, as I pointed out above, can the term \"measure of\ncurvature\" be properly applied to space without reference to a higher\ndimension, since free mobility is logically indispensable to the\nexistence of quantitative or metrical Geometry.\n\n=23.= The mathematical result of Riemann's dissertation may be summed\nup as follows. Assuming it possible to apply magnitude to space,\n_i.e._ to determine its elements and figures by means of algebraical\nquantities, it follows that space can be brought under the conception\nof a manifold, as a system of quantitatively determinable elements.\nOwing, however, to the peculiar nature of spatial measurement, the\nquantitative determination of space demands that magnitudes shall\nbe independent of place--in so far as this is not the case, our\nmeasurement will be necessarily inaccurate. If we now assume, as\nthe quantitative relation of distance between two elements, the\nsquare root of a quadratic function of the coordinates--a formula\nsubsequently proved by Helmholtz and Lie--then it follows, since\nmagnitudes are to be independent of place, that space must, within\nthe limits of observation, have a constant measure of curvature,\nor must, in other words, be homogeneous in all its parts. In the\ninfinitesimal, Riemann says (p. 267), observation could not detect\na departure from constancy on the part of the measure of curvature;\nbut he makes no attempt to show how Geometry could remain possible\nunder such circumstances, and the only Geometry he has constructed\nis based entirely on Free Mobility. I shall endeavour to prove, in\nChapter III., that any metrical Geometry, which should endeavour to\ndispense with this axiom, would be logically impossible. At present\nI will only point out that Riemann, in spite of his desire to prove\nthat all the axioms can be dispensed with, has nevertheless, in his\nmathematical work, retained three fundamental axioms, namely, Free\nMobility, the finite integral number of dimensions, and the axiom\nthat two points have a unique relation, namely distance. These, as we\nshall see hereafter, are retained, in actual mathematical work, by\nall metrical Metageometers, even when they believe, like Riemann and\nHelmholtz, that no axioms are philosophically indispensable.\n\n=24.= _Helmholtz_, the historically nearest follower of Riemann, was\nguided by a similar empirical philosophy, and arrived independently\nat a very similar method of formulating the axioms. Although\nHelmholtz published nothing on the subject until after Riemann's\ndeath, he had then only just seen Riemann's dissertation (which was\npublished posthumously), and had worked out his results, so far as\nthey were then completed, in entire independence both of Riemann\nand of Lobatchewsky. Helmholtz is by far the most widely read of all\nwriters on Metageometry, and his writings, almost alone, represent\nto philosophers the modern mathematical standpoint on this subject.\nBut his importance is much greater, in this domain, as a philosopher\nthan as a mathematician; almost his only original mathematical\nresult, as regards Geometry, is his proof of Riemann's formula for\nthe infinitesimal arc, and even this proof was far from rigid, until\nLie reformed it by his method of continuous groups. In this chapter,\ntherefore, only two of his writings need occupy us, namely the two\narticles in the _Wissenschaftliche Abhandlungen_, Vol. II., entitled\nrespectively \"Ueber die thats\u00e4chlichen Grundlagen der Geometrie,\"\n1866 (p. 610 ff.), and \"Ueber die Thatsachen, die der Geometrie zum\nGrunde liegen,\" 1868 (p. 618 ff.).\n\n=25.= In the first of these, which is chiefly philosophical,\nHelmholtz gives hints of his then uncompleted mathematical work,\nbut in the main contents himself with a statement of results. He\nannounces that he will prove Riemann's quadratic formula for the\ninfinitesimal arc; but for this purpose, he says, we have to _start_\nwith Congruence, since without it spatial measurement is impossible.\nNevertheless, he maintains that Congruence is proved by experience.\nHow we could, without the help of measurement, discover lapses\nfrom Congruence, is a point which he leaves undiscussed. He then\nenunciates the four axioms which he considers essential to Geometry,\nas follows:\n\n(1) _As regards continuity and dimensions._ In a space of _n_\ndimensions, a point is uniquely determined by the measurement of _n_\ncontinuous variables (coordinates).\n\n(2) _As regards the existence of moveable rigid bodies._ Between the\n2_n_ coordinates of any point-pair of a rigid body, there exists\nan equation which is the same for all congruent point-pairs. By\nconsidering a sufficient number of point-pairs, we get more equations\nthan unknown quantities: this gives us a method of determining the\nform of these equations, so as to make it possible for them all to be\nsatisfied.\n\n(3) _As regards free mobility._ Every point can pass freely and\ncontinuously from one position to another. From (2) and (3) it\nfollows, that if two systems _A_ and _B_ can be brought into\ncongruence in any one position, this is also possible in every other\nposition.\n\n(4) _As regards independence of rotation in rigid bodies_\n(Monodromy). If (_n_ - 1) points of a body remain fixed, so that every\nother point can only describe a certain curve, then that curve is\nclosed.\n\nThese axioms, says Helmholtz, suffice to give, with the axiom of\nthree dimensions, the Euclidean and non-Euclidean systems as the only\nalternatives. That they _suffice_, mathematically, cannot be denied,\nbut they seem, in some respects, to go too far. In the first place,\nthere is no necessity to make the axiom of Congruence apply to actual\nrigid bodies--on this subject I have enlarged in Chapter II.[29]\nAgain, Free Mobility, as distinct from Congruence, hardly needs to\nbe specially formulated: what barrier could empty space offer to a\npoint's progress? The axiom is involved in the homogeneity of space,\nwhich is the same thing as the axiom of Congruence. Monodromy, also,\nhas been severely criticized; not only is it evident that it might\nhave been included in Congruence, but even from the purely analytical\npoint of view, Sophus Lie has proved it to be superfluous[30]. Thus\nthe axiom of Congruence, rightly formulated, includes Helmholtz's\nthird and fourth axioms and part of his second axiom. All the\nfour, or rather, as much of them as is relevant to Geometry, are\nconsequences, as we shall see hereafter, of the one fundamental\nprinciple of the relativity of position.\n\n=26.= The second article, which is mainly mathematical, supplies\nthe promised proof of the arc-formula, which is Helmholtz's most\nimportant contribution to Geometry. Riemann had _assumed_ this\nformula, as the simplest of a number of alternatives: Helmholtz\nproved it to be a necessary consequence of his axioms. The present\npaper begins with a short repetition of the first, including the\nstatement of the axioms, to which, at the end of the paper, two more\nare added, (5) that space has three dimensions, and (6) that space is\ninfinite. It is supposed in the text, as also in the first paper,\nthat the measure of curvature cannot be negative, and, consequently,\nthat an infinite space must be Euclidean. This error in both papers\nis corrected in notes, added after the appearance of Beltrami's paper\non negative curvature. It is a sample of the slightly unprofessional\nnature of Helmholtz's mathematical work on this subject, which\nelicits from Klein the following remarks[31]: \"Helmholtz is not a\nmathematician by profession, but a physicist and physiologist....\nFrom this non-mathematical quality of Helmholtz, it follows naturally\nthat he does not treat the mathematical portion of his work with the\nthoroughness which one would demand of a mathematician by trade (_von\nFach_).\" He tells us himself that it was the physiological study of\nvision which led him to the question of the axioms, and it is as a\nphysicist that he makes his axioms refer to actual rigid bodies.\nAccordingly, we find errors in his mathematics, such as the axiom\nof Monodromy, and the assumption that the measure of curvature must\nbe positive. Nevertheless, the proof of Riemann's arc-formula is\nextremely able, and has, on the whole, been substantiated by Lie's\nmore thorough investigations.\n\n=27.= Helmholtz's other writings on Geometry are almost wholly\nphilosophical, and will be discussed at length in Chapter II. For the\npresent, we may pass to the only other important writer of the second\nperiod, _Beltrami_. As his work is purely mathematical, and contains\nfew controverted points, it need not, despite its great importance,\ndetain us long.\n\nThe \"Saggio di Interpretazione della Geometria non-Euclidea[32],\"\nwhich is principally confined to two dimensions, interprets\nLobatchewsky's results by the characteristic method of the second\nperiod. It shows, by a development of the work of Gauss and\nMinding[33], that all the propositions in plane Geometry, which\nLobatchewsky had set forth, hold, within ordinary Euclidean space,\non surfaces of constant negative curvature. It is strange, as Klein\npoints out[34], that this interpretation, which was known to Riemann\nand perhaps even to Gauss, should have remained so long without\nexplicit statement. This is the more strange, as Lobatchewsky's\n\"G\u00e9om\u00e9trie Imaginaire\" had appeared in Crelle, Vol. XVII.[35],\nand Minding's article, from which the interpretation follows at\nonce, had appeared in Crelle, Vol. XIX. Minding had shewn that the\nGeometry of surfaces of constant negative curvature, in particular\nas regards geodesic triangles, could be deduced from that of the\nsphere by giving the radius a purely imaginary value _ia_[36]. This\nresult, as we have seen, had also been obtained by Lobatchewsky for\nhis Geometry, and yet it took thirty years for the connection to be\nbrought to general notice.\n\n=28.= In Beltrami's Saggio, straight lines are, of course, replaced\nby geodesics; his coordinates are obtained through a point-by-point\ncorrespondence with an auxiliary plane, in which straight lines\ncorrespond to geodesics on the surface. Thus geodesics have linear\nequations, and are always uniquely determined by two points.\nDistances on the surface, however, are not equal to distances on\nthe plane; thus while the surface is infinite, the corresponding\nportion of the plane is contained within a certain finite circle.\nThe distance of two points on the surface is a certain function of\nthe coordinates, not the ordinary function of elementary Geometry.\nThese relations of plane and surface are important in connection with\nCayley's theory of distance, which we shall have to consider next.\nIf we were to define distance on the plane as that function of the\ncoordinates which gives the corresponding distance on the surface, we\nshould obtain what Klein calls \"a plane with a hyperbolic system of\nmeasurement (_Massbestimmung_)\" in which Cayley's theory of distance\nwould hold. It is evident, however, that the ordinary notion of\ndistance has been presupposed in setting up the coordinate system, so\nthat we do not really get alternative Geometries on one and the same\nplane. The bearing of these remarks will appear more fully when we\ncome to consider Cayley and Klein.\n\n=29.= The value of Beltrami's Saggio, in his own eyes, lies in\nthe intelligible Euclidean sense which it gives to Lobatchewsky's\nplanimetry: the corresponding system of Solid Geometry, since it has\nno meaning for Euclidean space, is barely mentioned in this work.\nIn a second paper[37], however, almost contemporaneous with the\nfirst, he proceeds to consider the general theory of _n_-dimensional\nmanifolds of constant negative curvature. This paper is greatly\ninfluenced by Riemann's dissertation; it begins with the formula\nfor the linear element, and proves from this first, that Congruence\nholds for such spaces, and next, that they have, according to\nRiemann's definition, a constant negative measure of curvature. (It\nis instructive to observe, that both in this and in the former Essay,\ngreat stress is laid on the necessity of the Axiom of Congruence.)\n\nThis work has less philosophical interest than the former, since it\ndoes little more than repeat, in a general form, the results which\nthe Saggio had obtained for two dimensions--results which sink,\nwhen extended to _n_ dimensions, to the level of mere mathematical\nconstructions. Nevertheless, the paper is important, both as a\nrestoration of negative curvature, which had been overlooked\nby Helmholtz, and as an analytical treatment of Lobatchewsky's\nresults--a treatment which, together with the Saggio, at last\nrestored to them the prominence they deserved.\n\n\nThird Period.\n\n=30.= The third period differs radically, alike in its methods and\naims, and in the underlying philosophical ideas, from the period\nwhich it replaced. Whereas everything, in the second period, turned\non measurement, with its apparatus of Congruence, Free Mobility,\nRigid Bodies, and the rest, these vanish completely in the third\nperiod, which, swinging to the opposite extreme, regards quantity\nas a perfectly irrelevant category in Geometry, and dispenses with\ncongruence and the method of superposition. The ideas of this period,\nunfortunately, have found no exponent so philosophical as Riemann or\nHelmholtz, but have been set forth only by technical mathematicians.\nMoreover the change of fundamental ideas, which is immense, has\nnot brought about an equally great change in actual procedure; for\nthough spatial quantity is no longer a part of projective Geometry,\nquantity is still employed, and we still have equations, algebraic\ntransformations, and so on. This is apt to give rise to confusion,\nespecially in the mind of the student, who fails to realise that the\nquantities used, so far as the propositions are really projective,\nare mere names for points, and not, as in metrical Geometry, actual\nspatial magnitudes.\n\nNevertheless, the fundamental difference between this period\nand the former must strike any one at once. Whereas Riemann\nand Helmholtz dealt with metrical ideas, and took, as their\nfoundations, the measure of curvature and the formula for the linear\nelement--both purely metrical--the new method is erected on the\nformulae for transformation of coordinates required to express a\ngiven collineation. It begins by reducing all so-called metrical\nnotions--distance, angle, etc.--to projective forms, and obtains,\nfrom this reduction, a methodological unity and simplicity before\nimpossible. This reduction depends, however, except where the\nspace-constant is negative, upon imaginary figures--in Euclid, the\ncircular points at infinity; it is moreover purely symbolic and\nanalytical, and must be regarded as philosophically irrelevant.\nAs the question concerning the import of this reduction is of\nfundamental importance to our theory of Geometry, and as Cayley, in\nhis Presidential Address to the British Association in 1883, formally\nchallenged philosophers to discuss the use of imaginaries, on which\nit depends, I will treat this question at some length. But first let\nus see how, as a matter of mathematics, the reduction is effected.\n\n=31.= We shall find, throughout this period, that almost\nevery important proposition, though misleading in its obvious\ninterpretation, has nevertheless, when rightly interpreted, a wide\nphilosophical bearing. So it is with the work of _Cayley_, the\npioneer of the projective method.\n\nThe projective formula for angles, in Euclidean Geometry, was first\nobtained by Laguerre, in 1853. This formula had, however, a perfectly\nEuclidean character, and it was left for Cayley to generalize\nit so as to include both angles and distances in Euclidean and\nnon-Euclidean systems alike[38].\n\n_Cayley_ was, to the last, a staunch supporter of Euclidean\n_space_, though he believed that non-Euclidean _Geometries_ could\nbe applied, within Euclidean space, by a change in the definition\nof distance[39]. He has thus, in spite of his Euclidean orthodoxy,\nprovided the believers in the possibility of non-Euclidean spaces\nwith one of their most powerful weapons. In his \"Sixth Memoir upon\nQuantics\" (1859), he set himself the task of \"establishing the notion\nof distance upon purely descriptive principles.\" He showed that, with\nthe ordinary notion of distance, it can be rendered projective by\nreference to the circular points and the line at infinity, and that\nthe same is true of angles[40]. Not content with this, he suggested\na new definition of distance, as the inverse sine or cosine of a\ncertain function of the coordinates; with this definition, the\nproperties usually known as metrical become projective properties,\nhaving reference to a certain conic, called by Cayley the Absolute.\n(The circular points are, analytically, a degenerate conic, so that\nordinary Geometry forms a particular case of the above.) He proves\nthat, when the Absolute is an _imaginary_ conic, the Geometry so\nobtained for two dimensions is spherical Geometry. The correspondence\nwith Lobatchewsky, in the case where the Absolute is _real_, is not\nworked out: indeed there is, throughout, no evidence of acquaintance\nwith non-Euclidean systems. The importance of the memoir, to Cayley,\nlies entirely in its proof that metrical is only a branch of\ndescriptive Geometry.\n\n=32.= The connection of Cayley's Theory of Distance with Metageometry\nwas first pointed out by Klein[41]. Klein showed in detail that, if\nthe Absolute be real, we get Lobatchewsky's (hyperbolic) system; if\nit be imaginary, we get either spherical Geometry or a new system,\nanalogous to that of Helmholtz, called by Klein elliptic; if the\nAbsolute be an imaginary point-pair, we get parabolic Geometry, and\nif, in particular, the point-pair be the circular points, we get\nordinary Euclid. In elliptic Geometry, two straight lines in the same\nplane meet in only one point, not two as in Helmholtz's system. The\ndistinction between the two kinds of Geometry is difficult, and will\nbe discussed later.\n\n=33.= Since these systems are all obtained from a Euclidean plane,\nby a mere alteration in the definition of distance, Cayley and\nKlein tend to regard the whole question as one, not of the nature\nof space, but of the definition of distance. Since this definition,\non their view, is perfectly arbitrary, the philosophical problem\nvanishes--Euclidean _space_ is left in undisputed possession, and\nthe only problem remaining is one of convention and mathematical\nconvenience[42]. This view has been forcibly expressed by Poincar\u00e9:\n\"What ought one to think,\" he says, \"of this question: Is the\nEuclidean Geometry true? The question is nonsense.\" Geometrical\naxioms, according to him, are mere conventions: they are \"definitions\nin disguise[43].\" Thus Klein blames Beltrami for regarding his\nauxiliary plane as merely auxiliary, and remarks that, if he had\nknown Cayley's Memoir, he would have seen the relation between\nthe plane and the pseudosphere to be far more intimate than he\nsupposed[44]. A view which removes the problem entirely from the\narena of philosophy demands, plainly, a full discussion. To this\ndiscussion we will now proceed.\n\n=34.= The view in question has arisen, it would seem, from a natural\nconfusion as to the nature of the coordinates employed. Those who\nhold the view have not adequately realised, I believe, that their\ncoordinates are not _spatial_ quantities, as in metrical Geometry,\nbut mere conventional signs, by which different points can be\ndistinctly designated. There is no reason, therefore, until we\nalready have metrical Geometry, for regarding one function of the\ncoordinates as a better expression of distance than another, so\nlong as the fundamental addition-equation[45] is preserved. Hence,\nif our coordinates are regarded as adequate for all Geometry, an\nindeterminateness arises in the expression of distance, which can\nonly be avoided by a convention. But projective coordinates--so our\nargument will contend--though perfectly adequate for all projective\nproperties, and entirely free from any metrical presupposition, are\ninadequate to express metrical properties, just because they have\nno metrical presupposition. Thus where metrical properties are in\nquestion, Beltrami remains justified as against Klein; the reduction\nof metrical to projective properties is only apparent, though\nthe independence of these last, as against metrical Geometry, is\nperfectly real.\n\n=35.= But what are projective coordinates, and how are they\nintroduced? This question was not touched upon in Cayley's Memoir,\nand it seemed, therefore, as if a logical error were involved in\nusing coordinates to define distance. For coordinates, in all\nprevious systems, had been deduced from distance; to use any existing\ncoordinate system in defining distance was, accordingly, to incur\na vicious circle. Cayley mentions this difficulty in a note, where\nhe only remarks, however, that he had regarded his coordinates\nas numbers arbitrarily assigned, on some system not further\ninvestigated, to different points. The difficulty has been treated at\nlength by Sir R. Ball (Theory of the Content, Trans. R. I. A. 1889),\nwho urges that if the values of our coordinates already involve the\nusual measure of distance, then to give a new definition, while\nretaining the usual coordinates, is to incur a contradiction. He\nsays (op. cit. p. 1): \"In the study of non-Euclidean Geometry I have\noften felt a difficulty which has, I know, been shared by others. In\nthat theory it seems as if we try to replace our ordinary notion of\ndistance between two points by the logarithm of a certain anharmonic\nratio[46]. But this ratio itself involves the notion of distance\nmeasured in the ordinary way. How, then, can we supersede our old\nnotion of distance by the non-Euclidean notion, inasmuch as the very\ndefinition of the latter involves the former?\"\n\n=36.= This objection is valid, we must admit, so long as anharmonic\nratio is defined in the ordinary metrical manner. It would be\nvalid, for example, against any attempt to found a new definition\nof distance on Cremona's account of anharmonic ratio[47], in\nwhich it appears as a metrical property unaltered by projective\ntransformation. If a logical error is to be avoided, in fact, all\nreference to spatial magnitude of any kind must be avoided; for all\nspatial magnitude, as will be shown hereafter[48], is logically\ndependent on the fundamental magnitude of distance. Anharmonic\nratio and coordinates must alike be defined by purely descriptive\nproperties, if the use afterwards made of them is to be free from\nmetrical presuppositions, and therefore from the objections of Sir R.\nBall.\n\nSuch a definition has been satisfactorily given by Klein[49],\nwho appeals, for the purpose, to v. Staudt's quadrilateral\nconstruction[50]. By this construction, which I have reproduced in\noutline in Chapter III. Section A, \u00a7 112 ff., we obtain a purely\ndescriptive definition of harmonic and anharmonic ratio, and, given\na pair of points, we can obtain the harmonic conjugate to any\nthird point on the same straight line. On this construction, the\nintroduction of projective coordinates is based. Starting with any\nthree points on a straight line, we assign to them arbitrarily the\nnumbers 0, 1, \u221e. We then find the harmonic conjugate to the first\nwith respect to 1, \u221e, and assign to it the number 2. The object of\nassigning this number rather than any other, is to obtain the value\n-1 for the anharmonic ratio of the four numbers corresponding to the\nfour points[51]. We then find the harmonic conjugate to the point\n1, with respect to 2, \u221e, and assign to it the number 3; and so on.\nKlein has shown that by this construction, we can obtain any number\nof points, and can construct a point corresponding to any given\nnumber, fractional or negative. Moreover, when two sets of four\npoints have the same anharmonic ratio, descriptively defined[52],\nthe corresponding numbers also have the same anharmonic ratio. By\nintroducing such a numerical system on two straight lines, or on\nthree, we obtain the coordinates of any point in a plane, or in\nspace. By this construction, which is of fundamental importance\nto projective Geometry, the logical error, upon which Sir R. Ball\nbases his criticism, is satisfactorily avoided. Our coordinates\nare introduced by a purely descriptive method, and involve no\npresupposition whatever as to the measurement of distance.\n\n=37.= With this coordinate system, then, to define distance as a\ncertain function of the coordinates is not to be guilty of a vicious\ncircle. But it by no means follows that the definition of distance is\narbitrary. All reference to distance has been hitherto excluded, to\navoid metrical ideas; but when distance is introduced, metrical ideas\ninevitably reappear, and we have to remember that our coordinates\ngive no information, _prim\u00e2 facie_, as to any of these metrical\nideas. It is open to us, of course, if we choose, to continue to\nexclude distance in the ordinary sense, as the quantity of a finite\nstraight line, and to define the _word_ distance in any way we\nplease. But the conception, for which the word has hitherto stood,\nwill then require a new name, and the only result will be a confusion\nbetween the _apparent_ meaning of our propositions, to those who\nretain the associations belonging to the old sense of the word, and\nthe _real_ meaning, resulting from the new sense in which the word is\nused.\n\nThis confusion, I believe, has actually occurred, in the case of\nthose who regard the question between Euclid and Metageometry as\none of the definition of distance. Distance is a quantitative\nrelation, and as such presupposes identity of quality. But projective\nGeometry deals only with quality--for which reason it is called\ndescriptive--and cannot distinguish between two figures which are\nqualitatively alike. Now the meaning of qualitative likeness,\nin Geometry, is the possibility of mutual transformation by a\ncollineation[53]. Any two pairs of points on the same straight line,\ntherefore, are qualitatively alike; their only qualitative relation\nis the straight line, which both pairs have in common; and it is\nexactly the qualitative identity of the relations of the two pairs,\nwhich enables the difference of their relations to be exhaustively\ndealt with by quantity, as a difference of distance. But where\nquantity is excluded, any two pairs of points on the same straight\nline appear as alike, and even any two sets of three: for any three\npoints on a straight line can be projectively transformed into any\nother three. It is only with _four_ points in a line that we acquire\na projective property distinguishing them from other sets of four,\nand this property is anharmonic ratio, descriptively defined. The\nprojective Geometer, therefore, sees no reason to give a name to the\nrelation between two points, in so far as this relation is anything\nover and above the unlimited straight line on which they lie; and\nwhen he introduces the notion of distance, he defines it, in the\nonly way in which projective principles allow him to define it, as a\nrelation between _four_ points. As he nevertheless wishes the word\nto give him the power of distinguishing between different _pairs_\nof points, he agrees to take two out of the four points as fixed.\nIn this way, the only variables in distance are the two remaining\npoints, and distance appears, therefore, as a function of _two_\nvariables, namely the coordinates of the two variable points. When we\nhave further defined our function so that distance may be additive,\nwe have a function with many of the properties of distance in the\nordinary sense. This function, therefore, the projective Geometer\nregards as the only proper definition of distance.\n\nWe can see, in fact, from the manner in which our projective\ncoordinates were introduced, that _some_ function of these\ncoordinates must express distance in the ordinary sense. For\nthey were introduced serially, so that, as we proceeded from the\nzero-point towards the infinity-point, our coordinates continually\ngrew. To every point, a definite coordinate corresponded: to the\ndistance between two variable points, therefore, as a function\ndependent on no other variables, must correspond some definite\nfunction of the coordinates, since these are themselves functions of\ntheir points. The function discussed above, therefore, must certainly\ninclude distance in the ordinary sense.\n\nBut the arbitrary and conventional nature of distance, as maintained\nby Poincar\u00e9 and Klein, arises from the fact that the two fixed\npoints, required to determine our distance in the projective sense,\nmay be arbitrarily chosen, and although, when our choice is once\nmade, any two points have a definite distance, yet, according as we\nmake that choice, distance will become a different function of the\ntwo variable points. The ambiguity thus introduced is unavoidable on\nprojective principles; but are we to conclude, from this, that it\nis really unavoidable? Must we not rather conclude that projective\nGeometry cannot adequately deal with distance? If _A_, _B_, _C_, be\nthree different points on a line, there must be _some_ difference\nbetween the relation of _A_ to _B_ and of _A_ to _C_, for otherwise,\nowing to the qualitative identity of all points, _B_ and _C_ could\nnot be distinguished. But such a difference involves a relation,\nbetween _A_ and _B_, which is independent of other points on the\nline; for unless we have such a relation, the other points cannot\nbe distinguished as different. Before we can distinguish the two\nfixed points, therefore, from which the projective definition\nstarts, we must already suppose some relation, between any two\npoints on our line, in which they are independent of other points;\nand this relation is distance in the ordinary sense[54]. When we\nhave measured this quantitative relation by the ordinary methods\nof metrical Geometry, we can proceed to decide what base-points\nmust be chosen, on our line, in order that the projective function\ndiscussed above may have the same value as ordinary distance. But\nthe choice of these base-points, when we are discussing distance\nin the ordinary sense, is not arbitrary, and their introduction is\nonly a technical device. Distance, in the ordinary sense, remains\na relation between _two_ points, not between _four_; and it is the\nfailure to perceive that the projective sense differs from, and\ncannot supersede, the ordinary sense, which has given rise to the\nviews of Klein and Poincar\u00e9. The question is not one of convention,\nbut of the irreducible metrical properties of space. To sum up:\nQuantities, as used in projective Geometry, do not stand for spatial\nmagnitudes, but are conventional symbols for purely qualitative\nspatial relations. But distance, _qu\u00e2_ quantity, presupposes identity\nof quality, as the condition of quantitative comparison. Distance in\nthe ordinary sense is, in short, that quantitative relation, between\ntwo points on a line, by which their difference from other points\ncan be defined. The projective definition, however, being unable to\ndistinguish a collection of less than four points from any other on\nthe same straight line, makes distance depend on two other points\nbesides those whose relation it defines. No name remains, therefore,\nfor distance in the ordinary sense, and many projective Geometers,\nhaving abolished the name, believe the thing to be abolished also,\nand are inclined to deny that _two_ points have a unique relation at\nall. This confusion, in projective Geometry, shows the importance of\na name, and should make us chary of allowing new meanings to obscure\none of the fundamental properties of space.\n\n=38.= It remains to discuss the manner in which non-Euclidean\nGeometries result from the projective definition of distance, as also\nthe true interpretation to be given to this view of Metageometry. It\nis to be observed that the projective methods which follow Cayley\ndeal throughout with a Euclidean plane, on which they introduce\ndifferent measures of distance. Hence arises, in any interpretation\nof these methods, an apparent subordination of the non-Euclidean\nspaces, as though these were less self-subsistent than Euclid's.\nThis subordination is not intended in what follows; on the contrary,\nthe correlation with Euclidean space is regarded as valuable,\nfirst, because Euclidean space has been longer studied and is more\nfamiliar, but secondly, because this correlation proves, when truly\ninterpreted, that the other spaces are self-subsistent. We may\nconfine ourselves chiefly, in discussing this interpretation, to\ndistances measured along a single straight line. But we must be\ncareful to remember that the metrical definition of distance--which,\naccording to the view here advocated, is the only adequate\ndefinition--is the same in Euclidean and in non-Euclidean spaces; to\nargue in its favour is not, therefore, to argue in favour of Euclid.\n\nThe projective scheme of coordinates consists of a series of numbers,\nof which each represents a certain anharmonic ratio and denotes one\nand only one point, and which increase uniformly with the distance\nfrom a fixed origin, until they become infinite on reaching a certain\npoint. Now Cayley showed that, in Euclidean Geometry, distance may\nbe expressed as the limit of the logarithm of the anharmonic ratio\nof the two points and the (coincident) points at infinity on their\nstraight line; while, if we assumed that the points at infinity were\ndistinct, we obtained the formula for distance in hyperbolic or\nspherical Geometry, according as these points were real or imaginary.\nHence it follows that, with the projective definition of distance,\nwe shall obtain precisely the formulae of hyperbolic, parabolic or\nspherical Geometry, according as we choose the point, to which the\nvalue +\u221e is assigned, at a finite, infinite or imaginary distance\n(in the ordinary sense) from the point to which we assign the value\n0. Our straight line remains, all the while, an ordinary Euclidean\nstraight line. But we have seen that the projective definition of\ndistance fits with the true definition only when the two fixed\npoints to which it refers are suitably chosen. Now the ordinary\nmeaning of distance is required in non-Euclidean as in Euclidean\nGeometries--indeed, it is only in metrical properties that these\nGeometries differ. Hence our _Euclidean_ straight line, though it\nmay serve to illustrate other Geometries than Euclid's, can only be\ndealt with correctly by Euclid. Where we give a different definition\nof distance from Euclid's, we are still in the domain of purely\nprojective properties, and derive no information as to the metrical\nproperties of our straight line. But the importance, to Metageometry,\nof this new interpretation, lies in the fact that, having\nindependently established the metrical formulae of non-Euclidean\nspaces, we find, as in Beltrami's Saggio, that these spaces can\nbe related, by a homographic correspondence, with the points of\nEuclidean space; and that this can be effected in such a manner as\nto give, for the distance between two points of our non-Euclidean\nspace, the hyperbolic or spherical measure of distance for the\ncorresponding points of Euclidean space.\n\n=39.= On the whole, then, a modification of Sir R. Ball's view,\nwhich is practically a generalized statement of Beltrami's method,\nseems the most tenable. He imagines what, with Grassmann, he calls\na Content, _i.e._ a perfectly general three-dimensional manifold,\nand then correlates its elements, one by one, with points in\nEuclidean space. Thus every element of the Content acquires, as its\ncoordinates, the ordinary Euclidean coordinates of the corresponding\npoint in Euclidean space. By means of this correlation, our\ncalculations, though they refer to the Content, are carried on, as in\nBeltrami's Saggio, in ordinary Euclidean space. Thus the confusion\ndisappears, but with it, the supposed Euclidean interpretation also\ndisappears. Sir R. Ball's Content, if it is to be a space at all,\nmust be a space radically different from Euclid's[55]; to speak, as\nKlein does, of ordinary planes with hyperbolic or elliptic measures\nof distance, is either to incur a contradiction, or to forego any\nmetrical meaning of distance. Instead of ordinary planes, we have\nsurfaces like Beltrami's, of constant measure of curvature; instead\nof Euclid's space, we have hyperbolic or spherical space. At the\nsame time, it remains true that we can, by Klein's method, give a\nEuclidean meaning to every symbolic proposition in non-Euclidean\nGeometry. For by substituting, for distance, the logarithm above\nalluded to, we obtain, from the non-Euclidean result, a result which\nfollows from the ordinary Euclidean axioms. This correspondence\nremoves, once for all, the possibility of a lurking contradiction in\nMetageometry, since, to a proposition in the one, corresponds one\nand only one proposition in the other, and contradictory results in\none system, therefore, would correspond to contradictory results in\nthe other. Hence Metageometry cannot lead to contradictions, unless\nEuclidean Geometry, at the same moment, leads to corresponding\ncontradictions. Thus the Euclidean plane with hyperbolic or elliptic\nmeasure of distance, though either contradictory or not metrical\nas an independent notion, has, as a help in the interpretation of\nnon-Euclidean results, a very high degree of utility.\n\n=40.= We have still to discuss Klein's third kind of non-Euclidean\nGeometry, which he calls elliptic. The difference between this and\nspherical Geometry is difficult to grasp, but it may be illustrated\nby a simpler example. A plane, as every one knows, can be wrapped,\nwithout stretching, on a cylinder, and straight lines in the plane\nbecome, by this operation, geodesics on the cylinder. The Geometries\nof the plane and the cylinder, therefore, have much in common.\nBut since the generating circle of the cylinder, which is one of\nits geodesics, is finite, only a portion of the plane is used up\nin wrapping it once round the cylinder. Hence, if we endeavour to\nestablish a point-to-point correspondence between the plane and the\ncylinder, we shall find an infinite series of points on the plane\nfor a single point on the cylinder. Thus it happens that geodesics,\nthough on the plane they have only one point in common, may on\nthe cylinder have an infinite number of intersections. Somewhat\nsimilar to this is the relation between the spherical and elliptic\nGeometries. To any one point in elliptic space, two points correspond\nin spherical space. Thus geodesics, which in spherical space may have\ntwo points in common, can never, in elliptic space, have more than\none intersection.\n\nBut Klein's method can only prove that elliptic Geometry holds of\nthe ordinary Euclidean plane with elliptic measure of distance.\nKlein has made great endeavours to enforce the distinction between\nthe spherical and elliptic Geometries[56], but it is not immediately\nevident that the latter, as distinct from the former, is valid.\n\nIn the first place, Klein's elliptic Geometry, which arises as\none of the alternative metrical systems on a Euclidean plane or\nin a Euclidean space, does not by itself suffice, if the above\ndiscussion has been correct, to prove the possibility of an elliptic\nspace, _i.e._ of a space having a point-to-point correspondence\nwith the Euclidean space, and having as the ordinary distance\nbetween two of its points the elliptic definition of the distance\nbetween corresponding points of the Euclidean space. To prove this\npossibility, we must adopt the direct method of Newcomb (Crelle's\nJournal, Vol. 83). Now in the first place Newcomb has not proved that\nhis postulates are self-consistent; he has only failed to prove that\nthey are contradictory[57]. This would leave elliptic space in the\nsame position in which Lobatchewsky and Bolyai left hyperbolic space.\nBut further there seems to be, at first sight, in _two_-dimensional\nelliptic space, a positive contradiction. To explain this, however,\nsome account of the peculiarities of the elliptic plane will be\nnecessary.\n\n[Illustration]\n\nThe elliptic plane, regarded as a figure in three-dimensional\nelliptic space, is what is called a double surface[58], _i.e._ as\nNewcomb says (_loc. cit._ p. 298): \"The two sides of a complete plane\nare not distinct, as in a Euclidean surface.... If ... a being should\ntravel to distance 2_D_, he would, on his return, find himself on\nthe opposite surface to that on which he started, and would have\nto repeat his journey in order to return to his original position\nwithout leaving the surface.\" Now if we imagine a _two_-dimensional\nelliptic space, the distinction between the sides of a plane becomes\nunmeaning, since it only acquires significance by reference to the\nthird dimension. Nevertheless, some such distinction would be forced\nupon us. Suppose, for example, that we took a small circle provided\nwith an arrow, as in the figure, and moved this circle once round the\nuniverse. Then the sense of the arrow would be reversed. We should\nthus be forced, either to regard the new position as distinct from\nthe former, which transforms our plane into a spherical plane, or to\nattribute the reversal of the arrow to the action of a motion which\nrestores our circle to its original place. It is to be observed that\nnothing short of moving round the universe would suffice to reverse\nthe sense of the arrow. This reversal _seems_ like an action of\nempty space, which would force us to regard the points which, from a\nthree-dimensional point of view, are coincident though opposite, as\nreally distinct, and so reduce the elliptic to the spherical plane.\nBut motion, not space, really causes the change, and the elliptic\nplane is therefore not proved to be impossible. The question is not,\nhowever, of any great philosophic importance.\n\n=41.= In connection with the reduction of metrical to projective\nGeometry, we have one more topic for discussion. This is the\ngeometrical use of imaginaries, by means of which, except in the\ncase of hyperbolic space, the reduction is effected. I have already\ncontended, on other grounds, that this reduction, in spite of its\nimmense technical importance, and in spite of the complete logical\nfreedom of projective Geometry from metrical ideas, is _purely_\ntechnical, and is not philosophically valid. The same conclusion will\nappear, if we take up Cayley's challenge at the British Association,\nin his Presidential Address of 1883.\n\nIn this address, Professor Cayley devoted most of his time to\nnon-Euclidean systems. Non-Euclidean _spaces_, he declared, seemed to\nhim mistaken _\u00e0 priori_[59]; but non-Euclidean _Geometries_, here as\nin his mathematical works, were accepted as flowing from a change in\nthe definition of distance. This view has been already discussed, and\nneed not, therefore, be further criticised here. What I wish to speak\nabout, is the question with which Cayley himself opened his address,\nnamely, the geometrical use and meaning of imaginary quantities. From\nthe manner in which he spoke of this question, it becomes imperative\nto treat it somewhat at length. For he said (pp. 8-9):\n\n\"... The notion which is the really fundamental one (and I cannot\ntoo strongly emphasize the assertion) underlying and pervading the\nwhole notion of modern analysis and Geometry, [is] that of imaginary\nmagnitude in analysis, and of imaginary space (or space as the _locus\nin quo_ of imaginary points and figures) in Geometry: I use in each\ncase the word imaginary as including real.... Say even the conclusion\nwere that the notion belongs to mere technical mathematics, or has\nreference to nonentities in regard to which no science is possible,\nstill it seems to me that (as a subject of philosophical discussion)\nthe notion ought not to be thus ignored; it should at least be shown\nthat there is a right to ignore it.\"\n\n=42.= This right it is now my purpose to demonstrate. But for fear\nnon-mathematicians should miss the point of Cayley's remark (which\nhas sometimes been erroneously supposed to refer to non-Euclidean\nspaces), I may as well explain, at the outset, that this question\nis radically distinct from, and only indirectly connected with, the\nvalidity or import of Metageometry. An imaginary quantity is one\nwhich involves \u221a-1: its most general form is _a_ + \u221a-1_b_ where _a_\nand _b_ are real; Cayley uses the word imaginary so as to include\nreal, in order to cover the special case where _b_ = 0. It will\nbe convenient, in what follows, to exclude this wider meaning,\nand assume that _b_ is not zero. An imaginary point is one whose\ncoordinates involve \u221a-1, _i.e._ whose coordinates are imaginary\nquantities. An imaginary curve is one whose points are imaginary--or,\nin some special uses, one whose equation contains imaginary\ncoefficients. The mathematical subtleties to which this notion leads\nneed not be here discussed; the reader who is interested in them will\nfind an excellent elementary account of their geometrical uses in\nKlein's Nicht-Euklid, II. pp. 38-46. But for our present purpose, we\nmay confine ourselves to imaginary points. If these are found to have\na merely technical import, and to be destitute of any philosophical\nmeaning, then the same will hold of any collection of imaginary\npoints, _i.e._ of any imaginary curve or surface.\n\nThat the notion of imaginary points is of supreme importance in\nGeometry, will be seen by any one who reflects that the circular\npoints are imaginary, and that the reduction of metrical to\nprojective Geometry, which is one of Cayley's greatest achievements,\ndepends on these points. But to discuss adequately their\nphilosophical import is difficult to me, since I am unacquainted with\nany satisfactory philosophy of imaginaries in pure Algebra. I will\ntherefore adopt the most favourable hypothesis, and assume that no\nobjection can be successfully urged against this use. Even on this\nhypothesis, I think, no case can be made out for imaginary points in\nGeometry.\n\nIn the first place, we must exclude, from the imaginary points\nconsidered, those whose coordinates are only imaginary with certain\nspecial systems of coordinates. For example, if one of a point's\ncoordinates be the tangent from it to a sphere, this coordinate will\nbe imaginary for any point inside the sphere, and yet the point is\nperfectly real. A point, then, is only to be called imaginary, when,\nwhatever real system of coordinates we adopt, one or more of the\nquantities expressing these coordinates remains imaginary. For this\npurpose, it is mathematically sufficient to suppose our coordinates\nCartesian--a point whose Cartesian coordinates are imaginary, is a\ntrue imaginary point in the above sense.\n\nTo discuss the meaning of such a point, it is necessary to consider\nbriefly the fundamental nature of the correspondence between a\npoint and its coordinates. Assuming that elementary Geometry has\nproved--what I think it does satisfactorily prove--that spatial\nrelations are susceptible of quantitative measurement, then a given\npoint will have, with a suitable system of coordinates, in a space\nof _n_ dimensions, _n_ quantitative relations to the fixed spatial\nfigure forming the axes of coordinates, and these _n_ quantitative\nrelations will, under certain reservations, be unique--_i.e._, no\nother point will have the same quantities assigned to it. (With many\npossible coordinate systems, this latter condition is not realized:\nbut for that very reason they are inconvenient, and employed only in\nspecial problems.) Thus given a coordinate system, and given any set\nof quantities, these quantities, _if they determine a point at all_,\ndetermine it uniquely. But, by a natural extension of the method, the\nabove reservation is dropped, and it is assumed that to _every_ set\nof quantities some point must correspond. For this assumption there\nseems to me no vestige of evidence. As well might a postman assume\nthat, because every house in a street is uniquely determined by its\nnumber, therefore there must be a house for every imaginable number.\nWe must know, in fact, that a given set of quantities can be the\ncoordinates of some point in space, before it is legitimate to give\nany spatial significance to these quantities: and this knowledge,\nobviously, cannot be derived from operations with coordinates alone,\non pain of a vicious circle. We must, to return to the above analogy,\nknow the number of houses in Piccadilly, before we know whether a\ngiven number has a corresponding house or not; and arithmetic alone,\nhowever subtly employed, will never give us this information.\n\nThus the distinction which is important is, not the distinction\nbetween real and imaginary quantities, but between quantities to\nwhich points correspond and quantities to which no points correspond.\nWe can conventionally agree to denote real points by imaginary\ncoordinates, as in the Gaussian method of denoting by the single\nquantity (_a_ + \u221a-1_b_) the point whose ordinary coordinates are\n_a_, _b_. But this does not touch Cayley's meaning. Cayley means\nthat it is of great utility in mathematics to regard, as points\nwith a real existence in space, the assumed spatial correlates of\nquantities which, with the coordinate system employed, have no\ncorrelates in every-day space; and that this utility is supposed,\nby many mathematicians, to indicate the validity of so fruitful an\nassumption. To fix our ideas, let us consider Cartesian axes in\nthree-dimensional Euclidean space. Then it appears, by inspection,\nthat a point may be situated at any distance to right or left of any\nof the three coordinate planes; taking this distance as a coordinate,\ntherefore, it appears that real points correspond to all quantities\nfrom -\u221e to +\u221e. The same appears for the other two coordinates;\nand since elementary Geometry proves their variations mutually\nindependent, we know that one and only one real point corresponds\nto any three real quantities. But we also know, from the exhaustive\nmethod pursued, that all space is covered by the range of these three\nvariable quantities: a fresh set of quantities, therefore, such as is\nintroduced by the use of imaginaries, possesses no spatial correlate,\nand can be supposed to possess one only by a convenient fiction.\n\n=43.= The fact that the fiction _is_ convenient, however, may\nbe thought to indicate that it is more than a fiction. But this\npresumption, I think, can be easily explained away. For all the\nfruitful uses of imaginaries, in Geometry, are those which begin\nand end with real quantities, and use imaginaries only for the\nintermediate steps. Now in all such cases, we have a real spatial\ninterpretation at the beginning and end of our argument, where alone\nthe spatial interpretation is important: in the intermediate links,\nwe are dealing in a purely algebraical manner with purely algebraical\nquantities, and may perform any operations which are algebraically\npermissible. If the quantities with which we end are capable of\nspatial interpretation, then, and only then, our result may be\nregarded as geometrical. To use geometrical language, in any other\ncase, is only a convenient help to the imagination. To speak, for\nexample, of projective properties which refer to the circular points,\nis a mere _memoria technica_ for purely algebraical properties;\nthe circular points are not to be found in space, but only in the\nauxiliary quantities by which geometrical equations are transformed.\nThat no contradictions arise from the geometrical interpretation of\nimaginaries, is not wonderful: for they are interpreted solely by the\nrules of Algebra, which we may admit as valid in their application\nto imaginaries. The perception of space being wholly absent,\nAlgebra rules supreme, and no inconsistency can arise. Wherever,\nfor a moment, we allow our ordinary spatial notions to intrude, the\ngrossest absurdities do arise--every one can see that a circle, being\na closed curve, cannot get to infinity. The metaphysician, who should\ninvent anything so preposterous as the circular points, would be\nhooted from the field. But the mathematician may steal the horse with\nimpunity.\n\nFinally, then, only a knowledge of space, not a knowledge of Algebra,\ncan assure us that any given set of quantities will have a spatial\ncorrelate, and in the absence of such a correlate, operations with\nthese quantities have no geometrical import. This is the case with\nimaginaries in Cayley's sense, and their use in Geometry, great as\nare its technical advantages, and rigid as is its technical validity,\nis wholly destitute of philosophical importance.\n\n=44.= We have now, I think, discussed most of the questions\nconcerning the scope and validity of the projective method. We\nhave seen that it is independent of all metrical presuppositions,\nand that its use of coordinates does not involve the assumption\nthat spatial magnitudes are measured or expressed by them. We have\nseen that it is able to deal, by its own methods alone, with the\nquestion of the qualitative likeness of geometrical figures, which\nis logically prior to any comparison as to quantity, since quantity\npresupposes qualitative likeness. We have seen also that, so far as\nits legitimate use extends, it applies equally to all homogeneous\nspaces, and that its criterion of an independently possible\nspace--the determination of a straight line by two points[60]--is\nnot subject to the qualifications and limitations which belong, as\nwe have seen in the case of the cylinder, to the metrical criterion\nof constant curvature. But we have also seen that, when projective\nGeometry endeavours to grapple with spatial magnitude, and bring\ndistance and the measurement of angles beneath its sway, its success,\nthough technically valid and important, is philosophically an\napparent success only. Metrical Geometry, therefore, if quantity is\nto be applied to space at all, remains a separate, though logically\nsubsequent branch of Mathematics.\n\n=45.= It only remains to say a few words about _Sophus Lie_. As a\nmathematician, as the inventor of a new and immensely powerful method\nof analysis, he cannot be too highly praised. Geometry is only one\nof the numerous subjects to which his theory of continuous groups\napplies, but its application to Geometry has made a revolution in\nmethod, and has rendered possible, in such problems as Helmholtz's, a\ntreatment infinitely more precise and exhaustive than any which was\npossible before.\n\nThe general definition of a group is as follows: If we have any\nnumber of independent variables _x{1}x{2}...x{n}_, and\nany series of transformations of these into new variables--the\ntransformations being defined by equations of specified forms,\nwith parameters varying from one transformation to another--then\nthe series of transformations form a _group_, if the successive\napplication of any two is equivalent to a single member of the\noriginal series of transformations. The group is _continuous_, when\nwe can pass, by infinitesimal gradations within the group, from any\none of the transformations to any other.\n\nNow, in Geometry, the result of two successive motions or\ncollineations of a figure can always be obtained by a single motion\nor collineation, and any motion or collineation can be built up\nof a series of infinitesimal motions or collineations. Moreover\nthe analytical expression of either is a certain transformation\nof the coordinates of all the points of the figure[61]. Hence the\ntransformations determining a motion or a collineation are such\nas to form a continuous group. But the question of the projective\nequivalence of two figures, to which all projective Geometry is\nreducible, must always be dealt with by a collineation; and the\nquestion of the equality of two figures, to which all metrical\nGeometry is reducible, must always be decided by a motion such as\nto cause superposition; hence the whole subject of Geometry may\nbe regarded as a theory of the continuous groups which define all\npossible collineations and motions.\n\nNow Sophus Lie has developed, at great length, the purely analytical\ntheory of groups; he has therefore, by this method of formulating\nthe problem, a very powerful weapon ready for the attack. In two\npapers \"On the foundations of Geometry[62],\" undertaken at Klein's\nurgent request, he takes premisses which roughly correspond to those\nof Helmholtz, omitting Monodromy, and applies the theory of groups to\nthe deduction of their consequences[63]. Helmholtz's work, he says,\ncan hardly be looked upon as _proving_ its conclusions, and indeed\nthe more searching analysis of the group-theory reveals several\npossibilities unknown to Helmholtz. Nevertheless, as a pioneer,\ndevoid of Lie's machinery, Helmholtz deserves, I think, more praise\nthan Lie is willing to give him[64].\n\nLie's method is perfectly exhaustive; omitting the premiss of\nMonodromy, the others show that a body has six degrees of freedom,\n_i.e._ that the group giving all possible motions of a body will\nhave six independent members; if we keep one point fixed, the\nnumber of independent members is reduced to three. He then, from\nhis general theory, enumerates all the groups which satisfy this\ncondition. In order that such a group should give possible motions,\nit is necessary, by Helmholtz's second axiom, that it should leave\ninvariant some function of the coordinates of any two points. This\neliminates several of the groups previously enumerated, each of which\nhe discusses in turn. He is thus led to the following results:\n\nI. _In two dimensions_, if free mobility is to hold _universally_,\nthere are no groups satisfying Helmholtz's first three axioms, except\nthose which give the ordinary Euclidean and non-Euclidean motions;\nbut if it is to hold only _within a certain region_, there is also\na possible group in which the curve described by any point in a\nrotation is not closed, but an equiangular spiral. To exclude this\npossibility, Helmholtz's axiom of Monodromy is required.\n\nII. _In three dimensions_, the results go still more against\nHelmholtz. Assuming free mobility only _within a certain region_, we\nhave to distinguish two cases: _Either_ free mobility holds, within\nthat region, absolutely without exception, _i.e._ when one point is\nheld fast, _every_ other point within the region can move freely\nover a surface: in this case the axiom of Monodromy is unnecessary,\nand the first three axioms suffice to define our group as that of\nEuclidean and non-Euclidean motions. _Or_ free mobility, within the\nspecified region, holds only of every point _of general position_,\nwhile the points of a certain line, when one point is fixed, are only\nable to move on that line, not on a surface: when this is the case,\nother groups are possible, and can only be excluded by Helmholtz's\nfourth axiom.\n\nHaving now stated the purely mathematical results of Lie's\ninvestigations, we may return to philosophical considerations, by\nwhich Helmholtz's work was mainly motived. It becomes obvious,\nnot only that exceptions within a certain region, but also that\nlimitation to a certain region, of the axiom of Free Mobility,\nare philosophically quite impossible and inconceivable. How can a\ncertain line, or a certain surface, form an impassable barrier in\nspace, or have any mobility different in kind from that of all other\nlines or surfaces? The notion cannot, in philosophy, be permitted\nfor a moment, since it destroys that most fundamental of all the\naxioms, the homogeneity of space. We not only may, therefore, but\nmust take Helmholtz's axiom of Free Mobility in its very strictest\nsense; the axiom of Monodromy thus becomes mathematically, as well\nas philosophically, superfluous. This is, from a philosophical\nstandpoint, the most important of Lie's results.\n\n=46.= I have now come to the end of my history of Metageometry.\nIt has not been my aim to give an exhaustive account of even the\nimportant works on the subject--in the third period, especially, the\nnames of Poincar\u00e9, Pasch, Cremona, Veronese, and others who might be\nmentioned, would have cried shame upon me, had I had any such object.\nBut I have tried to set forth, as clearly as I could, the principles\nat work in the various periods, the motives and results of successive\ntheories. We have seen how the philosophical motive, at first\npredominant, has been gradually extruded by the purely mathematical\nand technical spirit of most recent Geometers. At first, to discredit\nthe Transcendental Aesthetic seemed, to Metageometers, as important\nas to advance their science; but from the works of Cayley, Klein\nor Lie, no reader could gather that Kant had ever lived. We have\nalso seen, however, that as the interest _in_ philosophy waned, the\ninterest _for_ philosophy increased: as the mathematical results\nshook themselves free from philosophical controversies, they\nassumed gradually a stable form, from which further development,\nwe may reasonably hope, will take the form of growth, rather than\ntransformation. The same gradual development out of philosophy might,\nI believe, be traced in the infancy of most branches of mathematics;\nwhen philosophical motives cease to operate, this is, in general, a\nsign that the stage of uncertainty as to premisses is past, so that\nthe future belongs entirely to mathematical technique. When this\nstable stage has been attained, it is time for Philosophy to borrow\nof Science, accepting its final premisses as those imposed by a real\nnecessity of fact or logic.\n\n=47.= Now in discussing the systems of Metageometry, we have found\ntwo kinds, radically distinct and subject to different axioms. The\nhistorically prior kind, which deals with metrical ideas, discusses,\nto begin with, the conditions of Free Mobility, which is essential to\nall measurement of space. It finds the analytical expression of these\nconditions in the existence of a space-constant, or constant measure\nof curvature, which is equivalent to the homogeneity of space. This\nis its first axiom.\n\nIts second axiom states that space has a finite integral number of\ndimensions, _i.e._ in metrical terms, that the position of a point,\nrelative to any other figure in space, is uniquely determined by a\nfinite number of spatial magnitudes, called coordinates.\n\nThe third axiom of metrical Geometry may be called, to distinguish it\nfrom the corresponding projective axiom, the axiom of distance. There\nexists one relation, it says, between any two points, which can be\npreserved unaltered in a combined motion of both points, and which,\nin any motion of a system as one rigid body, is always unaltered.\nThis relation we call distance.\n\nThe above statement of the three essential axioms of metrical\nGeometry is taken from Helmholtz as amended by Lie. Lie's own\nstatement of the axioms, as quoted above, has been too much\ninfluenced by projective methods to give a historically correct\nrendering of the spirit of the second period; Helmholtz's statement,\non the other hand, requires, as Lie has shewn, very considerable\nmodifications. The above compromise may, therefore, I hope be taken\nas accepting Lie's corrections while retaining Helmholtz's spirit.\n\n=48.= But metrical Geometry, though it is historically prior, is\nlogically subsequent to projective Geometry. For projective Geometry\ndeals directly with that qualitative likeness, which the judgment of\nquantitative comparison requires as its basis. Now the above three\naxioms of metrical Geometry, as we shall see in Chapter III. Section\nB, do not presuppose measurement, but are, on the contrary, the\nconditions presupposed by measurement. Without these axioms, which\nare common to all three spaces, measurement would be impossible;\nwith them, so I shall contend, measurement is able, though only\nempirically, to decide approximately which of the three spaces is\nvalid of our actual world. But if these three axioms themselves\nexpress, not results, but conditions, of measurement, must they not\nbe equivalent to the statement of that qualitative likeness on which\nquantitative comparison depends? And if so, must we not expect to\nfind the same axioms, though perhaps under a different form, in\nprojective Geometry?\n\n=49.= This expectation will not be disappointed. The above three\naxioms, as we shall see hereafter, are one and all philosophically\nequivalent to the homogeneity of space, and this in turn is\nequivalent to the axioms of projective Geometry. The axioms of\nprojective Geometry, in fact, may be roughly stated thus:\n\nI. Space is continuous and infinitely divisible; the zero of\nextension, resulting from infinite division, is called a Point. All\npoints are qualitatively similar, and distinguished by the mere fact\nthat they lie outside one another.\n\nII. Any two points determine a unique figure, the straight line;\ntwo straight lines, like two points, are qualitatively similar, and\ndistinguished by the mere fact that they are mutually external.\n\nIII. Three points not in one straight line determine a unique figure,\nthe plane, and four points not in one plane determine a figure of\nthree dimensions. This process may, so far as can be seen _\u00e0 priori_,\nbe continued, without in any way interfering with the possibility\nof projective Geometry, to five or to _n_ points. But projective\nGeometry requires, as an axiom, that the process should stop with\nsome positive integral number of points, after which, any fresh point\nis contained in the figure determined by those already given. If the\nprocess stops with (_n_ + 1) points, our space is said to have _n_\ndimensions.\n\nThese three axioms, it will be seen, are the equivalents of the\nthree axioms of metrical Geometry[65], expressed without reference\nto quantity. We shall find them to be deducible, as before, from\nthe homogeneity of space, or, more generally still, from the\npossibility of experiencing externality. They will therefore appear\nas _\u00e0 priori_, as essential to the existence of any Geometry and to\nexperience of an external world as such.\n\n=50.= That some logical necessity is involved in these axioms might,\nI think, be inferred as probable, from their historical development\nalone. For the systems of Metageometry have not, in general, been set\nup as more likely to fit facts than the system of Euclid; with the\nexception of Z\u00f6llner, for example, I know of no one who has regarded\nthe fourth dimension as required to explain phenomena. As regards the\nspace-constant again, though a _small_ space-constant is regarded\nas empirically possible, it is not usually regarded as probable;\nand the finite space-constants, with which Metageometry is equally\nconversant, are not usually thought even possible, as explanations\nof empirical fact[66]. Thus the motive has been throughout not one\nof fact, but one of logic. Does not this give a strong presumption,\nthat those axioms which are retained, are retained because they are\nlogically indispensable? If this be so, the axioms common to Euclid\nand Metageometry will be _\u00e0 priori_, while those peculiar to Euclid\nwill be empirical. After a criticism of some differing theories of\nGeometry, I shall proceed, in Chapters III. and IV., to the proof and\nconsequences of this thesis, which will form the remainder of the\npresent work.\n\nFOOTNOTES:\n\n[5] V. M\u00e9moires de l'Acad\u00e9mie royale des Sciences de l'lnstitut de\nFrance, T. XII. 1833, for a full statement of his results, with\nreferences to former writings.\n\n[6] This bolder method, it appears, had been suggested, nearly a\ncentury earlier, by an Italian, Saccheri. His work, which seems to\nhave remained completely unknown until Beltrami rediscovered it in\n1889, is called \"Euclides ab omni naevo vindicatus, etc.\" Mediolani,\n1733. (See Veronese, Grundz\u00fcge der Geometrie, German translation,\nLeipzig, 1894, p. 636.) His results included spherical as well as\nhyperbolic space; but they alarmed him to such an extent that he\ndevoted the last half of his book to disproving them.\n\n[7] Klein's first account of elliptic Geometry, as a result of\nCayley's projective theory of distance, appeared in two articles\nentitled \"Ueber die sogenannte Nicht-Euklidische Geometrie, I,\nII,\" Math. Annalen 4, 6 (1871-2). It was afterwards independently\ndiscovered by Newcomb, in an article entitled \"Elementary Theorems\nrelating to the geometry of a space of three dimensions, and of\nuniform positive curvature in the fourth dimension,\" Crelle's\nJournal f\u00fcr die reine und angewandte Mathematik, Vol. 83 (1877). For\nan account of the mathematical controversies concerning elliptic\nGeometry, see Klein's \"Vorlesungen \u00fcber Nicht-Euklidische Geometrie,\"\nG\u00f6ttingen 1893, I. p. 284 ff. A bibliography of the relevant\nliterature up to the year 1878 was given by Halsted in the American\nJournal of Mathematics, Vols. 1, 2.\n\n[8] Veronese (op. cit. p. 638) denies the priority of Gauss in the\ninvention of a non-Euclidean system, though he admits him to have\nbeen the first to regard the axiom of parallels as indemonstrable.\nHis grounds for the former assertion seem scarcely adequate: on the\nevidence against it, see Klein, Nicht-Euklid, I. pp. 171-174.\n\n[9] V. Briefwechsel mit Schumacher, Bd. II. p. 268.\n\n[10] f. Helmholtz, Wiss. Abh. II. p. 611.\n\n[11] Crelle's Journal, 1837.\n\n[12] Theorie der Parallellinien, Berlin, 1840. Republished, Berlin,\n1887. Translated by Halsted, Austin, Texas, U.S.A. 4th edition, 1892.\n\n[13] Frischauf, Absolute Geometrie, nach Johann Bolyai, Leipzig,\n1872. Halsted, The Science Absolute of Space, translated from the\nLatin, 4th edition, Austin, Texas, U.S.A. 1896.\n\n[14] Both Lobatchewsky and Bolyai, as Veronese remarks, start rather\nfrom the point-pair than from distance. See Frischauf, Absolute\nGeometrie, Anhang.\n\n[15] Compare Stallo, Concepts of Modern Physics, p. 248.\n\n[16] Gesammelte Werke, pp. 255-268.\n\n[17] On the history of this word, see Stallo, Concepts of Modern\nPhysics, p. 258. It was used by Kant, and adapted by Herbart to\nalmost the same meaning as it bears in Riemann. Herbart, however,\nalso uses the word _Reihenform_ to express a similar idea. See\nPsychologie als Wissenschaft, I. \u00a7 100 and II. \u00a7 139, where Riemann's\nanalogy with colours is also suggested.\n\n[18] Compare Erdmann's \"Gr\u00f6ssenbegriff vom Raum.\"\n\n[19] Compare Veronese, op. cit. p. 642: \"Riemann ist in seiner\nDefinition des Begriffs Gr\u00f6sse dunkel.\" See also Veronese's whole\nfollowing criticism.\n\n[20] Vortr\u00e4ge und Reden, Vol. II. p. 18.\n\n[21] Cf. Klein, Nicht-Euklid, I. p. 160.\n\n[22] Since we are considering the curvature at a point, we are only\nconcerned with the first infinitesimal elements of the geodesics that\nstart from such a point.\n\n[23] Disquisitiones generales circa superficies curvas, Werke, Bd.\nIV. SS. 219-258, 1827.\n\n[24] Nevertheless, the Geometries of different surfaces of equal\ncurvature are liable to important differences. For example, the\ncylinder is a surface of zero curvature, but since its lines of\ncurvature in one direction are finite, its Geometry coincides with\nthat of the plane only for lengths smaller than the circumference of\nits generating circle (see Veronese, op. cit. p. 644). Two geodesics\non a cylinder may meet in many points. For surfaces of zero curvature\non which this is not possible, the identity with the plane may\nbe allowed to stand. Otherwise, the identity extends only to the\nproperties of figures not exceeding a certain size.\n\n[25] For we may consider two different parts of the same surface as\ncorresponding parts of different surfaces; the above proposition then\nshows that a figure can be reproduced in one part when it has been\ndrawn in another, if the measures of curvature correspond in the two\nparts.\n\n[26] Crelle, Vols, XIX., XX., 1839-40.\n\n[27] In this formula, _u_, _v_ may be the lengths of lines, or the\nangles between lines, drawn on the surface, and having thus no\nnecessary reference to a third dimension.\n\n[28] In what follows, I have given rather Klein's exposition of\nRiemann, than Riemann's own account. The former is much clearer\nand fuller, and not substantially different in any way. V. Klein,\nNicht-Euklid, I. pp. 206 ff.\n\n[29] See \u00a7\u00a7 69-73.\n\n[30] Grundlagen der Geometrie, I. and II., Leipziger Berichte, 1890;\nv. end of present chapter, \u00a7 45.\n\n[31] Nicht-Euklid, I. pp. 258-9.\n\n[32] Giornale di Matematiche, Vol. VI., 1868. Translated into\nFrench by J. Ho\u00fcel in the \"Annales Scientifiques de l'\u00c9cole Normale\nSup\u00e9rieure,\" Vol. VI. 1869.\n\n[33] Crelle's Journal, Vols. XIX. XX., 1839-40.\n\n[34] Nicht-Euklid, I. p. 190.\n\n[35] This article is more trigonometrical and analytical than the\nGerman book, and therefore makes the above interpretation peculiarly\nevident.\n\n[36] Such surfaces are by no means particularly remote. One of them,\nfor example, is formed by the revolution of the common Tractrix\n\n x = asin \u03c6, y = a(log tan \u03c6/2 + cos \u03c6).\n\n[37] \"Teoria fondamentale degli spazii di curvatura costanta,\" Annali\ndi Matematica, II. Vol. 2, 1868-9. Also translated by J. Ho\u00fcel, _loc.\ncit._\n\n[38] See Klein, Nicht-Euklid, I. p. 47 ff., and the references there\ngiven.\n\n[39] See quotation below, from his British Association Address.\n\n[40] Compare the opening sentence, due to Cayley, of Salmon's Higher\nPlane Curves.\n\n[41] V. Nicht-Euklid, I. Chaps. I. and II.\n\n[42] See p. 9 of Cayley's address to the Brit. Ass. 1883. Also a\nquotation from Klein in Erdmann's Axiome der Geometrie, p. 124 note.\n\n[43] Nature, Vol. XLV. p. 407.\n\n[44] Nicht-Euklid, I. p. 200.\n\n[45] I.e. the equation _AB_ + _BC_ = _AC_, for three points in one\nstraight line.\n\n[46] The formula substituted by Klein for Cayley's inverse sine or\ncosine. The two are equivalent, but Klein's is mathematically much\nthe more convenient.\n\n[47] Elements of Projective Geometry, Second Edition, Oxford, 1893,\nChap. IX.\n\n[48] Chap. III. Section B.\n\n[49] See Nicht-Euklid, I. p. 338 ff.\n\n[50] See his Geometrie der Lage, \u00a7 8, Harmonische Gebilde.\n\n[51] The anharmonic ratio of four numbers, _p_, _q_, _r_, _s_, is\ndefined as\n\n (p - q).(r - s) / (p - r).(q - s).\n\n[52] _I.e._ as transformable into each other by a collineation. See\nChap. III. Sec. A, \u00a7 110.\n\n[53] See Chap. III. Sec. A.\n\n[54] It follows from this, that the reduction of metrical to\nprojective properties, even when, as in hyperbolic Geometry, the\nAbsolute is real, is only apparent, and has a merely technical\nvalidity.\n\n[55] Sir R. Ball does not regard his non-Euclidean content as a\npossible space (_v. op. cit._ p. 151). In this important point I\ndisagree with his interpretation, holding such a content to be a\nspace as possible, _\u00e0 priori_, as Euclid's, and perhaps actually true\nwithin the margin due to errors of observation.\n\n[56] See Nicht-Euklid, I. p. 97 ff. and p. 292 ff.\n\n[57] Newcomb says (_loc. cit._ p. 293): \"The system here set forth is\nfounded on the following three postulates.\n\n\"1. I assume that space is triply extended, unbounded, without\nproperties dependent either on position or direction, and possessing\nsuch planeness in its smallest parts that both the postulates of the\nEuclidean Geometry, and our common conceptions of the relations of\nthe parts of space are true for every indefinitely small region in\nspace.\n\n\"2. I assume that this space is affected with such curvature that a\nright line shall always return into itself at the end of a finite and\nreal distance 2_D_ without losing, in any part of its course, that\nsymmetry with respect to space on all sides of it which constitutes\nthe fundamental property of our conception of it.\n\n\"3. I assume that if two right lines emanate from the same point,\nmaking the indefinitely small angle _a_ with each other, their\ndistance apart at the distance _r_ from the point of intersection\nwill be given by the equation\n\n s = 2aD/\u03c0 sin r\u03c0/2D.\n\nThe right line thus has this property in common with the Euclidean\nright line that two such lines intersect only in a single point.\nIt may be that the number of points in which two such lines can\nintersect admit of being determined from the laws of curvature,\nbut not being able so to determine it, I assume as a postulate the\nfundamental property of the Euclidean right line.\"\n\nIt is plain that in the absence of the determination spoken of, the\npossibility of elliptic space is not established. It may be possible,\nfor example, to prove that, in a space where there is a maximum to\ndistance, there must be an infinite number of straight lines joining\ntwo points of maximum distance. In this event, elliptic space would\nbecome impossible.\n\n[58] For an elucidation of this term, see Klein, Nicht-Euklid, I. p.\n99 ff.\n\n[59] Cf. p. 9 of Report: \"My own view is that Euclid's twelfth axiom,\nin Playfair's form of it, does not need demonstration, but is part\nof our notion of space, of the physical space of our experience,\nbut which is the representation lying at the bottom of all external\nexperience.\"\n\n[60] The exception to this axiom, in spherical space, presupposes\nmetrical Geometry, and does not destroy the validity of the axiom for\nprojective Geometry. See Chap. III. Sec. B, \u00a7 171.\n\n[61] Mathematicians of Lie's school have a habit, at first somewhat\nconfusing, of speaking of motions of space instead of motions of\nbodies, as though space as a whole could move. All that is meant is,\nof course, the equivalent motion of the coordinate axes, _i.e._ a\nchange of axes in the usual elementary sense.\n\n[62] \"Ueber die Grundlagen der Geometrie,\" Leipziger Berichte, 1890.\nThe problem of these two papers is really metrical, since it is\nconcerned, not with collineations in general, but with motions. The\nproblem, however, is dealt with by the projective method, motions\nbeing regarded as collineations which leave the Absolute unchanged.\nIt seemed impossible, therefore, to discuss Lie's work, until some\naccount had been given of the projective method.\n\n[63] Lie's premisses, to be accurate, are the following:\n\nLet\n\n x{1} = f(x, y, z, a{1}, a{2}...)\n x{2} = \u03c6(x, y, z, a{1}, a{2}...)\n x{3} = \u03c8(x, y, z, a{1}, a{2}...)\n\ngive an infinite family of real transformations of space, as to which\nwe make the following hypotheses:\n\nA. The functions f, \u03c6, \u03c8, are _analytical_ functions of\n\n x, y, z, a{1}, a{2}....\n\nB. Two points x{1}y{1}z{1}, x{2}y{2}z{2} possess an\ninvariant, _i.e._\n\n \u03a9(x{1}, y{1}, z{1}, x{2}, y{2}, z{2}) =\n \u03a9(x{1\u2032}, y{1\u2032}, z{1\u2032}, x{2\u2032}, y{2\u2032}, z{2\u2032})\n\nwhere x{1\u2032}..., x{2\u2032}..., are the transformed coordinates of\nthe two points.\n\nC. Free Mobility: _i.e._, any point can be moved into any other\nposition; when one point is fixed, any other point of general\nposition can take up \u221e^{2} positions; when two points are fixed, any\nother of general position can take up \u221e^{1} positions; when three, no\nmotion is possible--these limitations being results of the equations\ngiven by the invariant \u03a9.\n\n[64] On this point, cf. Klein, H\u00f6here Geometrie, G\u00f6ttingen, 1893, II.\npp. 225-244, especially pp. 230-1.\n\n[65] Axiom II. of the metrical triad corresponds to Axiom III. of the\nprojective, and _vice vers\u00e2_.\n\n[66] Cf. Helmholtz, Wiss. Abh. Vol. II. p. 640, note: \"Die Bearbeiter\nder Nicht-Euklidischen Geometrie (haben) deren objective Wahrheit nie\nbehauptet.\"\n\n\n\n\nCHAPTER II.\n\nCRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL THEORIES OF GEOMETRY.\n\n\n=51.= We have now traced the mathematical development of the theory\nof geometrical axioms, from the first revolt against Euclid to the\npresent day. We may hope, therefore, to have at our command the\ntechnical knowledge required for the philosophy of the subject. The\nimportance of Geometry, in the theories of knowledge which have\narisen in the past, can scarcely be exaggerated. In Descartes, we\nfind the whole theory of method dominated by analytical Geometry, of\nwhose fruitfulness he was justly proud. In Spinoza, the paramount\ninfluence of Geometry is too obvious to require comment. Among\nmathematicians, Newton's belief in absolute space was long supreme,\nand is still responsible for the current formulation of the laws of\nmotion. Against this belief on the one hand, and against Leibnitz's\ntheory of space on the other, and not, as Caird has pointed out[67],\nagainst Hume's empiricism, was directed that keystone of the Critical\nPhilosophy, the Kantian doctrine of space. Thus Geometry has been,\nthroughout, of supreme importance in the theory of knowledge.\n\nBut in a criticism of representative modern theories of Geometry,\nwhich is designed to be, not a history of the subject, but an\nintroduction to, and defence of, the views of the author, it will\nnot be necessary to discuss any more ancient theory than that of\nKant. Kant's views on this subject, true or false, have so dominated\nsubsequent thought, that whether they were accepted or rejected,\nthey seemed equally potent in forming the opinions, and the manner of\nexposition, of almost all later writers.\n\n\nKant.\n\n=52.= It is not my purpose, in this chapter, to add to the voluminous\nliterature of Kantian criticism, but only to discuss the bearing of\nMetageometry on the argument of the Transcendental Aesthetic, and\nthe aspect under which this argument must be viewed in a discussion\nof Geometry[68]. On this point several misunderstandings seem to me\nto have had wide prevalence, both among friends and foes, and these\nmisunderstandings I shall endeavour, if I can, to remove.\n\nIn the first place, what does Kant's doctrine mean for Geometry?\nObviously not the aspect of the doctrine which has been attacked by\npsychologists, the \"Kantian machine-shop\" as James calls it--at any\nrate, if this can be clearly separated from the logical aspect. The\nquestion whether space is given in sensation, or whether, as Kant\nmaintained, it is given by an intuition to which no external matter\ncorresponds, may for the present be disregarded. If, indeed, we\nheld the view which seems crudely to sum up the standpoint of the\nCritique, the view that all certain knowledge is self-knowledge,\nthen we should be committed, if we had decided that Geometry was\napodeictic, to the view that space is subjective. But even then, the\npsychological question could only arise when the epistemological\nquestion had been solved, and could not, therefore, be taken into\naccount in our first investigation. The question before us is\nprecisely the question whether, or how far, Geometry is apodeictic,\nand for the moment we have only to investigate this question, without\nfear of psychological consequences.\n\n=53.= Now on this question, as on almost all questions in the\nAesthetic or the Analytic, Kant's argument is twofold. On the one\nhand, he says, Geometry is known to have apodeictic certainty:\ntherefore space must be _\u00e0 priori_ and subjective. On the other hand,\nit follows, from grounds independent of Geometry, that space is\nsubjective and _\u00e0 priori_; therefore Geometry must have apodeictic\ncertainty. These two arguments are not clearly distinguished in the\nAesthetic, but a little analysis, I think, will disentangle them.\nThus in the first edition, the first two arguments deduce, from\nnon-geometrical grounds, the apriority of space; the third deduces\nthe apodeictic certainty of Geometry, and maintains, conversely,\nthat no other view can account for this certainty[69]; the last two\narguments only maintain that space is an intuition, not a concept.\nIn the second edition, the double argument is clearer, the apriority\nof space being proved independently of Geometry in the metaphysical\ndeduction, and deduced from the certainty of Geometry, as the only\npossible explanation of this, in the transcendental deduction. In the\nProlegomena, the latter argument alone is used, but in the Critique\nboth are employed.\n\n=54.= Now it must be admitted, I think, that Metageometry has\ndestroyed the legitimacy of the argument from Geometry to space; we\ncan no longer affirm, on purely geometrical grounds, the apodeictic\ncertainty of Euclid. But unless Metageometry has done more than\nthis--unless it has proved, what I believe it alone cannot prove,\nthat Euclid has _not_ apodeictic certainty--then Kant's other line\nof argument retains what force it may ever have had. The actual\nspace we know, it may say, is admittedly Euclidean, and is proved,\nwithout any reference to Geometry, to be _\u00e0 priori_; _hence_ Euclid\nhas apodeictic certainty, and non-Euclid stands condemned. To this\nit is no answer to urge, with the Metageometers, that non-Euclidean\nsystems are _logically_ self-consistent; for Kant is careful to argue\nthat geometrical reasoning, by virtue of our intuition of space, is\nsynthetic, and cannot, though _\u00e0 priori_, be upheld by the principle\nof contradiction alone[70]. Unless non-Euclideans can prove, what\nthey have certainly failed to prove up to the present, that we can\nframe an _intuition_ of non-Euclidean spaces, Kant's position cannot\nbe upset by Metageometry alone, but must also be attacked, if it is\nto be successfully attacked, on its purely philosophical side.\n\n=55.= For such an attack, two roads lie open: either we may disprove\nthe first two arguments of the Aesthetic, or we may criticize, from\nthe standpoint of general logic, the Kantian doctrine of synthetic\n_\u00e0 priori_ judgments and their connection with subjectivity. Both\nthese attacks, I believe, could be conducted with some success;\nbut if we are to disprove the apodeictic certainty of Geometry,\none or other is essential, and both, I believe, will be found only\npartially successful. It will be my aim to prove, in discussing\nthese two lines of attack, (1) that the distinction of synthetic and\nanalytic judgments is untenable, and further, that the principle of\ncontradiction can only give fruitful results on the assumption that\nexperience in general, or, in a particular science, some special\nbranch of experience, is to be formally possible; (2) that the first\ntwo arguments of the Transcendental Aesthetic suffice to prove,\nnot Euclidean space, but _some_ form of externality--which may be\nsensational or intuitional, but not merely conceptual--a necessary\nprerequisite of experience of an external world. In the third and\nfourth chapters, I shall contend, as a result of these conclusions,\nthat those axioms, which Euclid and Metageometry have in common,\ncoincide with those properties of any form of externality which are\ndeducible, by the principle of contradiction, from the possibility of\nexperience of an external world. These properties, then, may be said,\nthough not quite in the Kantian sense, to be _\u00e0 priori_ properties of\nspace, and as to these, I think, a modified Kantian position may be\nmaintained. But the question of the subjective or objective nature\nof space may be left wholly out of account during the course of this\ndiscussion, which will gain by dealing exclusively with logical, as\nopposed to psychological points of view.\n\n=56.= (1) _Kant's logical position._ The doctrine of synthetic and\nanalytic judgments--at any rate if this is taken as the corner-stone\nof Epistemology--has been so completely rejected by most modern\nlogicians[71], that it would demand little attention here, but\nfor the fact that an enthusiastic French Kantian, M. Renouvier,\nhas recently appealed to it, with perfect confidence, on the very\nquestion of Geometry[72]. And it must be owned, with M. Renouvier,\nthat if such judgments existed, in the Kantian sense, non-Euclidean\nGeometry, which makes no appeal to intuition, could have nothing to\nsay against them. M. Renouvier's contention, therefore, forces us\nbriefly to review the arguments against Kant's doctrine, and briefly\nto discuss what logical canon is to replace it.\n\nEvery judgment--so modern logic contends--is both synthetic and\nanalytic; it combines parts into a whole, and analyses a whole into\nparts[73]. If this be so, the distinction of analysis and synthesis,\nwhatever may be its importance in pure Logic, can have no value in\nEpistemology. But such a doctrine, it must be observed, allows full\nscope to the principle of contradiction: this criterion, since all\njudgments, in one aspect at least, are analytic, is applicable to\nall judgments alike. On the other hand, the whole which is analysed\nmust be supposed already given, before the parts can be mutually\ncontradictory: for only by connection in a given whole can two parts\nor adjectives be incompatible. Thus the principle of contradiction\nremains barren until we already have some judgments, and even some\ninference: for the parts may be regarded, to some extent, as an\ninference from the whole, or _vice vers\u00e2_. When once the arch of\nknowledge is constructed, the parts support one another, and the\nprinciple of contradiction is the keystone: but until the arch is\nbuilt, the keystone remains suspended, unsupported and unsupporting,\nin the empty air. In other words, knowledge once existent can be\nanalysed, but knowledge which should have to win every inch of the\nway against a critical scepticism, could never begin, and could never\nattain that circular condition in which alone it can stand.\n\nBut Kant's doctrine, if true, is designed to restrain a critical\nscepticism even where it might be effective. Certain fundamental\npropositions, he says, are not deducible from logic, _i.e._\ntheir contradictories are not self-contradictory; they combine a\nsubject and predicate which cannot, in any purely logical way,\nbe shewn to have any connection, and yet these judgments have\napodeictic certainty. But concerning such judgments, Kant is\ngenerally careful not to rely upon the mere subjective conviction\nthat they are undeniable: he proves, with every precaution, that\nwithout them experience would be impossible. Experience consists\nin the combination of terms which formal logic leaves apart, and\npresupposes, therefore, certain judgments by which a framework is\nmade for bringing such terms together. Without these judgments--so\nKant contends--all synthesis and all experience would be impossible.\nIf, therefore, the detail of the Kantian reasoning be sound, his\nresults may be obtained by the principle of contradiction _plus_ the\npossibility of experience, as well as by his distinction of synthetic\nand analytic judgments.\n\nLogic, at the present day, arrogates to itself at once a wider and a\nnarrower sphere than Kant allowed to it. Wider, because it believes\nitself capable of condemning any false principle or postulate;\nnarrower, because it believes that its law of contradiction,\nwithout a given whole or a given hypothesis, is powerless, and\nthat two terms, _per se_, though they may be different, cannot be\ncontradictories, but acquire this relation only by combination\nin a whole about which something is known, or by connection with\na postulate which, for some reason, must be preserved. Thus no\njudgment, _per se_, is either analytic or synthetic, for the\nseverance of a judgment from its context robs it of its vitality, and\nmakes it not truly a judgment at all. But in its proper context it\nis neither purely synthetic nor purely analytic; for while it is the\nfurther determination of a given whole, and thus in so far analytic,\nit also involves the emergence of _new_ relations within this whole,\nand is so far synthetic.\n\n=57.= We may retain, however, a distinction roughly corresponding to\nthe Kantian _\u00e0 priori_ and _\u00e0 posteriori_, though less rigid, and\nmore liable to change with the degree of organisation of knowledge.\nKant usually endeavoured to prove, as observed above, that his\nsynthetic _\u00e0 priori_ propositions were necessary prerequisites of\nexperience; now although we cannot retain the term synthetic, we\ncan retain the term _\u00e0 priori_, for those assumptions, or those\npostulates, from which alone the possibility of experience follows.\nWhatever can be deduced from these postulates, without the aid of the\nmatter of experience, will also, of course, be _\u00e0 priori_. From the\nstandpoint of general logic, the laws of thought and the categories,\nwith the indispensable conditions of their applicability, will be\nalone _\u00e0 priori_; but from the standpoint of any special science,\nwe may call _\u00e0 priori_ whatever renders possible the experience\nwhich forms the subject-matter of our science. In Geometry, to\nparticularize, we may call _\u00e0 priori_ whatever renders possible\nexperience of externality as such.\n\nIt is to be observed that this use of the term is at once more\nrationalistic and less precise than that of Kant. Kant would seem\nto have supposed himself immediately aware, by inspection, that\nsome knowledge was apodeictic, and its subject-matter, therefore,\n_\u00e0 priori_: but he did not always deduce its apriority from any\nfurther principle. Here, however, it is to be shown, before admitting\napriority, that the falsehood of the judgment in question would not\nbe effected by a mere change in the _matter_ of experience, but only\nby a change which should render some branch of experience formally\nimpossible, _i.e._ inaccessible to our methods of cognition. The\nabove use is also less precise, for it varies according to the\nspecialization of the experience we are assuming possible, and with\nevery progress of knowledge some new connection is perceived, two\npreviously isolated judgments are brought into logical relation, and\nthe _\u00e0 priori_ may thus, at any moment, enlarge its sphere, as more\nis found deducible from fundamental postulates.\n\n=58.= (2) _Kant's arguments for the apriority of space._ Having now\ndiscussed the logical canon to be used as regards the _\u00e0 priori_, we\nmay proceed to test Kant's arguments as regards space. The argument\nfrom Geometry, as remarked above, is upset by Metageometry, at least\nso far as those properties are concerned, which belong to Euclid\nbut not to non-Euclidean spaces; as regards the common properties\nof both kinds of space, we cannot decide on their apriority till we\nhave discussed the consequences of denying them, which will be done\nin Chapter III. As regards the two arguments which prove that space\nis an intuition, not a concept, they would call for much discussion\nin a special criticism of Kant, but here they may be passed by\nwith the obvious comment that infinite homogeneous Euclidean space\nis a concept, not an intuition--a concept invented to explain an\nintuition, it is true, but still a pure concept[74]. And it is this\npure concept which, in all discussions of Geometry, is primarily to\nbe dealt with; the intuition need only be referred to where it throws\nlight on the functions or the nature of the concept. The second\nof Kant's arguments, that we can imagine empty space, though not\nthe absence of space, is false if it means a space without matter\nanywhere, and irrelevant if it merely means a space between matters\nand regarded as empty[75]. The only argument of importance, then,\nis the first argument. But I must insist, at the outset, that our\nproblem is purely logical, and that all psychological implications\nmust be excluded to the utmost possible extent. Moreover, as will be\nproved in Chapter IV., the proper function of space is to distinguish\nbetween different presented things, not between the Self and the\nobject of sensation or perception. The argument then becomes the\nfollowing: consciousness of a world of mutually external things\ndemands, in presentations, a cognitive but non-inferential element\nleading to the discrimination of the objects presented. This element\nmust be non-inferential, for from whatever number or combination of\npresentations, which did not of themselves demand diversity in their\nobjects, I could never be led to infer the mutual externality of\ntheir objects. Kant says: \"In order that sensations may be ascribed\nto something external to me ... and similarly in order that I may\nbe able to present them as outside and beside one another, ... the\npresentation of space must be already present.\" But this goes rather\ntoo far: in the first place, the question should be only as to the\nmutual externality of presented things, not as to their externality\nto the Self[76]; and in the second place, things will appear mutually\nexternal if I have the presentation of _any_ form of externality,\nwhether Euclidean or non-Euclidean. Whatever may be true of the\n_psychological_ scope of this argument--whose validity is here\nirrelevant--the _logical_ scope extends, not to Euclidean space, but\nonly to any form of externality which could exist intuitively, and\npermit knowledge, in beings with our laws of thought, of a world of\ndiverse but interrelated things.\n\nMoreover externality, to render the scope of the argument wholly\nlogical, must not be left with a sensational or intuitional meaning,\nthough it must be supposed given in sensation or intuition. It must\nmean, in this argument, the fact of Otherness[77], the fact of being\ndifferent from some other thing: it must involve the distinction\nbetween different things, and must be that element, in a cognitive\nstate, which leads us to discriminate constituent parts in its\nobject. So much, then, would appear to result from Kant's argument,\nthat experience of diverse but interrelated things demands, as a\nnecessary prerequisite, some sensational or intuitional element, in\nperception, by which we are led to attribute complexity to objects\nof perception[78]; that this element, in its isolation may be called\na form of externality; and that those properties of this form, if\nany such be found, which can be deduced from its mere function of\nrendering experience of interrelated diversity possible, are to\nbe regarded as _\u00e0 priori_. What these properties are, and how the\nvarious lines of argument here suggested converge to a single result,\nwe shall see in Chapters III. and IV.\n\n=59.= In the philosophers who followed Kant, Metaphysics, for the\nmost part, so predominated over Epistemology, that little was added\nto the theory of Geometry. What was added, came indirectly from\nthe one philosopher who stood out against the purely ontological\nspeculations of his time, namely _Herbart_. Herbart's actual views on\nGeometry, which are to be found chiefly in the first section of his\n_Synechologie_, are not of any great value, and have borne no great\nfruit in the development of the subject. But his psychological theory\nof space, his construction of extension out of series of points, his\ncomparison of space with the tone and colour-series, his general\npreference for the discrete above the continuous, and finally his\nbelief in the great importance of classifying space with other forms\nof series (_Reihenformen_[79]), gave rise to many of Riemann's\nepoch-making speculations, and encouraged the attempt to explain the\nnature of space by its analytical and quantitative aspect alone[80].\nThrough his influence on Riemann, he acquired, indirectly, a great\nimportance in geometrical philosophy. To Riemann's dissertation,\nwhich we have already discussed in its mathematical aspect, we must\nnow return, considering, this time, only its philosophical views.\n\n\nRiemann.\n\n=60.= The aim of Riemann's dissertation, as we saw in Chapter I.,\nwas to define space as a species of manifold, _i.e._ as a particular\nkind of collection of magnitudes. It was thus assumed, to begin with,\nthat spatial figures could be regarded as magnitudes, and the axioms\nwhich emerged, accordingly, determined only the particular place of\nthese among the many algebraically possible varieties of magnitudes.\nThe resulting formulation of the axioms--while, from the mathematical\nstandpoint of metrical Geometry, it was almost wholly laudable--must,\nfrom the standpoint of philosophy, be regarded, in my opinion, as a\n_petitio principii_. For when we have arrived at regarding spatial\nfigures as magnitudes, we have already traversed the most difficult\npart of the ground. The axioms of metrical Geometry--and it is\nmetrical Geometry, exclusively, which is considered in Riemann's\nEssay--will appear, in Chapter III., to be divisible into two\nclasses. Of these, the first class--which contains the axioms common\nto Euclid and Metageometry, the only axioms seriously discussed by\nRiemann--are not the results of measurement, nor of any conception\nof magnitude, but are conditions to be fulfilled before measurement\nbecomes possible. The second class only--those which express the\ndifference between Euclidean and non-Euclidean spaces--can be\ndeduced as results of measurement or of conceptions of magnitude.\nAs regards the first class, on the contrary, we shall see that\nthe relativity of position--by which space is distinguished from\nall other known manifolds, except time--leads logically to the\nnecessity of three of the most distinctive axioms of Geometry, and\nyet this relativity cannot be called a deduction from conceptions of\nmagnitude. In analytical Geometry, owing to the fact that coordinate\nsystems start from points, and hence build up lines and surfaces, it\nis easy to suppose that points can be given independently of lines\nand of each other, and thus the relativity of position is lost sight\nof. The error thus suggested by mathematics was probably reinforced\nby Herbart's theory of space, which, by its serial character, as\nwe have seen, appeared to him to facilitate a construction out of\nsuccessive points, and to which Riemann acknowledges his indebtedness\nboth in his Dissertation and elsewhere. The same error reappears\nin Helmholtz, in whom it is probably due wholly to the methods of\nanalytical Geometry. It is a striking fact that, throughout the\nwritings of these two men, there is not, so far as I know, one\nallusion to the relativity of position, that property of space\nfrom which, as our next chapter will shew, the richest quarry of\nconsequences can be extracted. This is not a result of any conception\nof magnitude, but follows from the nature of our space-intuition; yet\nno one, surely, could call it empirical, since it is bound up in the\nvery possibility of locating things _there_ as opposed to _here_.\n\n=61.= Indeed we can see, from a purely logical consideration of\nthe judgment of quantity, that Riemann's manner of approaching the\nproblem can never, by legitimate methods, attain to a philosophically\nsound formulation of the axioms. For quantity is a result of\ncomparison of two qualitatively similar objects, and the judgment of\nquantity neglects altogether the qualitative aspect of the objects\ncompared. Hence a knowledge of the essential properties of space can\nnever be obtained from judgments of quantity, which neglect these\nproperties, while they yet presuppose them. As well might one hope\nto learn the nature of man from a census. Moreover, the judgment of\nquantity is the result of comparison, and therefore presupposes\nthe possibility of comparison. To know whether, or by what means,\ncomparison is possible, we must know the qualities of the things\ncompared and of the medium in which comparison is effected; while to\nknow that _quantitative_ comparison is possible, we must know that\nthere is a qualitative identity between the things compared, which\nagain involves a previous qualitative knowledge. When spatial figures\nhave once been reduced to quantity, their quality has already been\nneglected, as known and similar to the quality of other figures. To\nhope, therefore, for the qualities of space, from a comparison of its\nexpression as pure quantity with other pure quantities, is an error\nnatural to an analytical geometer, but an error, none the less, from\nwhich there is no return to the qualitative basis of spatial quantity.\n\n=62.= We must entirely dissent, therefore, from the disjunction which\nunderlies Riemann's philosophy of space. Either the axioms must be\nconsequences of general conceptions of magnitude, he thinks, or\nelse they can only be proved by experience (p. 255). Whatever _can_\nbe derived from general conceptions of magnitude, we may retort,\ncannot be an _\u00e0 priori_ adjective of space: for all the necessary\nadjectives of space are presupposed in any judgment of spatial\nquantity, and cannot, therefore, be consequences of such a judgment.\nRiemann's disjunction, accordingly, since one of its alternatives is\nobviously impossible, really begs the question. In formulating the\naxioms of metrical Geometry, our question should be: What axioms,\n_i.e._ what adjectives of space, must be presupposed, in order that\nquantitative comparison of the parts of space may be possible at\nall? And only when we have determined these conditions, which are\n_\u00e0 priori_ necessary to any quantitative science of space, does the\nsecond question arise: what inferences can we draw, as to space,\nfrom the observed results of this quantitative science, _i.e._ of\nthis measurement of spatial figures? The conditions of measurement\nthemselves, though not results of any conception of magnitude, will\nbe _\u00e0 priori_, if it can be shown that, without them, experience of\nexternality would be impossible.\n\nAfter this initial protest against Riemann's general philosophical\nposition, let us proceed to examine, in detail, his use of the notion\nof a manifold.\n\n=63.= In the first place there is, if I am not mistaken, considerable\nobscurity in the definition of a manifold, of which an almost verbal\nrendering was given in Chapter I. What is meant, to begin with, by\na general conception capable of various determinations? Does not\nthis property belong to all conceptions? It affords, certainly,\na basis for counting, but if continuous quantity is to arise, we\nmust, surely, have some less discrete formulation. It might afford\na basis, for example, for the distinction of points in projective\nGeometry, but projective Geometry has nothing to do with quantity.\nSomething more fluid and flexible than a conception, one would think,\nis necessary as the basis of continua. Then, again, what is meant by\na quantum of a manifold? In space, the answer is obvious: what is\nmeant is a piece of volume. But how about Riemann's other continuous\nmanifold, colour? Does a quantum of colour mean a single line in the\nspectrum, or a band of finite thickness? In either case, what are\nthe magnitudes to be compared? And how is superposition necessary,\nor even possible? A colour is fixed by its position in the spectrum:\ntwo lines in the same spectrum cannot be superposed, and two lines in\ndifferent spectra need not be--their positions in their respective\nspectra suffice, or even, roughly, their immediate sense-quality. The\nfact is, Riemann had space in his mind from the start, and many of\nthe properties, which he enunciates as belonging to all manifolds,\nbelong, as a matter of fact, only to space. It is far from clear what\nthe magnitudes are which the various determinations make possible.\nDo these magnitudes measure the elements of the manifold, or the\nrelations between elements? This is surely a very fundamental point,\nbut it is one which Riemann never touches on. In the former case,\nthe superposition which he speaks of becomes unnecessary, since the\nmagnitude is inherent in the element considered. We do not require\nsuperposition to measure quantities corresponding to different tones\nor colours; these can be discovered by analysis of single tones or\ncolours. With space, on the other hand, if we seek for elements,\nwe can find none except points, and no analysis of a point will\nfind magnitudes inherent in it--such magnitudes are a fiction of\ncoordinate Geometry. The magnitudes which space deals with, as we\nshall see in Chapter III., are relations between points, and it is\nfor this reason that superposition is essential to space-measurement.\nThere is no inherent quality in a single point, as there is in a\nsingle colour, by which it can be quantitatively distinguished from\nanother. Thus the conception of a manifold, as defined by Riemann,\neither does not include colours, or does not involve superposition as\nthe only means of measurement. From this dilemma there is no escape.\n\n=64.= But if \"measurement _consists_ in a superposition of the\nmagnitudes compared\" (p. 256), does it not follow immediately that\nmeasurement is logically possible _only_ where such superposition\nleaves the magnitudes unchanged? And therefore that measurement, as\nabove defined, involves, as an _\u00e0 priori_ condition, that magnitudes\nare unchanged by motion? This consequence is not drawn by Riemann;\nindeed he proceeds immediately (pp. 256-7) to consider what he calls\na general portion of the doctrine of magnitude (_Gr\u00f6ssenlehre_),\nindependent of measurement. But how is any doctrine of magnitude\npossible, in which the magnitudes cannot be measured? The reason\nof the confusion is, that Riemann's definition of measurement is\napplicable to no single manifold except space, since it depends\non the noteworthy property that what we measure in Geometry is\nnot points, but relations between points, and the latter, though\nnot the former, may of course be unaltered by motion. Let us try,\nin illustration, to apply Riemann's definition of measurement to\ncolours. We must remember that motion, in dealing with the colour\nmanifold, means--not motion in space but--motion in the colour\nmanifold itself. Now since every point of the colour manifold is\ncompletely determined by three magnitudes, which are given in fact,\nand cannot be arbitrarily chosen, it is plain that measurement\nby superposition--involving, as it does, motion, and therefore\nchange in these determining magnitudes--is totally out of the\nquestion. The superposition of one colour on another, as a means\nof measurement, is sheer nonsense. And yet measurement is possible\nin the colour-manifold, by means of Helmholtz's law of mixture\n(_Mischungsgesetz_); but the measurement is of every separate\nelement, not of the relations between elements, and is thus radically\ndifferent from space-measurement[81]. The elements are not, like\npoints in space, qualitatively alike, and distinguished by the\nmere fact of their mutual externality. What we have, in colours, is\nthree fundamental qualitatively distinct elements, out of certain\nproportions of which we can build up all the other elements of the\nmanifold--each of the resulting elements having the same combination\nof qualitative diversity and similarity as the three original\nelements. But in space, what could we make of such a procedure? Given\nthree points, how are we to combine them in certain proportions? The\nphrase is meaningless. If some one makes the obvious retort, that we\nhave to combine lines, not points, my rejoinder is equally obvious.\nTo begin with, lines are not elements. Metaphysically, space has _no_\nelements, being, as the sequel will show, mere relations between\nnon-spatial elements. Mathematically, this fact exhibits itself in\nthe self-contradictory notion of the point, or zero magnitude in\nspace, as the limit in our vain search for spatial elements. But\neven if we allow the line to pass as the spatial element, what does\nthe combination of three lines in definite proportions give us? It\ngives us, simply, the coordinates of a _point_. Here again we see a\ngreat difference between the colour and space-manifolds. In colours,\nthe combination of magnitudes gives a new magnitude of the same\nkind; in space, it defines, not a magnitude at all, but a would-be\nelement of a different kind from the defining magnitudes. In the\ntone-manifold, we should find still different conditions. Here, no\none of the measuring magnitudes can vanish without the tone vanishing\ntoo, and all three are so bound up together, in the single resulting\nsensation, that none can exist without a finite quantity of the\nothers. They are all qualitatively different, both from each other,\nand from any possible tone, being constituents of it, as mass and\nvelocity are constituents of momentum. All these different conditions\nrequire to be examined, before a manifold can be completely defined;\nand until we have conducted such an examination in detail, we cannot\npronounce as to the _\u00e0 priori_ or empirical nature of the laws of the\nmanifold. As regards space, I have attempted such an examination in\nthe third and fourth chapters of this Essay.\n\n=65.= I do not wish to deny, however, the great value of the\nconception of space as a manifold. On the contrary, this conception\nseems to have become essential to any treatment of the question.\nI only wish to urge that the purely algebraical treatment of any\nmanifold, important as it may be in deducing fresh consequences from\nknown premisses, tends rather to conceal than to make clear the\nbasis of the premisses themselves, and is therefore misleading in a\nphilosophical investigation. For mathematics, where quantity reigns\nsupreme, Riemann's conception has proved itself abundantly fruitful;\nfor philosophy, on the contrary, where quantity appears rather as\na cloak to conceal the qualities it abstracts from, the conception\nseems to me more productive of error and confusion than of sound\ndoctrine.\n\nWe are thus brought back to the point from which we started, namely,\nthe falsity of Riemann's initial disjunction, and the consequent\nfallacy in his proof of the empirical nature of the axioms. His\nphilosophy is chiefly vitiated, to my mind, by this fallacy, and\nby the uncritical assumption that a metrical coordinate system can\nbe set up independently of any axioms as to space-measurement[82].\nRiemann has failed to observe, what I have endeavoured to prove in\nthe next chapter, that, unless space had a strictly constant measure\nof curvature, Geometry would become impossible; also that the absence\nof constant measure of curvature involves absolute position, which is\nan absurdity. Hence he is led to the conclusion that all geometrical\naxioms are empirical, and may not hold in the infinitesimal, where\nobservation is impossible. Thus he says (p. 267): \"Now the empirical\nconceptions, on which spatial measurements are based, the conceptions\nof the rigid body and the light-ray, appear to lose their validity\nin the infinitesimal: it is therefore quite conceivable that\nthe relations of spatial magnitudes in the infinitesimal do not\ncorrespond to the presuppositions of Geometry, and this would, in\nfact, have to be assumed, as soon as it would enable us to explain\nthe phenomena more simply.\" From this conclusion I must entirely\ndissent. In very large spaces, there might be a departure from\nEuclid; for they depend upon the axiom of parallels, which is not\ncontained in the axiom of Free Mobility; but in the infinitesimal,\ndepartures from Euclid could only be due to the absence of Free\nMobility, which, as I hope my third chapter will show, is once for\nall impossible.\n\n\nHelmholtz.\n\n=66.= Helmholtz, like Riemann, was important both in the mathematics\nand in the philosophy of Geometry. From the mathematical point\nof view, his work has been already considered in Chapter I.; the\nconsideration of his philosophy, which must occupy us here, will\nbe a more serious task. Like Riemann, he endeavoured to prove that\nall the axioms are empirical, and like Riemann, he based his proof\nchiefly on Metageometry. He had an additional resource, however,\nin the physiology of the senses, which first led him to reject the\nTranscendental Aesthetic, and enabled him to attack Kant from the\npsychological as well as the mathematical side[83].\n\nThe principal topics, for a criticism of Helmholtz, are three: First,\nhis criterion of the _\u00e0 priori_; second, his discussion with Land as\nto the \"imaginability\" of non-Euclidean spaces; third--and this is by\nfar the most important of the three--his theory of the dependence of\nGeometry on Mechanics. Let us discuss these three points successively.\n\n=67.= Helmholtz's criterion of apriority is difficult to discover,\nas he never, to my knowledge, gives a precise statement of it.\nFrom his discussion of physical and transcendental Geometry[84],\nhowever, it would appear that he regards as empirical whatever\napplies to empirical matter. For he there maintains, that even if\nspace were an _\u00e0 priori_ form, yet any Geometry, which aimed at an\napplication to Physics, would, since the actual places of bodies are\nnot known _\u00e0 priori_, be necessarily empirical[85]. It seems the\nmore probable that he regards this as a possible criterion, as it\nis adopted, in several passages, by his disciple Erdmann[86], and\nso strange a test could hardly be accepted by a philosopher, unless\nhe had found it in his master. I have called this a strange test,\nbecause it seems to me completely to ignore the work of the Critical\nPhilosophy. For if there is one thing which, one might have hoped,\nhad been made sufficiently clear by Kant's Critique, it is this,\nthat knowledge which is _\u00e0 priori_, being itself the condition of\npossible experience, applies--and in Kant's view, applies only--to\nempirical matter. Helmholtz and Erdmann, therefore, in setting up\nthis test without discussion, simply ignore the existence of Kant\nand the possibility of a transcendental argument. Helmholtz assumes\nalways that empirical knowledge must be wholly empirical, that there\ncan be no _\u00e0 priori_ conditions of the experience in question,\nthat experience will always be possible, and may give any kind of\nresult. Thus in discussing \"physical\" Geometry, he assumes that the\npossibility of empirical measurement involves no _\u00e0 priori_ axioms,\nand that no _\u00e0 priori_ element can be contained in the process. This\nassumption, as we shall see in Chapter III., is quite unwarrantable:\ncertain properties of _space_, in fact, are involved in the\npossibility of measuring _matter_. In spite of the fact, therefore,\nthat we apply measurement to empirical matter, and that our results\nare therefore empirical, there may well be an _\u00e0 priori_ element\nin measurement, which is presupposed in its possibility. Such a\ncriterion, therefore, must pronounce everything empirical, but must\nitself be pronounced worthless.\n\nAnother and a better criterion, it is true, is also to be found in\nHelmholtz, and has also been adopted by Erdmann. Whatever might,\nby a different experience, have been rendered different--so this\ncriterion contends--must itself be dependent on experience, and so\nempirical. This criterion seems perfectly sound, but Helmholtz's use\nof it is usually vitiated by his neglecting to prove the possibility\nof the different experience in question. He says, for example, that\nif our experience showed us only bodies which changed their shapes\nin motion, we should not arrive at the axiom of Congruence, which\nhe pronounces accordingly to be empirical. But I shall endeavour\nto prove, in Chapter III., that without the axiom of Congruence,\nexperience of spatial magnitude would be impossible. If my proof\nbe correct, it follows that no experience can ever reveal spatial\nmagnitudes which contradict this axiom--a possibility which Helmholtz\nnowhere discusses, in setting up his hypothetical experience. Thus\nthis second criterion, though perfectly sound, requires always\nan accompanying transcendental argument, as to the conditions of\npossible experience. But this accompaniment is seldom to be found in\nHelmholtz.\n\n=68.= One of the few cases, in which Helmholtz has attempted such\nan accompaniment, occurs in connection with our second point, the\nimaginability of non-Euclidean spaces. The argument on this point\nwas elicited by Helmholtz's Kantian opponents, who maintained that\nthe merely logical possibility of these spaces was irrelevant, since\nthe basis of Geometry was not logic, but intuition. The axioms,\nthey said, are synthetic propositions, and their contraries are,\ntherefore, not self-contradictory; they are nevertheless apodeictic\npropositions, since no other _intuition_ than the Euclidean is\npossible to us[87]. I have already criticized this line of argument\nin the beginning of the present chapter. Helmholtz's criticism,\nhowever, was different: admitting the internal consistency of\nthe argument, he denied one of its premisses. We _can_ imagine\nnon-Euclidean spaces, he said, though their unfamiliarity makes\nthis difficult. From this view it followed, of course, that Kant's\nargument, even if it were formally valid, could not prove the\napriority of Euclidean space in particular, but only of that general\nspace which included Euclid and non-Euclid alike[88].\n\nAlthough I agree with Helmholtz in thinking the distinction between\nEuclidean and non-Euclidean spaces empirical, I cannot think his\nargument on the \"imaginability\" of the latter a very happy one. The\nvalidity of any proof must turn, obviously, on the definition of\nimaginability. The definition which Helmholtz gives in his answer\nto Land is as follows: Imaginability requires \"die vollst\u00e4ndige\nVorstellbarkeit derjenigen Sinneseindr\u00fccke, welche das betreffende\nObject in uns nach den bekannten Gesetzen unserer Sinnesorgane\nunter allen denkbaren Bedingungen der Beobachtung erregen, und\nwodurch es sich von anderen \u00e4hnlichen Objecten unterscheiden w\u00fcrde\"\n(Wiss. Abh. II. p. 644). This definition is not very clear, owing\nto the ambiguity of the word \"_Vorstellbarkeit_.\" The following\ndefinition seems less ambiguous: \"Wenn die Reihe der Sinneseindr\u00fccke\nvollst\u00e4ndig und eindeutig angegeben werden kann, muss man m. E.\ndie Sache f\u00fcr _anschaulich vorstellbar_ erkl\u00e4ren\" (Vortr\u00e4ge und\nReden, II. p. 234). This makes clear, what also appears from his\nmanner of proof, that he regards things as imaginable which can be\n_described_ in conceptual terms. Such, as Land remarks (Mind, Vol.\nII. p. 45), \"is not the sense required for argumentation in this\ncase.\" That Land's criticism is just, is shown by Helmholtz's proof\nfor non-Euclidean spaces, for it consists only in an analogy to\nthe volume inside a sphere, which is mathematically obtained thus:\nWe take the symbols representing magnitudes in \"pseudo-spherical\"\n(hyperbolic) space, and give them a new Euclidean meaning; thus all\nour symbolic propositions become capable of two interpretations,\none for pseudo-spherical space, and one for the volume inside a\nsphere. It is, however, sufficiently obvious that this procedure,\nthough it enables us to _describe_ our new space, does not enable\nus to _imagine_ it, in the sense of calling up images of the way\nthings would look in it. We really derive, from this analogy, no\nmore knowledge than a man born blind may derive, as to light,\nfrom an analogy with heat. The dictum \"Nihil est in intellectu\nquod non fuerit ante in sensu,\" would unquestionably be true, if\nfor _intellect_ we were to substitute _imagination_; it is vain,\ntherefore, _if_ our actual space be Euclidean, to hope for a power of\n_imagining_ a non-Euclidean space. What Helmholtz might, I believe\nwith perfect truth, have urged against Land, is that the image we\nactually have of space is not sufficiently accurate to exclude, in\nthe actual space we know, all possibility of a slight departure from\nthe Euclidean type. But in maintaining that we cannot imagine, though\nwe can conceive and describe, a space different from that we actually\nhave, Land is, in my opinion, unquestionably in the right. For a\npure Kantian, who maintains, with Land, that none of the axioms can\nbe proved, this question is of great importance. But if, as I have\nmaintained, some of the axioms are susceptible of a transcendental\nproof, while the others can be verified empirically, the question\nis freed from psychological implications, and the imaginability or\nnon-imaginability of metageometrical spaces becomes unimportant.\n\n=69.= We come now to the third and most important question, the\nrelation of Geometry to Mechanics. There are three senses in which\nHelmholtz's appeal to rigid bodies may be taken: the first, I think,\nis the sense in which he originally intended it; the second seems\nto be the sense which he adopted in his defence against Land; while\nthe third is admitted by Land, and will be admitted in the following\nargument. These three senses are as follows:\n\n(1) It may be asserted that the actual meaning of the axiom of Free\nMobility lies in the assertion of empirical rigid bodies, and that\nthe two propositions are equivalent to one another. This is certainly\nfalse.\n\n(2) The axiom of Free Mobility, it may be said, is logically\ndistinguishable from the assertion of rigid bodies, and may even be\nnot empirical; but it is barren, even for pure Geometry, without the\naid of measures, which must themselves be empirical rigid bodies.\nThis sense is more plausible than the first, but I believe we can\nshow that, in this sense also, the proposition is false.\n\n(3) For pure Geometry and the abstract study of space, it may be\nsaid, Free Mobility, as applied to an abstract geometrical matter,\ngives a sufficient possibility of quantitative comparison; but\nthe moment we extend our results to mixed mathematics, and apply\nthem to empirically given matter, we require also, as measures,\nempirically given rigid bodies, or bodies, at least, whose departures\nfrom rigidity are empirically known. In this sense, I admit, the\nproposition is correct[89].\n\nIn discussing these three meanings, I shall not confine myself\nstrictly to the text of Helmholtz or Land: if I endeavoured to do so,\nI should be met by the difficulty that neither of them defines the\n_\u00e0 priori_, and that each is too much inclined, in my opinion, to\ntest it by psychological criteria. I shall, therefore, take the three\nmeanings in turn, without laying stress on their historical adequacy\nto the views of Land or Helmholtz.\n\n=70.= (1) Congruence may be taken to mean--as Helmholtz would\ncertainly seem to desire--that we find actual bodies, in our\nmechanical experience, to preserve their shapes with approximate\nconstancy, and that we infer, from this experience, the homogeneity\nof space. This view, in my opinion, radically misconceives the\nnature of measurement, and of the axioms involved in it. For what\nis meant by the non-rigidity of a body? We mean, simply, that\nit has changed its shape. But this involves the possibility of\ncomparison with its former shape, in other words, of measurement.\nIn order, therefore, that there may be any question of rigidity\nor non-rigidity, the measurement of spatial magnitudes must be\nalready possible. It follows that measurement cannot, without a\nvicious circle, be itself derived from experience of rigid bodies.\nGeometrical measurement, in fact, is the comparison of spatial\nmagnitudes, and such comparison involves, as will be proved at length\nin Chapter III., the homogeneity of space. This is, therefore, the\nlogical prerequisite of all experience of rigid bodies, and cannot\nbe the result of such experience. Without the homogeneity of space,\nthe very notion of rigidity or non-rigidity could not exist, since\nthese mean, respectively, the constancy or inconstancy of spatial\nmagnitude in pieces of matter, and both alike, therefore, presuppose\nthe possibility of spatial measurement. From the homogeneity of\nspace, we learn that a body, when it moves, will not change its shape\nwithout some physical cause; that it actually does not change its\nshape, is never asserted, and is indeed known to be false. As soon as\nmeasurement is possible, actual changes of shape can be estimated,\nand their empirical causes can be sought. But if space were not\nhomogeneous, measurement would be impossible, constant shape would\nbe a meaningless phrase, and rigidity could never be experienced.\nCongruence asserts, in short, that a body can, so far as mere space\nis concerned, move without change of shape; rigidity asserts that\nit actually does so move--a very different proposition, involving\nobviously, as its logical prius, the former geometrical proposition.\n\nThis argument may be summed up by the following disjunction: If\nbodies change their shapes in motion--and to some extent, since no\nbody is perfectly rigid, they must all do so--then one of two cases\nmust occur. _Either_ the changes of shape, as bodies move from\nplace to place, follow no geometrical law, are not, for instance,\nfunctions of the amount or direction of motion; in which case the\nlaw of causation requires that they should not be effects of the\nchange of place, but of some simultaneous non-geometrical change,\nsuch as temperature. _Or_ the changes are regular, and the shape _S_\nbecomes, in a new position _p_, _Sf_(_p_). In this case, the law of\nconcomitant variations leads us to attribute the change of shape\nto the mere motion, and shape thus becomes a function of absolute\nposition. But this is absurd, for position _means_ merely a relation\nor set of relations; it is impossible, therefore, that mere position\nshould be able to effect changes in a body. Position is one term in a\nrelation, not a thing _per se_; it cannot, therefore, act on a thing,\nnor exist by itself, apart from the other terms of the relation. Thus\nHelmholtz's view, that Congruence depends on the existence of rigid\nbodies, must, since it involves absolute position, be condemned as\na logical fallacy. Congruence, in fact, as I shall prove more fully\nin Chapter III., is an _\u00e0 priori_ deduction from the relativity of\nposition.\n\n=71.= (2) The above argument seems to me to answer satisfactorily\nHelmholtz's contention in the precise form which he first gave it.\nThe axiom of Congruence, we must agree, is logically distinguishable\nfrom the existence of rigid bodies. Nevertheless some reference\nto matter is logically involved in Geometry[90], but whether this\nreference makes Geometry empirical, or does not, rather, show an _\u00e0\npriori_ element in dynamics, is a further question.\n\nThe reference to matter is necessitated by the homogeneity of empty\nspace. For so long as we leave matter out of account, one position\nis perfectly indistinguishable from another, and a science of\nthe relations of positions is impossible. Indeed, before spatial\nrelations can arise at all, the homogeneity of empty space must\nbe destroyed, and this destruction must be effected by matter.\nThe blank page is useless to the geometer until he defaces its\nhomogeneity by lines in ink or pencil. No spatial figures, in short,\nare conceivable, without a reference to a not purely spatial matter.\nAgain, if Congruence is ever to be used, there must be motion: but\na purely geometrical point, being defined solely by its spatial\nattributes, cannot be supposed to move without a contradiction in\nterms. What moves, therefore, must be matter. Hence, in order that\nmotion may afford a test of equality, we must have some _matter_\nwhich is known to be unaffected throughout the motion, that is, we\nmust have some rigid bodies. And the difficulty is, that these bodies\nmust not only undergo no change due solely to the nature of space,\nbut must, further, be unchanged by their changing relation to other\nbodies. And here we have a requisite which can no longer be fulfilled\n_\u00e0 priori_: which, indeed, we know to be, in strictness, untrue.\nFor the forces acting on a body depend upon its spatial relations\nto other bodies, and changing forces are liable to produce changing\nconfiguration. Hence, it would seem, actual measurement must be\npurely empirical, and must depend on the degree of rigidity to be\nobtained, during the process of measurement, in the bodies with which\nwe are conversant.\n\nThis conclusion, I believe, is valid of all actual measurement.\nBut the possibility of such empirical and approximate rigidity,\nI must insist, depends on the _\u00e0 priori_ law that _mere_ motion,\napart from the action of other matter, cannot effect a change of\nshape. For without this law, the effect of other matter would not be\ndiscoverable; the laws of motion would be absurd, and Physics would\nbe impossible. Consider the second law, for example: How could we\nmeasure the change of motion, if motion itself produced a change\nin our measures? Or consider the law of gravitation: How could we\nestablish the inverse square, unless we were able, independently\nof Dynamics, to measure distances? The whole science of Dynamics,\nin short, is fundamentally dependent on Geometry, and but for the\nindependent possibility of measuring spatial magnitudes, none of the\nmagnitudes of Dynamics could be measured. Time, force, and mass are\nalike measured by spatial correlates: these correlates are given, for\ntime, by the first law, for force and mass, by the second and third.\nIt is true, then, that an empirical element appears unavoidably in\nall actual measurement, inasmuch as we can only know empirically\nthat a given piece of matter preserves its shape throughout the\nnecessary change of dynamical relations to other matter involved in\nmotion; but it is further true that, for Geometry--which regards\nmatter simply as supplying the necessary breach in the homogeneity\nof space, and the necessary term for spatial relations, not as the\nbearer of forces which change the configuration of other material\nsystems--for Geometry, which deals with this abstract and merely\nkinematical matter, rigidity is _\u00e0 priori_, in so far as the only\nchanges with which it is cognizant--changes of mere position,\nnamely--are incapable of affecting the shapes of the imaginary and\nabstract bodies with which it deals. To use a scholastic distinction,\nwe may say that matter is the _causa essendi_ of space, but Geometry\nis the _causa cognoscendi_ of Physics. Without a Geometry independent\nof Physics, Physics itself, which necessarily assumes the results of\nGeometry, could never arise; but when Geometry is used in Physics, it\nloses some of its _\u00e0 priori_ certainty, and acquires the empirical\nand approximate character which belongs to all accounts of actual\nphenomena.\n\n=72.= (3) This argument leads us to Land's distinction of physical\nand geometrical rigidity. The distinction may be expressed--and\nI think it is better expressed--by distinguishing between\nthe conceptions of matter proper to Dynamics and to Geometry\nrespectively. In Dynamics, we are concerned with matter as subject\nto and as causing motion, as affected by and as exerting _force_. We\nare therefore concerned with the changes of spatial configuration to\nwhich material systems are liable: the description and explanation\nof these changes is the proper subject-matter of all Dynamics. But\nin order that such a science may exist, it is obviously necessary\nthat spatial configuration should be already measurable. If this\nwere not the case, motion, acceleration and force would remain\nperfectly indeterminate. Geometry, therefore, must already exist\nbefore Dynamics becomes possible: to make Geometry dependent for its\npossibility on the laws of motion or any of their consequences, is\na gross \u1f55\u03c3\u03c4\u03b5\u03c1\u03bf\u03bd \u03c0\u03c1\u03cc\u03c4\u03b5\u03c1\u03bf\u03bd. Nevertheless, as we have seen, some sort\nof matter is essential to Geometry. But this geometrical matter is a\nmore abstract and wholly different matter from that of Dynamics. In\norder to study space by itself, we reduce the properties of matter\nto a bare minimum: we avoid entirely the category of causation,\nso essential to Dynamics, and retain nothing, in our matter, but\nits spatial adjectives[91]. The kind of rigidity affirmed of this\nabstract matter--a kind which suffices for the theory of our\nscience, though not for its application to the objects of daily\nlife--is purely geometrical, and asserts no more than this: That\nsince our matter is devoid, _ex hypothesi_, of causal properties,\nthere remains nothing, in mere empty space, which is capable of\nchanging the configuration of any geometrical system. A change of\nabsolute position, it asserts, is nothing; therefore the only real\nchange involved in motion is a change of relation to other matter;\nbut such other matter, for the purposes of our science, is regarded\nas destitute of causal powers; hence no change can occur, in the\nconfiguration of our system, by the mere effect of motion through\nempty space. The necessity of such a principle may be shown by a\nsimple _reductio ad absurdum_, as follows. A motion of translation\nof the universe as a whole, with constant direction and velocity, is\ndynamically negligeable; indeed it is, philosophically, no motion at\nall, for it involves no change in the condition or mutual relations\nof the things in the universe. But if our geometrical rigidity\nwere denied, the change in the parameter of space might cause all\nbodies to change their shapes owing to the mere change of absolute\nposition, which is obviously absurd.\n\nTo make quite plain the function of rigid bodies in Geometry, let us\nsuppose a liquid geometer in a liquid world. We cannot suppose the\nliquid perfectly homogeneous and undifferentiated, in the first place\nbecause such a liquid would be indistinguishable from empty space,\nin the second place because our geometer's body--unless he be a\ndisembodied spirit--will itself constitute a differentiation for him.\nWe may therefore assume\n\n \"dim beams,\n Which amid the streams\n Weave a network of coloured light,\"\n\nand we may suppose this network to form the occasion for our\ngeometer's reflections. Then he will be able to imagine a network\nin which the lines are straight, or circular, or parabolic, or any\nother shape, and he will be able to infer that such a network, if it\ncan be woven in one part of the fluid, can be woven in another. This\nwill form sufficient basis for his deductions. The superposition he\nis concerned with--since not actual equality, but only the formal\nconditions of equality, are the subject-matter of Geometry--is purely\nideal, and is unaffected by the impossibility of congealing any\nactual network. But in order to apply his Geometry to the exigencies\nof life, he would need some standard of comparison between actual\nnetworks, and here, it is true, he would need either a rigid body,\nor a knowledge of the conditions under which similar networks arose.\nMoreover these conditions, being necessarily empirical, could hardly\nbe known apart from previous measurement. Hence for applied, though\nnot for pure Geometry, one rigid body at least seems essential.\n\n=73.= The utility, for Dynamics, of our abstract geometrical matter,\nis sufficiently evident. For having, by its means, a power of\ndetermining the configurations of material systems in whatever part\nof space, and knowing that changes of configuration are not due to\nmere change of place, we are able to attribute these changes to the\naction of other matter, and thus to establish the notion of force,\nwhich would be impossible if change of shape might be due to empty\nspace.\n\nThus, to conclude: Geometry requires, if it is to be _practically_\npossible, some body or bodies which are either rigid (in the\ndynamical sense), or known to undergo some definite changes of shape\naccording to some definite law. (These changes, we may suppose,\nare known by the laws of Physics, which have been experimentally\nestablished, and which throughout assume the truth of Geometry.)\nOne or more such bodies are necessary to applied Geometry--but only\nin the sense in which rulers and compasses are necessary. They are\nnecessary as, in making the Ordnance Survey, an elaborate apparatus\nwas necessary for measuring the base line on Salisbury Plain. But\nfor the _theory_ of Geometry, geometrical rigidity suffices, and\ngeometrical rigidity means only that a shape, which is possible in\none part of space, is possible in any other. The empirical element\nin practice, arising from the purely empirical nature of physical\nrigidity, is comparable to the empirical inaccuracies arising from\nthe failure to find straight lines or circles in the world--which no\none but Mill has regarded as rendering Geometry itself empirical or\ninaccurate. But to make Geometry await the perfection of Physics,\nis to make Physics, which depends throughout on Geometry, forever\nimpossible. As well might we leave the formation of numbers until we\nhad counted the houses in Piccadilly.\n\n\nErdmann.\n\n=74.= In connection with Riemann and Helmholtz, it is natural to\nconsider Erdmann's philosophical work on their theories[92]. This is\ncertainly the most important book on the subject which has appeared\nfrom the philosophical side, and in spite of the fact that, like\nthe whole theory of Riemann and Helmholtz, it is inapplicable to\nprojective Geometry, it still deserves a very full discussion.\n\nErdmann agrees throughout with the conclusions of Riemann and\nHelmholtz, except on a few points of minor importance; and his views,\nas this agreement would lead one to expect, are ultra-empirical.\nIndeed his logic seems--though I say this with hesitation--to be\nincompatible with any system but that of Mill: there is apparently\nno distinction, to him, between the general and the universal, and\nconsequently no concept not embodied in a series of instances. Such\na theory of logic, to my mind, vitiates most of his work, as it\nvitiated Riemann's philosophy[93]. This general criticism will find\nabundant illustration in the course of our account of Erdmann's views.\n\n=75.= After a general introduction, and a short history of the\ndevelopment of Metageometry, Erdmann proceeds, in his second\nchapter, to discuss what are the axioms of Euclidean Geometry. The\narithmetical axioms, as they are called, he leaves aside, as applying\nto magnitude in general; what we want here, he says, is a definition\nof space, for which the geometrical axioms are alone relevant.\nBut a definition of space, he says--following Riemann--demands a\ngenus of which space shall be a species, and this, since our space\nis psychologically unique, can only be furnished by analytical\nmathematics (p. 36). Now the space-forms dealt with by Geometry are\nmagnitudes, and conceptions of magnitude are everywhere applied in\nGeometry. But before Riemann, only particular determinations of space\ncould be exhibited as magnitudes, and thus the desired definition\nwas impossible to obtain. Now, however, we can subsume space as a\nwhole under a general conception of magnitude, and thus obtain,\nbesides the space-intuition and the space-conception, a third form,\nnamely, the conception of space as a magnitude (_Gr\u00f6ssenbegriff\nvom Raum_, pp. 38-39). The definition of this will give us the\ncomplete, but not redundant, system of axioms, which could not be\nobtained by transforming the general intuition of space into the\nspace-conception, for want of a plurality of instances (p. 40).\n\n=76.= Before considering the subsequent method of definition, let\nus reflect on the theories involved in the above account of the\nconception of space as a magnitude. In the first place, it is assumed\nthat conceptions cannot be formed unless we have a series of separate\nobjects from which to abstract a common property--in other words,\nthat the universal is always the general. In the second place, it\nis assumed that all definition is classification under a genus. In\nthe third place, the conception of magnitude, if I am not mistaken,\nis fundamentally misunderstood when it is supposed applicable to\nspace as a whole. But in the fourth place, even if such a conception\nexisted, it could give none of the essential properties of space. Let\nus consider these four points successively.\n\n=77.= As regards the first point, it is to be observed that people\ncertainly had some conception of space before Riemann invented\nthe notion of a manifold, and that this conception was certainly\nsomething other than the common qualities of all the points, lines\nor figures in space. In the second place, Erdmann's view would\nmake it impossible to conceive God, unless one were a polytheist,\nor the universe--unless, like Leibnitz, one imagined a series of\npossible worlds, set over against God, and none of them, therefore,\na true Universe--or, to take an instance more likely to appeal to an\nempiricist, the necessarily unique centre of mass of the material\nuniverse. Any universal, in short, which is a bond or unity between\nthings, and not merely a common property among independent objects,\nbecomes impossible on Erdmann's view. We cannot, therefore, unless\nwe adopt Mill's philosophy intact, regard the conception of space\nas demanding a series of instances from which to abstract. But even\nif we did so regard it, Riemann's manifolds would leave us without\nresources. For Euclidean space still appears as unique, at the end of\nhis series of determinations. We have instances of manifolds, but not\ninstances of Euclidean space. Thus if Erdmann's theory of conceptions\nwere correct, he would still be left searching in vain for the\nconception of Euclidean space.\n\n=78.= The second point, the view that all definition is\nclassification, is closely allied to the first, and the two together\nplunge us into the depths of scholastic formal logic. The same\ninstances of things which could not, on Erdmann's view, be conceived,\nmay now be adduced as things which cannot be defined. Whatever was\nsaid above applies here also, and the point need not, therefore, be\nfurther discussed[94].\n\n=79.= As regards the third point, the impossibility of applying\nconceptions of magnitude to space as a whole, a longer argument will\nbe necessary, for we are concerned, here, with the whole question\nof the logical nature of judgments of magnitude. As we had before\ntoo much comparison for our needs, so we have now too little. I will\nendeavour to explain this point, which is of great importance, and\nunderlies, I think, most of the philosophical fallacies of Riemann's\nschool.\n\nA judgment of magnitude is always a judgment of comparison, and what\nis more, the comparison is never concerned with quality, but only\nwith quantity. Quality, in the judgment of magnitude, is supposed\nidentical, in the object whose magnitude is stated, and in the unit\nwith which it is compared. But quality, except in pure number, and in\npure quantity as dealt with by the Calculus, is always present, and\nis partly absorbed into quantity, partly untouched by the judgment of\nmagnitude. As Bosanquet says (Logic, Vol. I. p. 124); \"Quantitative\ncomparison is not _prima facie_ coordinate with qualitative,\nbut rather stands in its place as the _effect of comparison on\nquality_, which so far as comparable _becomes quantity_, and so\nfar as incomparable furnishes the distinction of parts essential\nto the quantitative whole\" (italics in the original). Thus, if we\nare to regard space as a magnitude, we must be able to adduce all\nthose series of instances of which Erdmann speaks, and which, for\nthe conception of space, seemed irrelevant. But it remains to be\nproved that the comparison, which we _can_ institute between various\nspaces, is capable of expression in a quantitative form. Rather it\nwould seem that the difference of quality is such as to preclude\nquantitative comparison between different spaces, and therefore also\nto preclude all judgments of magnitude about space as a whole. Here\nan exception might seem to be demanded by non-Euclidean spaces,\nwhose space-constants give a definite magnitude, inherent in space\nas a whole, and therefore, one might think, characterizing space as\na magnitude. But this is a mistake. For the space-constant, in such\nspaces, is the ultimate unit, the fixed term in all quantitative\ncomparison; it is itself, therefore, destitute of quantity, since\nthere is no independently given magnitude with which to compare it.\nA non-Euclidean world, in which the space-constant and all lines\nand figures were suddenly multiplied in a constant ratio, would be\nwholly unchanged; the lines, as measured against the space-constant,\nwould have the same magnitude as before, and the space-constant\nitself, having no outside standard of comparison, would be destitute\nof quantity, and therefore not subject to change of quantity. Such an\nenlargement of a non-Euclidean world, in other words, is unmeaning;\nand this proves how inapplicable is the notion of quantity to space\nas a whole.\n\nIt might be objected that this only proves the absence of\nquantitative difference between different spaces of positive\nspace-constant, or between those of negative space-constant: the\nquantitative difference persists, it might be said, between those\nof positive curvature in general and those of negative curvature\nin general, or between both together and Euclidean space. This I\nentirely deny. There is no qualitatively similar unit, in the three\nkinds of space, by which quantitative comparison could be effected.\nThe straight lines of one space cannot be put into the other: the\ntwo straight lines, in one space, whose product is the reciprocal\nof the measure of curvature, have no corresponding curves in the\nother space, and the measures of curvature cannot, therefore,\nbe quantitatively compared with each other. That the one may be\nregarded as positive, the other negative, I admit, but their values\nare indeterminate, and the units in the two cases are qualitatively\ndifferent. A debt of \u00a3300 may be represented as the asset of -\u00a3300,\nand the height of the Eiffel Tower is +300 metres; but it does\nnot follow that the two are quantitatively comparable. So with\nspace-constants: the space-constant is itself the unit for magnitudes\nin its own space, and differs qualitatively from the space-constant\nof another kind of space.\n\nAgain, to proceed to a more philosophical argument, two different\nspaces cannot co-exist in the same world: we may be unable to decide\nbetween the alternatives of the disjunction, but they remain, none\nthe less, absolutely incompatible alternatives. Hence we cannot get\nthat coexistence of two spaces which is essential to comparison. The\nfact seems to be that Erdmann, in his admiration for Riemann and\nHelmholtz, has fallen in with their mathematical bias, and assumed,\nas mathematicians naturally tend to assume, that quantity is\neverywhere and always applicable and adequate, and can deal with more\nthan the mere comparison of things whose qualities are already known\nas similar[95].\n\n=80.= This suggests the fourth and last of the above points, that the\n_qualities_ of space, even if space could be successfully regarded\nas a magnitude, would have to be entirely omitted in such a manner\nof regarding it, and that, therefore, none of its important or\nessential properties would emerge from such treatment. For to regard\nspace as a magnitude involves, as we saw, a comparison with something\nqualitatively similar, and an abstraction from the similar qualities.\nTo some extent and by the help of certain doubtful arguments, such a\ncomparison is instituted by Riemann and Erdmann; but when they have\ninstituted it, they forget all about the common qualities on which\nits possibility depends. But these are precisely the fundamental\nproperties of space, and those from which, as I shall endeavour to\nprove in Chapter III., the axioms common to Euclid and Metageometry\nfollow _\u00e0 priori_. Such are the dangers of the quantitative bias.\n\n=81.= After this protest against the initial assumptions in Erdmann's\ndeduction of space, let us return to consider the manner, in which\nthis deduction is carried out. Here there will be less ground\nfor criticism, as the deduction, given its presuppositions, is,\nI think, as good as such a deduction can be. To define space as\na magnitude, he says, let us start with two of its most obvious\nproperties, continuity and the three dimensions. Tones and colours\nafford other instances of a manifold with these two properties, but\ndiffer from space in that their dimensions are not homogeneous and\ninterchangeable. To designate this difference, Erdmann introduces\na useful pair of terms: in the general case, he calls a manifold\n_n_-determined (n-_bestimmt_); in the case where, as in space, the\ndimensions are homogeneous, he calls the manifold _n_-extended\n(n-_ausgedehnt_). Manifolds of the latter sort he calls extents\n(_Ausgedehntheiten_). That the difference between the two kinds\nis one of quality, not of quantity, he seems not to perceive; he\nalso overlooks the fact that, in the second kind, from its very\ndefinition, the axiom of Congruence must hold, on account of the\nqualitative similarity of different parts. In spite of this fact, he\ndefines space as an extent, and then regards Congruence as empirical,\nand as possibly false in the infinitesimal. This is the more strange,\nas he actually proves (p. 50) that measurement is impossible, in an\nextent, unless the parts are independent of their place, and can be\ncarried about unaltered as measures. In spite of this, he proceeds\nimmediately to discuss whether the measure of curvature is constant\nor variable, without investigating how, in the latter case, Geometry\ncould exist. We cannot know, he says, from geometrical superposition,\nthat geometrical bodies are independent of place, for if their\ndimensions altered in motion according to any fixed law, two bodies\nwhich could be superposed in one place could be superposed in any\nother. That such a hypothesis involves absolute position, and denies\nthe qualitative similarity of the parts of space, which he declares\n(p. 171) to be the principle of his theory of Geometry, is nowhere\nperceived. But what is more, his notion that magnitude is something\nabsolute, independent of comparison, has prevented him from seeing\nthat such a hypothesis is unmeaning. He says himself that, even on\nthis hypothesis, a geometrical body can be defined as one whose\npoints retain constant distances from each other, for, since we have\nno absolute measure, measurement could not reveal to us the change of\nabsolute magnitude (p. 60). But is not this a _reductio ad absurdum_?\nFor magnitude is nothing apart from comparison, and the comparison\nhere can only be effected by superposition; if, then, as on the above\nhypothesis, superposition always gives the same result, by whatever\nmotion it is effected, there is no sense in speaking of magnitudes as\nno longer equal when separated: absolute magnitude is an absurdity,\nand the magnitude resulting from comparison does not differ from that\nwhich would result if the dimensions of bodies were unchanged in\nmotion. Therefore, since magnitude is only intelligible as the result\nof comparison, the dimensions of bodies _are_ unchanged in motion,\nand the suggested hypothesis is unmeaning. On this subject I shall\nhave more to say in Chapter III.[96]\n\n=82.= This hypothesis, however, is not introduced for its own sake,\nbut only to usher in the Helmholtzian _deus ex machina_, Mechanics.\nFor Mechanics proves--so Erdmann confidently continues--that\nrigidity must hold, not merely as to ratios, in the above restricted\ngeometrical sense, but as to absolute magnitudes (p. 62). Hence we\nget at last true Congruence, empirical as Mechanics is empirical, and\nimpossible to prove apart from Mechanics. I have already criticized\nHelmholtz's view of the dependence of Geometry on Mechanics, and\nneed not here speak of it at length. It is a pity that Erdmann has\nin no way specified the procedure by which Mechanics decides the\ngeometrical alternatives--indeed he seems to rely on the _ipse dixit_\nof Helmholtz. How, if Geometry would be totally unable to discover a\nchange in dimensions of the kind suggested, the Laws of Motion, which\nthroughout depend on Geometry, should be able to discover it if it\nexisted, I am wholly at a loss to understand. Uniform motion in a\nstraight line, for example, presupposes geometrical measurement; if\nthis measurement is mistaken, what Mechanics imagines to be uniform\nmotion is not really such, but Mechanics can never discover the\ndiscrepancy. If the Laws of Motion had been regarded as _\u00e0 priori_,\nGeometry might possibly have been reinforced by them; but so long\nas they are empirical, they presuppose geometrical measurement, and\ncannot therefore condition or affect it.\n\nErdmann's conclusion, in the second chapter, is that Congruence is\nprobable, but cannot be verified in the infinitesimal; that its truth\ninvolves the actual existence of rigid bodies (though, by the way,\nwe know these to be, strictly speaking, non-existent), that rigid\nbodies are freely moveable, and do not alter their size in rotation\n(Helmholtz's Monodromy); that the axiom of three dimensions is\ncertain, since small errors are impossible; and that the remaining\naxioms of Euclid--those of the straight line and of parallels--are\napproximately, if not accurately, true of our actual space (pp.\n78, 83). He does not discuss how Congruence, on the above view,\nis compatible with the atomic theory, or even with the observed\ndeformations of approximately rigid bodies; nor how, if space, as he\nassumes, is homogeneous, rigid bodies can fail to be freely moveable\nthrough space. The axioms are all lumped together as empirical, and\nit appears, in the following chapters, that Erdmann regards their\nempirical nature as sufficiently proved by their applicability to\nempirical material (cf. pp. 159, 165)--a strange criterion, which\nwould prove the same conclusion, with equal facility, of Arithmetic\nand of the laws of thought.\n\n=83.= The third chapter, on the philosophical consequences of\nMetageometry, need not be discussed at length, since it deals rather\nwith space than with Geometry. At the same time, it will be worth\nwhile to treat briefly of Erdmann's criterion of apriority. On this\nsubject it is very difficult to discover his meaning, since it\nseems to vary with the topic he is discussing. Thus at one time (p.\n147) he rejects most emphatically the Kantian connection of the _\u00e0\npriori_ and the subjective[97], and yet at another time (p. 96) he\nregards every presentation of external things as partly _\u00e0 priori_,\npartly empirical, merely because such a presentation is due to an\ninteraction between ourselves and things, and is therefore partly\ndue to subjective activity, partly due to outside objects. Hence,\nhe says, the distinction is not between different presentations,\nbut between different aspects of one and the same presentation.\nThis seems to return wholly to the Kantian psychological criterion\nof subjectivity, with the added disadvantage that it makes the\ndistinction, like that of analytic and synthetic, epistemologically\nworthless. And yet he never hesitates to pronounce every piece\nof knowledge in turn empirical. The fact seems to be, that where\nhe wants a more logical criterion, he adopts a modification of\nHelmholtz's criterion for sensations. If space be an _\u00e0 priori_\nform, he says, no experience could possibly change it (p. 108);\nbut this Metageometry has proved not to be the case, since we can\nintuit the perceptions which non-Euclidean space would give us (p.\n115). I have criticised this argument in discussing Helmholtz; at\npresent we are concerned with Erdmann's criterion of apriority. The\nsubjectivity-criterion--though he certainly uses it in discussing the\napriority of space, and solemnly decides, by its means, that space\nis both _\u00e0 priori_ and empirical since a change either in us or in\nthe outer world could change it (p. 97)--would seem, like several of\nhis other tests, to be a lapse on his part: the criterion which he\nmeans to use is Helmholtz's. This criterion, I think, with a slight\nchange of wording, might be accepted; it seems to me a necessary,\nbut not a sufficient condition. The _\u00e0 priori_, we may say, is not\nonly that which no experience can change, but that without which\nexperience would become impossible. It is the omission to discuss\nthe conditions which render geometrical (and mechanical) experience\npossible, to my mind, which vitiates the empirical conclusions of\nHelmholtz and Erdmann. Why certain conditions should be necessary\nfor experience--whether on account of the constitution of the mind,\nor for some other reason--is a further question, which introduces\nthe relation of the _\u00e0 priori_ to the subjective. But in discussing\nthe question as to what knowledge is _\u00e0 priori_, as opposed to the\nquestion concerning the further consequences of apriority, it is\nwell to keep to the purely logical criterion, and so preserve our\nindependence of psychological controversies. The fact, if it be a\nfact, that the world might be such as to defy our attempts to know\nit, will not, with the above criterion, invalidate the conclusion\nthat certain elements in knowledge are _\u00e0 priori_; for whether\nfulfilled or not, they remain necessary conditions for the existence\nof any knowledge at all.\n\n=84.= With this caution as to the meaning of apriority, we shall\nfind, I think, that the conclusions of Erdmann's final chapter, on\nthe principles of a theory of Geometry, are largely invalidated\nby the diversity and inadequacy of his tests of the _\u00e0 priori_.\nHe begins by asserting, in conformity with the quantitative bias\nnoticed above, that the question as to the nature of geometrical\naxioms is completely analogous to the corresponding question of the\nfoundations of pure mathematics (p. 138). This is, I think, a radical\nerror: for the function of the axioms seems to be, to establish that\nqualitative basis on which, as we saw, all qualitative comparison\nmust rest. But in pure mathematics, this qualitative basis is\nirrelevant, for we deal there with pure quantity, _i.e._ with the\nmerely quantitative result of quantitative comparison, wherever it is\npossible, independently of the qualities underlying the comparison.\nGeometry, as Grassmann insists[98], ought not to be classed with\npure mathematics, for it deals with a matter which is given to the\nintellect, not created by it. The axioms give the means by which\nthis matter is made amenable to quantity, and cannot, therefore, be\nthemselves deduced from purely quantitative considerations.\n\nLeaving this point aside, however, let us return to Erdmann.\nHe distinguishes, within space, a form and a matter: the form\nis to contain the properties common to all extents, the matter\nthe properties which distinguish space from other extents. This\ndistinction, he says, is purely logical, and does not correspond\nwith Kant's: matter and form, for Erdmann, are alike empirical. The\naxioms and definitions of Geometry, he says, deal exclusively with\nthe matter of space. It seems a pity, having made this distinction,\nto put it to so little use: after a few pages, it is dropped, and\nno epistemological consequences are drawn from it. The reason is, I\nthink, that Erdmann has not perceived how much can be deduced from\nhis definition of an extent, as a manifold in which the dimensions\nare homogeneous and interchangeable. For this property suffices to\nprove the complete homogeneity of an extent, and hence--from the\nabsence of qualitative differences among elements--the relativity of\nposition and the axiom of Congruence. This deduction will be made at\nlength in the sequel[99]; at present, I have only to observe that\nevery extent, on this view, possesses all the properties (except\nthe three dimensions) common to Euclidean and non-Euclidean spaces.\nThe axioms which express these properties, therefore, apply to the\nform of space, and follow from homogeneity alone, which Erdmann\nallows (p. 171) as the principle of any theory of space. The above\ndistinction of form and matter, therefore, corresponds, when its\nfull consequences are deduced, to the distinction between the axioms\nwhich follow from the homogeneity of space and those which do not.\nSince, then, homogeneity is equivalent to the relativity of position,\nand the relativity of position is of the very essence of a form of\nexternality, it would seem that his distinction of form and matter\ncan also be made coextensive with the distinction of the _\u00e0 priori_\nand empirical in Geometry. On this subject, I shall have more to say\nin Chapter III.\n\nIn the remainder of the chapter, Erdmann insists that the straight\nline, etc., though not abstracted from experience, which nowhere\npresents straight lines, must yet, as applicable to admittedly\nempirical sciences, be empirical (p. 159)--a criterion which he\nappears to employ only when all other grounds for an empirical\nopinion fail, and one which, obviously, can never refuse to do its\nwork, since all elements of knowledge are susceptible of employment\non some empirical material. He also defines the straight line (p.\n155) as a line of constant curvature zero, as though curvature could\nbe measured independently of the straight line. Even the arithmetical\naxioms are declared empirical (p. 165), since in a world where things\nwere all hopelessly different from one another, these axioms could\nnot be applied. After this reminder of Mill, we are not surprised, a\nfew pages later (p. 172), at a vague appeal to \"English logicians\"\nas having proved Geometry to be an inductive science. Nevertheless,\nErdmann declares, almost on the last page of his book (p. 173), that\nGeometry is distinguished from all other sciences by the homogeneity\nof its material: a principle of which no single application occurs\nthroughout his book, and which, as we shall see in Chapter III.,\nflatly contradicts the philosophical theories advocated throughout\nhis preceding pages.\n\nOn the whole, then, it cannot be said that Erdmann has done much to\nstrengthen the philosophical position of Riemann and Helmholtz. I\nhave criticized him at length, because his book has the appearance\nof great thoroughness, and because it is undoubtedly the best\ndefence extant of the position which it takes up. We shall now have\nthe opposite task to perform, in defending Metageometry, on its\nmathematical side, from the attacks of Lotze and others, and in\nvindicating for it that measure of philosophical importance--far\ninferior, indeed, to the hopes of Erdmann--which it seems really to\npossess.\n\n\nLotze.\n\n=85.= Lotze's argument as regards Geometry[100]--which follows a\nmetaphysical argument as to the ontological nature of space, and\nassumes the results of this argument--consists of two parts: the\nfirst discusses the various meanings logically assignable (pp.\n233-247) to the proposition that other spaces than Euclid's are\npossible, and the second criticizes, in detail, the procedure of\nMetageometry. The first of these questions is very important, and\ndemands considerable care as to the logical import of a judgment\nof possibility. Although Lotze's discussion is excellent in many\nrespects, I cannot persuade myself that he has hit on the only true\nsense in which non-Euclidean spaces are possible. I shall endeavour\nto make good this statement in the following pages.\n\n=86.= Lotze opens with a somewhat startling statement, which,\nthough philosophically worthy to be true, does not appear to be\nhistorically borne out. Euclidean Geometry has been chiefly shaken,\nhe says, by the Kantian notion of the exclusive subjectivity of\nspace--if space is only our private form of intuition, to which\nthere exists no analogue in the objective world, then other beings\nmay have other spaces, without supposing any difference in the world\nwhich they arrange in these spaces (p. 233). This certainly seems\na legitimate deduction from the subjectivity of space, which, so\nfar from establishing the universal validity of Euclid, establishes\nhis validity only after an empirical investigation of the nature of\nspace as intuited by Tom, Dick or Harry. But as a matter of fact,\nthose who have done most to further non-Euclidean Geometry--with the\nexception of Riemann, who was a disciple of Herbart--have usually\ninherited from Newton a na\u00efve realism as regards absolute space.\nI might instance the passage quoted from Bolyai in Chapter I., or\nClifford, who seems to have thought that we actually see the images\nof things on the retina[101], or again Helmholtz's belief in the\ndependence of Geometry on the behaviour of rigid bodies. This belief\nled to the view that Geometry, like Physics, is an experimental\nscience, in which objective truth can be attained, it is true, but\nonly by empirical methods. However, Lotze's ground for uncertainty\nabout Euclid is a philosophically tenable ground, and it will be\ninstructive to observe the various possibilities which arise from it.\n\nIf space is only a subjective form--so Lotze opens his\nargument--other beings may have a different form. If this corresponds\nto a different world, the difference, he says, is uninteresting:\nfor our world alone is relevant to any metaphysical discussion. But\nif this different space corresponds to the same world which we know\nunder the Euclidean form, then, in his opinion, we get a question of\ngenuine philosophic interest. And here he distinguishes two cases:\n_either_ the relations between things, which are presented to these\nhypothetical beings under the form of some different space, are\nrelations which do not appear to us, or at any rate do not appear\nspatial; _or_ they are the same relations which appear to us as\nfigures in Euclidean space (p. 235). The first possibility would be\nillustrated, he says, by beings to whom the tone or colour-manifolds\nappeared extended; but we cannot, in his opinion, imagine a manifold,\nsuch as is required for this case, to have its dimensions homogeneous\nand comparable _inter se_, and therefore the contents of the various\npresentations constituting such a manifold could not be combined\ninto a single content containing them all. But the possibility of\nsuch a combination is of the essence of anything worth calling a\nspace: therefore the first of the above possibilities is unmotived\nand uninteresting. Lotze's conclusion on this point, I think, is\nundeniable, but I doubt whether his argument is very cogent. However,\nas this possibility has no connection with that contemplated by\nnon-Euclideans, it is not worth while to discuss it further.\n\nThe second possibility also, Lotze thinks, is not that of\nMetageometry, but in truth it comes nearer to it than any of the\nother possibilities discussed. If a non-Euclidean were at the same\ntime a believer in the subjectivity of space, he would have to be\nan adherent of this view. Let us see more precisely what the view\nis. In Book II., Chapter I., Lotze has accepted the argument of the\nTranscendental Aesthetic, but rejected that of the mathematical\nantinomies: he has decided that space is, as Kant believed,\nsubjective, but possesses nevertheless, what Kant denied it, an\nobjective counterpart. The relation of presented space to its\nobjective counterpart, as conceived by Lotze, is rather hard to\nunderstand. It seems scarcely to resemble the relation of sensation\nto its object--_e.g._ of light to ether-vibrations--for if it did,\nspace would not be in any peculiar sense subjective. It seems rather\nto resemble the relation of a perceived bodily motion to the state\nof mind of the person willing the motion. However this may be, the\nobjective counterpart of space is supposed to consist of certain\nimmediate interactions of monads, who experience the interactions\nas modifications of their internal states. Such interactions, it\nis plain, do not form the subject-matter of Geometry, which deals\nonly with our resulting perceptions of spatial figures. Now if\nLotze's construction of space be correct, there seems certainly\nno reason why these resulting perceptions should not, for one and\nthe same interaction between monads, be very different in beings\ndifferently constituted from ourselves. But if they were different,\nsays Lotze, they would have to be utterly different--as different,\nfor example, as the interval between two notes is from a straight\nline. The possibility is, therefore, in his opinion, one about which\nwe can know nothing, and one which must remain always a mere empty\nidea. This seems to me to go too far: for whatever the objective\ncounterpart may be, any argument which gives us information about\nit must, when reversed, give us information about any possible form\nof intuition in which this counterpart is presented. The argument\nwhich Lotze has used in his former chapter, for example, deducing,\nfrom the relativity of position, the merely relational nature of\nthe objective counterpart, allows us, conversely, to infer, from\nthis relational nature, the complete relativity of position in any\npossible space-intuition--unless, indeed, it bore a wholly deceitful\nrelation to those interactions of monads which form its objective\ncounterpart. But the complete relativity of position, as I shall\nendeavour to establish in Chapter III., suffices to prove that our\nGeometry must be Euclidean, elliptic, spherical or pseudo-spherical.\nWe have, therefore, it would seem, very considerable knowledge,\non Lotze's theory of space, of the manner in which what appears\nto us as space _must_ appear to any beings with our laws of\nthought. We cannot know, it is true, what _psychological_ theory\nof space-perception would apply to such beings: they might have a\nsense different from any of ours, and they might have no sense in\nany way resembling ours, but yet their Geometry would have points of\nresemblance to ours, as that of the blind coincides with that of the\nseeing. If space has any objective counterpart whatever, in short,\nand if any inference is possible, as Lotze holds it to be, from space\nto its counterpart, then a converse argument is also possible, though\nit may give some only of the qualities of Euclidean space, since some\nonly of these qualities may be found to have a necessary analogue in\nthe counterpart.\n\n=87.= Admitting, then, in Lotze's sense, the subjectivity of space,\nthe above possibility does not seem so empty as he imagines. He\ndiscusses it briefly, however, in order to pass on to what he regards\nas the real meaning of Metageometry. In this he is guilty of a\nmathematical mistake, which causes much irrelevant reasoning. For\nhe believes that Metageometry constructs its spaces out of straight\nlines and angles in all respects similar to Euclid's, whence he\nderives an easy victory in proving that these elements can lead only\nto the one space. In this he has been misled by the phraseology of\nnon-Euclideans, as well as by Euclid's separation of definitions\nand axioms. For the fact is, of course, that straight lines are\nonly fully defined when we add to the formal definition the axioms\nof the straight line and of parallels. Within Euclidean space,\nEuclid's definition suffices to distinguish the straight line from\nall other curves; the two axioms referred to are then absorbed into\nthe definition of space. But apart from the restriction to Euclidean\nspace, the definition has to be supplemented by the two axioms, in\norder to define completely the Euclidean straight line. Thus Lotze\nhas misconceived the bearing of non-Euclidean constructions, and\nhas simply missed the point in arguing as he does. The possibility\ncontemplated by a non-Euclidean, if it fell under any of Lotze's\ncases, would fall under the second case discussed above.\n\n=88.= But the bearing of Metageometry is really, I think, different\nfrom anything imagined by Lotze; and as few writers seem clear on\nthis point, I will enter somewhat fully into what I conceive to be\nits purpose.\n\nIn the first place, there are some writers--notably Clifford--who,\nbeing na\u00efve realists as regards space, hold that our evidence is\nwholly insufficient, as yet, to decide as to its nature in the\ninfinite or in the infinitesimal (cf. Essays, Vol. I. p. 320):\nthese writers are not concerned with any possibility of beings\ndifferent from ourselves, but simply with the everyday space we\nknow, which they investigate in the spirit of a chemist discussing\nwhether hydrogen is a metal, or an astronomer discussing the nebular\nhypothesis.\n\nBut these are a minority: most, more cautious, admit that our space,\nso far as observation extends, is Euclidean, and if not accurately\nEuclidean, must be only slightly spherical or pseudo-spherical. Here\nagain, it is the space of daily life which is under discussion,\nand here further the discussion is, I think, independent of any\nphilosophical assumption as to the nature of our space-intuition. For\neven if this be purely subjective, the translation of an intuition\ninto a conception can only be accomplished approximately, within\nthe errors of observation incident to self-analysis; and until the\nintuition of space has become a conception, we get no scientific\nGeometry. The apodeictic certainty of the axiom of parallels shrinks\nto an unmotived subjective conviction, and vanishes altogether in\nthose who entertain non-Euclidean doubts. To reinforce the Euclidean\nfaith, reason must now be brought to the aid of intuition; but\nreason, unfortunately, abandons us, and we are left to the mercy of\napproximate observations of stellar triangles--a meagre support,\nindeed, for the cherished religion of our childhood.\n\n=89.= But the possibility of an inaccuracy so slight, that our\nfinest instruments and our most distant parallaxes show no trace\nof it, would trouble men's minds no more than the analogous chance\nof inaccuracy in the law of gravitation, were it not for the\nphilosophical import of even the slenderest possibility in this\nsphere. And it is the philosophical bearing of Metageometry alone,\nI think, which constitutes its real importance. Even if, as we\nwill suppose for the moment, observation had established, beyond\nthe possibility of doubt, that our space might be safely regarded\nas Euclidean, still Metageometry would have shown a philosophical\npossibility, and on this ground alone it could claim, I think, very\nnearly all the attention which it at present deserves.\n\nBut what is this possibility? A thing is possible, according to\nBradley (Logic, p. 187), when it would follow from a certain number\nof conditions, some of which are known to be realized. Now the\nconditions to which a form of externality must conform, in order to\nbe affirmed, are: first, of course, that it should be experienced,\nor legitimately inferred from something experienced; but secondly,\nthat it should conform to certain logical conditions, detailed in\nChapter III., which may be summed up in the relativity of position.\nNow what Metageometry has done, in any case, is to suggest the proof\nthat the second of these conditions is fulfilled by non-Euclidean\nspaces. Euclid is affirmed, therefore, on the ground of immediate\nexperience alone, and his truth, as unmediated by logical necessity,\nis merely assertorical, or, if we prefer it, empirical. This is\nthe most important sense, it seems to me, in which non-Euclidean\nspaces are possible. They are, in short, a step in a philosophical\nargument, rather than in the investigation of fact: they throw light\non the nature of the grounds for Euclid, rather than on the actual\nconformation of space[102]. This import of Metageometry is denied by\nLotze, on the ground that non-Euclidean logic is faulty, a ground\nwhich he endeavours, by much detail and through many pages, to make\ngood--with what success, we will now proceed to examine.\n\n=90.= Lotze's attack on Metageometry--although it remains, so\nfar as I know, the best hostile criticism extant, and although\nits arguments have become part of the regular stock-in-trade of\nEuclidean philosophers--contains, if I am not mistaken, several\nmisunderstandings due to insufficient mathematical knowledge of the\nsubject. As these misunderstandings have been widely spread among\nphilosophers, and cannot be easily removed except by a critic who has\ngone into non-Euclidean Geometry with some care, it seems desirable\nto discuss Lotze's strictures point by point.\n\n[Illustration]\n\n=91.= The mathematical criticism begins (\u00a7 131) with a somewhat\nquestion-begging definition of parallel straight lines. Two straight\nlines _a\u03b1_, _b\u03b2_, according to this definition, are parallel\nwhen--_a_ and _b_ being arbitrary points on the two lines--if _a\u03b1_\n= _b\u03b2_, then _ab_ = _\u03b1\u03b2_, where _\u03b1_, _\u03b2_ are two other points on the\ntwo straight lines respectively. This definition--which contains\nEuclid's axiom and definition combined in a very convenient and\nenticing form--is of course thoroughly suitable to Euclidean\nGeometry, and leads immediately to all the Euclidean propositions\nabout parallels. But it is perhaps more honest to follow Euclid's\ncourse; when an axiom is thus buried in a definition, it is apt to\nseem, since definitions are supposed to be arbitrary, as though the\ndifficulty had been overcome, while in reality, the possibility of\nparallels, as above defined, involves the very point in question,\nnamely, the disputed axiom of parallels. For what this axiom asserts\nis simply the existence of lines conforming to Lotze's definition.\nThe deduction of the principal propositions on parallels, with\nwhich Lotze follows up his definition, is of course a very simple\nproceeding--a proceeding, however, in which the first step begs the\nquestion.\n\n=92.= The next argument for the apriority of Euclidean Geometry has,\noddly enough, an exactly opposite bearing, although it is a great\nfavourite with opponents of Metageometry. Measurements of stellar\ntriangles, and all similar attempts at an empirical determination\nof the space-constant are, according to Lotze, beside the mark; for\nany observed departure from two right angles, or any finite annual\nparallax for distant stars, would be attributed to some new kind of\nrefraction, or, as in the case of aberration, to some other physical\ncause, and never to the geometrical nature of space. This is a\nstrong argument for the empirical validity of Euclid, but as an\nargument for the apodeictic certainty of the orthodox system, it has\nan opposite tendency. For observations of the kind contemplated would\nhave to be due to departures from Euclidean straightness, hitherto\nunknown, on the part of stellar light-rays. Such departure could,\nin certain cases, be accounted for by a finite space-constant, but\nit could also, probably, be accounted for by a change in Optics,\nfor example, by attributing refractive properties to the ether.\nSuch properties could only exist if ether were of varying density,\nif (say) it were denser in the neighbourhood of any of the heavenly\nbodies. But such an assumption would, I believe, destroy the utility\nof ether for Physics; a slight alteration in our Geometry, so slight\nas not appreciably to affect distances within the Solar System,\nwould probably be in the end, therefore, should such errors ever\nbe discovered, a simpler explanation than any that Physics could\noffer. But this is not the point of my contention. The point is\nthat, if the physical explanation, as Lotze holds, be possible in\nthe above case, the converse must also hold: it must be possible to\nexplain the present phenomena by supposing ether refractive and space\nnon-Euclidean. From this conclusion there is no escape. If every\nconceivable behaviour of light-rays can be explained, within Euclid,\nby physical causes, it must also be possible, by a suitable choice\nof hypothetical physical causes, to explain the actual phenomena\nas belonging to a non-Euclidean space. Such a hypothesis would be\nrightly rejected by Science, for the present, on account of its\nunnecessary complexity. Nevertheless it would remain, for philosophy,\na possibility to be reckoned with, and the choice could only be\ndecided upon empirical grounds of simplicity. It may well be doubted\nwhether, in the world we know, the phenomena could be attributed\nto a distinctly non-Euclidean space, but this conclusion follows\ninevitably from the contention that no phenomena could force us to\nassume such a space. Lotze's argument, therefore, if pushed home,\ndisproves his own view, and puts Euclidean space, as an empirical\nexplanation of phenomena, on a level with luminiferous ether[103].\n\n=93.= Lotze now proceeds (\u00a7 132) to a detailed criticism of\nHelmholtz, whom he regards as a typical exponent of Metageometry.\nIt is possible that, at the time when he wrote, Helmholtz really\ndid occupy this position; but it is unfortunate that, in the minds\nof philosophers, he should still continue to do so, after the very\nmaterial advances brought about by the projective treatment of the\nsubject. It is also unfortunate that his somewhat careless attempts\nto popularise mathematical results have so often been disposed of,\nwithout due attention to his more technical and solid contributions.\nThus his romances about Flatland and Sphereland--at best only\nfairy-tale analogies of doubtful value--have been attacked as if they\nformed an essential feature of Metageometry.\n\nBut to proceed to particulars: Lotze readily allows that the\nFlatlanders would set up Plane Geometry, as we know it, but refuses\nto admit that the Spherelanders could, without inferring the third\ndimension, set up a two-dimensional spherical Geometry which should\nbe free from contradictions. I will endeavour to give a free\nrendering of Lotze's argument on this point.\n\n[Illustration]\n\nSuppose, he says, a north and south pole, _N_ and _S_, arbitrarily\nfixed, and an equator _EW_. Suppose a being, _B_, capable of\nimpressions only from things on the surface of the sphere, to move\nin a meridian _NBS_. Let _B_ start from some point _a_, and finally,\nafter describing a great circle, return to the same point _a_. If\n_a_ is known only by the quality of the impression it makes on _B_,\n_B_ may imagine he has not reached the same point _a_, but another\nsimilar point _a\u2032_, bearing a relation to _a_ similar to that of\nthe octave in singing: he might even not arrange his impressions\nspatially at all. In order that this may occur, we require the\nfurther assumption, that every difference in the above-mentioned\nfeelings (as he describes the meridian) may be presented as a spatial\ndistance between two places. Even now, _B_ may think he is describing\na Euclidean straight line, containing similar points at certain\nintervals. Allowing, however, that he realizes the identity of _a_\nwith his initial position, he will now seem, by motion in a straight\nline, to have returned to the point from which he started, for his\nmotion cannot, without the third dimension, seem to him other than\nrectilinear.\n\nUp to this point, there seems little ground for objection, except,\nperhaps, to the idea of a straight line with periodical similar\npoints--if _B_ were as philosophical as, in these discussions,\nwe usually suppose him to be, he would probably object to this\ninterpretation of his experiences, on the ground that it regards\nempty space as something independent of the objects in it. It is\nworth pointing out, also, that _B_ would not need to describe the\nwhole circle, in order suddenly to find himself home again with his\nold friends. Accurate measurements of small triangles would suffice\nto determine his space-constant, and show him the length of a great\ncircle (or straight line, as he would call it). We must admit, also,\nthat so hypothetical a being as _B_ might form no space-intuition at\nall, but as he is introduced solely for the purposes of the analogy,\nit is convenient to allow him all possible qualifications for his\npost. But these points do not touch the kernel of the argument, which\nlies in the statement that such a straight line, returning into\nitself after a finite time, would appear to _B_ as an \"unendurable\ncontradiction,\" and thus force him, for logical though not for\nsensational purposes, into the assumption of a third dimension. This\nassertion seems to me quite unwarranted: the whole of Metageometry\nis a solid array in disproof of it. Helmholtz's argument is, it must\nbe remembered, only an analogy, and the contradiction would exist\n_only_ for a Euclidean. A complete _three_-dimensional Geometry has,\nwe have seen in Chapter I., been developed on the assumption that\nstraight lines are of finite length. A _constant_ value for the\nmeasure of curvature, as our discussion of Riemann showed, involves\nneither reference to the fourth dimension, nor any kind of internal\ncontradiction. This fact disproves Lotze's contention, which arises\nsolely from inability to divest his imagination of Euclidean ideas.\n\nLotze next attacks Helmholtz for the assertion that _B_ would know\nnothing of parallel lines--parallel _straight_ lines, as the context\nshows, he meant to say[104]. Lotze, however, takes him as meaning,\napparently, mere curves of constant distance from a given straight\nline, which are part of the regular stock-in-trade of Metageometry.\nParallels of latitude, in the geographical sense, would not--with\nthe exception of the equator--appear to _B_ as straight lines, but\nas circles. _Great_ circles he _would_ call straight, and this fact\nseems to have misled Lotze into thinking _all_ circles were to be\ntreated as straight lines. Parallels of latitude, therefore, though\n_B_ might call them parallels, would not invalidate Helmholtz's\ncontention, which applies only to straight lines.\n\nThe argument that such small circles would be parallel, which we\nhave just disposed of, is only the preface to another proof that _B_\nwould need a third dimension. Let us call two of these parallels\nof latitude _l{n}_ and _l{s}_, and let them be equidistant from\nthe equator, one in the northern, one in the southern hemisphere.\nConsecutive tangent planes, along these parallels, converge, in\nthe one case northwards, in the other southwards. Either _B_ could\nbecome aware of their difference, says Lotze, or he could not. In\nthe former case, which he regards as the more probable, he easily\nproves that _B_ would infer a third dimension. But this alternative\nis, I think, wholly inadmissible. Tangent planes, like Euclidean\nplanes in general, would have no meaning to _B_; unless, indeed,\nhe were a metageometrician, which, with all his metaphysical and\nmathematical subtlety, the argument supposes him not to be--and to\nsuch a supposition Lotze, surely, is the last person who has a right\nto object. Lotze's attempted proof that this is the right alternative\nrests, if I understand him aright, on a sheer error in ordinary\nspherical Geometry. _B_ would observe, he says, that the meridians\nmade smaller angles with his path towards the nearer than towards the\nfurther pole--as a matter of fact, they would be simply perpendicular\nto his path in both directions. What Lotze means is, perhaps, that\nall the meridians would meet sooner in one direction than in the\nother, and this, of course, is true. But the poles, in which the\nmeridians meet, would appear to _B_ as the centres of the respective\nparallels, while the parallels themselves would appear to be circles.\nNow I am at a loss to see what difficulty would arise, to _B_, in\nsupposing two different circles to have different centres[105]. We\nmust, therefore, take the first alternative, that _B_ would have no\nsort of knowledge as to the direction in which the tangent planes\nconverged. Here Lotze attempts, if I have not misunderstood him, to\nprove a _reductio ad absurdum_: _B_ would think, he says, that he\nwas describing two paths wholly the same in direction, and then he\n_might_ regard both paths as circles in a plane. It may be observed\nthat direction, when applied to a circle as a whole, is meaningless;\nindeed direction, in all Metageometry, can only mean, even when\napplied to straight lines, direction towards a point. To speak of\ntwo lines, which do not meet, as having the same direction, is a\nsurreptitious introduction of the axiom of parallels. Apart from\nthis, I cannot conceive any objection, on _B_'s part, to such a\nview--one should say _must_, not _might_. The whole argumentation,\ntherefore, unless its obscurity has led me astray, must be pronounced\nfruitless and inconclusive.\n\n=94.= After this preliminary discussion of Sphereland, Lotze proceeds\nto the question of a fourth dimension, and thence to spherical and\npseudo-spherical space. As before, he appears to know only the\nmore careless and popular utterances of Helmholtz and Riemann, and\nto have taken no trouble to understand even the foundations of\nmathematical Metageometry. By this neglect, much of what he says\nis rendered wholly worthless. To begin with, he regards, as the\npurpose of Helmholtz's fairy tale, the suggestion of a possible\nfourth dimension, whereas the real purpose was quite the opposite--to\nmake intelligible a purely three-dimensional non-Euclidean space.\nHelmholtz introduced Flatland only because its relation to Sphereland\nis analogous to the relation of ours to spherical space[106]. But\nLotze says: The Flatlanders would find no difficulty in a third\ndimension, since it would in no way contradict their own Geometry,\nwhile the people in Sphereland, from the contradictions in their\ntwo-dimensional system, would already have been led to it. The\nlatter contention I have already tried to answer; the former has an\nodd sound, in view of the attempt, a few pages later, to prove _\u00e0\npriori_ that all forms of intuition, in any way analogous to space,\n_must_ have three dimensions. One cannot help suspecting that the\nFlatlanders, with two instead of three dimensions, would make a\nsimilar attempt. But to return to Lotze's argument: Neither analogy\ncan be used, he says, to prove that we ought perhaps to set up a\nfourth dimension, since, for us, no contradictions or otherwise\ninexplicable phenomena exist. The only people, so far as I know, who\nhave used this analogy, are Dr Abbot and a few Spiritualists--the\nformer in joke, the latter to explain certain phenomena more simply\nexplained, perhaps, by Maskelyne and Cooke. But although Lotze's\nconclusion in this matter is sound, and one with which Helmholtz\nmight have agreed, his arguments, to my mind, are irrelevant and\nunconvincing. There is this difference, he says, between us and the\nSpherelanders: the latter were logically forced to a new dimension,\nand found it possible; we are not forced to it, and find it, in our\nspace, impossible. I have contended that, on the contrary, nothing\nwould force the Spherelanders to assume a third dimension, while they\nwould find it impossible exactly as we find a fourth impossible--not\nlogically, that is to say, but only as a presentable construction in\ngiven space.\n\nAfter a somewhat elephantine piece of humour, about socialistic\nwhales in a four-dimensional sea of Fourrier's _eau sucr\u00e9e_, Lotze\nproceeds to a proof, by logic, that every form of intuition, which\nembraces the whole system of ordered relations of a coexisting\nmanifold, _must_ have three dimensions. One might object, on _\u00e0\npriori_ grounds, to any such attempt: what belongs to pure intuition\ncould hardly, one would have thought, be determined by _\u00e0 priori_\nreasoning[107]. I will not, however, develop this argument here,\nbut endeavour to point out, as far as its obscurity will allow, the\nparticular fallacy of the proof in question.\n\n[Illustration]\n\nLotze's argument is as follows. In this discussion, though our\nterminology is necessarily taken from space, we are really concerned\nwith a much more general conception. We assume, in order to preserve\nthe homogeneity of dimensions, that the difference (distance) between\nany two elements (points) of our manifold--to borrow Riemann's\nword--is of the same kind as, and commensurable with, the difference\nbetween any other two elements. Let us take a series of elements at\nsuccessive distances _x_ such that the distance between any two is\nthe sum of the distances between intermediate elements. Such a series\ncorresponds to a straight line, which is taken as the _x_-axis. Then\na series _OY_ is called perpendicular to the _x_-axis _OX_, when\nthe distances of any element _y_, on _OY_, from +_mx_ and -_mx_ are\nequal. By our hypothesis, these distances are comparable with, and\nqualitatively similar to, _x_ and _y_. So long as _OY_ is defined\nonly by relation to _OX_, it is conceptually unique. But now let\nus suppose the same relation as that between _OX_ and _OY_, to be\npossible between _OY_ and a new series _OZ_; we then get a third\nseries _OZ_ perpendicular to _OY_, and again conceptually unique, so\nlong as it is defined by relation to _OY_ alone. We might proceed,\nin the same way, to a fourth line _OU_ perpendicular to _OZ_. But it\nis necessary, for our purposes, that _OZ_ should be perpendicular to\n_OX_ as well as _OY_. Without this condition, _OZ_ might extend into\nanother world, and have no corresponding relation to _OX_--this is a\npossibility only excluded by our unavoidable spatial images. At this\npoint comes the crux of the argument. _That OZ_, says Lotze, which,\nbesides being perpendicular to _OY_, is also perpendicular to _OX_,\nmust be among the series of _OY_'s, for these were defined only by\nperpendicularity to _OX_. _Hence_, he concludes, there can only be\neven a third dimension if _OZ_ coincides with one, and--as soon as\n_OX_ is considered fixed--with _only_ one, of the many members of the\n_OY_ series.\n\nIn this argument it is difficult--to me at any rate--to see any\nforce at all. The only way I can account for it is, to suppose that\nLotze has neglected the possibility of any but single infinities.\nOn this interpretation, the argument might be stated thus: There is\nan infinite series of continuously varying _OY_'s; to the common\nproperty of these, we add another property, which will divide their\ntotal number by infinity. The remaining _OZ_, therefore, must be\nuniquely determined. The same form of argument, however, would prove\nthat two surfaces can only cut one another in a single point, and\nnumberless other absurdities. The fact is, that infinities may be of\ndifferent orders. For example, the number of points in a line may be\ntaken as a single infinity, and so may the number of lines in a plane\nthrough any point; hence, by multiplication, the number of points in\na plane is a double infinity, \u221e^{2}, and if we divide this number\nby a single infinity, we get still an infinite number left. Thus\nLotze's argument assumes what he has to prove, that the number of\nlines perpendicular to a given line, through any point, is a single\ninfinity, which is equivalent to the axiom of three dimensions. The\nwhole passage is so obscure, that its meaning may have escaped me. It\nis obvious _\u00e0 priori_, however, as I pointed out in the beginning,\nthat any proof of the axiom must be fallacious somewhere, and the\nabove interpretation of the argument is the only one I have been able\nto find.\n\n=95.= The rest of the Chapter is devoted to an attack on spherical\nand pseudo-spherical space, on the ground that they interfere with\nthe homogeneity of the three dimensions, and with the similarity\nof all parts of space. This is simply false. Such spaces, like the\nsurface of a sphere, _are_ exactly alike throughout. Lotze shows,\nhere and elsewhere, that he has not taken the pains to find out\nwhat Metageometry really is. I hold myself, and have tried to prove\nin this Essay, that Congruence is an _\u00e0 priori_ axiom, without\nwhich Geometry would be impossible; but the wish to uphold this\naxiom is, as Lotze ought to have known, the precise motive which\nled Metageometry to limit itself to spaces of constant measure of\ncurvature. We see here the importance of distinguishing between\nHelmholtz the philosopher and Helmholtz the mathematician. Though the\nphilosopher wished to dispense with Congruence, the mathematician, as\nwe saw in Chapter I., retained and strongly emphasized it. A little\nlater Lotze shows, again, how he has been misled by the unfortunate\nanalogy of Sphereland. A spherical _surface_, he says, he can\nunderstand; but how are we to pass from this to a spherical space?\nEither this surface is the whole of our space, as in Sphereland, or\nit generates space by a gradually growing radius. Such concentric\nspheres, as Lotze triumphantly points out, of course generate\nEuclidean space. His disjunction, however, is utterly and entirely\nfalse, and could never have been suggested by any one with even a\nsuperficial knowledge of Metageometry. This point is less laboured\nthan the former, which, in all its nakedness, is thus re-stated in\nthe last sentence of the Chapter: \"I cannot persuade myself that one\ncould, without the elements of homogeneous space, even form or define\nthe presentation of heterogeneous spaces, or of such as had variable\nmeasures of curvature.\" As though such spaces were ever set up by\nnon-Euclidean mathematics!\n\nIn conclusion, Lotze expresses a hope that Philosophy, on this\npoint, will not allow itself to be imposed upon by Mathematics. I\nmust, instead, rejoice that Mathematics has not been imposed upon by\nPhilosophy, but has developed freely an important and self-consistent\nsystem, which deserves, for its subtle analysis into logical and\nfactual elements, the gratitude of all who seek for a philosophy of\nspace.\n\n=96.= The objections to non-Euclidean Geometry which have just been\ndiscussed fall under four heads:\n\nI. Non-Euclidean spaces are not homogeneous; Metageometry therefore\nunduly reifies space.\n\nII. They involve a reference to a fourth dimension.\n\nIII. They cannot be set up without an implicit reference to Euclidean\nspace, or to the Euclidean straight line, on which they are therefore\ndependent.\n\nIV. They are self-contradictory in one or more ways.\n\nThe reader who has followed me in regarding these four objections as\nfallacious, will have no difficulty in disposing of any other critic\nof Metageometry, as these are the only mathematical arguments, so\nfar as I know, ever urged against non-Euclideans[108]. The logical\nvalidity of Metageometry, and the mathematical possibility of\nthree-dimensional non-Euclidean spaces, will therefore be regarded,\nthroughout the remainder of the work, as sufficiently established.\n\n=97.= Two other objections may, indeed, be urged against\nMetageometry, but these are rather of a philosophical than of a\nstrictly mathematical import. The first of these, which has been\nmade the base of operations by Delb\u0153uf, applies equally to all\nnon-Euclidean spaces. The second, which has not, so far as I know,\nbeen much employed, but yet seems to me deserving of notice, bears\ndirectly against spaces of positive curvature alone; but if it could\ndiscredit these, it might throw doubt on the method by which all\nalike are obtained. The two objections are:\n\nI. Space must be such as to allow of similarity, _i.e._ of the\nincrease or diminution, in a constant ratio, of all the lines in a\nfigure, without change of angles; whereas in non-Euclid, lines, like\nangles, have absolute magnitude.\n\nII. Space must be infinite, whereas spherical and elliptic spaces are\nfinite.\n\nI will discuss the first objection in connection with Delb\u0153uf's\narticles referred to above. The second, which has not, to my\nknowledge, been widely used in criticism, will be better deferred to\nChapter III.\n\n\nDelb\u0153uf.\n\n=98.= M. Delb\u0153uf's four articles in the Revue Philosophique contain\nmuch matter that has already been dealt with in the criticism of\nLotze, and much that is irrelevant for our present purpose. The only\npoint, which I wish to discuss here, is the question of absolute\nmagnitude, as it is called--the question, that is, whether the\npossibility of similar but unequal geometrical figures can be known\n_\u00e0 priori_[109].\n\nIn discussing this question, it is important, to begin with, to\ndistinguish clearly the sense in which absolute magnitude _is_\nrequired in non-Euclidean Geometry, from another sense, in which\nit would be absurd to regard any magnitude as absolute. Judgments\nof magnitude can only result from comparison, and if Metageometry\nrequired magnitudes which could be determined without comparison, it\nwould certainly deserve condemnation. But this is not required. All\nwe require is, that it shall be impossible, while the rest of space\nis unaffected, to alter the magnitude of any figure, as compared with\nother figures, while leaving the relative internal magnitudes of its\nparts unchanged. This construction, which is possible in Euclid,\nis impossible in Metageometry. We have to discuss whether such an\nimpossibility renders non-Euclidean spaces logically faulty.\n\nM. Delb\u0153uf's position on this axiom--which he calls the postulate\nof homogeneity[110]--is, that all Geometry must presuppose it, and\nthat Metageometry, consequently, though logically sound, is logically\nsubsequent to Euclid, and can only make its constructions within a\nEuclidean \"homogeneous\" space (Rev. Phil. Vol. XXXVII., pp. 380-1).\nHe would appear to think, nevertheless, that homogeneity (in his\nsense) is learnt from experience, though on this point he is not\nvery explicit. (See Vol. XXXVIII., p. 129.) No _\u00e0 priori_ proof, at\nany rate, is offered in his articles. As a result of experience,\nevery one would admit, similarity is known to be possible within the\nlimits of observation; but the fact that this possibility extends to\nOrdnance maps, which deal with a spherical surface, should make us\nchary of inferring, from such a datum, the certainty of Euclid for\nlarge spaces. Moreover if homogeneity be empirical, Metageometry,\nwhich dispenses with it, is not necessarily in _logical_ dependence\nupon Euclid, since homogeneity and isogeneity are _logically_\nseparable. I shall assume, therefore, as the only contention which\ncan be interesting to our argument, that homogeneity is regarded as\n_\u00e0 priori_, and as logically essential to Geometry.\n\n=99.= Now we saw, in discussing Erdmann's views of the judgment of\nquantity, that in non-Euclidean space, as in Euclidean, a change\nof all spatial magnitudes, in the same ratio, would be no change\nat all; the ratios of all magnitudes to the space-constant would\nbe unchanged, and the space-constant, as the ultimate standard of\ncomparison, cannot, in any intelligible sense, be said to have any\nparticular magnitude. The absolute magnitudes of Metageometry,\ntherefore, are absolute only as against any other _particular_\nmagnitude, not as against other magnitudes in general. If this were\nnot the case, the comparative nature of the judgment of magnitude\nwould be contradicted, and metrical Metageometry would become absurd.\nBut as it is, the difference from Euclid consists only in this: that\nin Metageometry we have, while in Euclid we have not, a standard of\ncomparison involved in the nature of our space as a whole, which we\ncall the space-constant. We have to discuss whether the assertion of\nsuch a standard involves an undue reification of space.\n\nI do not believe that this is the case. For an undue reification of\nspace would only arise, if we were no longer able to regard position\nas wholly relative, and as geometrically definable only by departure\nfrom other positions. But the relativity of position, as we have\nabundantly seen, is preserved by all spaces of constant curvature--in\nall of these, positions can only be defined, geometrically, by\nrelations to fresh positions[111]. This series of definitions may\nlead to an infinite regress, but it may also, as in spherical space,\nform a vicious circle, and return again to the position from which\nit started. No reification of space, no independent existence of\nmere relations, seems involved in such a procedure. The whole of\nMetageometry, in short, is a proof that the relativity of position\nis compatible with absolute magnitude, in the only sense required\nby non-Euclidean spaces. We must conclude, therefore, that there\nis nothing incompatible, in a denial of homogeneity (in Delb\u0153uf's\nsense), either with the relational nature of space, or with the\ncomparative nature of magnitude. This last _\u00e0 priori_ objection to\nMetageometry, therefore, cannot be maintained, and the issue must be\ndecided on empirical grounds alone.\n\n=100.= The foundations of Geometry have been the subject of much\nrecent speculation in France, and this seems to demand some notice.\nBut in spite of the splendid work which the French have done on the\nallied question of number and continuous quantity, I cannot persuade\nmyself that they have succeeded in greatly advancing the subject\nof geometrical philosophy. The chief writers have been, from the\nmathematical side, _Calinon_ and _Poincar\u00e9_, from the philosophical,\n_Renouvier_ and _Delb\u0153uf_; as a mediator between mathematics and\nphilosophy, _Lechalas_.\n\n_Calinon_, in an interesting article on the geometrical\nindeterminateness of the universe, maintains that any Geometry may be\napplied to the actual world by a suitable hypothesis as to the course\nof light-rays. For the earth only is known to us otherwise than by\nOptics, and the earth is an infinitesimal part of the universe.\nThis line of argument has been already discussed in connection with\nLotze, but Calinon adds a new suggestion, that the space-constant may\nperhaps vary with the time. This would involve a causal connection\nbetween space and other things, which seems hardly conceivable, and\nwhich, if regarded as possible, must surely destroy Geometry, since\nGeometry depends throughout on the irrelevance of Causation[112].\nMoreover, in all operations of measurement, some time is spent;\nunless we knew that space was unchanging throughout the operation,\nit is hard to see how our results could be trustworthy, and how,\nconsequently, a change in the parameter could be discovered. The\nsame difficulties would arise, in fact, as those which result from\nsupposing space not homogeneous.\n\n_Poincar\u00e9_ maintains that the question, whether Euclid or\nMetageometry should be accepted, is one of convenience and\nconvention, not of truth; axioms are definitions in disguise, and the\nchoice between definitions is arbitrary. This view has been discussed\nin Chapter I., in connection with Cayley's theory of distance, on\nwhich it depends.\n\n_Lechalas_ is a philosophical disciple of Calinon. He is a\nrationalist of the pre-Kantian type, but a believer in the validity\nof Metageometry. He holds that Geometry can dispense with all purely\nspatial postulates, and work with axioms of magnitude alone[113],\nwhich, in his opinion, are purely analytic. The principle of\ncontradiction, to him, is the sole and only test of truth; we make\nlong chains of reasoning from our premisses to see if contradictions\nwill emerge. It might be objected that this view, though it saves\ngeneral Geometry from being logically empirical, leaves it only\nempirically logical; this must, in fact, be the fate of every piece\nof _\u00e0 priori_ knowledge, if M. Lechalas's were the only test of\ntruth. However, he concludes that general Geometry is apodeictic,\nwhile the space of our actual world, like all other phenomena, is\ncontingent.\n\n_Delb\u0153uf_ criticizes non-Euclidean space from an ultra-realist\nstandpoint: he holds that _real_ space is neither homogeneous nor\nisogeneous, but that _conceived_ space, as abstracted from real\nspace, has both these properties. He offers no justification for\nhis real space, which seems to be maintained in the spirit of na\u00efve\nrealism, nor does he show how he has acquired his intimate knowledge\nof its constitution[114]. His arguments against Metageometry, in so\nfar as they are not repetitions of Lotze, have been discussed above.\n\n_Renouvier_, finally, is a pure Kantian, of the most orthodox type.\nHis views as to the importance, for Geometry, of the distinction\nbetween synthetic and analytic judgments, have been discussed, in\nconnection with Kant, at the beginning of the present Chapter[115].\n\n=101.= Before beginning the constructive argument of the next\nChapter, let us endeavour briefly to sum up the theories which\nhave been polemically advocated throughout the criticisms we have\njust concluded. We agreed to accept, with Kant, necessity for any\npossible experience as the test of the _\u00e0 priori_, but we refused,\nfor the present, to discuss the connection of the _\u00e0 priori_ with\nthe subjective, regarding the purely logical test as sufficient for\nour immediate purpose. We also refused to attach importance to the\ndistinction of analytic and synthetic, since it seemed to apply, not\nto different judgments, but only to different aspects of any judgment.\n\nWe then discussed Riemann's attempt to identify the empirical element\nin Geometry with the element not deducible from ideas of magnitude,\nand we decided that this identification was due to a confusion as\nto the nature of magnitude. For judgments of magnitude, we said,\nrequire always some qualitative basis, which is not quantitatively\nexpressible.\n\nIn criticizing Helmholtz, we decided that Mechanics logically\npresupposes Geometry, though space presupposes matter; but that the\nmatter which space presupposes, and to which Geometry indirectly\nrefers, is a more abstract matter than that of Mechanics, a matter\ndestitute of force and of causal attributes, and possessed only\nof the purely spatial attributes required for the possibility\nof spatial figures. But we conceded that Geometry, when applied\nto mixed mathematics or to daily life, demands more than this,\ndemands, in fact, some means of discovering, in the more concrete\nmatter of Mechanics, either a rigid body, or a body whose departure\nfrom rigidity follows some empirically discoverable law. _Actual_\nmeasurement, therefore, we agreed to regard as empirical.\n\nOur conclusions, as regards the empiricism of Riemann and Helmholtz,\nwere reinforced by a criticism of Erdmann. We then had an opposite\ntask to perform, in defending Metageometry against Lotze. Here we\nsaw that there are two senses in which Metageometry is possible. The\nfirst concerns our actual space, and asserts that it may have a very\nsmall space-constant; the second concerns philosophical theories\nof space, and asserts a purely logical possibility, which leaves\nthe decision to experience. We saw also that Lotze's mathematical\nstrictures arose from insufficient knowledge of the subject, and\ncould all be refuted by a better acquaintance with Metageometry.\n\nFinally, we discussed the question of absolute magnitude, and found\nin it no logical obstacle to non-Euclidean spaces. Our conclusion,\nthen, in so far as we are as yet entitled to a conclusion, is\nthat all spaces with a space-constant are _\u00e0 priori_ justifiable,\nand that the decision between them must be the work of experience.\nSpaces without a space-constant, on the other hand, spaces, that is,\nwhich are not homogeneous throughout, we found logically unsound\nand impossible to know, and therefore to be condemned _\u00e0 priori_.\nThe constructive proof of this thesis will form the argument of the\nfollowing chapter.\n\nFOOTNOTES:\n\n[67] The Critical Philosophy of Kant, Vol. I. p. 287.\n\n[68] For a discussion of Kant from a less purely mathematical\nstandpoint, see Chap. IV.\n\n[69] Cf. Vaihinger's Commentar, II. pp. 202, 265. Also p. 336 ff.\n\n[70] E.g. second edition, p. 39: \"So werden auch alle geometrischen\nGrunds\u00e4tze, z. B. dass in einem Triangel zwei Seiten zusammen gr\u00f6sser\nsind als die dritte, niemals aus allgemeinen Begriffen von Linie\nund Triangel, sondern aus der Anschauung, und zwar _\u00e0 priori_ mit\napodiktischer Gewissheit abgeleitet.\"\n\n[71] Cf. Bradley's Logic, Bk. III. Pt. I. Chap. VI.; Bosanquet's\nLogic, Bk. I. Chap. I. pp. 97-103.\n\n[72] Philosophie de la R\u00e8gle et du Compas, Ann\u00e9e Philosophique, II.\npp. 1-66.\n\n[73] I have stated this doctrine dogmatically, as a proof would\nrequire a whole treatise on Logic. I accept the proofs offered by\nBradley and Bosanquet, to which the reader is referred.\n\n[74] For a further discussion of this point, see Chaps. III. and IV.\n\n[75] See Chap. IV. for a discussion of this argument.\n\n[76] See Chap. IV. \u00a7 185.\n\n[77] An Otherness of substance, rather than of attribute, is here\nintended; an Otherness which may perhaps be called real as opposed to\nlogical diversity.\n\n[78] This proposition will be argued at length in Chap. IV.\n\n[79] See Psychologie als Wissenschaft, I. Section III. Chap. VII.;\nII. Section I. Chap. III. and Section II. Chap. III. Compare also\nSynechologie, Section I. Chaps. II. and III.\n\n[80] On the influence of Herbart on Riemann, compare Erdmann, Die\nAxiome der Geometrie, p. 30.\n\n[81] I do not mean that measurement of colours is effected without\nreference to their relations, since all measurement is essentially\ncomparison. But in colours, it is the elements which are compared,\nwhile in space, it is the relations between elements.\n\n[82] For a discussion of this point, see Chap. III. Sec. B, \u00a7 176.\n\n[83] The works of Helmholtz on geometrical philosophy comprise, in\naddition to the articles quoted in Chap. I., the following articles:\n\"Ursprung und Sinn der geometrischen Axiome, gegen Land,\" Wiss. Abh.\nVol. II. p. 640, 1878. (Also Mind, Vol. III.: an answer to Land in\nMind, Vol. II.) \"Ursprung und Bedeutung der geometrischen Axiome,\"\n1870, Vortr\u00e4ge und Reden, Vol. II. p. 1. (Also Mind, Vol. I.) Two\nAppendices to \"Die Thatsachen in der Wahrnehmung,\" entitled: II. \"Der\nRaum kann transcendental sein, ohne dass es die Axiome sind\"; and\nIII. \"Die Anwendbarkeit der Axiome auf die physische Welt,\" 1878,\nVortr\u00e4ge und Reden, Vol. II. p. 256 ff.\n\nThe two Appendices last mentioned are popularizings and expansions of\nthe article in Mind, Vol. III. The most widely read, though also, to\nmy mind, the least valuable, of all Helmholtz's writings on Geometry,\nis the article in Mind, Vol. I. This contains the famous and much\nmisunderstood analogies of Flatland and Sphereland, which will be\ndiscussed, and as far as possible defended, in answering Lotze's\nattack on Metageometry--an attack based, apparently, almost entirely\non this one popular article. The present discussion, therefore, may\nbe confined almost entirely to Mind, Vol. III., and the philosophical\nportions of the two papers quoted in Chap. I., _i.e._ to the articles\nin Wiss. Abh. Vol. II. pp. 610-660. His other works are popular, and\nimportant only because of the large public to which they appeal.\n\n[84] In the answer to Land, Mind, Vol. III. and Wiss. Abh. II. p. 640.\n\n[85] See also Die Thatsachen in der Wahrnehmung, Zusatz II., Der Raum\nkann transcendental sein, ohne dass es die Axiome sind. Vortr\u00e4ge und\nReden, Vol. II.\n\n[86] See below, criticism of Erdmann, \u00a7 84.\n\n[87] See Prof. Land, in Mind, Vol. II.\n\n[88] See concluding paragraph of Helmholtz's article in Mind, Vol.\nIII.\n\n[89] Cf. Veronese, Grundz\u00fcge der Geometrie (German translation), p.\nix. Also pp. xxxiv, 304, and Note II. pp. 692-4.\n\n[90] See Chap. IV. \u00a7 197 ff.\n\n[91] Cf. the opinion of Bolyai, quoted by Erdmann, Axiome, p. 26; cf.\nalso ib. p. 60.\n\n[92] Die Axiome der Geometrie: Eine philosophische Untersuchung der\nRiemann-Helmholtz'schen Raumtheorie, Leipzig, 1877.\n\n[93] On the influence of Mill, cf. Stallo, Concepts of Modern\nPhysics, p. 216.\n\n[94] This view seems to be derived, through Riemann, from Herbart.\nSee Psych. als Wiss. ed. Hart. Vol. V. p. 262.\n\n[95] The same irreducibility of space to mere magnitude is proved by\nKant's hands and spherical triangles, in which a difference persists\nin spite of complete quantitative equality.\n\n[96] See \u00a7\u00a7 146-7.\n\n[97] \"Jeder Versuch, Kant's Lehre von der Apriorit\u00e4t als\ndes subjectiven, von aller Erfahrung absolut unabh\u00e4ngigen\nErkenntnissfactors, trotzdem zu halten, ist deshalb von voruherein\naussichtslos.\"\n\n[98] Ausdehnungslehre von 1844, 2nd edition, pp. xxii. xxiii.\n\n[99] See \u00a7 129 ff.\n\n[100] Metaphysik, Book II. Chap. II. My references are to the\noriginal.\n\n[101] See Lectures and Essays, Vol. I. p. 261.\n\n[102] On the meaning of geometrical possibility, cf. Veronese,\nGrundz\u00fcge der Geometrie (German translation), pp. xi.-xiii.\n\n[103] Compare Calinon, \"Sur l'Ind\u00e9termination g\u00e9om\u00e9trique de\nl'Univers,\" Revue Philosophique, 1893, Vol. XXXVI. pp. 595-607.\n\n[104] Vortr\u00e4ge und Reden, Vol. II. p. 9: \"Parallele Linien w\u00fcrden\ndie Bewohner der Kugel gar nicht kennen. Sie w\u00fcrden behaupten, dass\njede beliebige zwei _geradeste_ Linien, geh\u00f6rig verlangert, sich\nschliesslich nicht nur in einem, sondern in zwei Punkten schneiden\nm\u00fcssten.\" (The italics are mine.) The omission of _straight_ in such\nphrases is a frequent laxity of mathematicians.\n\n[105] It has been suggested to me that Lotze regards the meridians\nas projected on to a plane, as in a map. If this be so, there is an\nobviously illegitimate introduction of the third dimension.\n\n[106] This is proved by Helmholtz's remark at the end of a detailed\nattempt to make spherical and pseudo-spherical spaces imaginable\n(l.c. p. 28): \"Anders ist es mit den drei Dimensionen des Raumes. Da\nalle unsere Mittel sinnlicher Anschauung sich nur auf einen Raum von\ndrei Dimensionen erstrecken, und die vierte Dimension nicht bloss\neine Ab\u00e4nderung von Vorhandenem, sondern etwas vollkommen Neues w\u00e4re,\nso befinden wir uns schon wegen unserer k\u00f6rperlichen Organisation in\nder absoluten Unm\u00f6glichkeit, uns eine Anschauungsweise einer vierten\nDimension vorzustellen.\"\n\n[107] Cf. Grassmann, Ausdehnungslehre von 1844, 2nd Edition, p. xxiii.\n\n[108] See especially Stallo, Concepts of Modern Physics,\nInternational Science Series, Vol. XLII. Chaps. XIII. and\nXIV.; Renouvier, \"Philosophie de la r\u00e8gle et du compas,\" Ann\u00e9e\nPhilosophique, II.; Delb\u0153uf, \"L'ancienne et les nouvelles\ng\u00e9om\u00e9tries,\" Revue Philosophique, Vols. XXXVI.-XXXIX.\n\n[109] M. Delb\u0153uf deserves credit for having based Euclid, already in\n1860, in his \"Prol\u00e9gom\u00e8nes Philosophiques de la G\u00e9om\u00e9trie,\" on this\naxiom--certainly a better basis, at first sight, than the axiom of\nparallels.\n\n[110] This meaning of homogeneity must not be confounded with the\nsense in which I have used the word. In Delb\u0153uf's sense, it means\nthat figures may be similar though of different sizes; in my sense it\nmeans that figures may be similar though in different places. This\nproperty of space is called by Delb\u0153uf isogeneity.\n\n[111] For a full proof of this proposition, see Chap. III.\n\n[112] See Chap. III., especially \u00a7 133.\n\n[113] For a criticism of this view, see the above discussions on\nRiemann and Erdmann.\n\n[114] Cf. Couturat, \"De l'Infini Math\u00e9matique,\" Paris, F\u00e9lix Alcan,\n1896, p. 544.\n\n[115] The following is a list of the most important recent French\nphilosophical writings on Geometry, so far as I am acquainted with\nthem.\n\n Andrade: \"Les bases exp\u00e9rimentales de la g\u00e9om\u00e9trie euclidienne\";\n Rev. Phil. 1890, II., and 1891, I.\n\n Bonnel: \"Les hypoth\u00e8ses dans la g\u00e9om\u00e9trie\"; Gauthier-Villars,\n 1897.\n\n L'Abb\u00e9 de Broglie: \"La g\u00e9om\u00e9trie non-euclidienne,\" two articles;\n Annales de Phil. Chr\u00e9t. 1890.\n\n Calinon: \"Les espaces g\u00e9om\u00e9triques\"; Rev. Phil. 1889, I., and\n 1891, II. \"Sur l'ind\u00e9termination g\u00e9om\u00e9trique de l'univers\"; ib.\n 1893, II.\n\n Couturat: \"L'Ann\u00e9e Philosophique de F. Pillon,\" Rev. de M\u00e9t. et\n de Morale, Jan. 1893.\n\n \"Note sur la g\u00e9om\u00e9trie non-euclidienne et la relativit\u00e9 de\n l'espace\"; ib., May, 1893.\n\n \"\u00c9tudes sur l'espace et le temps,\" ib. Sep. 1896.\n\n Delb\u0153uf: \"L'ancienne et les nouvelles g\u00e9om\u00e9tries,\" four articles;\n Rev. Phil. 1893-5.\n\n Lechalas: \"La g\u00e9om\u00e9trie g\u00e9n\u00e9rale\"; Crit. Phil. 1889.\n\n \"La g\u00e9om\u00e9trie g\u00e9n\u00e9rale et les jugements synth\u00e9tiques \u00e0 priori\"\n and \"Les bases exp\u00e9rimentales de la g\u00e9om\u00e9trie\"; Rev. Phil. 1890,\n II.\n\n \"M. Delb\u0153uf et Le probl\u00e8me des mondes semblables\"; ib. 1894, I.\n\n \"Note sur la g\u00e9om\u00e9trie non-euclidienne et le principe de\n similitude\"; Rev. de M\u00e9t. et de Morale, March, 1893.\n\n \"La courbure et la distance en g\u00e9om\u00e9trie g\u00e9n\u00e9rale\"; ib., March,\n 1896.\n\n \"La g\u00e9om\u00e9trie g\u00e9n\u00e9rale et l'intuition\"; Annales de Phil. Chr\u00e9t.,\n 1890.\n\n \"Etude sur l'espace et le temps\"; Paris, Alcan, 1896.\n\n Liard: \"Des d\u00e9finitions g\u00e9om\u00e9trie et des d\u00e9finitions empiriques,\"\n 2nd ed.; Paris, Alcan, 1888.\n\n Mansion: \"Premiers principes de la m\u00e9tag\u00e9om\u00e9trie\"; two\n articles in Rev. N\u00e9o-Scholastique, 1896. Separately published,\n Gauthier-Villars, 1896.\n\n Milhaud: \"La g\u00e9om\u00e9trie non-euclidienne et la th\u00e9orie de la\n connaissance\"; Rev. Phil. 1888, I.\n\n Poincar\u00e9: \"Non-Euclidian Geometry\"; Nature, Vol. XLV., 1891-2.\n\n \"L'espace et la g\u00e9om\u00e9trie\"; Rev. de M\u00e9t. et de Morale, Nov. 1895.\n\n \"R\u00e9sponse \u00e0 quelques critiques,\" ib. Jan. 1897.\n\n Renouvier: \"Philosophie de la r\u00e8gle et du compas\"; Crit. Phil.,\n 1889, and L'Ann\u00e9e Phil., II^{me} ann\u00e9e, 1891.\n\n Sorel: \"Sur la g\u00e9om\u00e9trie non-euclidienne\"; Rev. Phil., 1891, I.\n\n Tannery: \"Th\u00e9orie de la connaissance math\u00e9matique\"; Rev. Phil.,\n 1894, II.\n\n\n\n\nCHAPTER III.\n\n\nSection A.\n\nTHE AXIOMS OF PROJECTIVE GEOMETRY.\n\n=102.= Projective Geometry proper, as we saw in Chapter I., does not\nemploy the conception of magnitude, and does not, therefore, require\nthose axioms which, in the systems of the second or metrical period,\nwere required solely to render possible the application of magnitude\nto space. But we saw, also, that Cayley's reduction of metrical\nto projective properties was purely technical and philosophically\nirrelevant. Now it is in metrical properties alone--apart from\nthe exception to the axiom of the straight line, which itself,\nhowever, presupposes metrical properties[116]--that non-Euclidean\nand Euclidean spaces differ. The properties dealt with by projective\nGeometry, therefore, in so far as these are obtained without the use\nof imaginaries, are properties common to all spaces. Finally, the\ndifferences which appear between the Geometries of different spaces\nof the same curvature--_e.g._ between the Geometries of the plane\nand the cylinder--are differences in projective properties[117].\nThus the necessity which arises, in metrical Geometry, for further\nqualifications besides those of constant curvature, disappears when\nour general space is defined by purely projective properties.\n\n=103.= We have good ground for expecting, therefore, that the axioms\nof projective Geometry will be the simplest and most complete\nexpression of the indispensable requisites of any geometrical\nreasoning: and this expectation, I hope, will not be disappointed.\nProjective Geometry, in so far as it deals only with the properties\ncommon to all spaces, will be found, if I am not mistaken, to be\nwholly _\u00e0 priori_, to take nothing from experience, and to have, like\nArithmetic, a creature of the pure intellect for its object. If this\nbe so, it is that branch of pure mathematics which Grassmann, in his\n_Ausdehnungslehre_ of 1844, felt to be possible, and endeavoured, in\na brilliant failure, to construct without any appeal to the space of\nintuition.\n\n=104.= But unfortunately, the task of discovering the axioms of\nprojective Geometry is far from easy. They have, as yet, found\nno Riemann or Helmholtz to formulate them philosophically. Many\ngeometers have constructed systems, which they intended to be, and\nwhich, with sufficient care in interpretation, really are, free from\nmetrical presuppositions. But these presuppositions are so rooted\nin all the very elements of Geometry, that the task of eliminating\nthem demands a reconstruction of the whole geometrical edifice. Thus\nEuclid, for example, deals, from the start, with spatial equality--he\nemploys the circle, which is necessarily defined by means of\nequality, and he bases all his later propositions on the congruence\nof triangles as discussed in Book I.[118] Before we can use any\nelementary proposition of Euclid, therefore, even if this expresses a\nprojective property, we have to prove that the property in question\ncan be deduced by projective methods. This has not, in general,\nbeen done by projective geometers, who have too often assumed, for\nexample, that the quadrilateral construction--by which, as we saw in\nChap. I., they introduce projective coordinates--or anharmonic ratio,\nwhich is _prim\u00e2 facie_ metrical, could be satisfactorily established\non their principles. Both these assumptions, however, can be\njustified, and we may admit, therefore, that the claims of projective\nGeometry to logical independence of measurement or congruence are\nvalid. Let us see, then, how it proceeds.\n\n=105.= In the first place, it is important to realize that\nwhen coordinates are used, in projective Geometry, they are not\ncoordinates in the ordinary metrical sense, _i.e._ the numerical\nmeasures of certain spatial magnitudes. On the contrary, they are a\nset of numbers, arbitrarily but systematically assigned to different\npoints, like the numbers of houses in a street, and serving only,\nfrom a philosophical standpoint, as convenient designations for\npoints which the investigation wishes to distinguish. But for the\nbrevity of the alphabet, in fact, they might, as in Euclid, be\nreplaced by letters. How they are introduced, and what they mean, has\nbeen discussed in Chapter I. Here we have only to repeat a caution,\nwhose neglect has led to much misunderstanding.\n\n=106.= The distinction between various points, then, is not a result,\nbut a condition, of the projective coordinate system. The coordinate\nsystem is a wholly extraneous, and merely convenient, set of marks,\nwhich in no way touches the essence of projective Geometry. What we\nmust begin with, in this domain, is the possibility of distinguishing\nvarious points from one another. This may be designated, with\nVeronese, as the first axiom of Geometry[119]. How we are to define\na point, and how we distinguish it from other points, is for the\nmoment irrelevant; for here we only wish to discover the nature\nof projective Geometry, and the kind of properties which it uses\nand demonstrates. How, and with what justification, it uses and\ndemonstrates them, we will discuss later.\n\n=107.= Now it is obvious that a mere collection of points,\ndistinguished one from another, cannot found a Geometry: we must\nhave some idea of the manner in which the points are interrelated,\nin order to have an adequate subject-matter for discussion. But\nsince all ideas of quantity are excluded, the relations of points\ncannot be relations of distance in the ordinary sense, nor even, in\nthe sense of ordinary Geometry, anharmonic ratios, for anharmonic\nratios are usually defined as the ratios of four distances, or of\nfour sines, and are thus quantitative. But since all quantitative\ncomparison presupposes an identity of quality, we may expect to find,\nin projective Geometry, the qualitative substrata of the metrical\nsuperstructure.\n\nAnd this, we shall see, is actually the case. We have not distance,\nbut we _have_ the straight line; we have not quantitative anharmonic\nratio, but we _have_ the property, in any four points on a line,\nof being the intersections with the rays of a given pencil. And\nfrom this basis, we can build up a qualitative science of abstract\nexternality, which is projective Geometry. How this happens, I shall\nnow proceed to show.\n\n=108.= All geometrical reasoning is, in the last resort, circular:\nif we start by assuming points, they can only be defined by the\nlines or planes which relate them; and if we start by assuming lines\nor planes, they can only be defined by the points through which\nthey pass. This is an inevitable circle, whose ground of necessity\nwill appear as we proceed. It is, therefore, somewhat arbitrary to\nstart either with points or with lines, as the eminently projective\nprinciple of duality mathematically illustrates; nevertheless we will\nelect, with most geometers, to start with points[120]. We suppose,\ntherefore, as our datum, a set of discrete points, for the moment\nwithout regard to their interconnections. But since connections are\nessential to any reasoning about them as a system, we introduce, to\nbegin with, the axiom of the straight line. Any two of our points,\nwe say, lie on a line which those two points completely define.\nThis line, being determined by the two points, may be regarded as\na relation of the two points, or an adjective of the system formed\nby both together. This is the only purely qualitative adjective--as\nwill be proved later--of a system of two points. Now projective\nGeometry can only take account of qualitative adjectives, and can\ndistinguish between different points only by their relations to\nother points, since all points, _per se_, are qualitatively similar.\nHence it comes that, for projective Geometry, when two points only\nare given, they are qualitatively indistinguishable from any two\nother points on the same straight line, since any two such other\npoints have the same qualitative relation. Reciprocally, since one\nstraight line is a figure determined by any two of its points, and\nall points are qualitatively similar, it follows that all straight\nlines are qualitatively similar. We may regard a point, therefore, as\ndetermined by two straight lines which meet in it, and the point,\non this view, becomes the only qualitative relation between the two\nstraight lines. Hence, if the point only be regarded as given, the\ntwo straight lines are qualitatively indistinguishable from any other\npair through the point.\n\n=109.= The extension of these two reciprocal principles is the\nessence of all projective transformations, and indeed of all\nprojective Geometry. The fundamental operations, by which figures\nare projectively transformed, are called projection and section. The\nvarious forms of projection and section are defined in Cremona's\n\"Projective Geometry,\" Chapter I., from which I quote the following\naccount.\n\n\"_To project from a fixed point S_ (the _centre of projection_) a\nfigure (_ABCD_ ... _abcd_ ...) composed of points and straight lines,\nis to construct the straight lines or _projecting rays SA_, _SB_,\n_SC_, _SD_, ... and the planes (_projecting planes_) _Sa_, _Sb_,\n_Sc_, _Sd_, ... We thus obtain a new figure composed of straight\nlines and planes which all pass through the centre _S_.\n\n\"_To cut by a fixed plane \u03c3 (transversal plane_) a figure (_\u03b1\u03b2\u03b3\u03b4_ ...\n_abcd_ ...) made up of planes and straight lines, is to construct the\nstraight lines or _traces \u03c3\u03b1, \u03c3\u03b2, \u03c3\u03b3_ ... and the points or _traces\n\u03c3a, \u03c3b, \u03c3c_....[121] By this means we obtain a new figure\ncomposed of straight lines and points lying in the plane _\u03c3_.\n\n\"_To project from a fixed straight line s_ (the _axis_) a figure\n_ABCD_ composed of points, is to construct the planes _sA_, _sB_,\n_sC_.... The figure thus obtained is composed of planes which all\npass through the axis _s_.\n\n\"_To cut by a fixed straight line s_ (a _transversal_) a figure\n_\u03b1\u03b2\u03b3\u03b4_ ... composed of planes, is to construct the points _s\u03b1_, _s\u03b2_,\n_s\u03b3_.... In this way a new figure is obtained, composed of points all\nlying on the fixed transversal _s_.\n\n\"If a figure is composed of straight lines _a_, _b_, _c_ ... which\nall pass through a fixed point or _centre S_, it can be _projected_\nfrom a straight line or _axis s_ passing through _S_; the result is a\nfigure composed of planes _sa_, _sb_, _sc_....\n\n\"If a figure is composed of straight lines _a_, _b_, _c_ ...\nall lying in a fixed plane, it may be cut by a straight line\n(transversal) _s_ lying in the same plane; the figure which results\nis formed by the points _sa_, _sb_, _sc_....\"\n\n=110.= The successive application, to any figure, of two reciprocal\noperations of projection and section, is regarded as producing a\nfigure protectively indistinguishable from the first, provided only\nthat the dimensions of the original figure were the same as those\nof the resulting figure, that, for example, if the second operation\nbe section by a plane, the original figure shall have been a plane\nfigure. The figures obtained from a given figure, by projection\nor section alone, are related to that figure by the principle of\nduality, of which we shall have to speak later on.\n\nI shall endeavour to show, in what follows, first, in what sense\nfigures obtained from each other by projective transformation are\nqualitatively alike; secondly, what axioms, or adjectives of space,\nare involved in the principle of projective transformation; and\nthirdly, that these adjectives must belong to any form of externality\nwith more than one dimension, and are, therefore, _\u00e0 priori_\nproperties of any possible space.\n\nFor the sake of simplicity, I shall in general confine myself to two\ndimensions. In so doing, I shall introduce no important difference of\nprinciple, and shall greatly simplify the mathematics involved.\n\n=111.= The two mathematically fundamental things in projective\nGeometry are anharmonic ratio, and the quadrilateral construction.\nEverything else follows mathematically from these two. Now what is\nmeant, in projective Geometry, by anharmonic ratio?\n\n[Illustration]\n\nIf we start from anharmonic ratio as ordinarily defined, we are\nmet by the difficulty of its quantitative nature[122]. But among\nthe properties deduced from this definition, many, if not most,\nare purely qualitative. The most fundamental of these is that, if\nthrough any four points in a straight line we draw four straight\nlines which meet in a point, and if we then draw a new straight line\nmeeting these four, the four new points of intersection have the\nsame anharmonic ratio as the four points we started with. Thus, in\nthe figure, _abcd_, _a\u2032b\u2032c\u2032d\u2032_, _a\u2033b\u2033c\u2033d\u2033_, all have the same\nanharmonic ratio. The reciprocal relation holds for the anharmonic\nratio of four straight lines. Here we have, plainly, the required\nbasis for a qualitative definition. The definition must be as follows:\n\nTwo sets of four points each are defined as having the same\nanharmonic ratio, when (1) each set of four lies in one straight\nline, and (2) corresponding points of different sets lie two by two\non four straight lines through a single point, or when both sets have\nthis relation to any third set[123]. And reciprocally: Two sets of\nfour straight lines are defined as having the same anharmonic ratio\nwhen (1) each set of four passes through a single point, and (2)\ncorresponding lines of different sets pass, two by two, through four\npoints in one straight line, or when both sets have this relation to\nany third set.\n\nTwo sets of points or of lines, which have the same anharmonic ratio,\nare treated by projective Geometry as equivalent: this qualitative\nequivalence replaces the quantitative equality of metrical Geometry,\nand is obviously included, by its definition, in the above account of\nprojective transformations in general.\n\n=112.= We have next to consider the quadrilateral construction[124].\nThis has a double purpose: first, to define the important special\ncase known as a harmonic range; and secondly, to afford an\nunambiguous and exhaustive method of assigning different numbers to\ndifferent points. This last method has, again, a double purpose:\nfirst, the purpose of giving a convenient symbolism for describing\nand distinguishing different points, and of thus affording a means\nfor the introduction of analysis; and secondly, of so assigning\nthese numbers that, if they had the ordinary metrical significance,\nas distances from some point on the numbered straight line, they\nwould yield -1 as the anharmonic ratio of a harmonic range, and\nthat, if four points have the same anharmonic ratio as four\nothers, so have the corresponding numbers. This last purpose is\ndue to purely technical motives: it avoids the confusion with our\npreconceptions which would result from any other value for a harmonic\nrange; it allows us, when metrical interpretations of projective\nresults are desired, to make these interpretations without tedious\nnumerical transformations, and it enables us to perform projective\ntransformations by algebraical methods. At the same time, from the\nstrictly projective point of view, as observed above, the numbers\nintroduced have a purely conventional meaning; and until we pass to\nmetrical Geometry, no reason can be shown for assigning the value -1\nto a harmonic range. With this preliminary, let us see in what the\nquadrilateral construction consists.\n\n[Illustration]\n\n=113.= A harmonic range, in elementary Geometry, is one whose\nanharmonic ratio is -1, or one in which the three segments formed by\nthe four points are in harmonic progression, or again, one in which\nthe ratio of the two internal segments is equal to the ratio of the\ntwo external segments. If _a_, _b_, _c_, _d_ be the four points, it\nis easily seen that these definitions are equivalent to one another:\nthey give respectively:\n\n (ab/bc)/(ad/dc) = -1, (1/ab) - (1/ac) = (1/ac) - (1/ad),\n\n and (ab/bc) = (ad/cd).\n\nBut as they are all quantitative, they cannot be used for our present\npurpose. Nor are any definitions which involve bisection of lines or\nangles available. We must have a definition which proceeds entirely\nby the help of straight lines and points, without measurement of\ndistances or angles. Now from the above definitions of a harmonic\nrange, we see that _a_, _b_, _c_, _d_ have the same anharmonic ratio\nas _c_, _b_, _a_, _d_. This gives us the property we require for our\ndefinition. For it shows that, in a harmonic range, we can find a\nprojective transformation which will interchange _a_ and _c_. This is\na necessary and sufficient condition for a harmonic range, and the\nquadrilateral construction is the general method for giving effect to\nit.\n\n[Illustration]\n\nGiven any three points _A_, _B_, _D_ in one straight line, the\nquadrilateral construction finds the point _C_ harmonic to _A_ with\nrespect to _B_, _D_ by the following method: Take any point _O_\noutside the straight line _ABD_, and join it to _B_ and _D_. Through\n_A_ draw any straight line cutting _OD_, _OB_ in _P_ and _Q_. Join\n_DQ_, _BP_, and let them intersect in _R_. Join _OR_, and let _OR_\nmeet _ABD_ in _C_. Then _C_ is the point required.\n\nTo prove this, let _DRQ_ meet _OA_ in _T_, and draw _AR_, meeting\n_OD_ in _S_. Then a projective transformation of _A_, _B_, _C_, _D_\nfrom _R_ on to _OD_ gives the points _S_, _P_, _O_, _D_, which,\nprojected from _A_ on to _DQ_, give _R_, _Q_, _T_, _D_. But these\nagain, projected from _O_ on to _ABD_, give _C_, _B_, _A_, _D_.\nHence _A_, _B_, _C_, _D_ can be projectively transformed into _C_,\n_B_, _A_, _D_, and therefore form a harmonic range. From this point,\nthe proof that the construction is unique and general follows\nsimply[125].\n\nThe introduction of numbers, by this construction, offers no\ndifficulties of principle--except, indeed, those which always\nattend the application of number to continua--and may be studied\nsatisfactorily in Klein's Nicht-Euklid (I. p. 337 ff.). The principle\nof it is, to assign the numbers 0, 1, \u221e to _A_, _B_, _D_ and\ntherefore the number 2 to _C_, in order that the differences _AB_,\n_AC_, _AD_ may be in harmonic progression. By taking _B_, _C_, _D_ as\na new triad corresponding to _A_, _B_, _D_, we find a point harmonic\nto _B_ with respect to _C_, _D_ and assign to it the number 3, and so\non. In this way, we can obtain any number of points, and we are sure\nof having no number and no point twice over, so that our coordinates\nhave the essential property of a unique correspondence with the\npoints they denote, and _vice versa_.\n\n=114.= The point of importance in the above construction,\nhowever, and the reason why I have reproduced it in detail, is\nthat it proceeds entirely by means of the general principles of\ntransformation enunciated above. From this stage onwards, everything\nis effected by means of the two fundamental ideas we have just\ndiscussed, and everything, therefore, depends on our general\nprinciple of projective equivalence. This principle, as regards two\ndimensions, may be stated more simply than in the passage quoted from\nCremona. It starts, in two dimensions, from the following definitions:\n\nTo project the points _A_, _B_, _C_, _D_ ... from a centre _O_, is to\nconstruct the straight lines _OA_, _OB_, _OC_, _OD_....\n\nTo cut a number of straight lines _a_, _b_, _c_, _d_ ... by a\ntransversal _s_, is to construct the points _sa_, _sb_, _sc_,\n_sd_....[126]\n\nThe successive application of these two operations, provided the\noriginal figure consisted of points on one straight line or of\nstraight lines through one point, gives a figure projectively\nindistinguishable from the former figure; and hence, by extension,\nif any points in one straight line in the original figure lie in one\nstraight line in the derived figure, and reciprocally for straight\nlines through points, the two operations have given projectively\nsimilar figures. This general principle may be regarded as consisting\nof two parts, according to the order of the operations: if we begin\nwith projection and end with section, we transform a figure of\npoints into another figure of points; by the converse order, we\ntransform a figure of lines into another figure of lines.\n\n=115.= Before we can be clear as to the meaning of our principle, we\nmust have some notion as to our definition of points and straight\nlines. But this definition, in projective Geometry, cannot be given\nwithout some discussion of the principle of duality, the mathematical\nform of the philosophical circle involved in geometrical definitions.\n\nConfining ourselves for the moment to two dimensions, the principle\nasserts, roughly speaking, that any theorem, dealing with lines\nthrough a point and points on a line, remains true if these two\nterms, wherever they occur, are interchanged. Thus: two points\nlie on one straight line which they completely determine; and two\nstraight lines meet in one point, which they completely determine.\nThe four points of intersection of a transversal with four lines\nthrough a point have an anharmonic ratio independent of the\nparticular transversal; and the four lines joining four points on one\nstraight line to a fifth point have an anharmonic ratio independent\nof that fifth point. So also our general principle of projective\ntransformation has two sides: one in which points move along fixed\nlines, and one in which lines turn about fixed points.\n\nThis duality suggests that any definition of points must be effected\nby means of the straight line, and any definition of the straight\nline must be effected by means of points. When we take the third\ndimension into account, it is true, the duality is no longer so\nsimple; we have now to take account also of the plane, but this only\nintroduces a circle of three terms, which is scarcely preferable to\na circle of two terms. We now say: Three points, or a line and a\npoint, determine a plane: but conversely, three planes, or a line\nand plane, determine a point. We may regard the straight line as a\nrelation between two of its points, but we may also regard the point\nas a relation between two straight lines through it. We may regard\nthe plane as a relation between three points, or between a point and\na line, but we may also regard the point as a relation between three\nplanes, or between a line and a plane, which meet in it.\n\n=116.= How are we to get outside this circle? The fact is that, in\npure Geometry, we cannot get outside it. For space, as we shall see\nmore fully hereafter, is nothing but relations; if, therefore, we\ntake any spatial figure, and seek for the terms between which it is a\nrelation, we are compelled, in Geometry, to seek these terms within\nspace, since we have nowhere else to seek them, but we are doomed,\nsince anything purely spatial is a mere relation, to find our terms\nmelting away as we grasp them.\n\nThus the relativity of space, while it is the essence of the\nprinciple of duality, at the same time renders impossible the\nexpression of that principle, or of any other principle of pure\nGeometry, in a manner which shall be free from contradictions.\nNevertheless, if we are to advance at all with our analysis of\ngeometrical reasoning and with our definitions of lines and points,\nwe must, for a while, ignore this contradiction; we must argue\nas though it did not exist, so as to free our science from any\ncontradictions which are not inevitable.\n\n=117.= In accordance with this procedure, then, let us define our\npoints as the terms of spatial relations, regarding whatever is not\na point as a relation between points. What, on this view, must our\npoints be taken to be? Obviously, if extension is mere relativity,\nthey must be taken to contain no extension; but if they are to\nsupply the terms for spatial relations, _e.g._ for straight lines,\nthese relations must exhibit them as the terms of the figures they\nrelate. In other words, since what can really be taken, without\ncontradiction, as the term of a spatial relation, is unextended, we\nmust take, as the term to be used in Geometry, where we cannot go\noutside space, the least spatial thing which Geometry can deal with,\nthe thing which, though _in_ space, _contains_ no space; and this\nthing we define as the point[127].\n\nNeglecting, then, the fundamental contradiction in this definition,\nthe rest of our definitions follow without difficulty. The straight\nline is the relation between two points, and the plane is the\nrelation between three. These definitions will be argued and defended\nat length in section B of this Chapter[128], where we can discuss at\nthe same time the alternative metrical definitions; for our present\npurpose, it is sufficient to observe that projective Geometry, from\nthe first, regards the straight line as determined by two points, and\nthe plane as determined by three, from which it follows, if we take\npoints as possible terms for spatial relations, that the straight\nline and the plane may be regarded as relations between two and three\npoints respectively. If we agree on these definitions, we can proceed\nto discuss the fundamental principle of projective Geometry, and to\nanalyse the axioms implicated in its truth.\n\n=118.= Projective Geometry, we have seen, does not deal with\nquantity, and therefore recognizes no difference where the\ndifference is purely quantitative. Now quantitative comparison\ndepends on a recognized identity of quality; the recognition of\nqualitative identity, therefore, is logically prior to quantity, and\npresupposed by every judgment of quantity. Hence all figures, whose\ndifferences can be exhaustively described by quantity, _i.e._ by\npure measurement, must have an identity of quality, and this must be\nrecognizable without appeal to quantity. It follows that, by defining\nthe word quality in geometrical matters, we shall discover what\nsets of figures are projectively indiscernible. If our definition\nis correct, it ought to yield the general projective principle with\nwhich we set out.\n\n=119.= We agreed to regard points as the terms of spatial relations,\nand we agreed that different points could be distinguished. But\nwe postponed the discussion of the conditions under which this\ndistinction could be effected. This discussion will yield us the\ndefinition of quality and the proof of our general projective\nprinciple.\n\nPoints, to begin with, have been defined as nothing but the terms for\nspatial relations. They have, therefore, no intrinsic properties;\nbut are distinguished solely by means of their relations. Now the\nrelation between two points, we said, is the straight line on which\nthey lie. This gives that identity of quality for all pairs of points\non the same straight line, which is required both by our projective\nprinciple and by metrical Geometry. (For only where there is identity\nof quality can quantity be properly applied.) If only two points are\ngiven, they cannot, without the use of quantity, be distinguished\nfrom any two other points on the same straight line; for the\nqualitative relation between any two such points is the same as for\nthe original pair, and only by a difference of relation can points be\ndistinguished from one another.\n\nBut conversely, one straight line is nothing but the relation\nbetween two of its points, and all points are qualitatively alike.\nHence there can be nothing to distinguish one straight line from\nanother except the points through which it passes, and these are\ndistinguished from other points only by the fact that it passes\nthrough them. Thus we get the reciprocal transformation: if we are\ngiven only one point, any pair of straight lines through that point\nis qualitatively indistinguishable from any other. This again is, on\nthe one hand, the basis of the second part of our general projective\nprinciple, and on the other hand the condition of applying quantity,\nin the measurement of angles, to the departure of two intersecting\nstraight lines.\n\n=120.= We can now see the reason for what may have hitherto seemed a\nsomewhat arbitrary fact, namely, the necessity of _four_ collinear\npoints for anharmonic ratio. Recurring to the quadrilateral\nconstruction and the consequent introduction of number, we see\nthat anharmonic ratio is an intrinsic projective relation of four\ncollinear points or concurrent straight lines, such that given three\nterms and the relation, the fourth term can be uniquely determined\nby projective methods. Now consider first a pair of points. Since\nall straight lines are projectively equivalent, the relation between\none pair of points is precisely equivalent to that between another\npair. Given one point only, therefore, no projective relation, to\nany second point, can be assigned, which shall in any way limit our\nchoice of the second point. Given two points, however, there is such\na relation--the third point may be given collinear with the first\ntwo. This limits its position to one straight line, but since two\npoints determine nothing but one straight line, the third point\ncannot be further limited. Thus we see why no intrinsic projective\nrelation can be found between three points, which shall enable us,\nfrom two, uniquely to determine the third. With three given collinear\npoints, however, we have more given than a mere straight line, and\nthe quadrilateral construction enables us uniquely to determine any\nnumber of fresh collinear points. This shows why anharmonic ratio\nmust be a relation between four points, rather than between three.\n\n=121.= We can now prove, I think, that two figures, which are\nprojectively related, are qualitatively similar. Let us begin with\na collection of points on a straight line. So long as these are\nconsidered without reference to other points or figures, they are\nall qualitatively similar. They can be distinguished by immediate\nintuition, but when we endeavour, without quantity, to distinguish\nthem conceptually, we find the task impossible, since the only\nqualitative relation of any two of them, the straight line, is\nthe same for any other two. But now let us choose, at hap-hazard,\nsome point outside the straight line. The points of our line now\nacquire new adjectives, namely their relations to the new point,\n_i.e._ the straight lines joining them to this new point. But these\nstraight lines, reciprocally, alone define our external point,\nand all straight lines are qualitatively similar. If we take some\nother external point, therefore, and join it to the same points\nof our original straight line, we obtain a figure in which, so\nlong as quantity is excluded, there is no conceptual difference\nfrom the former figure. Immediate intuition can distinguish the\ntwo figures, but qualitative discrimination cannot do so. Thus we\nobtain a projective transformation of four lines into four other\nlines, as giving a figure qualitatively indistinguishable from the\noriginal figure. A similar argument applies to the other projective\ntransformations. Thus the only reason, within projective Geometry,\nfor not regarding projective figures as actually identical, is the\nintuitive perception of difference of position. This is fundamental,\nand must be accepted as a _datum_. It is presupposed in the\ndistinction of various points, and forms the very life of Geometry.\nIt is, in fact, the essence of the notion of a form of externality,\nwhich notion forms the subject-matter of projective Geometry.\n\n=122.= We may now sum up the results of our analysis of projective\nGeometry, and state the axioms on which its reasoning is based. We\nshall then have to prove that these axioms are necessary to any form\nof externality, with which we shall pass, from mere analysis, to a\ntranscendental argument.\n\nThe axioms which have been assumed in the above analysis, and which,\nit would seem, suffice to found projective Geometry, may be roughly\nstated as follows:\n\nI. We can distinguish different parts of space, but all parts are\nqualitatively similar, and are distinguished only by the immediate\nfact that they lie outside one another.\n\nII. Space is continuous and infinitely divisible; the result of\ninfinite division, the zero of extension, is called a _point_[129].\n\nIII. Any two points determine a unique figure, called a straight\nline, any three in general determine a unique figure, the plane.\nAny four determine a corresponding figure of three dimensions, and\nfor aught that appears to the contrary, the same may be true of any\nnumber of points. But this process comes to an end, sooner or later,\nwith some number of points which determine the whole of space. For\nif this were not the case, no number of relations of a point to a\ncollection of given points could ever determine its relation to fresh\npoints, and Geometry would become impossible[130].\n\nThis statement of the axioms is not intended to have any exclusive\nprecision: other statements equally valid could easily be made. For\nall these axioms, as we shall see hereafter, are philosophically\ninterdependent, and may, therefore, be enunciated in many ways. The\nabove statement, however, includes, if I am not mistaken, everything\nessential to projective Geometry, and everything required to prove\nthe principle of projective transformation. Before discussing the\napriority of these axioms, let us once more briefly recapitulate the\nends which they are intended to attain.\n\n=123.= From the exclusively mathematical standpoint, as we have\nseen, projective Geometry discusses only what figures can be obtained\nfrom each other by projective transformations, _i.e._ by the\noperations of projection and section. These operations, in all their\nforms, presuppose the point, straight line, and plane[131], whose\nnecessity for projective Geometry, from the purely mathematical point\nof view, is thus self-evident from the start. But philosophically,\nprojective Geometry has, as we saw, a wider aim. This wider aim,\nwhich gives, to the investigation of projectively equivalent figures,\nits chief importance, consists in the determination of qualitative\nspatial similarity, in the determination, that is, of all the figures\nwhich, when any one figure is given, can be distinguished from the\ngiven figure, so long as quantity is excluded, only by the mere fact\nthat they are external to it.\n\n=124.= Now when we consider what is involved in such absolute\nqualitative equivalence, we find at once, as its most obvious\nprerequisite, the perfect homogeneity of space. For it is assumed\nthat a figure can be completely defined by its internal relations,\nand that the external relations, which constitute its position,\nthough they suffice to distinguish it from other figures, in\nno way affect its internal properties, which are regarded as\nqualitatively identical with those of figures with quite different\nexternal relations. If this were not the case, anything analogous\nto projective transformation would be impossible. For such\ntransformation always alters the position, _i.e._ the external\nrelations, of a figure, and could not, therefore, if figures were\ndependent on their relations to other figures or to empty space,\nbe studied without reference to other figures, or to the absolute\nposition of the original figure. We require for our principle,\nin short, what may be called the mutual passivity and reciprocal\nindependence of two parts or figures of space.\n\nThis passivity and this independence involve the homogeneity of\nspace, or its equivalent, the relativity of position. For if\nthe internal properties of a figure are the same, whatever its\nexternal relations may be, it follows that all parts of space are\nqualitatively similar, since a change of external relation is a\nchange in the part of space occupied. It follows, also, that all\nposition is relative and extrinsic, _i.e._, that the position of a\npoint, or the part of space occupied by a figure, is not, and has\nno effect upon, any intrinsic property of the point or figure, but\nis exclusively a relation to other points or figures in space, and\nremains without effect except where such relations are considered.\n\n=125.= The homogeneity of space and the relativity of position,\ntherefore, are presupposed in the qualitative spatial comparison\nwith which projective Geometry deals. The latter, as we saw, is\nalso the basis of the principle of duality. But these properties,\nas I shall now endeavour to prove, belong of necessity to any form\nof externality, and are thus _\u00e0 priori_ properties of all possible\nspaces. To prove this, however, we must first define the notion of a\nform of externality in general.\n\nLet us observe, to begin with, that the distinction between\nEuclidean and non-Euclidean Geometries, so important in metrical\ninvestigations, disappears in projective Geometry proper. This\nsuggests that projective Geometry, though originally invented as\nthe science of Euclidean space, and subsequently of non-Euclidean\nspaces also, deals really with a wider conception, a conception which\nincludes both, and neglects the attributes in which they differ. This\nconception I shall speak of as a form of externality.\n\n=126.= In Grassmann's profound philosophical introduction to his\n_Ausdehnungslehre_ of 1844, he suggested that Geometry, though\nimproperly regarded as pure, was really a branch of applied\nmathematics, since it dealt with a subject-matter not created,\nlike number, by the intellect, but given to it, and therefore not\nwholly subject to its laws alone. But it must be possible--so he\ncontended--to construct a branch of pure mathematics, a science, that\nis, in which our object should be wholly a creature of the intellect,\nwhich should yet deal, as Geometry does, with extension--extension\nas conceived, however, not as empirically perceived in sensation or\nintuition.\n\nFrom this point of view, the controversy between Kantians and\nanti-Kantians becomes wholly irrelevant, since the distinction\nbetween pure and mixed mathematics does not lie in the distinction\nbetween the subjective and the objective, but between the purely\nintellectual on the one hand, and everything else on the other.\nNow Kant had contended, with great emphasis, that space was not an\nintellectual construction, but a subjective intuition. Geometry,\ntherefore, with Grassmann's distinction, belongs to mixed mathematics\nas much on Kant's view as on that of his opponents. And Grassmann's\ndistinction, I contend, is the more important for Epistemology,\nand the one to be adopted in distinguishing the _\u00e0 priori_ from\nthe empirical. For what is merely intuitional can change, without\nupsetting the laws of thought, without making knowledge formally\nimpossible: but what is purely intellectual cannot change, unless the\nlaws of thought should change, and all our knowledge simultaneously\ncollapse. I shall therefore follow Grassmann's distinction in\nconstructing an _\u00e0 priori_ and purely conceptual form of externality.\n\n=127.= The pure doctrine of extension, as constructed by Grassmann,\nneed not be discussed--it included much empirical material, and was\nphilosophically a failure. But his principles, I think, will enable\nus to prove that projective Geometry, abstractly interpreted, is\nthe science which he foresaw, and deals with a matter which can be\nconstructed by the pure intellect alone. If this be so, however,\nit must be observed that projective Geometry, for the moment, is\nrendered purely hypothetical[132]. All necessary truth, as Bradley\nhas shown, is hypothetical[133], and asserts, _prim\u00e2 facie_, only\nthe ground on which rests the necessary connection of premisses\nand conclusion. If we construct a mere conception of externality,\nand thus abandon our actually given space, the result of our\nconstruction, until we return to something actually given, remains\nwithout existential import--if there _be_ experienced externality, it\nasserts, then there must be a form of externality with such and such\nproperties. That there must be experienced externality, Kant's first\nargument about space proves, I think, to those who admit experience\nof a world of diverse but interrelated things. But this is a question\nwhich belongs to the next Chapter.\n\nWhat we have to do here is, not to discuss whether there is a form of\nexternality, but whether, if there be such a form, it must possess\nthe properties embodied in the axioms of projective Geometry. Now\nfirst of all, what do we mean by such a form?\n\n=128.= In any world in which perception presents us with various\nthings, with discriminated and differentiated contents, there must\nbe, in perception, at least one \"principle of differentiation[134],\"\nan element, that is, by which the things presented are distinguished\nas various. This element, taken in isolation, and abstracted from the\ncontent which it differentiates, we may call a form of externality.\nThat it must, when taken in isolation, appear as a form, and not as\na mere diversity of material content, is, I think, fairly obvious.\nFor a diversity of material content cannot be studied apart from\nthat material content; what we wish to study here, on the contrary,\nis the bare possibility of such diversity, which forms the residuum,\nas I shall try to prove hereafter[135], when we abstract from any\nsense-perception all that is distinctive of its particular matter.\nThis possibility, then, this principle of bare diversity, is our form\nof externality. How far it is necessary to assume such a form, as\ndistinct from interrelated things, I shall consider later on[136].\nFor the present, since space, as dealt with by Geometry, is certainly\na form of this kind, we have only to ask: What properties must such a\nform, when studied in abstraction, necessarily possess?\n\n=129.= In the first place, externality is an essentially relative\nconception--nothing can be external to itself. To be external to\nsomething is to be another with some relation to that thing. Hence,\nwhen we abstract a form of externality from all material content,\nand study it in isolation, position will appear, of necessity,\nas purely relative--a position can have no intrinsic quality, for\nour form consists of pure externality, and externality contains\nno shadow or trace of an intrinsic quality. Thus we obtain our\nfundamental postulate, the relativity of position, or, as we may put\nit, the complete absence, on the part of our form, of any vestige of\nthinghood.\n\nThe same argument may also be stated as follows: If we abstract the\nconception of externality, and endeavour to deal with it _per se_, it\nis evident that we must obtain an object alike destitute of elements\nand of totality. For we have abstracted from the diverse matter\nwhich filled our form, while any element, or any whole, would retain\nsome of the qualities of a matter. Either an element or a whole, in\nfact, would have to be a thing not external to itself, and would\nthus contain something not pure externality. Hence arise infinite\ndivisibility, with the self-contradictory notion of the point, in the\nsearch for elements, and unbounded extension, with the contradiction\nof an infinite regress or a vicious circle, in the search for a\ncompleted whole. Thus again, our form contains neither elements nor\ntotality, but only endless relations--the terms of these relations\nbeing excluded by our abstraction from the matter which fills our\nform.\n\n=130.= In like manner we can deduce the homogeneity of our form. The\ndiversity of content, which was possible only within the form of\nexternality, has been abstracted from, leaving nothing but the bare\npossibility of diversity, the bare principle of differentiation,\nitself uniform and undifferentiated. For if diversity presupposes\nsuch a form, the form cannot, unless it were contained in a fresh\nform, be itself diverse or differentiated.\n\nOr we may deduce the same property from the relativity of position.\nFor any quality in one position, by which it was marked out from\nanother, would be necessarily more or less intrinsic, and would\ncontradict the pure relativity. Hence all positions are qualitatively\nalike, _i.e._ the form is homogeneous throughout.\n\n=131.= From what has been said of homogeneity and relativity,\nfollows one of the strangest properties of a form of externality.\nThis property is, that the relation of externality between any two\nthings is infinitely divisible, and may be regarded, consequently,\nas made up of an infinite number of the would-be elements of our\nform, or again as the sum of two relations of externality[137]. To\nspeak of dividing or adding relations may well sound absurd--indeed\nit reveals the impropriety of the word relation in this connexion.\nIt is difficult, however, to find an expression which shall be less\nimproper. The fact seems to be, that externality is not so much a\nrelation as bare relativity, or the bare possibility of a relation.\nOn this subject, I shall enlarge in Chapter IV.[138] At this point\nit is only important to realize, what the subsequent argument will\nassume, that the relation--if we may so call it--of externality\nbetween two or more things must, since our form is homogeneous, be\ncapable of continuous alteration, and must, since our infinitely\ndivisible form is constituted by such relations, be capable of\ninfinite division. But the result of infinite division is defined\nas the element of our form. (Our form has no elements, but we have\nto imagine elements in order to reason about it, as will be shown\nmore fully in Chapter IV.) Hence it follows, that every relation\nof externality may be regarded, for scientific purposes, as an\ninfinite congeries of elements, though philosophically, the relations\nalone are valid, and the elements are a self-contradictory result\nof hypostatizing the form of externality. This way of regarding\nrelations of externality is important in understanding the meaning of\nsuch ideas as three or four collinear points.\n\nAs this point is difficult and important, I will repeat, in somewhat\ngreater detail, the explanation of the manner in which straight\nlines and planes come to be regarded as congeries of points. From\nthe strictly projective standpoint, though all other figures _are_\nmerely a collection of any required number of points, lines or\nplanes, given by some projective construction, straight lines and\nplanes themselves are given integrally, and are not to be considered\nas divisible or composed of parts. To say that a point lies on\na straight line means, for projective Geometry proper, that the\nstraight line is a relation between this and some other point.\nHere the points concerned, if our statement is to be freed from\ncontradictions, must be regarded, if I may use such an expression, as\n_real_ points--_i.e._ as unextended material centres[139]. Straight\nlines and planes are then relations between these material atoms.\nThey are relations, however, which may undergo a metrical alteration\nwhile remaining projectively unchanged. When the projective relation\nbetween the two points _A_, _B_ is the same as that between the two\npoints _A_, _C_, while the metrical relation (distance) is different,\nthe three points _A_, _B_, _C_ are said to be collinear. Now the\nmetrical manner of regarding spatial figures demands that they should\nbe hypostatized, and no longer regarded as mere relations. For when\nwe regard a quantity as extensive, _i.e._ as divisible into parts,\nwe necessarily regard it as more than a mere relation or adjective,\nsince no mere relation or adjective can be divided. For quantitative\ntreatment, therefore, spatial relations must be hypostatized[140].\nWhen this is done, we obtain, as we saw above, a homogeneous\nand infinitely divisible form of externality. We find now that\ndistance, for example, may be continuously altered without changing\nthe straight line on which it is measured. We thus obtain, on the\nstraight line in question, a continuous series of points, which,\nsince it is continuous, we regard as constituting our straight line.\nIt is thus solely from the hypostatizing of relations, which metrical\nGeometry requires, that the view of straight lines and planes as\n_composed_ of points arises, and it is from this hypostatizing that\nthe difficulties of metrical Geometry spring.\n\n=132.= The next step, in defining a form of externality, is obtained\nfrom the idea of _dimensions_. Positions, we have seen, are defined\nsolely by their relations to other positions. But in order that\nsuch definition may be possible, a finite number of relations must\nsuffice, since infinite numbers are philosophically inadmissible. A\nposition must be definable, therefore, if knowledge of our form is\nto be possible at all, by some finite integral number of relations\nto other positions. Every relation thus necessary for definition\nwe call a dimension. Hence we obtain the proposition: _Any form of\nexternality must have a finite integral number of dimensions_.\n\n=133.= The above argument, it may be urged, has overlooked a\npossibility. It has used a transcendental argument, so an opponent\nmay contend, without sufficiently proving that knowledge about\nexternality must be possible without reference to the matters\nexternal to each other. The definition of a position may be\nimpossible, so long as we neglect the matter which fills the form,\nbut may become possible when this matter is taken into account.\nSuch an objection can, I think, be successfully met, by a reference\nto the passivity and homogeneity of our form. For any dependence\nof the definition of a position on the particular matter filling\nthat position, would involve some kind of interaction between the\nmatter and its position, some effect of the diverse content on\nthe homogeneous form. But since the form is totally destitute of\nthinghood, perfectly impassive, and perfectly void of differences\nbetween its parts, any such effect is inconceivable. An effect on\na position would have to alter it in some way, but how could it be\naltered? It has no qualities except those which make it the position\nit is, as opposed to other positions; it cannot change, therefore,\nwithout becoming a different position. But such a change contradicts\nthe law of identity. Hence it is not the position which has changed,\nbut the content which has moved in the form. Thus it must be\npossible, if knowledge of our form can be obtained at all, to obtain\nthis knowledge in logical independence of the particular matter which\nfills it. The above argument, therefore, granted the possibility of\nknowledge in the department in question, shows the necessity of a\nfinite integral number of dimensions.\n\n=134.= Let us repeat our original argument in the light of this\nelucidation. A position is completely defined when, and only when,\nenough relations are known to enable us to determine its relation\nto any fresh known position. Only by relations within the form of\nexternality, as we have just seen, and never by relations which\ninvolve a reference to the particular matter filling the form,\ncan such a definition be effected. But the possibility of such a\ndefinition follows from the Law of Excluded Middle, when this law is\ninterpreted to mean, as Bosanquet makes it mean, that \"Reality ... is\na system of reciprocally determinate parts[141].\" For this implies\nthat, given the relations of a part _A_ to other parts _B_, _C_ ...,\na sufficient wealth of such relations throws light on the relations\nof _B_ to _C_, etc. If this were not the case, the parts _A_, _B_,\n_C_ ... could not be said to form such a system; for in such a\nsystem, to define _A_ is to define, at the same time, all the other\nmembers, and to give an adjective to _A_, is to give an adjective to\n_B_ and _C_. But the relations between positions are, when we restore\nthe matter from which the positions were abstracted, relations\nbetween the things occupying those positions, and these relations, we\nhave seen, can be studied without reference to the particular nature,\nin other respects, of the related things. It follows that, when we\napply the general principle of systematic unity to these relations in\nparticular, we find these relations to be dependent on each other,\nsince they are not dependent, for their definition, on anything else.\nThis gives the axiom of dimensions, in the above general form, as\nthe result, on our abstract geometrical level, of the relativity of\nposition and the law of excluded middle.\n\n=135.= Before proceeding further, it is necessary to discuss the\nimportant special case where a form of externality has only one\ndimension. Of the two such forms, given in experience, one, namely\ntime, presents an instance of this special case. But it may be shown,\nI think, that the function, in constituting the possibility of\nexperience, which we demand of such forms, could not be accomplished\nby a one-dimensional form alone. For in a one-dimensional form, the\nvarious contents may be arranged in a series, and cannot, without\ninterpenetration, change the order of contents in the series.\nBut interpenetration is impossible, since a form of externality\nis the mere expression of diversity among things, from which it\nfollows that things cannot occupy the same position in a form,\nunless there is another form by which to differentiate them. For\nwithout externality, there is no diversity[142]. Thus two bodies\nmay occupy the same space, but only at different times: two things\nmay exist simultaneously, but only at different places. A form of\none dimension, therefore, could not, by itself, allow that change\nof the relations of externality, by which alone a varied world\nof interrelated things can be brought into consciousness. In a\none-dimensional space, for example, only a single object, which must\nappear as a point, or two objects at most, one in front and one\nbehind, could ever be perceived. Thus two or more dimensions seem\nan essential condition of anything worth calling an experience of\ninterrelated things.\n\n=136.= It may be objected, to this argument, that its validity\ndepends upon the assumption that the change of a relation of\nexternality must be continuous. Both to make and to meet this\nobjection, in a manner which shall not imply time, seems almost\nimpossible. For we cannot speak of change, whether continuous or\ndiscrete, without imagining time. Let us, therefore, allow time to\nbe known, and discuss whether the temporal change, in any other form\nof externality, is necessarily continuous[143]. We must reply, I\nthink, that continuity is necessary. The change of relation, in our\nnon-temporal form, may be safely described as motion, and the law\nof Causality--since we have already assumed time--may be applied to\nthis motion. It then follows that discrete motion would involve a\nfinite effect from an infinitesimal cause, for a cause acting only\nfor a moment of time would be infinitesimal. It involves, also, a\nvalidity in the point of time, whereas what is valid in any form of\nexternality is not, as we have already seen, the infinitesimal and\nself-contradictory element resulting from infinite division, but the\nfinite relation which mathematics analyzes into vanishing elements.\nHence change must be continuous, and the possibility of serial\narrangement holds good.\n\nIn a one-dimensional form other than time, the same argument must\nhold. For something analogous to Causality would be necessary to\nexperience, and the relativity of the form would still necessarily\nhold. Hence, since only these two properties of time have been\nassumed, the above contention would remain valid of any second form\nwhose relations were correlated with those of the first, as the\nanalogue of Causality would require them to be.\n\n=137.= The next step in the argument, which assumes two or more\ndimensions, is concerned with the general analogues of straight\nlines and planes, _i.e._ with figures--which may be regarded either\nas relations between positions or as series of positions--uniquely\ndetermined by two or by three positions. If this step can be\nsuccessfully taken, our deduction of the above projective axioms will\nbe complete, and descriptive Geometry will be established as the\nabstract _\u00e0 priori_ doctrine of forms of externality.\n\nTo prove this contention, consider of what nature the relations\ncan be by which positions are defined. We have seen already that\nour form is purely relational and infinitely divisible, and that\npositions (points) are the self-contradictory outcome of the search\nfor something other than relations. What we really mean, therefore,\nby the relations defining a position, is, when we undo our previous\nabstraction, the relations of externality by which some thing is\nrelated to other things. But how, when we remain in the abstract\nform, must such relations appear?\n\n=138.= We have to prove that two positions must have a relation\nindependent of any reference to other positions. To prove this, let\nus recur to what was said, in connection with dimensions, as to the\npassivity and homogeneity of our form. Since positions are defined\nonly by relations, there must be relations, within the form, between\npositions. But if there are such relations, there must be a relation\nwhich is intrinsic to two positions. For to suppose the contrary,\nis to attribute an interaction or causal connection, of some kind,\nbetween those two positions and other positions--a supposition\nwhich the perfect homogeneity of our form renders absurd, since all\npositions are qualitatively similar, and cannot be changed without\nlosing their identity. We may put this argument thus: since positions\nare only defined by their relations, such definition could never\nbegin, unless it began with a relation between only two positions.\nFor suppose three positions _A_, _B_, _C_ were necessary, and gave\nrise to the relation _abc_ between the three. Then there would remain\nno means of defining the different pairs _BC_, _CA_, _AB_, since the\nonly relation defining them would be one common to all three pairs.\nNothing would be gained, in this case, by reference to fresh points,\nfor it follows, from the homogeneity and passivity of the form, that\nthese fresh points could not affect the internal relations of our\ntriad, which relations, if they can give definiteness at all, must\ngive it without the aid of external reference. Two positions must,\ntherefore, if definition is to be possible, have some relation which\nthey by themselves suffice to define. Precisely the same argument\napplies to three positions, or to four; the argument loses its scope\nonly when we have exhausted the dimensions of the form considered.\nThus, in three dimensions, five positions have no fresh relation,\nnot deducible from those already known, for by the definition of\ndimensions, all the relations involved can be deduced from those of\nthe fourth point to the first three, together with those of the fifth\nto the first three.\n\nWe may give the argument a more concrete, and perhaps a more\nconvincing shape, by considering the matter arranged in our form. If\ntwo things are mutually external, they must since they belong to the\nsame world, have some relation of externality; there is, therefore,\na relation of externality between two things. But since our form is\nhomogeneous, the same relation of externality may subsist in other\nparts of the form, _i.e._ while the two things considered alter their\nrelations of externality to other things. The relation of externality\nbetween two things is, therefore, independent of other things. Hence,\nwhen we return to the abstract language of the form, two positions\nhave a relation determined by those two positions alone, and\nindependent of other positions.\n\nPrecisely the same argument applies to the relations of three\npositions, and in each case the relation must appear in the form as\nnot a mere inference from the positions it relates. For relations,\nas we have seen, actually constitute a form of externality, and are\nnot mere inferences from terms, which are nowhere to be found in the\nform[144].\n\nTo sum up: Since position is relative, two positions must have\n_some_ relation to each other; and since our form of externality\nis homogeneous, this relation can be kept unchanged while the two\npositions change their relations to other positions. Hence their\nrelation is intrinsic, and independent of other positions. Since\nthe form is a mere complex of relations, the relation in question\nmust, if the form is sensuous or intuitive, be itself sensuous or\nintuitive, and not a mere inference. In this case, a unique relation\nmust be a unique figure--in spatial terms, the straight line joining\nthe two points.\n\n=139.= With this, our deduction of projective Geometry from the\n_\u00e0 priori_ conceptual properties of a form of externality is\ncompleted. That such a form, when regarded as an independent thing,\nis self-contradictory, has been abundantly evident throughout the\ndiscussion. But the science of the form has been founded on the\nopposite way of regarding it: we have held it throughout to be a mere\ncomplex of relations, and have deduced its properties exclusively\nfrom this view of it. The many difficulties, in applying such an _\u00e0\npriori_ deduction to intuitive space, and in explaining, as logical\nnecessities, properties which appear as sensuous or intuitional data,\nmust be postponed to Chapter IV. For the present, I wish to point out\nthat projective Geometry is wholly _\u00e0 priori_; that it deals with an\nobject whose properties are logically deduced from its definition,\nnot empirically discovered from data; that its definition, again, is\nfounded on the possibility of experiencing diversity in relation,\nor multiplicity in unity; and that our whole science, therefore, is\nlogically implied in, and deducible from, the possibility of such\nexperience.\n\n=140.= In metrical Geometry, on the contrary, we shall find a very\ndifferent result. Although the geometrical conditions which render\nspatial measurement possible, will be found identical, except for\nslight differences in the form of statement, with the _\u00e0 priori_\naxioms discussed above, yet the actual measurement--which deals\nwith actually given space, not the mere intellectual construction\nwe have been just discussing--gives results which can only be known\nempirically and approximately, and can be deduced by no necessity\nof thought. The Euclidean and non-Euclidean spaces give the various\nresults which are _\u00e0 priori_ possible; the axioms peculiar to\nEuclid--which are properly not axioms, but empirical results of\nmeasurement--determine, within the errors of observation, which of\nthese _\u00e0 priori_ possibilities is realized in our actual space. Thus\nmeasurement deals throughout with an empirically given matter, not\nwith a creature of the intellect, and its _\u00e0 priori_ elements are\nonly the conditions presupposed in the possibility of measurement.\nWhat these conditions are, we shall see in the second section of this\nchapter.\n\n\nSection B.\n\nTHE AXIOMS OF METRICAL GEOMETRY.\n\n=141.= We have now reviewed the axioms of projective Geometry, and\nhave seen that they are _\u00e0 priori_ deductions from the fact that\nwe can experience externality, _i.e._ a coexistent multiplicity of\ndifferent but interrelated things. But projective Geometry, in spite\nof its claims, is not the whole science of space, as is sufficiently\nproved by the fact that it cannot discriminate between Euclidean and\nnon-Euclidean spaces[145]. For this purpose, spatial measurement\nis required: metrical Geometry, with its quantitative tests, can\nalone effect the discrimination. For all application of Geometry to\nphysics, also, measurement is required; the law of gravitation, for\nexample, requires the determination of actual distances. For many\npurposes, in short, projective Geometry is wholly insufficient: thus\nit is unable to distinguish between different kinds of conics, though\ntheir distinction is of fundamental importance in many departments of\nknowledge.\n\nMetrical Geometry is, then, a necessary part of the science of\nspace, and a part not included in descriptive Geometry. Its _\u00e0\npriori_ element, nevertheless, so far as this is spatial and not\narithmetical, is the same as the postulate of projective Geometry,\nnamely, the homogeneity of space, or its equivalent, the relativity\nof position. We can see, in fact, that the _\u00e0 priori_ element in\nboth is likely to be the same. For the _\u00e0 priori_ in metrical\nGeometry will be whatever is presupposed in the possibility of\nspatial measurement, _i.e._ of quantitative spatial comparison. But\nsuch comparison presupposes simply a known identity of quality,\nthe determination of which is precisely the problem of projective\nGeometry. Hence the conditions for the possibility of measurement, in\nso far as they are not arithmetical, will be precisely the same as\nthose for projective Geometry.\n\n=142.= Metrical Geometry, therefore, though distinct from projective\nGeometry, is not independent of it, but presupposes it, and arises\nfrom its combination with the extraneous idea of _quantity_.\nNevertheless the mathematical form of the axioms, in metrical\nGeometry, is slightly different from their form in projective\nGeometry. The homogeneity of space is replaced by its equivalent, the\naxiom of Free Mobility. The axiom of the straight line is replaced\nby the axiom of distance: Two points determine a unique quantity,\ndistance, which is unaltered in any motion of the two points as a\nsingle figure. This axiom, indeed, will be found to involve the axiom\nof the straight line--such a quantity could not exist unless the\ntwo points determined a unique curve--but its mathematical form is\nchanged. Another important change is the collapse of the principle\nof duality: quantity can be applied to the straight line, because\nit is divisible into similar parts, but cannot be applied to the\nindivisible point. We thus obtain a reason, which was wanting in\ndescriptive Geometry, for preferring points, as spatial elements,\nto straight lines or planes[146]. Finally, an entirely new idea is\nintroduced with quantity, namely, the idea of _Motion_. Not that we\nstudy motion, or that any of our results have reference to motion,\nbut that they cannot, though in projective Geometry they could, be\nobtained without at least an ideal motion of our figures through\nspace.\n\nLet us now examine in detail the prerequisites of spatial\nmeasurement. We shall find three axioms, without which such\nmeasurement would be impossible, but with which it is adequate to\ndecide, empirically and approximately, the Euclidean or non-Euclidean\nnature of our actual space. We shall find, further, that these three\naxioms can be deduced from the conception of a form of externality,\nand owe nothing to the evidence of intuition. They are, therefore,\nlike their equivalents the axioms of projective Geometry, _\u00e0 priori_,\nand deducible from the conditions of spatial experience. This\nexperience, accordingly, can never disprove them, since its very\nexistence presupposes them.\n\n\nI. _The Axiom of Free Mobility._\n\n=143.= Metrical Geometry, to begin with, may be defined as the\nscience which deals with the comparison and relations of spatial\nmagnitudes. The conception of magnitude, therefore, is necessary from\nthe start. Some of Euclid's axioms, accordingly, have been classed as\narithmetical, and have been supposed to have nothing particular to do\nwith space. Such are the axioms that equals added to or subtracted\nfrom equals give equals, and that things which are equal to the same\nthing are equal to one another. These axioms, it is said, are purely\narithmetical, and do not, like the others, ascribe an adjective to\nspace. As regards their use in arithmetic, this is of course true.\nBut if an arithmetical axiom is to be applied to spatial magnitudes,\nit must have some spatial import[147], and thus even this class is\nnot, in Geometry, _merely_ arithmetical. Fortunately, the geometrical\nelement is the same in all the axioms of this class--we can see at\nonce, in fact, that it can amount to no more than a definition of\nspatial magnitude[148]. Again, since the space with which Geometry\ndeals is infinitely divisible, a definition of spatial magnitude\nreduces itself to a definition of spatial equality, for, as soon as\nwe have this last, we can compare two spatial magnitudes by dividing\neach into a number of equal units, and counting the number of such\nunits in each[149]. The ratio of the number of units is, of course,\nthe ratio of the two magnitudes.\n\n=144.= We require, then, at the very outset, some criterion of\nspatial equality: without such a criterion metrical Geometry would\nbecome wholly impossible. It might appear, at first sight, as though\nthis need not be an axiom, but might be a mere definition. In part\nthis is true, but not wholly. The part which is merely a definition\nis given in Euclid's eighth axiom: \"Magnitudes which exactly coincide\nare equal.\" But this gives a sufficient criterion only when the\nmagnitudes to be compared already occupy the same position. When, as\nwill normally be the case, the two spatial magnitudes are external to\none another--as, indeed, must be the case, if they are distinct, and\nnot whole and part--the two magnitudes can only be made to coincide\nby a motion of one or both of them. In order, therefore, that our\ndefinition of spatial magnitude may give unambiguous results,\ncoincidence when superposed, if it can ever occur, must occur always,\nwhatever path be pursued in bringing it about. Hence, if mere motion\ncould alter shapes, our criterion of equality would break down.\nIt follows that the application of the conception of magnitude\nto figures in space involves the following axiom[150]: _Spatial\nmagnitudes can be moved from place to place without distortion_; or,\nas it may be put, _Shapes do not in any way depend upon absolute\nposition in space_.\n\nThe above axiom is the axiom of Free Mobility[151]. I propose to\nprove (1) that the denial of this axiom would involve logical and\nphilosophical absurdities, so that it must be classed as wholly _\u00e0\npriori_; (2) that metrical Geometry, if it refused this axiom, would\nbe unable, without a logical absurdity, to establish the notion of\nspatial magnitude at all. The conclusion will be, that the axiom\ncannot be proved or disproved by experience, but is an _\u00e0 priori_\ncondition of metrical Geometry. As I shall thus be maintaining a\nposition which has been much controverted, especially by Helmholtz\nand Erdmann, I shall have to enter into the arguments at some length.\n\n=145.= A. _Philosophical Argument._ The denial of the axiom involves\nabsolute position, and an action of mere space, _per se_, on things.\nFor the axiom does not assert that real bodies, as a matter of\nempirical fact, never change their shape in any way during their\npassage from place to place: on the contrary, we know that such\nchanges do occur, sometimes in a very noticeable degree, and always\nto some extent. But such changes are attributed, not to the change\nof place as such, but to physical causes: changes of temperature,\npressure, etc. What our axiom has to deal with is not actual material\nbodies, but geometrical figures[152], and it asserts that a figure\nwhich is possible in any one position in space is possible in every\nother. Its meaning will become clearer by reference to a case where\nit does not hold, say the space formed by the surface of an egg.\nHere, a triangle drawn near the equator cannot be moved without\ndistortion to the point, as it would no longer fit the greater\ncurvature of the new position: a triangle drawn near the point\ncannot be fitted on to the flatter end, and so on. Thus the method\nof superposition, such as Euclid employs in Book I. Prop. IV.,\nbecomes impossible; figures cannot be freely moved about, indeed,\ngiven any figure, we can determine a certain series of possible\npositions for it on the egg, outside which it becomes impossible.\nWhat I assert is, then, that there is a philosophic absurdity in\nsupposing space in general to be of this nature. On the egg we have\nmarked points, such as the two ends; the space formed by its surface\nis not homogeneous, and if things are moved about in it, it must of\nitself exercise a distorting effect upon them, quite independently of\nphysical causes; if it did not exercise such an effect, the things\ncould not be moved. Thus such a space would not be homogeneous, but\nwould have marked points, by reference to which bodies would have\nabsolute position, quite independently of any other bodies. Space\nwould no longer be passive, but would exercise a definite effect upon\nthings, and we should have to accommodate ourselves to the notion\nof marked points in empty space; these points being marked, not by\nthe bodies which occupied them, but by their effects on any bodies\nwhich might from time to time occupy them. This want of homogeneity\nand passivity is, however, absurd; space must, since it is a form of\nexternality, allow only of relative, not of absolute, position, and\nmust be completely homogeneous throughout. To suppose it otherwise,\nis to give it a thinghood which no form of externality can possibly\npossess. We must, then, on purely philosophical grounds, admit that a\ngeometrical figure which is possible anywhere is possible everywhere,\nwhich is the axiom of Free Mobility.\n\n=146.= B. _Geometrical Argument._ Let us see next what sort of\nGeometry we could construct without this axiom. The ultimate standard\nof comparison of spatial magnitudes must, as we saw in introducing\nthe axiom, be equality when superposed; but need we, from this\nequality, infer equality when separated? It has been urged by Erdmann\nthat, for the more immediate purposes of Geometry, this would be\nunnecessary[153]. We might construct a new Geometry, he thinks, in\nwhich sizes varied with motion on any definite law. Such a view,\nas I shall show below, involves a logical error as to the nature\nof magnitude. But before pointing this out, let us discuss the\ngeometrical consequences of assuming its truth. Suppose the length\nof an infinitesimal arc in some standard position were _ds_; then\nin any other position _p_ its length would be _ds.f(p)_, where the\nform of the function _f(p)_ must be supposed known. But how are we\nto determine the position _p_? For this purpose, we require _p_'s\ncoordinates, _i.e._, some measurement of distance from the origin.\nBut the distance from the origin could only be measured if we assumed\nour law _f(p)_ to measure it by. For suppose the origin to be _O_,\nand _Op_ to be a straight line whose length is required. If we have\na measuring rod with which we travel along the line and measure\nsuccessive infinitesimal arcs, the measuring rod will change its\nsize as we move, so that an arc which appears by the measure to be\n_ds_ will really be _f(s).ds_, where _s_ is the previously traversed\ndistance. If, on the other hand, we move our line _Op_ slowly through\nthe origin, and measure each piece as it passes through, our measure,\nit is true, will not alter, but now we have no means of discovering\nthe law by which any element has changed its length in coming to\nthe origin. Hence, until we assume our function _f(p)_, we have no\nmeans of determining _p_, for we have just seen that distances from\nthe origin can only be estimated by means of the law _f(p)_. It\nfollows that experience can neither prove nor disprove the constancy\nof shapes throughout motion, since, if shapes were not constant, we\nshould have to _assume_ a law of their variation before measurement\nbecame possible, and therefore measurement could not itself reveal\nthat variation to us[154].\n\nNevertheless, such an arbitrarily assumed law _does_, at first sight,\ngive a mathematically possible Geometry. The fundamental proposition,\nthat two magnitudes which can be superposed in any one position can\nbe superposed in any other, still holds. For two infinitesimal arcs,\nwhose lengths in the standard position are _ds{1}_ and _ds{2}_,\nwould, in any other position _p_, have lengths _f(p).ds{1}_ and\n_f(p).ds{2}_, so that their ratio would be unaltered. From this\nconstancy of ratio, as we know through Riemann and Helmholtz, the\nabove proposition follows. Hence all that Geometry requires, it would\nseem, as a basis for measurement, is an axiom that the alteration\nof shapes during motion follows a definite known law, such as that\nassumed above.\n\n=147.= There is, however, in such a view, as I remarked above, a\nlogical error as to the nature of magnitude. This error has been\nalready pointed out in dealing with Erdmann[155], and need only be\nbriefly repeated here. A judgment of magnitude is essentially a\njudgment of comparison: in unmeasured quantity, comparison as to\nthe mere more or less, but in measured magnitude, comparison as to\nthe precise how many times. To speak of differences of magnitude,\ntherefore, in a case where comparison cannot reveal them, is\nlogically absurd. Now in the case contemplated above, two magnitudes,\nwhich appear equal in one position, appear equal also when compared\nin another position. There is no sense, therefore, in supposing\nthe two magnitudes unequal when separated, nor in supposing,\nconsequently, that they have changed their magnitudes in motion.\nThis senselessness of our hypothesis is the logical ground of the\nmathematical indeterminateness as to the law of variation. Since,\nthen, there is no means of comparing two spatial figures, as regards\nmagnitude, except superposition, the only logically possible axiom,\nif spatial magnitude is to be self-consistent, is the axiom of Free\nMobility in the form first given above.\n\n=148.= Although this axiom is _\u00e0 priori_, its application to the\nmeasurement of actual bodies, as we found in discussing Helmholtz's\nviews, always involves an empirical element[156]. Our axiom, then,\nonly supplies the _\u00e0 priori_ condition for carrying out an operation\nwhich, in the concrete, is empirical--just as arithmetic supplies the\n_\u00e0 priori_ condition for a census. As this topic has been discussed\nat length in Chapter II., I shall say no more about it here.\n\n=149.= There remain, however, a few objections and difficulties to be\ndiscussed. First, how do we obtain equality in solids, and in Kant's\ncases of right and left hands, or of right and left-handed screws,\nwhere actual superposition is impossible? Secondly, how can we take\ncongruence as the only possible basis of spatial measurement, when we\nhave before us the case of time, where no such thing as congruence\nis conceivable? Thirdly, it might be urged that we can immediately\nestimate spatial equality by the eye, with more or less accuracy,\nand thus have a measure independent of congruence. Fourthly, how is\nmetrical Geometry possible on non-congruent surfaces, if congruence\nbe the basis of spatial measurement? I will discuss these objections\nsuccessively.\n\n=150.= (1) How do we measure the equality of solids? These could only\nbe brought into actual congruence if we had a fourth dimension to\noperate in[157], and from what I have said before of the absolute\nnecessity of this test, it might seem as though we should be left\nhere in utter ignorance. Euclid is silent on the subject, and in\nall works on Geometry it is assumed as self-evident that two cubes\nof equal side are equal. This assumption suggests that we are not\nso badly off as we should have been without congruence, as a test\nof equality in one or two dimensions; for now we can at least be\nsure that two cubes have all their sides and all their faces equal.\nTwo such cubes differ, then, in no sensible spatial quality save\nposition, for volume, in this case at any rate, is not a sensible\nquality. They are, therefore, as far as such qualities are concerned,\nindiscernible. If their places were interchanged, we might know the\nchange by their colour, or by some other non-geometrical property;\nbut so far as any property of which Geometry can take cognisance is\nconcerned, everything would seem as before. To suppose a difference\nof volume, then, would be to ascribe an effect to mere position,\nwhich we saw to be inadmissible while discussing Free Mobility.\nExcept as regards position, they are geometrically indiscernible, and\nwe may call to our aid the Identity of Indiscernibles to establish\ntheir agreement in the one remaining geometrical property of volume.\nThis may seem rather a strange principle to use in Mathematics,\nand for Geometry their equality is, perhaps, best regarded as\na definition; but if we demand a philosophical ground for this\ndefinition, it is, I believe, only to be found in the Identity of\nIndiscernibles. We can, without error, make our _definition_ of\nthree-dimensional equality rest on two-dimensional congruence. For\nsince direct comparison as to volume is impossible, we are at liberty\nto _define_ two volumes as equal, when all their various lines,\nsurfaces, angles and solid angles are congruent, since there remains,\nin such a case, no _measurable_ difference between the figures\ncomposing the two volumes. Of course, as soon as we have established\nthis one case of equality of volumes, the rest of the theory follows;\nas appears from the ordinary method of integrating volumes, by\ndividing them into small cubes.\n\nThus congruence _helps_ to establish three-dimensional equality,\nthough it cannot directly _prove_ such equality; and the same\nphilosophical principle, of the homogeneity of space, by which\ncongruence was proved, comes to our rescue here. But how about\nright-handed and left-handed screws? Here we can no longer apply the\nIdentity of Indiscernibles, for the two are very well discernible.\nBut as with solids, so here, Free Mobility can help us much. It\ncan enable us, by ordinary measurement, to show that the internal\nrelations of both screws are the same, and that the difference lies\nonly in their relation to other things in space. Knowing these\ninternal relations, we can calculate, by the Geometry which Free\nMobility has rendered possible, all the geometrical properties of\nthese screws--radius, pitch, etc.--and can show them to be severally\nequal in both. But this is all we require. Mediate comparison is\npossible, though immediate comparison is not. Both can, for instance,\nbe compared with the cylinder on which both would fit, and thus their\nequality can be proved. A precisely similar proof holds, of course,\nfor the other cases, right and left hands, spherical triangles, etc.\nOn the whole, these cases confirm my argument; for they show, as Kant\nintended them to show[158], the essential relativity of space.\n\n=151.= (2) As regards time, no congruence is here conceivable,\nfor to effect congruence requires always--as we saw in the case\nof solids--one more dimension than belongs to the magnitudes\ncompared. No day can be brought into temporal coincidence with any\nother day, to show that the two exactly cover each other; we are\ntherefore reduced to the arbitrary assumption that some motion or\nset of motions, given us in experience, is uniform. Fortunately,\nwe have a large set of motions which all roughly agree; the swing\nof the pendulum, the rotation and revolution of the earth and the\nplanets, etc. These do not exactly agree, but they lead us to the\nlaws of motion, by which we are able, on our arbitrary hypothesis,\nto estimate their small departures from uniformity; just as the\nassumption of Free Mobility enabled us to measure the departures\nof actual bodies from rigidity. But here, as there, another\npossibility is mathematically open to us, and can only be excluded\nby its philosophic absurdity; we might have assumed that the above\nset of approximately agreeing motions all had velocities which\nvaried approximately as some arbitrarily assumed function of the\ntime, _f(t)_ say, measured from some arbitrary origin. Such an\nassumption would still keep them as nearly synchronous as before,\nand would give an equally possible, though more complex, system\nof Mechanics; instead of the first law of motion, we should have\nthe following: A particle perseveres in its state of rest, or of\nrectilinear motion with velocity varying as _f(t)_, except in so far\nas it is compelled to alter that state by the action of external\nforces. Such a hypothesis _is_ mathematically possible, but, like the\nsimilar one for space, it is excluded logically by the comparative\nnature of the judgment of quantity, and philosophically by the fact\nthat it involves absolute time, as a determining agent in change,\nwhereas time can never, philosophically, be anything but a passive\nform, abstracted from change. I have introduced this parallel from\ntime, not as directly bearing on the argument, but as a simpler case\nwhich may serve to illustrate my reasoning in the more complex case\nof space. For since time, in mathematics, is one-dimensional, the\nmathematical difficulties are simpler than in Geometry; and although\nnothing accurately corresponds to congruence, there is a very similar\nmixture of mathematical and philosophical necessity, giving, finally,\na thoroughly definite axiom as the basis of time-measurement,\ncorresponding to congruence as the basis of space-measurement[159].\n\n=152.= (3) The case of time-measurement suggests the third of the\nabove objections to the absolute necessity of the axiom of Free\nMobility. Psycho-physics has shown that we have an approximate\npower, by means of what may be called the sense of duration, of\nimmediately estimating equal short times. This establishes a rough\nmeasure independent of any assumed uniform motion, and in space also,\nit may be said, we have a similar power of immediate comparison.\nWe can see, by immediate inspection, that the sub-divisions on\na foot rule are not grossly inaccurate; and so, it may be said,\nwe both have a measure independent of congruence, and also could\ndiscover, by experience, any gross departure from Free Mobility.\nAgainst this view, however, there is at the outset a very fundamental\npsychological objection. It has been urged that all our comparison\nof spatial magnitudes proceeds by ideal superposition. Thus James\nsays (Psychology, Vol. II. p. 152): \"Even where we only feel one\nsub-division to be vaguely larger or less, the mind must pass rapidly\nbetween it and the other sub-division, and receive the immediate\nsensible shock of the more,\" and \"so far as the sub-divisions of\na sense-space are to be _measured_ exactly against each other,\nobjective forms occupying one sub-division must be directly or\nindirectly superposed upon the other[160].\"\n\nEven if we waive this fundamental objection, however, others remain.\nTo begin with, such judgments of equality are only very rough\napproximations, and cannot be applied to lines of more than a certain\nlength, if only for the reason that such lines cannot well be seen\ntogether. Thus this method can only give us any security in our own\nimmediate neighbourhood, and could in no wise warrant such operations\nas would be required for the construction of maps &c., much less the\nmeasurement of astronomical distances. They might just enable us to\nsay that some lines were longer than others, but they would leave\nGeometry in a position no better than that of the Hedonical Calculus,\nin which we depend on a purely subjective measure. So inaccurate, in\nfact, is such a method acknowledged to be, that the foot-rule is as\nmuch a need of daily life as of science. Besides, no one would trust\nsuch immediate judgments, but for the fact that the stricter test of\ncongruence to some extent confirms them; if we could not apply this\ntest, we should have no ground for trusting them even as much as\nwe do. Thus we should have, here, no real escape from our absolute\ndependence upon the axiom of Free Mobility.\n\n=153.= (4) One last elucidatory remark is necessary before our proof\nof this axiom can be considered complete. We spoke above of the\nGeometry on an egg, where Free Mobility does not hold. What, I may\nbe asked, is there about a thoroughly non-congruent Geometry, more\nimpossible than this Geometry on the egg? The answer is obvious. The\nGeometry of non-congruent surfaces is _only_ possible by the use of\ninfinitesimals, and in the infinitesimal all surfaces become plane.\nThe fundamental formula, that for the length of an infinitesimal\narc, is only obtained on the assumption that such an arc may be\ntreated as a straight line, and that Euclidean Plane Geometry may be\napplied in the immediate neighbourhood of any point. If we had not\nour Euclidean measure, which could be moved without distortion, we\nshould have no method of comparing small arcs in different places,\nand the Geometry of non-congruent surfaces would break down. Thus\nthe axiom of Free Mobility, as regards three-dimensional space, is\nnecessarily implied and presupposed in the Geometry of non-congruent\nsurfaces; the possibility of the latter, therefore, is a dependent\nand derivative possibility, and can form no argument against the _\u00e0\npriori_ necessity of congruence as the test of equality.\n\n=154.= It is to be observed that the axiom of Free Mobility, as I\nhave enunciated it, includes also the axiom to which Helmholtz gives\nthe name of Monodromy. This asserts that a body does not alter its\ndimensions in consequence of a complete revolution through four\nright angles, but occupies at the end the same position as at the\nbeginning. The supposed mathematical necessity of making a separate\naxiom of this property of space has been disproved by Sophus Lie (v.\nChap. I. \u00a7 45); philosophically, it is plainly a particular case of\nFree Mobility[161], and indeed a particularly obvious case, for a\ntranslation really does make some change in a body, namely, a change\nin position, but a rotation through four right angles may be supposed\nto have been performed any number of times without appearing in the\nresult, and the absurdity of ascribing to space the power of making\nbodies grow in the process is palpable; everything that was said\nabove on congruence in general applies with even greater evidence to\nthis special case.\n\n=155.= The axiom of Free Mobility involves, if it is to be true, the\nhomogeneity of space, or the complete relativity of position. For if\nany shape, which is possible in one part of space, be always possible\nin another, it follows that all parts of space are qualitatively\nsimilar, and cannot, therefore, be distinguished by any intrinsic\nproperty. Hence positions in space, if our axiom be true, must be\nwholly defined by external relations, _i.e._ _Position is not an\nintrinsic, but a purely relative, property of things in space_. If\nthere could be such a thing as absolute position, in short, metrical\nGeometry would be impossible. This relativity of position is the\nfundamental postulate of all Geometry, to which each of the necessary\nmetrical axioms leads, and from which, conversely, each of these\naxioms can be deduced.\n\n=156.= This converse deduction, as regards Free Mobility, is not very\ndifficult, and follows from the argument of Section A[162], which\nI will briefly recapitulate. In the first place, externality is an\nessentially relative conception--nothing can be external to itself.\nTo be external to something is to be an other with some relation to\nthat thing. Hence, when we abstract a form of externality from all\nmaterial content, and study it in isolation, position will appear\nof necessity as purely relative--it can have no intrinsic quality,\nfor our form consists of pure externality, and externality contains\nno shadow or trace of an intrinsic quality. Hence we derive our\nfundamental postulate, the relativity of position. From this follows\nthe homogeneity of our form, for any quality in one position, which\nmarked out that position from another, would be necessarily more or\nless intrinsic, and would contradict the pure relativity. Finally\nFree Mobility follows from homogeneity, for our form would not be\nhomogeneous unless it allowed, in every part, shapes or systems\nof relations, which it allowed in any other part. Free Mobility,\ntherefore, is a necessary property of every possible form of\nexternality.\n\n=157.= In summing up the argument we have just concluded, we may\nexhibit it, in consequence of the two preceding paragraphs, in the\nform of a completed circle. Starting from the conditions of spatial\nmeasurement, we found that the comparison, required for measurement,\ncould only be effected by superposition. But we found, further, that\nthe result of such comparison will only be unambiguous, if spatial\nmagnitudes and shapes are unaltered by motion in space, if, in other\nwords, shapes do not depend upon absolute position in space. But this\naxiom can only be true if space is homogeneous and position merely\nrelative. Conversely, if position is assumed to be merely relative,\na change of magnitude in motion--involving as it does, the assertion\nof absolute position--is impossible, and our test of spatial equality\nis therefore adequate. But position in any form of externality must\nbe purely relative, since externality cannot be an intrinsic property\nof anything. Our axiom, therefore, is _\u00e0 priori_ in a double sense.\nIt is presupposed in all spatial measurement, and it is a necessary\nproperty of any form of externality. A similar double apriority, we\nshall see, appears in our other necessary axioms.\n\n\nII. _The Axiom of Dimensions[163]._\n\n=158.= We have seen, in discussing the axiom of Free Mobility, that\nall position is relative, that is, a position exists only by virtue\nof relations[164]. It follows that, if positions can be defined at\nall, they must be uniquely and exhaustively defined by some finite\nnumber of such relations. If Geometry is to be possible, it must\nhappen that, after enough relations have been given to determine a\npoint uniquely, its relations to any fresh known point are deducible\nfrom the relations already given. Hence we obtain, as an _\u00e0 priori_\ncondition of Geometry, logically indispensable to its existence, the\naxiom that _Space must have a finite integral number of Dimensions_.\nFor every relation required in the definition of a point constitutes\na dimension, and a fraction of a relation is meaningless. The number\nof relations required must be finite, since an infinite number of\ndimensions would be practically impossible to determine. If we\nremember our axiom of Free Mobility, and remember also that space\nis a continuum, we may state our axiom, for metrical Geometry, in\nthe form given by Helmholtz (v. Chap. I. \u00a7 25): \"In a space of n\ndimensions, the position of every point is uniquely determined by the\nmeasurement of n continuous independent variables (coordinates).[165]\"\n\n=159.= So much, then, is _\u00e0 priori_ necessary to metrical Geometry.\nThe restriction of the dimensions to three seems, on the contrary,\nto be wholly the work of experience[166]. This restriction cannot\nbe logically necessary, for as soon as we have formulated any\nanalytical system, it appears wholly arbitrary. Why, we are driven\nto ask, cannot we add a fourth coordinate to our _x_, _y_, _z_, or\ngive a geometrical meaning to _x^{4}_? In this more special form, we\nare tempted to regard the axiom of dimensions, like the number of\ninhabitants of a town, as a purely statistical fact, with no greater\nnecessity than such facts have.\n\nGeometry affords intrinsic evidence of the truth of my division of\nthe axiom of dimensions into an _\u00e0 priori_ and empirical portion.\nFor while the extension of the number of dimensions to four, or to\n_n_, alters nothing in plane and solid Geometry, but only adds a\nnew branch which interferes in no way with the old, _some_ definite\nnumber of dimensions is assumed in all Geometries, nor is it\npossible to conceive of a Geometry which should be free from this\nassumption[167].\n\n=160.= Let us, since the point seems of some interest, repeat our\nproof of the apriority of this axiom from a slightly different point\nof view. We will begin, this time, from the most abstract conception\nof space, such as we find in Riemann's dissertation, or in Erdmann's\nextents. We have here, an ordered manifold, infinitely divisible and\nallowing of Free Mobility[168]. Free Mobility involves, as we saw,\nthe power of passing continuously from any one point to any other, by\nany course which may seem pleasant to us; it involves, also, that,\nin such a course, no changes occur except changes of mere position,\n_i.e._, positions do not differ from one another in any qualitative\nway. (This absence of qualitative difference is the distinguishing\nmark of space as opposed to other manifolds, such as the colour- and\ntone-systems: in these, every element has a definite qualitative\nsensational value, whereas in space, the sensational value of a\nposition depends wholly on its spatial relation to our own body, and\nis thus not intrinsic, but relative.) From the absence of qualitative\ndifferences among positions, it follows logically that positions\nexist only by virtue of other positions; one position differs from\nanother just because they are two, not because of anything intrinsic\nin either. Position is thus defined simply and solely by relation to\nother positions. Any position, therefore, is completely defined when,\nand only when, enough such relations have been given to enable us to\ndetermine its relation to any new position, this new position being\ndefined by the same number of relations. Now, in order that such\ndefinition may be at all possible, a finite number of relations must\nsuffice. But every such relation constitutes a dimension. Therefore,\nif Geometry is to be possible, it is _\u00e0 priori_ necessary that space\nshould have a finite integral number of dimensions.\n\n=161.= The limitation of the dimensions to three is, as we have\nseen, empirical; nevertheless, it is not liable to the inaccuracy\nand uncertainty which usually belong to empirical knowledge. For\nthe alternatives which logic leaves to sense are discrete--if the\ndimensions are not three, they must be two or four or some other\nnumber--so that _small_ errors are out of the question[169]. Hence\nthe final certainty of the axiom of three dimensions, though in part\ndue to experience, is of quite a different order from that of (say)\nthe law of Gravitation. In the latter, a small inaccuracy might exist\nand remain undetected; in the former, an error would have to be so\nconsiderable as to be utterly impossible to overlook. It follows that\nthe certainty of our whole axiom, that the number of dimensions is\nthree, is almost as great as that of the _\u00e0 priori_ element, since\nthis element leaves to sense a definite disjunction of discrete\npossibilities.\n\n\nIII. _The Axiom of Distance._\n\n=162.= We have already seen, in discussing projective Geometry, that\ntwo points must determine a unique curve, the straight line. In\nmetrical Geometry, the corresponding axiom is, that two points must\ndetermine a unique spatial quantity, distance. I propose to prove,\nin what follows, (1) that if distance, as a quantity completely\ndetermined by two points, did not exist, spatial magnitude would\nnot be measurable; (2) that distance can only be determined by two\npoints, if there is an actual curve in space determined by those two\npoints; (3) that the existence of such a curve can be deduced from\nthe conception of a form of externality, and (4) that the application\nof quantity to such a curve necessarily leads to a certain magnitude,\nnamely distance, uniquely determined by any two points which\ndetermine the curve. The conclusion will be, if these propositions\ncan be successfully maintained, that the axiom of distance is _\u00e0\npriori_ in the same double sense as the axiom of Free Mobility,\n_i.e._ it is presupposed in the possibility of measurement, and it is\nnecessarily true of any possible form of externality.\n\n=163.= (1) The possibility of spatial measurement allows us to\ninfer the existence of a magnitude uniquely determined by any two\npoints. The proof of this depends on the axiom of Free Mobility, or\nits equivalent, the homogeneity of space. We have seen that these\nare involved in the possibility of spatial measurement; we may\nemploy them, therefore, in any argument as to the conditions of this\npossibility.\n\nNow to begin with, two points must, if Geometry is to be possible,\nhave _some_ relation to each other, for we have seen that such\nrelations alone constitute position or localization. But if two\npoints have a relation to each other, this must be an intrinsic\nrelation. For it follows, from the axiom of Free Mobility, that\ntwo points, forming a figure congruent with the given pair, can be\nconstructed in any part of space. If this were not possible, we have\nseen that metrical Geometry could not exist. But both the figures may\nbe regarded as composed of two points and their relation; if the\ntwo figures are congruent, therefore, it follows that the relation\nis quantitatively the same for both figures, since congruence is the\ntest of spatial equality. Hence the two points have a quantitative\nrelation, which is such that they can traverse all space in a\ncombined motion without in any way altering that relation. But in\nsuch a general motion, any external relation of the two points,\nany relation involving other points or figures in space, must be\naltered[170]. Hence the relation between the two points, being\nunaltered, must be an intrinsic relation, a relation involving no\nother point or figure in space; and this intrinsic relation we call\ndistance[171].\n\n=164.= It might be objected, to the above argument, that it involves\na _petitio principii_. For it has been assumed that the two points\nand their relation form a figure, to which other figures can be\ncongruent. Now if two points have no intrinsic relation, it would\nseem that they cannot form such a figure. The argument, therefore,\napparently assumes what it had to prove. Why, it may be asked, should\nnot three points be required, before we obtain any relation, which\nFree Mobility allows us to construct afresh in other parts of space?\n\nThe answer to this, as to the corresponding question in the first\nsection of this chapter, lies, I think, in the passivity of space,\nor the mutual independence of its parts. For it follows, from this\nindependence, that any figure, or any assemblage of points, may\nbe discussed without reference to other figures or points. This\nprinciple is the basis of infinite divisibility, of the use of\nquantity in Geometry, and of all possibility of isolating particular\nfigures for discussion. It follows that two points cannot be\ndependent, as to their relation, on any other points or figures, for\nif they were so dependent, we should have to suppose some action\nof such points or figures on the two points considered, which\nwould contradict the mutual independence of different positions.\nTo illustrate by an example: the relation of two given points does\nnot depend on the other points of the straight line on which the\ngiven points lie. For only through their relation, _i.e._ through\nthe straight line which they determine, can the other points of the\nstraight line be known to have any peculiar connection with the given\npair.\n\n=165.= But why, it may be asked, should there be only one such\nrelation between two points? Why not several? The answer to this lies\nin the fact that points are wholly constituted by relations, and have\nno intrinsic nature of their own[172]. A point is defined by its\nrelations to other points, and when once the relations necessary for\ndefinition have been given, no fresh relations to the points used in\ndefinition are possible, since the point defined has no qualities\nfrom which such relations could flow. Now one relation to any one\nother point is as good for definition as more would be, since however\nmany we had, they would all remain unaltered in a combined motion of\nboth points. Hence there can only be one relation determined by any\ntwo points.\n\n=166.= (2) We have thus established our first proposition--two points\nhave one and only one relation uniquely determined by those two\npoints. This relation we call their distance apart. It remains to\nconsider the conditions of the measurement of distance, _i.e._, how\nfar a unique value for distance involves a curve uniquely determined\nby the two points.\n\nIn the first place, some curve joining the two points is involved\nin the above notion of a combined motion of the two points, or of\ntwo other points forming a figure congruent with the first two.\nFor without some such curve, the two point-pairs cannot be known\nas congruent, nor can we have any test by which to discover when\na point-pair is moving as a single figure[173]. Distance must be\nmeasured, therefore, by some line which joins the two points. But\nneed this be a line which the two points completely determine?\n\n=167.= We are accustomed to the definition of the straight line\nas the _shortest_ distance between two points, which implies that\ndistance might equally well be measured by curved lines. This\nimplication I believe to be false, for the following reasons. When we\nspeak of the length of a curve, we can give a meaning to our words\nonly by supposing the curve divided into infinitesimal rectilinear\narcs, whose sum gives the length of an equivalent straight line;\nthus unless we presuppose the straight line, we have no means of\ncomparing the lengths of different curves, and can therefore never\ndiscover the applicability of our definition. It might be thought,\nperhaps, that some other line, say a circle, might be used as the\nbasis of measurement. But in order to estimate in this way the length\nof any curve other than a circle, we should have to divide the curve\ninto infinitesimal circular arcs. Now two successive points do not\ndetermine a circle, so that an arc of two points would have an\nindeterminate length. It is true that, if we exclude infinitesimal\nradii for the measuring circles, the lengths of the infinitesimal\narcs would be determinate, even if the circles varied, but that is\nonly because all the small circular arcs through two consecutive\npoints coincide with the straight line through those two points.\nThus, even with the help of the arbitrary restriction to a finite\nradius, all that happens is that we are brought back to the straight\nline. If, to mend matters, we take three consecutive points of our\ncurve, and reckon distance by the arc of the circle of curvature,\nthe notion of distance loses its fundamental property of being a\nrelation between _two_ points. For two consecutive points of the\narc could not then be said to have any corresponding distance\napart--three points would be necessary before the notion of distance\nbecame applicable. Thus the circle is not a possible basis for\nmeasurement, and similar objections apply, of course, with increased\nforce, to any other curve. All this argument is designed to show,\nin detail, the logical impossibility of measuring distance by any\ncurve not completely defined by the two points whose distance apart\nis required. If in the above we had taken distance as measured\nby circles of _given radius_, we should have introduced into its\ndefinition a relation to other points besides the two whose distance\nwas to be measured, which we saw to be a logical fallacy. Moreover,\nhow are we to know that all the circles have equal radii, until we\nhave an independent measure of distance?\n\n=168.= A straight line, then, is not the _shortest_ distance, but is\nsimply _the_ distance between two points--so far, this conclusion\nhas stood firm. But suppose we had two or more curves through\ntwo points, and that all these curves were congruent _inter se_.\nWe should then say, in accordance with the definition of spatial\nequality, that the lengths of all these curves were equal. Now\nit might happen that, although no one of the curves was uniquely\ndetermined by the two end-points, yet the common length of all the\ncurves was so determined. In this case, what would hinder us from\ncalling this common length the distance apart, although no unique\nfigure in space corresponded to it? This is the case contemplated by\nspherical Geometry, where, as on a sphere, antipodes can be joined by\nan infinite number of geodesics, all of which are of equal length.\nThe difficulty supposed is, therefore, not a purely imaginary one,\nbut one which modern Geometry forces us to face. I shall consequently\ndiscuss it at some length.\n\n=169.= To begin with, I must point out that my axiom is not quite\nequivalent to Euclid's. Euclid's axiom states that two straight lines\ncannot enclose a space, _i.e._, cannot have more than one common\npoint. Now if every two points, without exception, determine a unique\nstraight line, it follows, of course, that two different straight\nlines can have only one point in common--so far, the two axioms are\nequivalent. But it may happen, as in spherical space, that two points\n_in general_ determine a unique straight line, but fail to do so\nwhen they have to each other the special relation of being antipodes.\nIn such a system every pair of straight lines in the same plane meet\nin two points, which are each other's antipodes; but two points, _in\ngeneral_, still determine a unique straight line. We are still able,\ntherefore, to obtain distances from unique straight lines, except in\nlimiting cases; and in such cases, we can take any point intermediate\nbetween the two antipodes, join it by the _same_ straight line to\nboth antipodes, and measure its distance from those antipodes in the\nusual way. The sum of these distances then gives a unique value for\nthe distance between the antipodes.\n\nThus even in spherical space, we are greatly assisted by the axiom\nof the straight line; all linear measurement is effected by it, and\nexceptional cases can be treated, through its help, by the usual\nmethods for limits. Spherical space, therefore, is not so adverse\nas it at first appeared to be to the _\u00e0 priori_ necessity of the\naxiom. Nevertheless we have, so far, not attacked the kernel of the\nobjection which spherical space suggested. To this attack it is now\nour duty to proceed.\n\n=170.= It will be remembered that, in our _\u00e0 priori_ proof that\ntwo points must have one definite relation, we held it impossible\nfor those two points to have, to the rest of space, any relation\nwhich would be unaltered by motion. Now in spherical space, in the\nparticular case where the two points are antipodes, they _have_ a\nrelation, unaltered by motion, to the rest of space--the relation,\nnamely, that their distance is half the circumference of the\nuniverse. In our former discussion, we assumed that any relation\nto outside space must be a relation of position--and a relation of\nposition must be altered by motion. But with a finite space, in\nwhich we have absolute magnitude, another relation becomes possible,\nnamely, a relation of magnitude. Antipodal points, accordingly,\nlike coincident points, no longer determine a unique straight line.\nAnd it is instructive to observe that there is, in consequence, an\nambiguity in the expression for distance, like the ordinary ambiguity\nin angular measurement. If 1/k^{2} be the space constant, and _d_\nbe one value for the distance between two points, 2\u03c0kn \u00b1 d, where\n_n_ is any integer, is an equally good value. Distance is, in short,\na periodic function like angle. Thus such a state of things rather\nconfirms than destroys my contention, that distance depends on a\ncurve uniquely determined by two points. For as soon as we drop this\nunique determination, we see ambiguities creeping into our expression\nfor distance. Distance still has a set of _discrete_ values,\ncorresponding to the fact that, given one point, the straight line\nis uniquely determined for all other points but one, the antipodal\npoint. It is tempting to go on, and say: If through _every_ pair of\npoints there were an infinite number of the curves used in measuring\ndistance, distance would be able, for the same pair of points, to\ntake, not only a discrete series, but an infinite _continuous_ series\nof values.\n\n=171.= This, however, is mere speculation. I come now to the _pi\u00e8ce\nde r\u00e9sistance_ of my argument. The ambiguity in spherical space\narose, as we saw, from a relation of _magnitude_ to the rest of\nspace--such a relation being unaltered by a motion of the two points,\nand therefore falling outside our introductory reasoning. But what\nis this relation of magnitude? Simply a relation of the _distance_\nbetween the two points to a _distance_ given in the nature of the\nspace in question. It follows that such a relation _presupposes_ a\nmeasure of distance, and need not, therefore, be contemplated in\nany argument which deals with the _\u00e0 priori_ requisites for the\npossibility of definite distances[174].\n\n=172.= I have now shown, I hope conclusively, that spherical space\naffords no objection to the apriority of my axiom. Any two points\nhave one relation, their distance, which is independent of the\nrest of space, and this relation requires, as its measure, a curve\nuniquely determined by those two points. I might have taken the bull\nby the horns, and said: Two points _can_ have no relation but what\nis given by lines which join them, and therefore, if they have a\nrelation independent of the rest of space, there must be one line\njoining them which they completely determine. Thus James says[175]:\n\n\"Just as, in the field of quantity, the relation between two numbers\nis another number, so in the field of space the relations are facts\nof the same order with the facts they relate.... When we speak of\nthe relation of direction of two points towards each other, we mean\nsimply the sensation of the line that joins the two points together.\n_The line is the relation...._ The relation of position between the\ntop and bottom points of a vertical line is that line, and nothing\nelse.\"\n\nIf I had been willing to use this doctrine at the beginning, I\nmight have avoided all discussion. A unique relation between two\npoints _must_ in this case, involve a unique line between them. But\nit seemed better to avoid a doctrine not universally accepted, the\nmore so as I was approaching the question from the logical, not the\npsychological, side. After disposing of the objections, however, it\nis interesting to find this confirmation of the above theory from so\ndifferent a standpoint. Indeed, I believe James's doctrine could be\nproved to be a logical necessity, as well as a psychological fact.\nFor what sort of thing can a spatial relation between two distinct\npoints be? It must be something spatial, and it must, since points\nare wholly constituted by their relations, be something at least\nas real and tangible as the points it relates. There seems nothing\nwhich can satisfy these requirements, except a line joining them.\nHence, once more, a unique relation must involve a unique line. That\nis, linear magnitude is logically impossible, unless space allows of\ncurves uniquely determined by any two of their points.\n\n=173.= (3) But farther, the existence of curves uniquely determined\nby two points can be deduced from the nature of any form of\nexternality[176]. For we saw, in discussing Free Mobility, that this\naxiom, together with homogeneity and the relativity of position,\ncan be so deduced, and we saw in the beginning of our discussion on\ndistance, that the existence of a unique relation between two points\ncould be deduced from the homogeneity of space. Since position is\nrelative, we may say, any two points must have _some_ relation to\neach other: since our form of externality is homogeneous, this\nrelation can be kept unchanged while the two points move in the\nform, _i.e._, change their relations to other points; hence their\nrelation to each other is an intrinsic relation, independent of their\nrelations to other points. But since our form _is_ merely a complex\nof relations, a relation of externality must appear in the form, with\nthe same evidence as anything else in the form; thus if the form be\nintuitive or sensational, the relation must be immediately presented,\nand not a mere inference. Hence the intrinsic relation between two\npoints must be a unique figure in our form, _i.e._ in spatial terms,\nthe straight line joining the two points.\n\n=174.= (4) Finally, we have to prove that the existence of such a\ncurve necessarily leads, when quantity is applied to the relation\nbetween two points, to a unique magnitude, which those two points\ncompletely determine. With this, we shall be brought back to\ndistance, from which we started, and shall complete the circle of our\nargument.\n\nWe saw, in section A \u00a7 119, that the figure formed by two points is\nprojectively indistinguishable from that formed by any two other\npoints in the same straight line; the figure, in both cases, is,\nfrom the projective standpoint, simply the straight line on which\nthe two points lie. The difference of relation, in the two cases,\nis not qualitative, since projective Geometry cannot deal with it;\nnevertheless, there is some difference of relation. For instance, if\none point be kept fixed, while the other moves, there is obviously\nsome change of relation. This change, since all parts of the straight\nline are qualitatively alike, must be a change of quantity. If two\npoints, therefore, determine a unique figure, there must exist, for\nthe distinction between the various other points of this figure, a\nunique quantitative relation between the two determining points, and\ntherefore, since these points are arbitrary, between only two points.\nThis relation is _distance_, with which our argument began, and to\nwhich it at least returns.\n\n=175.= To sum up: If points are defined simply by relations to other\npoints, _i.e._, if all position is relative, _every point must have\nto every other point one, and only one, relation independent of\nthe rest of space. This relation is the distance between the two\npoints._ Now a relation between two points can only be defined by a\nline joining them--nay further, it may be contended that a relation\ncan only _be_ a line joining them. Hence a unique relation involves\na unique line, _i.e._, a line determined by any two of its points.\nOnly in a space which admits of such a line is linear magnitude a\nlogically possible conception. But when once we have established the\npossibility, _in general_, of drawing such lines, and therefore of\nmeasuring linear magnitudes, we may find that a certain magnitude has\na peculiar relation to the constitution of space. The straight line\nmay turn out to be of finite length, and in this case its length will\ngive a certain peculiar magnitude, the space-constant. Two antipodal\npoints, that is, points which bisect the entire straight line, will\nthen have a relation of magnitude which, though unaltered by motion,\nis rendered peculiar by a certain constant relation to the rest of\nspace. This peculiarity presupposes a measure of linear magnitude in\ngeneral, and cannot, therefore, upset the apriority of the axiom of\nthe straight line. But it destroys, for points having the peculiar\nantipodal relation to each other, the argument which proved that the\nrelation between two points could not, since it was unchanged by\nmotion, have reference to the rest of space. Thus it is intelligible\nthat, for such special points, the axiom breaks down, and an infinite\nnumber of straight lines are possible between them; but unless we had\nstarted with assuming the general validity of the axiom, we could\nnever have reached a position in which antipodal points could have\nbeen known to be peculiar, or, indeed, a position which would have\nenabled us to give any quantitative definition whatever of particular\npoints.\n\nDistance and the straight line, as relations uniquely determined\nby two points, are thus _\u00e0 priori_ necessary to metrical Geometry.\nBut further, they are properties which must belong to any form of\nexternality. Since their necessity for Geometry was deduced from\nhomogeneity and the relativity of position, and since these are\nnecessary properties of any form of externality, the same argument\nproves both conclusions. We thus obtain, as in the case of Free\nMobility, a double apriority: The axiom of Distance, and its\nimplication, the axiom of the Straight Line, are, on the one hand,\npresupposed in the possibility of spatial magnitude, and cannot,\ntherefore, be contradicted by any experience resulting from the\nmeasurement of space; while they are consequences, on the other hand,\nof the necessary properties of any form of externality which is to\nrender possible experience of an external world.\n\n=176.= In connection with the straight line, it will be convenient\nto discuss the conditions of a metrical coordinate system. The\nprojective coordinate system, as we have seen, aims only at a\nconvenient nomenclature for different points, and can be set up\nwithout introducing the notion of spatial quantity. But a metrical\ncoordinate system does much more than this. It defines every point\nquantitatively, by its quantitative spatial relations to a certain\ncoordinate figure. Only when the system of coordinates is thus\nmetrical, _i.e._, when every coordinate represents some spatial\nmagnitude, which is itself a relation of the point defined to some\nother point or figure--can operations with coordinates lead to a\nmetrical result. When, as in projective Geometry, the coordinates\nare not spatial magnitudes, no amount of transformation can give a\nmetrical result. I wish to prove, here, that a metrical coordinate\nsystem necessarily involves the straight line, and cannot, without a\nlogical fallacy, be set up on any other basis. The projective system\nof coordinates, as we saw, is entirely based on the straight line;\nbut the metrical system is more important, since its quantities\nembody actual information as to spatial magnitudes, which, in\nprojective Geometry, is not the case.\n\nIn the first place, a point's metrical coordinates constitute a\ncomplete quantitative definition of it; now a point can only be\ndefined, as we have seen, by its relations to other points, and\nthese relations can only be defined by means of the straight line.\nConsequently, any metrical system of coordinates must involve the\nstraight line, as the basis of its definitions of points.\n\nThis _\u00e0 priori_ argument, however, though I believe it to be quite\nsound, is not likely to carry conviction to any one persuaded of the\nopposite. Let us, therefore, examine metrical coordinate systems in\ndetail, and show, in each case, their dependence on the straight line.\n\nWe have already seen that the notion of distance is impossible\nwithout the straight line. We cannot, therefore, define our\ncoordinates in any of the ordinary ways, as the distances from three\nplanes, lines, points, spheres, or what not. Polar coordinates\nare impossible, since,--waiving the straightness of the radius\nvector--the length of the radius vector becomes unmeaning. Triangular\ncoordinates involve not only angles, which must in the limit be\nrectilinear, but straight lines, or at any rate some well-defined\ncurves. Now curves can only be metrically defined in two ways:\n_Either_ by relation to the straight line, as, _e.g._, by the\ncurvature at any point, _or_ by purely analytical equations, which\npresuppose an intelligible system of metrical coordinates. What\nmethods remain for assigning these arbitrary values to different\npoints? Nay, how are we to get any estimate of the difference--to\navoid the more special notion of distance--between two points?\nThe very notion of a point has become illusory. When we have a\ncoordinate system, we may define a point by its three coordinates;\nin the absence of such a system, we may define the notion of point\n_in general_ as the intersection of three surfaces or of two curves.\nHere we take surfaces and curves as notions which intuition makes\nplain, but if we wish them to give us a precise numerical definition\nof _particular_ points, we must specify the kind of surface or\ncurve to be used. Now this, as we have seen, is only possible when\nwe presuppose either the straight line, or a coordinate system. It\nfollows that every coordinate system presupposes the straight line,\nand is logically impossible without it.\n\n=177.= The above three axioms, we have seen, are _\u00e0 priori_ necessary\nto metrical Geometry. No others can be necessary, since metrical\nsystems, logically as unassailable as Euclid's, and dealing with\nspaces equally homogeneous and equally relational, have been\nconstructed by the metageometers, without the help of any other\naxioms. The remaining axioms of Euclidean Geometry--the axiom of\nparallels, the axiom that the number of dimensions is three, and\nEuclid's form of the axiom of the straight line (two straight lines\ncannot enclose a space)--are not essential to the possibility of\nmetrical Geometry, _i.e._, are not deducible from the fact that\na science of spatial magnitudes is possible. They are rather\nto be regarded as empirical laws, obtained, like the empirical\nlaws of other sciences, by actual investigation of the given\nsubject-matter--in this instance, experienced space.\n\n=178.= In summing up the distinctive argument of this Section,\nwe may give it a more general form, and discuss the conditions\nof measurement in any continuous manifold, _i.e._, the qualities\nnecessary to the manifold, in order that quantities in it may be\ndeterminable, not only as to the more or less, but as to the precise\n_how much_.\n\nMeasurement, we may say, is the application of number to continua,\nor, if we prefer it, the transformation of mere quantity into number\nof units. Using _quantity_ to denote the vague more or less, and\n_magnitude_ to denote the precise number of units, the problem of\nmeasurement may be defined as the transformation of quantity into\nmagnitude.\n\nNow a number, to begin with, is a whole consisting of smaller units,\nall of these units being qualitatively alike. In order, therefore,\nthat a continuous quantity may be expressible as a number, it must,\non the one hand, be itself a whole, and must, on the other hand,\nbe divisible into qualitatively similar parts. In the aspect of a\nwhole, the quantity is _intensive_; in the aspect of an aggregate of\nparts, it is _extensive_. A purely intensive quantity, therefore,\nis not numerable--a purely extensive quantity, if any such could\nbe imagined, would not be a single quantity at all, since it would\nhave to consist of wholly unsynthesized particulars. A measurable\nquantity, therefore, is a whole divisible into similar parts. But\na continuous quantity, if divisible at all, must be _infinitely_\ndivisible. For otherwise the points at which it could be divided\nwould form natural barriers, and so destroy its continuity. But\nfurther, it is not sufficient that there should be a possibility\nof division into mutually external parts; while the parts, to be\nperceptible as parts, must be mutually external, they must also, to\nbe knowable as _equal_ parts, be capable of overcoming their mutual\nexternality. For this, as we have seen, we require superposition,\nwhich involves Free Mobility and homogeneity--the absence of Free\nMobility in time, where all other requisites of measurement are\nfulfilled, renders direct measurement of time impossible. Hence\ninfinite divisibility, free mobility, and homogeneity are necessary\nfor the possibility of measurement in _any_ continuous manifold, and\nthese, as we have seen, are equivalent to our three axioms. These\naxioms are necessary, therefore, not only for spatial measurement,\nbut for all measurement. The only manifold given in experience, in\nwhich these conditions are satisfied, is space. All other exact\nmeasurement--as could be proved, I believe, for every separate\ncase--is effected, as we saw in the case of time, by reduction to\na spatial correlative. This explains the paramount importance, to\nexact science, of the mechanical view of nature, which reduces\nall phenomena to motions in time and space. For number is, of\nall conceptions, the easiest to operate with, and science seeks\neverywhere for an opportunity to apply it, but finds this opportunity\nonly by means of spatial equivalents to phenomena[177].\n\n=179.= We have now seen in what the _\u00e0 priori_ element of Geometry\nconsists. This _\u00e0 priori_ element may be defined as the axioms common\nto Euclidean and non-Euclidean spaces, as the axioms deducible from\nthe conception of a form of externality, or--in metrical Geometry--as\nthe axioms required for the possibility of measurement. It remains\nto discuss, in a final chapter, some questions of a more general\nphilosophic nature, in which we shall have to desert the firm ground\nof mathematics and enter on speculations which I put forward very\ntentatively, and with little faith in their ultimate validity. The\nchief questions for this final chapter will be two: (1) How is such\n_\u00e0 priori_ and purely logical necessity possible, as applied to an\nactually given subject-matter like space? (2) How can we remove the\ncontradictions which have haunted us in this chapter, arising out of\nthe relativity, infinite divisibility, and unbounded extension of\nspace? These two questions are forced upon us by the present chapter,\nbut as they open some of the fundamental problems of philosophy, it\nwould be rash to expect a conclusive or wholly satisfactory answer. A\nfew hints and suggestions may be hoped for, but a complete solution\ncould only be obtained from a complete philosophy, of which the\nprospects are far too slender to encourage a confident frame of mind.\n\n\nFOOTNOTES:\n\n[116] See infra, Axiom of Distance, in Sec. B. of this Chapter.\n\n[117] Thus on a cylinder, two geodesics, _e.g._ a generator and\na helix, may have any number of intersections--a very important\ndifference from the plane.\n\n[118] Cf. Cremona, Projective Geometry (Clarendon Press, 2nd ed.\n1893) p. 50: \"Most of the propositions in Euclid's Elements are\nmetrical, and it is not easy to find among them an example of a\npurely descriptive theorem.\"\n\n[119] Op. cit. p. 226.\n\n[120] Some ground for this choice will appear when we come to\nmetrical Geometry.\n\n[121] The straight line _\u03c3a_ denotes the straight line common to the\nplanes _\u03c3_ and _a_, the point _\u03c3a_ denotes the point common to the\nplane _\u03c3_ and the straight line _a_, and similarly for the rest of the\nnotation.\n\n[122] Cremona (op. cit. Chap. IX. p. 50) defines anharmonic ratio as\na metrical property which is unaltered by projection. This, however,\ndestroys the logical independence of projective Geometry, which can\nonly be maintained by a purely descriptive definition.\n\n[123] There is no corresponding property of _three_ points on a line,\nbecause they can be projectively transformed into any other three\npoints on the same line. See \u00a7 120.\n\n[124] Due to v. Staudt's \"Geometrie der Lage.\"\n\n[125] See Cremona, op. cit. Chapter VIII.\n\n[126] The corresponding definitions, for the two-dimensional manifold\nof lines through a point, follow by the principle of duality.\n\n[127] It is important to observe that this definition of the Point\nintroduces metrical ideas. Without metrical ideas, we saw, nothing\nappears to give the Point precedence of the straight line, or indeed\nto distinguish it conceptually from the straight line. A reference\nto quantity is therefore inevitable in defining the Point, if the\ndefinition is to be geometrical. A non-metrical definition would have\nto be also non-geometrical. See Chap. IV. \u00a7\u00a7 196-199.\n\n[128] \u00a7\u00a7 163-175.\n\n[129] On this axiom, however, compare \u00a7 131.\n\n[130] For the proof of this proposition, see Chap. III. Sec. B, Axiom\nof Dimensions.\n\n[131] The straight line and plane, in all discussions of general\nGeometry, are not necessarily Euclidean. They are simply figures\ndetermined, in general, by two and by three points respectively;\nwhether they conform to the axiom of parallels and to Euclid's form\nof the axiom of the straight line, is not to be considered in the\ngeneral definition.\n\n[132] That projective Geometry must have existential import, I shall\nattempt to prove in Chapter IV.\n\n[133] Logic, Book I. Chapter II.\n\n[134] Cf. Bradley's Logic, p. 63. It will be seen that the sense in\nwhich I have spoken of space as a principle of differentiation is not\nthe sense of a \"principle of individuation\" which Bradley objects to.\n\n[135] Chap. IV. \u00a7\u00a7 186-191.\n\n[136] Chap. IV. \u00a7 201 ff.\n\n[137] It is important to observe, however, that this way of regarding\nspatial relations is metrical; from the projective standpoint, the\nrelation between two points is the whole unbounded straight line on\nwhich they lie, and need not be regarded as divisible into parts or\nas built up of points.\n\n[138] \u00a7\u00a7 207, 208. Cf. Hegel, Naturphilosophie, \u00a7 254.\n\n[139] See Chap. IV. \u00a7\u00a7 196-199.\n\n[140] See a forthcoming article on \"The relations of number and\nquantity\" by the present writer in _Mind_, July, 1897.\n\n[141] Logic, Vol. II. Chap. VII. p. 211.\n\n[142] Real, as opposed to logical, diversity is throughout intended.\nDiverse aspects may coexist in a thing at one time and place, but two\ndiverse real things cannot so coexist.\n\n[143] On the insufficiency of time alone, see Chapter IV. \u00a7 191.\n\n[144] Geometrically, the axiom of the plane is, not that three\npoints determine a figure at all, which follows from the axiom of\nthe straight line, but that the straight line joining two casual\npoints of the plane lies wholly in the plane. This axiom requires a\nprojective method of constructing the plane, _i.e._ of finding all\nthe triads of points which determine the same projective figure as\nthe given triad. The required construction will be obtained if we\ncan find any projective figure determined by three points, and any\nprojective method of reaching other points which determine the same\nfigure.\n\n[Illustration]\n\nLet _O_, _P_, _Q_ be the three points whose projective relation\nis required. Then we have given us the three straight lines _PQ_,\n_QO_, _OP_. Metrically, the relation between these points is made\nup of the area, and the magnitude of the sides and angles, of the\ntriangle _OPQ_, just as the relation between two points is distance.\nBut projectively, the figure is unchanged when _P_ and _Q_ travel\nalong _OP_ and _OQ_, or when _OP_ and _OQ_ turn about _O_ in such a\nway as still to meet _PQ_. This is a result of the general principle\nof projective equivalence enunciated above (\u00a7\u00a7 108, 109). Hence the\nprojective relation between _O_, _P_, _Q_ is the same as that between\n_O_, _p_, _q_ or _O_, _P\u2032_, _Q\u2032_; that is, _p_, _q_ and _P\u2032_, _Q\u2032_\nlie in the plane _OPQ_. In this way, any number of points on the\nplane may be obtained, and by repeating the construction with fresh\ntriads, every point of the plane can be reached. We have to prove\nthat, when the plane is so constructed, the straight line joining any\ntwo points of the plane lies wholly in the plane.\n\nIt is evident, from the manner of construction, that any point of\n_PQ_, _OP_, _OQ_, _OP\u2032_ or _OQ\u2032_ lies in the plane. If we can prove\nthat any point of _pq_ lies in the plane, we shall have proved all\nthat is required, since _pq_ may be transformed, by successive\nrepetitions of the same construction, into any straight line joining\ntwo points of the plane. But we have seen that the same plane is\ndetermined by _O_, _p_, _q_ and by _O_, _P_, _Q_. The straight lines\n_PQ_, _pq_ have, therefore, the same relation to the plane. But _PQ_\nlies wholly in the plane; therefore _pq_ also lies wholly in the\nplane. Hence our axiom is proved.\n\n[145] A detailed proof has been given above, Chap. I. 3rd period. It\nis to be observed that any reference to infinitely distant elements\ninvolves metrical ideas.\n\n[146] Cf. Section A, \u00a7\u00a7 115-117.\n\n[147] Contrast Erdmann, op. cit. p. 138.\n\n[148] Cf. Erdmann, op. cit. p. 164.\n\n[149] Strictly speaking, this method is only applicable where the two\nmagnitudes are commensurable. But if we take infinite divisibility\nrigidly, the units can theoretically be taken so small as to obtain\nany required degree of approximation. The difficulty is the universal\none of applying to continua the essentially discrete conception of\nnumber.\n\n[150] Cf. Erdmann, op. cit. p. 50.\n\n[151] Also called the axiom of congruence. I have taken congruence to\nbe the _definition_ of spatial equality by superposition, and shall\ntherefore generally speak of the _axiom_ as Free Mobility.\n\n[152] For the sense in which these figures are to be regarded as\nmaterial, see criticism of Helmholtz, Chapter II. \u00a7\u00a7 69 ff.\n\n[153] Op. cit. p. 60.\n\n[154] The view of Helmholtz and Erdmann, that mechanical experience\nsuffices here, though geometrical experience fails us, has been\ndiscussed above, Chapter II. \u00a7\u00a7 73, 82.\n\n[155] Chapter II. \u00a7 81.\n\n[156] Chapter II. \u00a7 72.\n\n[157] Contrast Delb\u0153uf, L'ancienne et les nouvelles g\u00e9om\u00e9tries, II.\nRev. Phil. 1894, Vol. xxxvii. p. 354.\n\n[158] Prolegomena, \u00a7 13. See Vaihinger's Commentar, II. pp. 518-532\nesp. pp. 521-2. The above was Kant's whole purpose in 1768, but only\npart of his purpose in the Prolegomena, where the intuitive nature of\nspace was also to be proved.\n\n[159] On the subject of time measurement, cf. Bosanquet's Logic, Vol.\ni. pp. 178-183. Since time, in the above account, is measured by\nmotion, its measurement presupposes that of spatial magnitudes.\n\n[160] Cf. Stumpf. Ursprung der Raumvorstellung, p. 68.\n\n[161] As is Helmholtz's other axiom, that the possibility of\nsuperposition is independent of the course pursued in bringing it\nabout.\n\n[162] Cf. \u00a7\u00a7 129, 130.\n\n[163] This deduction is practically the same as that in Sec. A, but\nI have stated it here with more special reference to space and to\nmetrical Geometry.\n\n[164] The question: \"Relations to what?\" is a question involving\nmany difficulties. It will be touched on later in this chapter, and\nanswered, as far as possible, in the fourth chapter. For the present,\nin spite of the glaring circle involved, I shall take the relations\nas relations to other positions.\n\n[165] Wiss. Abh. Vol. II. p. 614.\n\n[166] Cp. Grassmann, Ausdehnungslehre von 1844, 2nd ed. p. XXIII.\n\n[167] Delb\u0153uf, it is true, speaks of Geometries with _m_/_n_\ndimensions, but gives no reference (Rev. Phil. T. xxxvi. p. 450).\n\n[168] In criticizing Erdmann, it will be remembered, we saw that Free\nMobility is a necessary property of his extents, though he does not\nregard it as such.\n\n[169] Cf. Riemann, Hypothesen welche der Geometrie zu Grunde liegen,\nGesammelte Werke, p. 266; also Erdmann, op. cit. p. 154.\n\n[170] This is subject, in spherical space, to the modification\npointed out below, in dealing with the exception to the axiom of the\nstraight line. See \u00a7\u00a7 168-171.\n\n[171] In speaking of distance at once as a quantity and as an\nintrinsic relation, I am anxious to guard against an apparent\ninconsistency. I have spoken of the judgment of quantity, throughout,\nas one of comparison; how, then, can a quantity be intrinsic? The\nreply is that, although measurement and the judgment of quantity\nexpress the result of comparison, yet the terms compared must exist\nbefore the comparison; in this case, the terms compared in measuring\ndistances, _i.e._ in comparing them _inter se_, are intrinsic\nrelations between points. Thus, although the _measurement_ of\ndistance involves a reference to other distances, and its expression\nas a magnitude requires such a reference, yet its existence does not\ndepend on any external reference, but exclusively on the two points\nwhose distance it is.\n\n[172] See the end of the argument on Free Mobility, \u00a7 155 ff.\n\n[173] In Frischauf's \"Absolute Geometrie nach Johann Bolyai,\" Anhang,\nthere is a series of definitions, starting from the sphere, as the\nlocus of congruent point-pairs when one point of the pair is fixed,\nand hence obtaining the circle and the straight line. From the above\nit follows, that the sphere so defined already involves a curve\nbetween the points of the point-pair, by which various point-pairs\ncan be known as congruent; and it will appear, as we proceed, that\nthis curve must be a straight line. Frischauf's definition by means\nof the sphere involves, therefore, a vicious circle, since the sphere\npresupposes the straight line, as the test of congruent point-pairs.\n\n[174] Nor in any argument which, like those of projective Geometry,\navoids the notion of magnitude or distance altogether. It follows\nthat the propositions of projective Geometry apply, without reserve,\nto spherical space, since the exception to the axiom of the straight\nline arises only on metrical ground.\n\n[175] Psychology, Vol. II. pp. 149-150.\n\n[176] This step in the argument has been put very briefly, since it\nis a mere repetition of the corresponding argument in Section A, and\nis inserted here only for the sake of logical completeness. See \u00a7 137\nff.\n\n[177] Cf. Hannequin, Essai critique sur l'hypoth\u00e8se des atomes,\nParis, 1895, passim.\n\n\n\n\nCHAPTER IV.\n\nPHILOSOPHICAL CONSEQUENCES.\n\n\n=180.= In the present chapter, we have to discuss two questions\nwhich, though scarcely geometrical, are of fundamental importance\nto the theory of Geometry propounded above. The first of these\nquestions is this: What relation can a purely logical and deductive\nproof, like that from the nature of a form of externality, bear to an\nexperienced subject-matter such as space? You have merely framed a\ngeneral conception, I may be told, containing space as a particular\nspecies, and you have then shown, what should have been obvious from\nthe beginning, that this general conception contained some of the\nattributes of space. But what ground does this give for regarding\nthese attributes as _\u00e0 priori_? The conception Mammal has some of the\nattributes of a horse; but are these attributes therefore _\u00e0 priori_\nadjectives of the horse? The answer to this obvious objection is so\ndifficult, and involves so much general philosophy, that I have kept\nit for a final chapter, in order not to interrupt the argument on\nspecially geometrical topics.\n\n=181.= I have already indicated, in general terms, the ground for\nregarding as _\u00e0 priori_ the properties of any form of externality.\nThis ground is transcendental, _i.e._ it is to be found in the\nconditions required for the possibility of experience. The form of\nexternality, like Riemann's manifolds, is a general class-conception,\nincluding time as well as Euclidean and non-Euclidean spaces. It\nis not motived, however, like the manifolds, by a _quantitative_\nresemblance to space, but by the fact that it fulfils, if it has\nmore than one dimension, all those functions which, in our actual\nworld, are fulfilled by space. But a form of externality, in order\nto accomplish this, must be, not a mere conception, but an actually\nexperienced intuition. Hence the conception of such a form is the\ngeneral _conception_, containing under it every logically possible\n_intuition_ which can fulfil the function actually fulfilled by\nspace. And this function is, to render possible experience of\ndiverse but interrelated things. Some form in sense-perception,\nthen, whose conception is included under our form of externality,\nis _\u00e0 priori_ necessary to experience of diversity in relation, and\nwithout experience of this, we should, as modern logic shows, have no\nexperience at all. This still leaves untouched the relation of the\n_\u00e0 priori_ to the subjective: the form of externality is necessary\nto experience, but is not, _on that account_, to be declared purely\nsubjective. Of course, necessity for experience can only arise from\nthe nature of the mind which experiences; but it does not follow that\nthe necessary conditions could be fulfilled, unless the objective\nworld had certain properties. The _ground_ of necessity, we may\nsafely say, arises from the mind; but it by no means follows that\nthe _truth_ of what is necessary depends only on the constitution of\nthe mind. Where this is not the case, our conclusion, when a piece\nof knowledge has been declared _\u00e0 priori_, can only be: Owing to the\nconstitution of the _mind_, experience will be impossible unless the\n_world_ accepts certain adjectives.\n\nSuch, in outline, will be the argument of the first half of this\nchapter, and such will be the justification for regarding as _\u00e0\npriori_ those axioms of Geometry, which were deduced above from the\nconception of a form of externality. For these axioms, and these\nonly, are necessarily true of any world in which experience is\npossible.\n\n=182[178].= The view suggested has, obviously, much in common with\nthat of the Transcendental Aesthetic. Indeed the whole of it, I\nbelieve, can be obtained by a certain limitation and interpretation\nof Kant's classic arguments. But as it differs, in many important\npoints, from the conclusions aimed at by Kant, and as the agreement\nmay easily seem greater than it is, I will begin by a brief\ncomparison, and endeavour, by reference to authoritative criticisms,\nto establish the legitimacy of my divergence from him.\n\n=183.= In the first place, the psychological element is much larger\nin Kant's thesis than in mine. I shall contend, it is true, that a\nform of externality, if it is to do its work, must not be a mere\nconception or a mere inference, but must be a given element in\nsense-perception--not, of course, originally given in isolation,\nbut discoverable, through analysis, by attention to the object of\nsense-perception[179]. But Kant contended, not only that this element\nis given, but also that it is subjective. Space, for him, is, on the\none hand, not conceptual, but on the other hand, not sensational.\nIt forms, for him, no part of the data of sense, but is added by a\nsubjective intuition, which he regards as not only logically, but\npsychologically, prior to objects in space[180].\n\nThis part of Kant's argument is wholly irrelevant for us. Whether a\nform of externality be given in sense, or in a pure intuition, is for\nus unimportant, since we neglect the question as to the connection\nof the _\u00e0 priori_ and the subjective; while the temporal priority of\nspace to objects in it has been generally recognized as irrelevant\nto Epistemology, and has often been regarded as forming no part of\nKant's thesis[181]. If we call intuitional whatever is given in\nsense-perception, then we may contend that a form of externality must\nbe intuitional; but whether it is a pure intuition, in Kant's sense,\nor not, is irrelevant to us, as is its priority to the objects in it.\n\nThat the non-sensational nature of space is no essential part of\nKant's _logical_ teaching, appears from an examination of his\nargument. He has made, in the introduction, the purely logical\ndistinction of matter and form, but has given to this distinction,\nin the very moment of suggesting it, a psychological implication.\nThis he does by the assertion that the form, in which the matter\nof sensations is ordered, cannot itself be sensational. From this\nassumption it follows, of course, that space cannot be sensational.\nBut the assumption is totally unsupported by argument, being set\nforth, apparently, as a self-evident axiom; it has been severely\ncriticized by Stumpf[182] and others[183], and has been described by\nVaihinger as a fatal _petitio principii_[184]; it is irrelevant to\nthe logical argument, when this argument is separated, as we have\nseparated it, from all connection with psychological subjectivity;\nand finally, it leaves us a prey to psychological theories of space,\nwhich have seemed, of late, but little favourable to the pure Kantian\ndoctrine.\n\n=184.= We have a right, therefore, in an epistemological inquiry, to\nneglect Kant's psychological teaching--in so far, at any rate, as it\ndistinguishes spatial intuition from sensation--and attend rather to\nthe logical aspect alone. That part of his psychological teaching,\nwhich maintains that space is not a mere conception, is, with certain\nlimitations, sufficiently evident as applied to actual space; but for\nus, it must be transformed into a much more difficult thesis, namely,\nthat _no_ form of externality, which renders experience of diversity\nin relation possible, can be merely conceptual. This question, to\nwhich we must return later, is no longer psychological, but belongs\nwholly to Epistemology.\n\n=185.= What, then, remains the kernel, for our purposes, of Kant's\nfirst argument for the apriority of space? His argument, in the form\nin which he gave it, is concerned with the eccentric projection\nof sensations. In order that I may refer sensations, he says,\nto something outside myself, I must already have the subjective\nspace-form in the mind. In this shape, as Vaihinger points out\n(Commentar, II. pp. 69, 165), the argument rests on a _petitio\nprincipii_, for only if sensations are necessarily non-spatial does\ntheir projection demand a subjective space-form. But, further, is the\nlogical apriority of space concerned with the externality of things\nto ourselves?\n\nSpace _seems_ to perform two functions: on the one hand, it reveals\nthings, by the eccentric projection of sensations, as external\nto the self, while, on the other hand, it reveals simultaneously\npresented things as mutually external. These two functions, though\noften treated as coordinate and almost equivalent[185], seem to me\nwidely different. Before we discuss the apriority of space, we must\ncarefully distinguish, I think, between these two functions, and\ndecide which of them we are to argue about.\n\nNow externality to the Self, it would seem, must necessarily raise\nthe whole question of the nature and limits of the Ego, and what is\nmore, it cannot be derived from spatial presentation, unless we give\nthe Self a definite position in space. But things acquire a position\nin space only when they can appear in sense-perception; we are\nforced, therefore, if we adopt this view of the function of space, to\nregard the Self as a phenomenon presented to sense-perception. But\nthis reduces externality to the Self to externality to the body. The\nbody, however, is a presented object like any other, and externality\nof objects to it is, therefore, a special case of the mutual\nexternality of presented things. Hence we cannot regard space as\ngiving, primarily at any rate, externality to the Self, but only the\nmutual externality of the things presented to sense-perception[186].\n\n=186.= This, then, is the kind of externality we are to expect\nfrom space, and our question must be: Would the existence of\ndiverse but interrelated things be unknowable, if there were not,\nin sense-perception, some form of externality? This is the crucial\nquestion, on which turns the apriority of our form, and hence of the\nnecessary axioms of Geometry.\n\n=187.= The converse argument to mine, the argument from the\nspatio-temporal element in perception to a world of interrelated\nbut diverse things, is developed at length in Bradley's Logic. It\nis put briefly in the following sentence (p. 44, note): \"If space\nand time are continuous, and if all appearance must occupy some time\nor space--and it is not hard to support both these _theses_--we can\nat once proceed to the conclusion, no mere particular exists. Every\nphenomenon will exist in more times or spaces than one; and against\nthat diversity will be itself an universal[187].\" The importance of\nthis fact appears, when we consider that, if any _mere_ particular\nexisted, all judgment and inference as to that particular would be\nimpossible, since all judgment and inference necessarily operate\nby means of universal. But all reality is constructed from the\n_This_ of immediate presentation, from which judgment and inference\nnecessarily spring. Owing, however, to the continuity and relativity\nof space and time, no _This_ can be regarded either as simple or\nas self-subsistent. Every _This_, on the one hand, can be analyzed\ninto _Thises_, and on the other hand, is found to be necessarily\nrelated to other things, outside the limits of the given object of\nsense-perception. This function of space and time is presupposed in\nthe following statement from Bosanquet's Logic (Vol. I. pp. 77-78):\n\"Reality is given for me _in_ present sensuous perception, and _in_\nthe immediate feeling of my own sentient existence that goes with\nit. The real world, as a definite organized system, is _for me_ an\nextension of this present sensation and self feeling by means of\njudgment, and it is the essence of judgment to effect and sustain\nsuch an extension.... The subject in every judgment of Perception\nis some given spot or point in sensuous contact with the percipient\nself. But, as all reality is continuous, the subject is not _merely_\nthis given spot or point.\"\n\n=188.= This doctrine of Bradley and Bosanquet is the converse of\nthe epistemological doctrine I have to advocate. Owing to the\ncontinuity and relativity of space and time, they say, we are able\nto construct a systematic world, by judgment and inference, out of\nthat fragmentary and yet necessarily complex existence which is\ngiven in sense-perception. My contention is, conversely, that since\nall knowledge is necessarily derived by an extension of the _This_\nof sense-perception, and since such extension is only possible if\nthe _This_ has that fragmentary and yet complex character conferred\nby a form of externality, therefore some form of externality,\ngiven with the _This_, is essential to all knowledge, and is thus\nlogically _\u00e0 priori_. Bradley's argument, if sound, already proves\nthis contention; for while, on the one hand, he uses no properties of\nspace and time but those which belong to every form of externality,\nhe proves, on the other hand, that judgment and inference require\nthe _This_ to be neither single nor self-subsistent. But I will\nendeavour, since the point is of fundamental importance, to\nreproduce the proof, in a form more suited than Bradley's to the\nepistemological question.\n\n=189.= The essence of my contention is that, if experience is to be\npossible, every sensational _This_ must, when attended to, be found,\non the one hand, resolvable into _Thises_, and on the other hand\ndependent, for some of its adjectives, on external reference. The\nsecond of these theses follows from the first, for if we take one\nof the _Thises_ contained in the first _This_, we get a new _This_\nnecessarily related to the other _Thises_ which make up the original\n_This_. I may, therefore, confine myself to the first proposition,\nwhich affirms that the object of perception must contain a diversity,\nnot only of conceptual content, but of existence, and that this can\nonly be known if sense-perception contains, as an element, some form\nof externality.\n\nMy premiss, in this argument, is that all knowledge involves a\nrecognition of diversity in relation, or, if we prefer it, of\nidentity in difference. This premiss I accept from Logic, as\nresulting from the analysis of judgment and inference. To prove\nsuch a premiss, would require a treatise on Logic; I must refer the\nreader, therefore, to the works of Bradley and Bosanquet on the\nsubject. It follows at once, from my premiss, that knowledge would be\nimpossible, unless the object of attention could be complex, _i.e._\nnot a _mere_ particular. Now could the mental object--_i.e._, in this\nconnection, the object of a cognition--be complex, if the object of\nimmediate perception were always simple?\n\n=190.= We might be inclined, at first sight, to answer this question\naffirmatively. But several difficulties, I think, would prevent\nsuch an answer. In the first place, knowledge must start from\nperception. Hence, either we could have no knowledge except of our\npresent perception, or else we must be able to contrast and compare\nit with some other perception. Now in the first case, since the\npresent perception, by hypothesis, is a mere particular, knowledge\nof it is impossible, according to our premiss. But in the second\ncase, the other perception, with which we compare our first, must\nhave occurred at some other time, and with time, we have at once a\nform of externality. But what is more, our present perception is\nno longer a mere particular. For the power of comparing it with\nanother perception involves a point of identity between the two, and\nthus renders both complex. Moreover, time must be continuous, and\nthe present, as Bradley points out, is no mere point of time[188].\nThus our present perception contains the complexity involved in\nduration throughout the specious present: its mere particularity\nand its simplicity are lost. Its self-subsistence is also lost, for\nbeyond the specious present, lie the past and the future, to which\nour present perception thus unavoidably refers us. Time at least,\ntherefore, is essential to that identity in difference, which all\nknowledge postulates.\n\n=191.= But we have derived, from all this, no ground for affirming a\nmultiplicity of real things, or a form of externality of more than\none dimension, which, we saw, was necessary for the truth of two out\nof our three axioms. This brings us to the question: Have we enough,\nwith time alone as a form of externality, for the possibility of\nknowledge?\n\nThis question we must, I think, answer in the negative. With time\nalone, we have seen, our presented object must be complex, but its\ncomplexity must, if I may use such a phrase, be merely adjectival.\nWithout a second form of externality, only one thing can be given at\none moment[189], and this one thing, therefore, must constitute the\nwhole of our world. The object of past perception must--since our one\nthing has nothing external to it, by which it could be created or\ndestroyed--be regarded as the same thing in a different state. The\ncomplexity, therefore, will lie only in the changing states of our\none thing--it will be adjectival, not substantival. Moreover we have\nthe following dilemma: Either the one thing must be ourselves, or\nelse self-consciousness could never arise. But the chief difficulty\nof such a world would lie in the changes of the thing. What could\ncause these changes, since we should know of nothing external to our\nthing? It would be like a Leibnitzian monad, without any God outside\nit to prearrange its changes. Causality, in such a world, could not\nbe applied, and change would be wholly inexplicable.\n\nHence we require also the possibility of a diversity of\nsimultaneously existing things, not merely of successive adjectives;\nand this, we have seen, cannot be given by time alone, but only by\na form of externality for simultaneous parts of one presentation.\nWe could never, in other words, infer the existence of diverse but\ninterrelated things, unless the object of sense-perception could have\nsubstantival complexity, and for such complexity we require a form\nof externality other than time. Such a form, moreover, as was shown\nin Chapter III., Section A (\u00a7 135), can only fulfil its functions\nif it has more than one dimension. In our actual world, this form\nis given by space; in any world, knowable to beings with our laws\nof thought, some such form, as we have now seen, must be given in\nsense-perception.\n\nThis argument may be briefly summed up, by assuming the doctrine\nof Bradley, that all knowledge is obtained by inference from the\n_This_ of sense-perception. For, if this be so, the _This_--in\norder that inference, which depends on identity in difference, may\nbe possible at all--must itself be complex, and must, on analysis,\nreveal adjectives having a reference beyond itself. But this, as was\nshown above, can only happen by means of a form of externality. This\nestablishes the \u00e0 priori axioms of Geometry, as necessarily having\nexistential import and validity in any intelligible world.\n\n=192.= The above argument, I hope, has explained why I hold it\npossible to deduce, from a mere conception like that of a form\nof externality, the logical apriority of certain axioms as to\nexperienced space. The Kantian argument--which was correct, if our\nreasoning has been sound, in asserting that real diversity, in our\nactual world, could only be known by the help of space--was only\nmistaken, so far as its purely logical scope extends, in overlooking\nthe possibility of other forms of externality, which could, if they\nexisted, perform the same task with equal efficiency. In so far as\nspace differs, therefore, from these other conceptions of possible\nintuitional forms, it is a mere experienced fact, while in so far as\nits properties are those which all such forms must have, it is _\u00e0\npriori_ necessary to the possibility of experience.\n\nI cannot hope, however, that no difficulty will remain, for the\nreader, in such a deduction, from abstract conceptions, of the\nproperties of an actual _datum_ in sense-perception. Let us consider,\nfor example, such a property as impenetrability. To suppose two\nthings simultaneously in the same position in a form of externality,\nis a logical contradiction; but can we say as much of actual space\nand time? Is not the impossibility, here, a matter of experience\nrather than of logic? Not if the above argument has been sound,\nI reply. For in that case, we infer real diversity, _i.e._ the\nexistence of different things, only from difference of position in\nspace or time. It follows, that to suppose two things in the same\npoint of space and time, is still a logical contradiction: not\nbecause we have constructed the data of sense out of logic, but\nbecause logic is dependent, as regards its application, on the nature\nof these data. This instance illustrates, what I am anxious to make\nplain, that my argument has not attempted to construct the living\nwealth of sense-perception out of \"bloodless categories,\" but only to\npoint out that, unless sense-perception contained a certain element,\nthese categories would be powerless to grapple with it.\n\n=193.= How we are to account for the fortunate realization of these\nrequirements--whether by a pre-established harmony, by Darwinian\nadaptation to our environment, by the subjectivity of the necessary\nelement in sense-perception, or by a fundamental identity and unity\nbetween ourselves and the rest of reality--is a further question,\nbelonging rather to metaphysics than to our present line of argument.\nThe _\u00e0 priori_, we have said throughout, is that which is necessary\nfor the possibility of experience, and in this we have a purely\nlogical criterion, giving results which only Logic and Epistemology\ncan prove or disprove. What is subjective in experience, on the\ncontrary, is primarily a question for psychology, and should be\ndecided on psychological grounds alone. When these two questions have\nbeen separately answered, but not till then, we may frame theories as\nto the connection of the _\u00e0 priori_ and the subjective; to allow such\ntheories to influence our decision, on either of the two previous\nquestions, is liable, surely, to confuse the issue, and prevent a\nclear discrimination between fundamentally different points of view.\n\n=194.= I come now to the second question with which this chapter\nhas to deal, the question, namely: What are we to do with the\ncontradictions which obtruded themselves in Chapter III., whenever we\ncame to a point which seemed fundamental? I shall treat this question\nbriefly, as I have little to add to answers with which we are all\nfamiliar. I have only to prove, first, that the contradictions are\ninevitable, and therefore form no objection to my argument; secondly,\nthat the first step in removing them is to restore the notion of\nmatter, as that which, in the data of sense-perception, is localized\nand interrelated in space.\n\n=195.= The contradictions in space are an ancient theme--as ancient,\nin fact, as Zeno's refutation of motion. They are, roughly, of two\nkinds, though the two kinds cannot be sharply divided. There are\nthe contradictions inherent in the notion of the continuum, and the\ncontradictions which spring from the fact that space, while it must,\nto be knowable, be pure relativity, must also, it would seem, since\nit is immediately experienced, be something more than mere relations.\nThe first class of contradictions has been encountered more\nfrequently in this essay, and is also, I think, the more definite,\nand the more important for our present purpose. I doubt, however,\nwhether the two classes are really distinct; for any continuum,\nI believe, in which the elements are not data, but intellectual\nconstructions resulting from analysis, can be shown to have the same\nrelational and yet not wholly relational character as belongs to\nspace.\n\nThe three following contradictions, which I shall discuss\nsuccessively, seem to me the most prominent in a theory of Geometry.\n\n(1) Though the parts of space are intuitively distinguished, no\nconception is adequate to differentiate them. Hence arises a\nvain search for elements, by which the differentiation could be\naccomplished, and for a whole, of which the parts of space are to be\ncomponents. Thus we get the point, or zero extension, as the spatial\nelement, and an infinite regress or a vicious circle in the search\nfor a whole.\n\n(2) All positions being relative, positions can only be defined by\ntheir relations, _i.e._ by the straight lines or planes through\nthem; but straight lines and planes, being all qualitatively similar,\ncan only be defined by the positions they relate. Hence, again, we\nget a vicious circle.\n\n(3) Spatial figures must be regarded as relations. But a relation\nis necessarily indivisible, while spatial figures are necessarily\ndivisible _ad infinitum_.\n\n=196.= (1) _Points._ The antinomy of the point--which arises wherever\na continuum is given, and elements have to be sought in it--is\nfundamental to Geometry. It has been given, perhaps unintentionally,\nby Veronese as the first axiom, in the form: \"There are different\npoints. All points are identical\" (_op. cit._ p. 226). We saw, in\ndiscussing projective Geometry, that straight lines and planes must\nbe regarded, on the one hand as relations between points, and on the\nother hand as made up of points[190]. We saw again, in dealing with\nmeasurement, how space must be regarded as infinitely divisible,\nand yet as mere relativity. But what is divisible and consists of\nparts, as space does, must lead at last, by continued analysis, to\na simple and unanalyzable part, as the unit of differentiation. For\nwhatever can be divided, and has parts, possesses some thinghood,\nand must, therefore, contain two ultimate units, the whole namely,\nand the smallest element possessing thinghood. But in space this is\nnotoriously not the case. After hypostatizing space, as Geometry\nis compelled to do, the mind imperatively demands elements, and\ninsists on having them, whether possible or not. Of this demand,\nall the geometrical applications of the infinitesimal calculus are\nevidence[191]. But what sort of elements do we thus obtain? Analysis,\nbeing unable to find any earlier halting-place, finds its elements\nin points, that is, in zero quanta of space. Such a conception is\na palpable contradiction, only rendered tolerable by its necessity\nand familiarity. A point must be spatial, otherwise it would not\nfulfil the function of a spatial element; but again it must contain\nno space, for any finite extension is capable of further analysis.\nPoints can never be given in intuition, which has no concern with the\ninfinitesimal: they are a purely conceptual construction, arising\nout of the need of terms between which spatial relations can hold.\nIf space be more than relativity, spatial relations must involve\nspatial relata; but no relata appear, until we have analyzed our\nspatial data down to nothing. The contradictory notion of the point,\nas a thing in space without spatial magnitude, is the only outcome\nof our search for spatial relata. This _reductio ad absurdum_ surely\nsuffices, by itself, to prove the essential relativity of space.\n\n=197.= Thus Geometry is forced, since it wishes to regard space\nas independent, to hypostatize its abstractions, and therefore to\ninvent a self-contradictory notion as the spatial element. A similar\nabsurdity appears, even more obviously, in the notion of a whole of\nspace. The antinomy may, therefore, be stated thus: Space, as we have\nseen throughout, must, if knowledge of it is to be possible, be mere\nrelativity; but it must also, if _independent_ knowledge of it, such\nas Geometry seeks, is to be possible, be something more than mere\nrelativity, since it is divisible and has parts. But we saw, in Chap.\nIII., Section A (\u00a7 133) that knowledge of a form of externality must\nbe logically independent of the particular matter filling the form.\nHow then are we to extricate ourselves from this dilemma?\n\nThe only way, I think, is, not to make Geometry dependent on\nPhysics, which we have seen to be erroneous[192], but to give every\ngeometrical proposition a certain reference to matter in general. And\nat this point an important distinction must be made. We have hitherto\nspoken of space as relational, and of spatial figures as relations.\nBut space, it would seem, is rather relativity than relations--itself\nnot a relation, it gives the bare possibility of relations between\ndiverse things[193]. As applied to a spatial figure, which can only\narise by a differentiation of space, and hence by the introduction\nof some differentiating matter, the word relation is, perhaps, less\nmisleading than any other; as applied to empty undifferentiated\nspace, it seems by no means an accurate description.\n\nBut a bare possibility cannot exist, or be given in sense-perception!\nWhat becomes, then, of the arguments of the first part of this\nchapter? I reply, it is not empty space, but spatial figures, which\nsense-perception reveals, and spatial figures, as we have just seen,\ninvolve a differentiation of space, and therefore a reference to the\nmatter which is in space. It is spatial figures, also, and not empty\nspace, with which Geometry has to deal. The antinomy discussed above\narises then--so it would seem--from the attempt to deal with empty\nspace, rather than with spatial figures and the matter to which they\nnecessarily refer.\n\n=198.= Let us see whether, by this change, we can overcome the\nantinomy of the point. Spatial figures, we shall now say, are\nrelations between the matter which differentiates empty space. Their\ndivisibility, which seemed to contradict their relational character,\nmay be explained in two ways: first, as holding of the figures\nconsidered as parts of empty space, which is itself not a relation;\nsecond, as denoting the possibility of continuous change in the\nrelation expressed by the spatial figure. These two ways are, at\nbottom, the same; for empty space is a possibility of relations, and\nthe figure, when viewed in connection with empty space, thus becomes\na _possible_ relation, with which other possible relations may be\ncontrasted or compared. But the second way of regarding divisibility\nis the better way, since it introduces a reference to the matter\nwhich differentiates empty space, without which, spatial figures,\nand therefore Geometry, could not exist. It is empty space, then--so\nwe must conclude--which gives rise to the antinomy in question; for\nempty space is a bare possibility of relations, undifferentiated and\nhomogeneous, and thus wholly destitute of parts or of thinghood.\nTo speak of parts of a possibility is nonsense; the parts and\ndifferentiations arise only through a reference to the matter which\nis differentiated in space.\n\n=199.= But what nature must we ascribe to this matter, which is\nto be involved in all geometrical propositions? In criticizing\nHelmholtz (Chap. II. \u00a7 73), it may be remembered, we decided that\nGeometry refers to a peculiar and abstract kind of matter, which is\nnot regarded as possessing any causal qualities, as exerting or as\nsubject to the action of forces. And this is the matter, I think,\nwhich we require for the needs of the moment. Not that we affirm,\nof course, that actual matter can be destitute of the properties\nwith which Physics is cognizant, but that we abstract from these\nproperties, as being irrelevant to Geometry. All that we require, for\nour immediate purpose, is a subject of that diversity which space\nrenders possible, or terms for those relations by which empty space,\nif space is to be studied at all, must be differentiated. But how\nmust a matter, which is to fulfil this function, be regarded?\n\nEmpty space, we have said, is a possibility of diversity in\nrelation, but spatial figures, with which Geometry necessarily\ndeals, are the actual relations rendered possible by empty space.\nOur matter, therefore, must supply the terms for these relations.\nIt must be differentiated, since such differentiation, as we have\nseen, is the special work of space. We must find, therefore, in\nour matter, that unit of differentiation, or atom[194], which in\nspace we could not find. This atom must be simple, _i.e._ it must\ncontain no real diversity; it must be a _This_ not resolvable into\n_Thises_. Being simple, it can contain no relations within itself,\nand consequently, since spatial figures are mere relations, it cannot\nappear as a spatial figure; for every spatial figure involves some\ndiversity of matter. But our atom must have spatial relations with\nother atoms, since to supply terms for these relations is its only\nfunction. It is also capable of having these relations, since it is\ndifferentiated from other atoms. Hence we obtain an unextended term\nfor spatial relations, precisely of the kind we require. So long as\nwe sought this term without reference to anything more than space,\nthe self-contradictory notion of the point was the only outcome\nof our search; but now that we allow a reference to the matter\ndifferentiated by space, we find at once the term which was needed,\nnamely, a non-spatial simple element, with spatial relations to other\nelements. To Geometry such a term will appear, owing to its spatial\nrelations, as a point; but the contradiction of the point, as we now\nsee, is a result only of the undue abstraction with which Geometry\ndeals.\n\n=200.= (2) _The circle in the definition of straight lines and\nplanes._ This difficulty need not long detain us, since we have\nalready, with the material atom, broken through the relativity\nwhich caused our circle. Straight lines, in the purely geometrical\nprocedure, are defined only by points, and points only by straight\nlines. But points, now, are replaced by material atoms: the duality\nof points and lines, therefore, has disappeared, and the straight\nline may be defined as the spatial relation between two unextended\natoms. These atoms have spatial adjectives, derived from their\nrelations to other atoms; but they have no _intrinsic_ spatial\nadjectives, such as could belong to them if they had extension or\nfigure. Thus straight lines and planes are the true spatial units,\nand points result only from the attempt to find, within space, those\nterms for spatial relations which exist only in a more than spatial\nmatter. Straight lines, planes and volumes are the spatial relations\nbetween two, three or four unextended atoms, and points are a merely\nconvenient geometrical fiction, by which possible atoms are replaced.\nFor, since space, as we saw, is a possibility, Geometry deals not\nwith actually realized spatial relations, but with the whole scheme\nof possible relations.\n\n=201.= (3) _Space is at once relational and more than relational._\nWe have already touched on the question how far space is other than\nrelations, but as this question is quite fundamental, as _relation_\nis an ambiguous and dangerous word, as I have made constant use of\nthe relativity of space without attempting to define a relation, it\nwill be necessary to discuss this antinomy at length.\n\n=202.= Now for this discussion it is essential to distinguish clearly\nbetween empty space and spatial figures. Empty space, as a form\nof externality, is not actual relations, but the possibility of\nrelations: if we ascribe existential import to it, as the ground,\nin reality, of all diversity in relation, we at once have space as\nsomething not itself relations, though giving the possibility of all\nrelations. In this sense, space is to be distinguished from spatial\norder. Spatial order, it may be said, presupposes space, as that in\nwhich this order is possible. Thus Stumpf says[195]: \"There is no\norder or relation without a positive absolute content, underlying\nit, and making it possible to order anything in this manner. Why and\nhow should we otherwise distinguish one order from another?... To\ndistinguish different orders from one another, we must everywhere\nrecognize a particular absolute content, in relation to which the\norder takes place. And so space, too, is not a mere order, but just\nthat by which the spatial order, side-by-sideness (_Nebeneinander_)\ndistinguishes itself from the rest.\"\n\nMay we not, then, resolve the antinomy very simply, by a reference to\nthis ambiguity of space? Bradley contends (Appearance and Reality,\npp. 36-7) that, on the one hand, space has parts, and is therefore\nnot mere relations, while on the other hand, when we try to say what\nthese parts are, we find them after all to be mere relations. But\ncannot the space which has parts be regarded as empty space, Stumpf's\nabsolute underlying content, which is not mere relations, while the\nparts, in so far as they turn out to be mere relations, are those\nrelations which constitute spatial order, not empty space? If this\ncan be maintained, the antinomy no longer exists.\n\nBut such an explanation, though I believe it to be a first step\ntowards a solution, will, I fear, itself demand almost as much\nexplanation as the original difficulty. For the connection of empty\nspace with spatial order is itself a question full of difficulty, to\nbe answered only after much labour.\n\n=203.= Let us consider what this empty space is. (I speak of \"empty\"\nspace without necessarily implying the absence of matter, but only\nto denote a space which is not a mere order of material things.)\nStumpf regards it as given in sense; Kant, in the last two arguments\nof his metaphysical deduction, argues that it is an intuition, not\na concept, and must be known before spatial order becomes possible.\nI wish to maintain, on the contrary, that it is wholly conceptual;\nthat space is given only as spatial order; that spatial relations,\nbeing given, appear as more than mere relations, and so become\nhypostatized; that when hypostatized, the whole collection of\nthem is regarded as contained in empty space; but that this empty\nspace itself, if it means more than the logical possibility of\nspace-relations, is an unnecessary and self-contradictory assumption.\nLet us begin by considering Kant's arguments on this point.\n\nLeibnitz had affirmed that space was only relations, while Newton\nhad maintained the objective reality of absolute space. Kant adopted\na middle course: he asserted absolute space, but regarded it as\npurely subjective. The assertion of absolute space is the object of\nhis second argument; for if space were mere relations between things,\nit would necessarily disappear with the disappearance of the things\nin it; but this the second argument denies[196]. Now spatial order\nobviously does disappear with matter, but absolute or empty space may\nbe supposed to remain. It is this, then, which Kant is arguing about,\nand it is this which he affirms to be a pure intuition, necessarily\npresupposed by spatial order[197].\n\n=204.= But can we agree in regarding empty space, the \"infinite\ngiven whole,\" as really given? Must we not, in spite of Kant's\nargument, regard it as wholly conceptual? It is not required, in\nthe first place, by the argument of the first half of this chapter,\nwhich required only that every _This_ of sense-perception should\nbe resolvable into _Thises_, and thus involved only an order among\n_Thises_, not anything given originally without reference to them\nat all. In the second place, Kant's two arguments[198] designed to\nprove that empty space is not conceptual, are inadequate to their\npurpose. The argument that the parts of space are not contained\n_under_ it, but _in_ it, proves certainly that space is not a general\nconception, of which spatial figures are the instances; but it by\nno means follows that empty space is not a conception. Empty space\nis undifferentiated and homogeneous; parts of space, or spatial\nfigures, arise only by reference to some differentiating matter, and\nthus belong rather to spatial order than to empty space. If empty\nspace be the pre-condition of spatial order, we cannot expect it\nto be connected with spatial relations as genus with species. But\nempty space may nevertheless be a universal conception; it may be\nrelated to spatial order as the state to the citizens. These are not\ninstances of the state, but are contained in it; they also, in a\nsense, presuppose it, for a man can only become a citizen by being\nrelated to other citizens in a state[199].\n\nThe uniqueness of space, again, seems hardly a valid argument for\nits intuitional nature; to regard it as an argument implies, indeed,\nthat all conceptions are abstracted from a series of instances--a\nview which has been criticized in Chapter II. (\u00a7 77), and need not be\nfurther discussed here[200]. There is no ground, therefore, in Kant's\ntwo arguments for the intuitional nature of empty space, which can be\nmaintained against criticism.\n\n=205.= Another ground for condemning empty space is to be found in\nthe mathematical antinomies. For it is no solution, as Lotze points\nout (Metaphysik, Bk. II. Chap. I., \u00a7 106), to regard empty space as\npurely subjective: contradictions in a necessary subjective intuition\nform as great a difficulty as in anything else. But these antinomies\narise only in connection with empty space, not with spatial order\nas an aggregate of relations. For only when space is regarded as\npossessed of some thinghood, can a whole or a true element be\ndemanded. This we have seen already in connection with the Point.\nWhen space is regarded, so far as it is valid, as only spatial order,\nunbounded extension and infinite divisibility both disappear. What\nis divided is not spatial relations, but matter; and if matter, as\nwe have seen that Geometry requires, consists of unextended atoms\nwith spatial relations, there is no reason to regard matter either as\ninfinitely divisible, or as consisting of atoms of finite extension.\n\n=206.= But whence arises, on this view, the paradox that we cannot\nbut regard space as having more or less thinghood, and as divisible\n_ad infinitum_? This must be explained, I think, as a psychological\nillusion, unavoidably arising from the fact that spatial relations\nare immediately presented. They thus have a peculiar psychical\nquality, as immediate experiences, by which quality they can be\ndistinguished from time-relations or any other order in which things\nmay be arranged. To Stumpf, whose problem is psychological, such a\npsychical quality would constitute an absolute underlying content,\nand would fully justify his thesis; to us, however, whose problem is\nepistemological, it would not do so, but would leave the _meaning_ of\nthe spatial element in sense-perception free from any implication\nof an absolute or empty space[201]. May we not, then, abandon empty\nspace, and say: Spatial order consists of _felt_ relations, and _qu\u00e2_\nfelt has, for Psychology, an existence not wholly resolvable into\nrelations, and unavoidably _seeming_ to be more than mere relations.\nBut when we examine the information, as to space, which we derive\nfrom sense-perception, we find ourselves plunged in contradictions,\nas soon as we allow this information to consist of more than\nrelations. This leaves spatial order alone in the field, and reduces\nempty space to a mere name for the logical possibility of spatial\nrelations.\n\n=207.= The apparent divisibility of the relations which constitute\nspatial order, then, may be explained in two ways, though these are\nat bottom equivalent. We may take the relation as considered in\nconnection with empty space, in which case it becomes more than a\nrelation; but being falsely hypostatized, it appears as a complex\nthing, necessarily composed of elements, which elements, however,\nnowhere emerge until we analyze the pseudo-thing down to nothing,\nand arrive at the point. In this sense, the divisibility of spatial\nrelations is an unavoidable illusion. Or again, we may take the\nrelation in connection with the material atoms it relates. In this\ncase, other atoms may be imagined, differently localized by different\nspatial relations. If they are localized on the straight line joining\ntwo of the original atoms, this straight line appears as divided by\nthem. But the original relation is not really divided: all that has\nhappened is, that two or more equivalent relations have replaced\nit, as two compounded relations of father and son may replace the\nequivalent relation of grandfather and grandson. These two ways of\nviewing the apparent divisibility are equivalent: for empty space, in\nso far as it is not illusion, is a name for the aggregate of possible\nspace-relations. To regard a figure in empty space as divided,\ntherefore, means, if it means anything, to regard two or more other\npossible relations as substituted for it, which gives the second way\nof viewing the question.\n\nThe same reference to matter, then, by which the antinomy of the\nPoint was solved, solves also the antinomy as to the relational\nnature of space. Space, if it is to be freed from contradictions,\nmust be regarded exclusively as spatial order, as relations between\nunextended material atoms. Empty space, which arises, by an\ninevitable illusion, out of the spatial element in sense-perception,\nmay be regarded, if we wish to retain it, as the bare principle of\nrelativity, the bare logical possibility of relations between diverse\nthings. In this sense, empty space is wholly conceptual; spatial\norder alone is immediately experienced.\n\n=208.= But in what sense does spatial order consist of relations?\nWe have hitherto spoken of externality as a relation, and in a\nsense such a manner of speaking is justified. Externality, when\npredicated of anything, is an adjective of that thing, and implies\na reference to some other thing. To this extent, then, externality\nis analogous to other relations; and only to this extent, in our\nprevious arguments, has it been regarded as a relation. But when we\ntake account of further qualities of relations, externality begins to\nappear, not so much as a relation, but rather as a necessary aspect\nor element in every relation. And this is borne out by the necessity,\nfor the existence of relations, of some given form of externality.\n\nEvery relation, we may say, involves a diversity between the\nrelated terms, but also some unity. Mere diversity does not give\na ground for that interaction, and that interdependence, which a\nrelation requires. Mere unity leaves the terms identical, and thus\ndestroys the reference of one to another required for a relation.\nMere externality, taken in abstraction, gives only the element of\ndiversity required for a relation, and is thus more abstract than any\nactual relation. But mere diversity does not give that indivisible\nwhole of which any actual relation must consist, and is thus, when\nregarded abstractly, not subject to the restrictions of ordinary\nrelations.\n\nBut with mere diversity, we seem to have returned to empty space, and\nabandoned spatial order. Mere diversity, surely, is either complete\nor non-existent; degrees of diversity, or a quantitative measure of\nit, are nonsense. We cannot, therefore, reduce spatial order to mere\ndiversity. Two things, if they occupy different positions in space,\nare necessarily diverse, but are as necessarily something more;\notherwise spatial order becomes unmeaning.\n\nEmpty space, then, in the above sense of the possibility of spatial\nrelations, contains only one aspect of a relation, namely the aspect\nof diversity; but spatial order, by its reference to matter, becomes\nmore concrete, and contains also the element of unity, arising out\nof the connection of the different material atoms. Spatial order,\nthen, consists of relations in the ordinary sense; its merely\nspatial element, however--if one may make such a distinction--the\nelement, that is, which can be abstracted from matter and regarded\nas constituting empty space, is only one aspect of a relation, but\nan aspect which, in the concrete, must be inseparably bound up\nwith the other aspect. Here, once more, we see the ground of the\ncontradictions in empty space, and the reason why spatial order is\nfree from these contradictions.\n\n\n_Conclusion._\n\n=209.= We have now completed our review of the foundations of\nGeometry. It will be well, before we take leave of the subject,\nbriefly to review and recapitulate the results we have won.\n\nIn the first chapter, we watched the development of a branch of\nMathematics designed, at first, only to establish the logical\nindependence of Euclid's axiom of parallels, and the possibility\nof a self-consistent Geometry which dispensed with it. We found\nthe further development of the subject entangled, for a while, in\nphilosophical controversy; having shown one axiom to be superfluous,\nthe geometers of the second period hoped to prove the same conclusion\nof all the others, but failed to construct any system free from three\nfundamental axioms. Being concerned with analytical and metrical\nGeometry, they tended to regard Algebra as _\u00e0 priori_, but held that\nthose properties of spatial magnitudes, which were not deducible from\nthe laws of Algebra, must be empirical. In all this, they aimed as\nmuch at discrediting Kant as at advancing Mathematics. But with the\nthird period, the interest in Philosophy diminishes, the opposition\nto Euclid becomes less marked, and most important of all, measurement\nis no longer regarded as fundamental, and space is dealt with by\ndescriptive rather than quantitative methods. But nevertheless,\nthree axioms, substantially the same as those retained in the second\nperiod, are still retained by all geometers.\n\nIn the second chapter, we endeavoured, by a criticism of some\ngeometrical philosophies, to prepare the ground for a constructive\ntheory of Geometry. We saw that Kant, in applying the argument of\nthe Transcendental Aesthetic to space, had gone too far, since\nits logical scope extended only to some form of externality in\ngeneral. We saw that Riemann, Helmholtz and Erdmann, misled by the\nquantitative bias, overlooked the qualitative substratum required\nby all judgments of quantity, and thus mistook the direction in\nwhich the necessary axioms of Geometry are to be found. We rejected,\nalso, Helmholtz's view that Geometry depends on Physics, because we\nfound that Physics must assume a knowledge of Geometry before it\ncan become possible. But we admitted, in Geometry, a reference to\nmatter--not, however, to matter as empirically known in Physics, but\nto a more abstract matter, whose sole function is to appear in space,\nand supply the terms for spatial relations. We admitted, however,\nbesides this, that all _actual_ measurement must be effected by means\nof _actual_ matter, and is only empirically possible, through the\nempirical knowledge of approximately rigid bodies. In criticizing\nLotze, we saw that the most important sense, in which non-Euclidean\nspaces are possible, is a philosophical sense, namely, that they are\nnot condemned by any _\u00e0 priori_ argument as to the necessity of space\nfor experience, and that consequently, if they are not affirmed,\nthis must be on empirical grounds alone. Lotze's strictures on the\nmathematical procedure of Metageometry we found to be wholly due to\nignorance of the subject.\n\nProceeding, in the third chapter, to a constructive theory of\nGeometry, we saw that projective Geometry, which has no reference to\nquantity, is necessarily true of any form of externality. Its three\naxioms--homogeneity, dimensions, and the straight line--were all\ndeduced from the conception of a form of externality, and, since some\nsuch form is necessary to experience, were all declared _\u00e0 priori_.\nIn metrical Geometry, on the contrary, we found an empirical element,\narising out of the alternatives of Euclidean and non-Euclidean space.\nThree _\u00e0 priori_ axioms, common to these spaces, and necessary\nconditions of the possibility of measurement, still remained; these\nwere the axiom of Free Mobility, the axiom that space has a finite\nintegral number of dimensions, and the axiom of distance. Except for\nthe new idea of motion, these were found equivalent to the projective\ntriad, and thus necessarily true of any form of externality. But\nthe remaining axioms of Euclid--the axiom of three dimensions, the\naxiom that two straight lines can never enclose a space, and the\naxiom of parallels--were regarded as empirical laws, derived from the\ninvestigation and measurement of our actual space, and true only, as\nfar as the last two are concerned, within the limits set by errors of\nobservation.\n\nIn the present chapter, we completed our proof of the apriority\nof the projective and equivalent metrical axioms, by showing the\nnecessity, for experience, of some form of externality, given by\nsensation or intuition, and not merely inferred from other data.\nWithout this, we said, a knowledge of diverse but interrelated\nthings, the corner-stone of all experience, would be impossible.\nFinally, we discussed the contradictions arising out of the\nrelativity and continuity of space, and endeavoured to overcome\nthem by a reference to matter. This matter, we found, must consist\nof unextended atoms, localized by their spatial relations, and\nappearing, in Geometry, as points. But the non-spatial adjectives\nof matter, we contended, are irrelevant to Geometry, and its\ncausal properties may be left out of account. To deal with the new\ncontradictions, involved in such a notion of matter, would demand\na fresh treatise, leading us, through Kinematics, into the domains\nof Dynamics and Physics. But to discuss the special difficulties\nof space is all that is possible in an essay on the Foundations of\nGeometry.\n\nFOOTNOTES:\n\n[178] Compare, with the following paragraphs, the admirable\ndiscussion in Mr Hobhouse's Theory of Knowledge (Methuen 1896), Part\nI. Chapter II.\n\n[179] I speak of sense-perception instead of sensation, so as not to\nprejudge the issue as to the sensational nature of space.\n\n[180] See Vaihinger's Commentar, II. pp. 86-7, 168-171.\n\n[181] See Caird, Critical Philosophy of Kant, Vol. I. p. 287.\n\n[182] Ursprung der Raumvorstellung, pp. 12-30.\n\n[183] See the references in Vaihinger's Commentar, II. p. 76 ff.\n\n[184] Commentar, II. p. 71 ff.\n\n[185] _E.g._ by Caird, _op. cit._ Vol. I. p. 286.\n\n[186] I have no wish to deny, however, that space is essential in the\nsubsequent distinction of Self and not-Self.\n\n[187] See also Book I. Chap II. passim; especially p. 51 ff. and pp.\n70-1.\n\n[188] Logic, p. 51 ff.\n\n[189] For the _This_, on such a hypothesis, has a purely temporal\ncomplexity, and is not resolvable into coexisting _Thises_.\n\n[190] Chapter III. Section A, (\u00a7 131).\n\n[191] Cf. Hannequin, Essai critique sur l'hypoth\u00e8se des atomes, Paris\n1895, Chap. I. Section III.; especially p. 43.\n\n[192] See Chapter II. \u00a7 69 ff.\n\n[193] See third antinomy below, \u00a7 201 ff.\n\n[194] This atom, of course, must not be confounded with the atom of\nChemistry.\n\n[195] Ursprung der Raumvorstellung, p. 15.\n\n[196] See Vaihinger's Commentar, II. pp. 189-190.\n\n[197] See ibid. p. 224 ff. for Kant's inconsistencies on this point.\n\n[198] The fourth and fifth in the first edition, the third and fourth\nin the second.\n\n[199] Cf. Vaihinger's Commentar, II. p. 218.\n\n[200] Cf. Vaihinger's Commentar, II. p. 207.\n\n[201] Cf. James, Psychology, Vol. II., p. 148 ff.\n\n\n\n\n Cambridge:\n\n PRINTED BY J. AND C. F. CLAY,\n AT THE UNIVERSITY PRESS.\n\n\n\n\n TRANSCRIBER'S NOTE\n\n Italic text is denoted by _underscores_.\n\n Bold text is denoted by =equal signs=.\n\n A superscript is denoted by ^{x}. For example, y^{4} indicates y to\n the power 4.\n\n A subscript is denoted by {x}. For example, a{ik} indicates the ik-th\n element of array a.\n\n Algebraic expressions in the original text usually had the variables in\n italic. The italic underscore markup has been removed for readability,\n so for example: _dx_{i}_ = \u03bb\u2032_d_\u2032_x_{i}_ + \u03bb\u2033_d\u2033x_{i}_\n has been changed to: dx{i} = \u03bb\u2032d\u2032x{i} + \u03bb\u2033d\u2033x{i}\n\n Obvious typographical errors and punctuation errors have been\n corrected after careful comparison with other occurrences within\n the text and consultation of external sources.\n\n Except for those changes noted below, all misspellings in the text,\n and inconsistent or archaic usage, have been retained. For example,\n co-exist, coexist; every-day, everyday; connexion; assertorial;\n apodeictic; premisses.\n\n \u00a7 82, 'so Erdmannn confidently' replaced by 'so Erdmann confidently'.\n \u00a7 150 Footnote [157], 'Delboeuf' replaced by 'Delb\u0153uf' for consistency.\n \u00a7 152, 'one subdivison must' replaced by 'one sub-division must'.\n \u00a7 159 Footnote [167], 'Delboeuf' replaced by 'Delb\u0153uf' for consistency.\n \u00a7 204, 'and homogenous;' replaced by 'and homogeneous;'.","title":"An essay on the foundations of geometry","tradition":"philosophy-ethics","translated_text":null,"word_count":78474}