TRANSCRIBER'S NOTE

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Obvious typographical errors and punctuation errors have been corrected after careful comparison with other occurrences within the text and consultation of external sources.

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THE

FOUNDATIONS OF GEOMETRY.

London: C. J. CLAY AND SONS,

CAMBRIDGE UNIVERSITY PRESS WAREHOUSE,
AVE MARIA LANE.

Glasgow: 263, ARGYLE STREET.

Leipzig: F. A. BROCKHAUS.
New York: THE MACMILLAN COMPANY.
Bombay: GEORGE BELL AND SONS.

AN ESSAY
ON THE
FOUNDATIONS OF GEOMETRY

BY

BERTRAND A. W. RUSSELL. M.A.

FELLOW OF TRINITY COLLEGE, CAMBRIDGE.

CAMBRIDGE:

AT THE UNIVERSITY PRESS.

1897

[All Rights reserved.]

Cambridge:
PRINTED BY J. AND C. F. CLAY,
AT THE UNIVERSITY PRESS.

PREFACE.

The present work is based on a dissertation submitted at the Fellowship Examination of Trinity College, Cambridge, in the year 1895. Section B of the third chapter is in the main a reprint, with some serious alterations, of an article in Mind (New Series, No. 17). The substance of the book has been given in the form of lectures at the Johns Hopkins University, Baltimore, and at Bryn Mawr College, Pennsylvania.

My chief obligation is to Professor Klein. Throughout the first chapter, I have found his "Lectures on non-Euclidean Geometry" an invaluable guide; I have accepted from him the division of Metageometry into three periods, and have found my historical work much lightened by his references to previous writers. In Logic, I have learnt most from Mr Bradley, and next to him, from Sigwart and Dr Bosanquet. On several important points, I have derived useful suggestions from Professor James's "Principles of Psychology."

My thanks are due to Mr G. F. Stout and Mr A. N. Whitehead for kindly reading my proofs, and helping me by many useful criticisms. To Mr Whitehead I owe, also, the inestimable assistance of constant criticism and suggestion throughout the course of construction, especially as regards the philosophical importance of projective Geometry.

Haslemere.

May, 1897.

TO

JOHN McTAGGART ELLIS McTAGGART

TO WHOSE DISCOURSE AND FRIENDSHIP IS OWING

THE EXISTENCE OF THIS BOOK.

[TABLE OF CONTENTS.]

INTRODUCTION.
OUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC, PSYCHOLOGY AND MATHEMATICS.

1. Geometry, throughout the 17th and 18th centuries, remained, in the war against empiricism, an impregnable fortress of the idealists. Those who held—as was generally held on the Continent—that certain knowledge, independent of experience, was possible about the real world, had only to point to Geometry: none but a madman, they said, would throw doubt on its validity, and none but a fool would deny its objective reference. The English Empiricists, in this matter, had, therefore, a somewhat difficult task; either they had to ignore the problem, or if, like Hume and Mill, they ventured on the assault, they were driven into the apparently paradoxical assertion that Geometry, at bottom, had no certainty of a different kind from that of Mechanics—only the perpetual presence of spatial impressions, they said, made our experience of the truth of the axioms so wide as to seem absolute certainty.

Here, however, as in many other instances, merciless logic drove these philosophers, whether they would or no, into glaring opposition to the common sense of their day. It was only through Kant, the creator of modern Epistemology, that the geometrical problem received a modern form. He reduced the question to the following hypotheticals: If Geometry has apodeictic certainty, its matter, i.e. space, must be à priori, and as such must be purely subjective; and conversely, if space is purely subjective, Geometry must have apodeictic certainty. The latter hypothetical has more weight with Kant, indeed it is ineradicably bound up with his whole Epistemology; nevertheless it has, I think, much less force than the former. Let us accept, however, for the moment, the Kantian formulation, and endeavour to give precision to the terms à priori and subjective.

2. One of the great difficulties, throughout this controversy, is the extremely variable use to which these words, as well as the word empirical, are put by different authors. To Kant, who was nothing of a psychologist, à priori and subjective were almost interchangeable terms[1]; in modern usage there is, on the whole, a tendency to confine the word subjective to Psychology, leaving à priori to do duty for Epistemology. If we accept this differentiation, we may set up, corresponding to the problems of these two sciences, the following provisional definitions: à priori applies to any piece of knowledge which, though perhaps elicited by experience, is logically presupposed in experience: subjective applies to any mental state whose immediate cause lies, not in the external world, but within the limits of the subject. The latter definition, of course, is framed exclusively for Psychology: from the point of view of physical Science all mental states are subjective. But for a Science whose matter, strictly speaking, is only mental states, we require, if we are to use the word to any purpose, some differentia among mental states, as a mark of a more special subjectivity on the part of those to which this term is applied.

Now the only mental states whose immediate causes lie in the external world are sensations. A pure sensation is, of course, an impossible abstraction—we are never wholly passive under the action of an external stimulus—but for the purposes of Psychology the abstraction is a useful one. Whatever, then, is not sensation, we shall, in Psychology, call subjective. It is in sensation alone that we are directly affected by the external world, and only here does it give us direct information about itself.

3. Let us now consider the epistemological question, as to the sort of knowledge which can be called à priori. Here we have nothing to do—in the first instance, at any rate—with the cause or genesis of a piece of knowledge; we accept knowledge as a datum to be analysed and classified. Such analysis will reveal a formal and a material element in knowledge. The formal element will consist of postulates which are required to make knowledge possible at all, and of all that can be deduced from these postulates; the material element, on the other hand, will consist of all that comes to fill in the form given by the formal postulates—all that is contingent or dependent on experience, all that might have been otherwise without rendering knowledge impossible. We shall then call the formal element à priori, the material element empirical.

4. Now what is the connection between the subjective and the à priori? It is a connection, obviously—if it exists at all—from the outside, i.e. not deducible directly from the nature of either, but provable—if it can be proved—only by a general view of the conditions of both. The question, what knowledge is à priori, must, on the above definition, depend on a logical analysis of knowledge, by which the conditions of possible experience may be revealed; but the question, what elements of a cognitive state are subjective, is to be investigated by pure Psychology, which has to determine what, in our perceptions, belongs to sensation, and what is the work of thought or of association. Since, then, these two questions belong to different sciences, and can be settled independently, will it not be wise to conduct the two investigations separately? To decree that the à priori shall always be subjective, seems dangerous, when we reflect that such a view places our results, as to the à priori, at the mercy of empirical psychology. How serious this danger is, the controversy as to Kant's pure intuition sufficiently shows.

5. I shall, therefore, throughout the present Essay, use the word à priori without any psychological implication. My test of apriority will be purely logical: Would experience be impossible, if a certain axiom or postulate were denied? Or, in a more restricted sense, which gives apriority only within a particular science: Would experience as to the subject-matter of that science be impossible, without a certain axiom or postulate? My results also, therefore, will be purely logical. If Psychology declares that some things, which I have declared à priori, are not subjective, then, failing an error of detail in my proofs, the connection of the à priori and the subjective, so far as those things are concerned, must be given up. There will be no discussion, accordingly, throughout this Essay, of the relation of the à priori to the subjective—a relation which cannot determine what pieces of knowledge are à priori, but rather depends on that determination, and belongs, in any case, rather to Metaphysics than to Epistemology.

6. As I have ventured to use the word à priori in a slightly unconventional sense, I will give a few elucidatory remarks of a general nature.

The à priori, since Kant at any rate, has generally stood for the necessary or apodeictic element in knowledge. But modern logic has shown that necessary propositions are always, in one aspect at least, hypothetical. There may be, and usually is, an implication that the connection, of which necessity is predicated, has some existence, but still, necessity always points beyond itself to a ground of necessity, and asserts this ground rather than the actual connection. As Bradley points out, "arsenic poisons" remains true, even if it is poisoning no one. If, therefore, the à priori in knowledge be primarily the necessary, it must be the necessary on some hypothesis, and the ground of necessity must be included as à priori. But the ground of necessity is, so far as the necessary connection in question can show, a mere fact, a merely categorical judgment. Hence necessity alone is an insufficient criterion of apriority.

To supplement this criterion, we must supply the hypothesis or ground, on which alone the necessity holds, and this ground will vary from one science to another, and even, with the progress of knowledge, in the same science at different times. For as knowledge becomes more developed and articulate, more and more necessary connections are perceived, and the merely categorical truths, though they remain the foundation of apodeictic judgments, diminish in relative number. Nevertheless, in a fairly advanced science such as Geometry, we can, I think, pretty completely supply the appropriate ground, and establish, within the limits of the isolated science, the distinction between the necessary and the merely assertorical.

7. There are two grounds, I think, on which necessity may be sought within any science. These may be (very roughly) distinguished as the ground which Kant seeks in the Prolegomena, and that which he seeks in the Pure Reason. We may start from the existence of our science as a fact, and analyse the reasoning employed with a view to discovering the fundamental postulate on which its logical possibility depends; in this case, the postulate, and all which follows from it alone, will be à priori. Or we may accept the existence of the subject-matter of our science as our basis of fact, and deduce dogmatically whatever principles we can from the essential nature of this subject-matter. In this latter case, however, it is not the whole empirical nature of the subject-matter, as revealed by the subsequent researches of our science, which forms our ground; for if it were, the whole science would, of course, be à priori. Rather it is that element, in the subject-matter, which makes possible the branch of experience dealt with by the science in question[2]. The importance of this distinction will appear more clearly as we proceed[3].

8. These two grounds of necessity, in ultimate analysis, fall together. The methods of investigation in the two cases differ widely, but the results cannot differ. For in the first case, by analysis of the science, we discover the postulate on which alone its reasonings are possible. Now if reasoning in the science is impossible without some postulate, this postulate must be essential to experience of the subject-matter of the science, and thus we get the second ground. Nevertheless, the two methods are useful as supplementing one another, and the first, as starting from the actual science, is the safest and easiest method of investigation, though the second seems the more convincing for exposition.

9. The course of my argument, therefore, will be as follows: In the first chapter, as a preliminary to the logical analysis of Geometry, I shall give a brief history of the rise and development of non-Euclidean systems. The second chapter will prepare the ground for a constructive theory of Geometry, by a criticism of some previous philosophical views; in this chapter, I shall endeavour to exhibit such views as partly true, partly false, and so to establish, by preliminary polemics, the truth of such parts of my own theory as are to be found in former writers. A large part of this theory, however, cannot be so introduced, since the whole field of projective Geometry, so far as I am aware, has been hitherto unknown to philosophers. Passing, in the third chapter, from criticism to construction, I shall deal first with projective Geometry. This, I shall maintain, is necessarily true of any form of externality, and is, since some such form is necessary to experience, completely à priori. In metrical Geometry, however, which I shall next consider, the axioms will fall into two classes: (1) Those common to Euclidean and non-Euclidean spaces. These will be found, on the one hand, essential to the possibility of measurement in any continuum, and on the other hand, necessary properties of any form of externality with more than one dimension. They will, therefore, be declared à priori. (2) Those axioms which distinguish Euclidean from non-Euclidean spaces. These will be regarded as wholly empirical. The axiom that the number of dimensions is three, however, though empirical, will be declared, since small errors are here impossible, exactly and certainly true of our actual world; while the two remaining axioms, which determine the value of the space-constant, will be regarded as only approximately known, and certain only within the errors of observation[4]. The fourth chapter, finally, will endeavour to prove, what was assumed in Chapter III., that some form of externality is necessary to experience, and will conclude by exhibiting the logical impossibility, if knowledge of such a form is to be freed from contradictions, of wholly abstracting this knowledge from all reference to the matter contained in the form.

I shall hope to have touched, with this discussion, on all the main points relating to the Foundations of Geometry.

FOOTNOTES:

[1] Cf. Erdmann, Axiome der Geometrie, p. 111: "Für Kant sind Apriorität und ausschliessliche Subjectivität allerdings Wechselbegriffe."

[2] I use "experience" here in the widest possible sense, the sense in which the word is used by Bradley.

[3] Where the branch of experience in question is essential to all experience, the resulting apriority may be regarded as absolute; where it is necessary only to some special science, as relative to that science.

[4] I have given no account of these empirical proofs, as they seem to be constituted by the whole body of physical science. Everything in physical science, from the law of gravitation to the building of bridges, from the spectroscope to the art of navigation, would be profoundly modified by any considerable inaccuracy in the hypothesis that our actual space is Euclidean. The observed truth of physical science, therefore, constitutes overwhelming empirical evidence that this hypothesis is very approximately correct, even if not rigidly true.

CHAPTER I.
A SHORT HISTORY OF METAGEOMETRY.

10. When a long established system is attacked, it usually happens that the attack begins only at a single point, where the weakness of the established doctrine is peculiarly evident. But criticism, when once invited, is apt to extend much further than the most daring, at first, would have wished.

"First cut the liquefaction, what comes last,

But Fichte's clever cut at God himself?"

So it has been with Geometry. The liquefaction of Euclidean orthodoxy is the axiom of parallels, and it was by the refusal to admit this axiom without proof that Metageometry began. The first effort in this direction, that of Legendre[5], was inspired by the hope of deducing this axiom from the others—a hope which, as we now know, was doomed to inevitable failure. Parallels are defined by Legendre as lines in the same plane, such that, if a third line cut them, it makes the sum of the interior and opposite angles equal to two right angles. He proves without difficulty that such lines would not meet, but is unable to prove that non-parallel lines in a plane must meet. Similarly he can prove that the sum of the angles of a triangle cannot exceed two right angles, and that if any one triangle has a sum equal to two right angles, all triangles have the same sum; but he is unable to prove the existence of this one triangle.

11. Thus Legendre's attempt broke down; but mere failure could prove nothing. A bolder method, suggested by Gauss, was carried out by Lobatchewsky and Bolyai[6]. If the axiom of parallels is logically deducible from the others, we shall, by denying it and maintaining the rest, be led to contradictions. These three mathematicians, accordingly, attacked the problem indirectly: they denied the axiom of parallels, and yet obtained a logically consistent Geometry. They inferred that the axiom was logically independent of the others, and essential to the Euclidean system. Their works, being all inspired by this motive, may be distinguished as forming the first period in the development of Metageometry.

The second period, inaugurated by Riemann, had a much deeper import: it was largely philosophical in its aims and constructive in its methods. It aimed at no less than a logical analysis of all the essential axioms of Geometry, and regarded space as a particular case of the more general conception of a manifold. Taking its stand on the methods of analytical metrical Geometry, it established two non-Euclidean systems, the first that of Lobatchewsky, the second—in which the axiom of the straight line, in Euclid's form, was also denied—a new variety, by analogy called spherical. The leading conception in this period is the measure of curvature, a term invented by Gauss, but applied by him only to surfaces. Gauss had shown that free mobility on surfaces was only possible when the measure of curvature was constant; Riemann and Helmholtz extended this proposition to n dimensions, and made it the fundamental property of space.

In the third period, which begins with Cayley, the philosophical motive, which had moved the first pioneers, is less apparent, and is replaced by a more technical and mathematical spirit. This period is chiefly distinguished from the second, in a mathematical point of view, by its method, which is projective instead of metrical. The leading mathematical conception here is the Absolute (Grundgebild), a figure by relation to which all metrical properties become projective. Cayley's work, which was very brief, and attracted little attention, has been perfected and elaborated by F. Klein, and through him has found general acceptance. Klein has added to the two kinds of non-Euclidean Geometry already known, a third, which he calls elliptic; this third kind closely resembles Helmholtz's spherical Geometry, but is distinguished by the important difference that, in it, two straight lines meet in only one point[7]. The distinctive mark of the spaces represented by both is that, like the surface of a sphere, they are finite but unbounded. The reduction of metrical to projective properties, as will be proved hereafter, has only a technical importance; at the same time, projective Geometry is able to deal directly with those purely descriptive or qualitative properties of space which are common to Euclid and Metageometry alike. The third period has, therefore, great philosophical importance, while its method has, mathematically, much greater beauty and unity than that of the second; it is able to treat all kinds of space at once, so that every symbolic proposition is, according to the meaning given to the symbols, a proposition in whichever Geometry we choose. This has the advantage of proving that further research cannot lead to contradictions in non-Euclidean systems, unless it at the same moment reveals contradictions in Euclid. These systems, therefore, are logically as sound as that of Euclid himself.

After this brief sketch of the characteristics of the three periods, I will proceed to a more detailed account. It will be my aim to avoid, as far as possible, all technical mathematics, and bring into relief only those fundamental points in the mathematical development, which seem of logical or philosophical importance.

First Period.

12. The originator of the whole system, Gauss, does not appear, as regards strictly non-Euclidean Geometry, in any of his hitherto published papers, to have given more than results; his proofs remain unknown to us. Nevertheless he was the first to investigate the consequences of denying the axiom of parallels[8], and in his letters he communicated these consequences to some of his friends, among whom was Wolfgang Bolyai. The first mention of the subject in his letters occurs when he was only 18; four years later, in 1799, writing to W. Bolyai, he enunciates the important theorem that, in hyperbolic Geometry, there is a maximum to the area of a triangle. From later writings it appears that he had worked out a system nearly, if not quite, as complete as those of Lobatchewsky and Bolyai[9].

It is important to remember, however, that Gauss's work on curvature, which was published, laid the foundation for the whole method of the second period, and was undertaken, according to Riemann and Helmholtz[10], with a view to an (unpublished) investigation of the foundations of Geometry. His work in this direction will, owing to its method, be better treated of under the second period, but it is interesting to observe that he stood, like many pioneers, at the head of two tendencies which afterwards diverged.

13. Lobatchewsky, a professor in the University of Kasan, first published his results, in their native Russian, in the proceedings of that learned body for the years 1829–1830. Owing to this double obscurity of language and place, they attracted little attention, until he translated them into French[11] and German[12]: even then, they do not appear to have obtained the notice they deserved, until, in 1868, Beltrami unearthed the article in Crelle, and made it the theme of a brilliant interpretation.

In the introduction to his little German book, Lobatchewsky laments the slight interest shown in his writings by his compatriots, and the inattention of mathematicians, since Legendre's abortive attempt, to the difficulties in the theory of parallels. The body of the work begins with the enunciation of several important propositions which hold good in the system proposed as well as in Euclid: of these, some are in any case independent of the axiom of parallels, while others are rendered so by substituting, for the word "parallel," the phrase "not intersecting, however far produced." Then follows a definition, intentionally framed so as to contradict Euclid's: With respect to a given straight line, all others in the same plane may be divided into two classes, those which cut the given straight line, and those which do not cut it; a line which is the limit between the two classes is called parallel to the given straight line. It follows that, from any external point, two parallels can be drawn, one in each direction. From this starting-point, by the Euclidean synthetic method, a series of propositions are deduced; the most important of these is, that in a triangle the sum of the angles is always less than, or always equal to two right angles, while in the latter case the whole system becomes orthodox. A certain analogy with spherical Geometry—whose meaning and extent will appear later—is also proved, consisting roughly in the substitution of hyperbolic for circular functions.

14. Very similar is the system of Johann Bolyai, so similar, indeed, as to make the independence of the two works, though a well-authenticated fact, seem all but incredible. Johann Bolyai first published his results in 1832, in an appendix to a work by his father Wolfgang, entitled; "Appendix, scientiam spatii absolute veram exhibens: a veritate aut falsitate Axiomatis XI. Euclidei (a priori haud unquam decidenda) independentem; adjecta ad casum falsitatis, quadratura circuli geometrica." Gauss, whose bosom friend he became at college and remained through life, was, as we have seen, the inspirer of Wolfgang Bolyai, and used to say that the latter was the only man who appreciated his philosophical speculations on the axioms of Geometry; nevertheless, Wolfgang appears to have left to his son Johann the detailed working out of the hyperbolic system. The works of both the Bolyai are very rare, and their method and results are known to me only through the renderings of Frischauf and Halsted[13]. Both as to method and as to results, the system is very similar to Lobatchewsky's, so that neither need detain us here. Only the initial postulates, which are more explicit than Lobatchewsky's, demand a brief attention. Frischauf's introduction, which has a philosophical and Newtonian air, begins by setting forth that Geometry deals with absolute (empty) space, obtained by abstracting from the bodies in it, that two figures are called congruent when they differ only in position, and that the axiom of Congruence is indispensable in all determination of spatial magnitudes. Congruence was to refer to geometrical bodies, with none of the properties of ordinary bodies except impenetrability (Erdmann, Axiome der Geometrie, p. 26). A straight line is defined as determined by two of its points[14], and a plane as determined by three. These premisses, with a slight exception as to the straight line, we shall hereafter find essential to every Geometry. I have drawn attention to them, as it is often supposed that non-Euclideans deny the axiom of Congruence, which, here and elsewhere, is never the case. The stress laid on this axiom by Bolyai is probably due to the influence of Gauss, whose work on the curvature of surfaces laid the foundation for the use made of congruence by Helmholtz.

15. It is important to remember that, throughout the period we have just reviewed, the purpose of hyperbolic Geometry is indirect: not the truth of the latter, but the logical independence of the axiom of parallels from the rest, is the guiding motive of the work. If, by denying the axiom of parallels while retaining the rest, we can obtain a system free from logical contradictions, it follows that the axiom of parallels cannot be implicitly contained in the others. If this be so, attempts to dispense with the axiom, like Legendre's, cannot be successful; Euclid must stand or fall with the suspected axiom. Of course, it remained possible that, by further development, latent contradictions might have been revealed in these systems. This possibility, however, was removed by the more direct and constructive work of the second period, to which we must now turn our attention.

Second Period.

16. The work of Lobatchewsky and Bolyai remained, for nearly a quarter of a century, without issue—indeed, the investigations of Riemann and Helmholtz, when they came, appear to have been inspired, not by these men, but rather by Gauss[15] and Herbart. We find, accordingly, very great difference, both of aim and method, between the first period and the second. The former, beginning with a criticism of one point in Euclid's system, preserved his synthetic method, while it threw over one of his axioms. The latter, on the contrary, being guided by a philosophical rather than a mathematical spirit, endeavoured to classify the conception of space as a species of a more general conception: it treated space algebraically, and the properties it gave to space were expressed in terms, not of intuition, but of algebra. The aim of Riemann and Helmholtz was to show, by the exhibition of logically possible alternatives, the empirical nature of the received axioms. For this purpose, they conceived space as a particular case of a manifold, and showed that various relations of magnitude (Massverhältnisse) were mathematically possible in an extended manifold. Their philosophy, which seems to me not always irreproachable, will be discussed in Chapter II.; here, while it is important to remember the philosophical motive of Riemann and Helmholtz, we shall confine our attention to the mathematical side of their work. In so doing, while we shall, I fear, somewhat maim the system of their thoughts, we shall secure a closer unity of subject, and a more compact account of the purely mathematical development. But there is, in my opinion, a further reason for separating their philosophy from their mathematics. While their philosophical purpose was, to prove that all the axioms of Geometry are empirical, and that a different content of our experience might have changed them all, the unintended result of their mathematical work was, if I am not mistaken, to afford material for an à priori proof of certain axioms. These axioms, though they believed them to be unnecessary, were always introduced in their mathematical works, before laying the foundations of non-Euclidean systems. I shall contend, in Chapter III., that this retention was logically inevitable, and was not merely due, as they supposed, to a desire for conformity with experience. If I am right in this, there is a divergence between Riemann and Helmholtz the philosophers, and Riemann and Helmholtz the mathematicians. This divergence makes it the more desirable to trace the mathematical development apart from the accompanying philosophy.

17. Riemann's epoch-making work, "Ueber die Hypothesen, welche der Geometrie zu Grande liegen[16]", was written, and read to a small circle, in 1854; owing, however, to some changes which he desired to make in it, it remained unpublished till 1867, when it was published by his executors. The two fundamental conceptions, on whose invention rests the historic importance of this dissertation, are that of a manifold, and that of the measure of curvature of a manifold. The former conception serves a mainly philosophical purpose, and is designed, principally, to exhibit space as an instance of a more general conception. On this aspect of the manifold, I shall have much to say in Chapter II.; its mathematical aspect, which alone concerns us here, is less complicated and less fruitful of controversy. The latter conception also serves a double purpose, but its mathematical use is the more prominent. We will consider these two conceptions successively.

18. (1) Conception of a manifold[17]. The general purpose of Riemann's dissertation is, to exhibit the axioms as successive steps in the classification of the species space. The axioms of Geometry, like the marks of a scholastic definition, appear as successive determinations of class-conceptions, ending with Euclidean space. We have thus, from the analytical point of view, about as logical and precise a formulation as can be desired—a formulation in which, from its classificatory character, we seem certain of having nothing superfluous or redundant, and obtain the axioms explicitly in the most desirable form, namely as adjectives of the conception of space. At the same time, it is a pity that Riemann, in accordance with the metrical bias of his time, regarded space as primarily a magnitude[18], or assemblage of magnitudes, in which the main problem consists in assigning quantities to the different elements or points, without regard to the qualitative nature of the quantities assigned. Considerable obscurity thus arises as to the whole nature of magnitude[19]. This view of Geometry underlies the definition of the manifold, as the general conception of which space forms a special case. This definition, which is not very clear, may be rendered as follows.

19. Conceptions of magnitude, according to Riemann, are possible there only, where we have a general conception, capable of various determinations (Bestimmungsweisen). The various determinations of such a conception together form a manifold, which is continuous or discrete, according as the passage from one determination to another is continuous or discrete. Particular bits of a manifold, or quanta, can be compared by counting when discrete, and by measurement when continuous. "Measurement consists in a superposition of the magnitudes to be compared. If this be absent, magnitudes can only be compared when one is part of another, and then only the more or less, not the how much, can be decided" (p. 256). We thus reach the general conception of a manifold of several dimensions, of which space and colours are mentioned as special cases. To the absence of this conception Riemann attributes the "obscurity" which, on the subject of the axioms, "lasted from Euclid to Legendre" (p. 254). And Riemann certainly has succeeded, from an algebraic point of view, in exhibiting, far more clearly than any of his predecessors, the axioms which distinguish spatial quantity from other quantities with which mathematics is conversant. But by the assumption, from the start, that space can be regarded as a quantity, he has been led to state the problem as: What sort of magnitude is space? rather than: What must space be in order that we may be able to regard it as a magnitude at all? He does not realise, either—indeed in his day there were few who realized—that an elaborate Geometry is possible which does not deal with space as a quantity at all. His definition of space as a species of manifold, therefore, though for analytical purposes it defines, most satisfactorily, the nature of spatial magnitudes, leaves obscure the true ground for this nature, which lies in the nature of space as a system of relations, and is anterior to the possibility of regarding it as a system of magnitudes at all.

But to proceed with the mathematical development of Riemann's ideas. We have seen that he declared measurement to consist in a superposition of the magnitudes to be compared. But in order that this may be a possible means of determining magnitudes, he continues, these magnitudes must be independent of their position in the manifold (p. 259). This can occur, he says, in several ways, as the simplest of which, he assumes that the lengths of lines are independent of their position. One would be glad to know what other ways are possible: for my part, I am unable to imagine any other hypothesis on which magnitude would be independent of place. Setting this aside, however, the problem, owing to the fact that measurement consists in superposition, becomes identical with the determination of the most general manifold in which magnitudes are independent of place. This brings us to Riemann's other fundamental conception, which seems to me even more fruitful than that of a manifold.

20. (2) Measure of curvature. This conception is due to Gauss, but was applied by him only to surfaces; the novelty in Riemann's dissertation was its extension to a manifold of n dimensions. This extension, however, is rather briefly and obscurely expressed, and has been further obscured by Helmholtz's attempts at popular exposition. The term curvature, also, is misleading, so that the phrase has been the source of more misunderstanding, even among mathematicians, than any other in Pangeometry. It is often forgotten, in spite of Helmholtz's explicit statement[20], that the "measure of curvature" of an n-dimensional manifold is a purely analytical expression, which has only a symbolic affinity to ordinary curvature. As applied to three-dimensional space, the implication of a four-dimensional "plane" space is wholly misleading; I shall, therefore, generally use the term space-constant instead[21]. Nevertheless, as the conception grew, historically, out of that of curvature, I will give a very brief exposition of the historical development of theories of curvature.

Just as the notion of length was originally derived from the straight line, and extended to other curves by dividing them into infinitesimal straight lines, so the notion of curvature was derived from the circle, and extended to other curves by dividing them into infinitesimal circular arcs. Curvature may be regarded, originally, as a measure of the amount by which a curve departs from a straight line; in a circle, which is similar throughout, this amount is evidently constant, and is measured by the reciprocal of the radius. But in all other curves, the amount of curvature varies from point to point, so that it cannot be measured without infinitesimals. The measure which at once suggests itself is, the curvature of the circle most nearly coinciding with the curve at the point considered. Since a circle is determined by three points, this circle will pass through three consecutive points of the curve. We have thus defined the curvature of any curve, plane or tortuous; for, since any three points lie in a plane, such a circle can always be described.

If we now pass to a surface, what we want is, by analogy, a measure of its departure from a plane. The curvature, as above defined, has become indeterminate, for through any point of the surface we can draw an infinite number of arcs, which will not, in general, all have the same curvature. Let us, then, draw all the geodesics joining the point in question to neighbouring points of the surface in all directions. Since these arcs form a singly infinite manifold, there will be among them, if they have not all the same curvature, one arc of maximum, and one of minimum curvature[22]. The product of these maximum and minimum curvatures is called the measure of curvature of the surface at the point under consideration. To illustrate by a few simple examples: on a sphere, the curvatures of all such lines are equal to the reciprocal of the radius of the sphere, hence the measure of curvature everywhere is the square of the reciprocal of the radius of the sphere. On any surface, such as a cone or a cylinder, on which straight lines can be drawn, these have no curvature, so that the measure of curvature is everywhere zero—this is the case, in particular, with the plane. In general, however, the measure of curvature of a surface varies from point to point.

Gauss, the inventor of this conception[23], proved that, in order that two surfaces may be developable upon each other—i.e. may be such that one can be bent into the shape of the other without stretching or tearing—it is necessary that the two surfaces should have equal measures of curvature at corresponding points. When this is the case, every figure which is possible on the one is, in general, possible on the other, and the two have practically the same Geometry[24]. As a corollary, it follows that a necessary condition, for the free mobility of figures on any surface, is the constancy of the measure of curvature[25]. This condition was proved to be sufficient, as well as necessary, by Minding[26].

21. So far, all has been plain sailing—we have been dealing with purely geometrical ideas in a purely geometrical manner—but we have not, as yet, found any sense of the measure of curvature, in which it can be extended to space, still less to an n-dimensional manifold. For this purpose, we must examine Gauss's method, which enables us to determine the measure of curvature of a surface at any point as an inherent property, quite independent of any reference to the third dimension.

The method of determining the measure of curvature from within is, briefly, as follows: If any point on the surface be determined by two coordinates, u, v, then small arcs of the surface are given by the formula

ds² = Edu² + 2Fdu dv + Gdv²,

where E, F, G are, in general, functions of u, v.[27] From this formula alone, without reference to any space outside the surface, we can determine the measure of curvature at the point u, v, as a function of E, F, G and their differentials with respect to u and v. Thus we may regard the measure of curvature of a surface as an inherent property, and the above geometrical definition, which involved a reference to the third dimension, may be dropped. But at this point a caution is necessary. It will appear in [Chap. III. (§ 176)], that it is logically impossible to set up a precise coordinate system, in which the coordinates represent spatial magnitudes, without the axiom of Free Mobility, and this axiom, as we have just seen, holds on surfaces only when the measure of curvature is constant. Hence our definition of the measure of curvature will only be really free from reference to the third dimension, when we are dealing with a surface of constant measure of curvature—a point which Riemann entirely overlooks. This caution, however, applies only in space, and if we take the coordinate system as presupposed in the conception of a manifold, we may neglect the caution altogether—while remembering that the possibility of a coordinate system in space involves axioms to be investigated later. We can thus see how a meaning might be found, without reference to any higher dimension, for a constant measure of curvature of three-dimensional space, or for any measure of curvature of an n-dimensional manifold in general.

22. Such a meaning is supplied by Riemann's dissertation, to which, after this long digression, we can now return. We may define a continuous manifold as any continuum of elements, such that a single element is defined by n continuously variable magnitudes. This definition does not really include space, for coordinates in space do not define a point, but its relations to the origin, which is itself arbitrary. It includes, however, the analytical conception of space with which Riemann deals, and may, therefore, be allowed to stand for the moment. Riemann then assumes that the difference—or distance, as it may be loosely called—between any two elements is comparable, as regards magnitude, to the difference between any other two. He assumes further, what it is Helmholtz's merit to have proved, that the difference ds between two consecutive elements can be expressed as the square root of a quadratic function of the differences of the coordinates: i.e.

ds² = Σ₁ⁿ Σ₁ⁿ a_ik dx_i.dx_k ,

where the coefficients a_ik are, in general, functions of the coordinates x₁ x₂ ... x_n. [28] The question is: How are we to obtain a definition of the measure of curvature out of this formula? It is noticeable, in the first place, that, just as in a surface we found an infinite number of radii of curvature at a point, so in a manifold of three or more dimensions we must find an infinite number of measures of curvature at a point, one for every two-dimensional manifold passing through the point, and contained in the higher manifold. What we have first to do, therefore, is to define such two-dimensional manifolds. They must consist, as we saw on the surface, of a singly infinite series of geodesics through the point. Now a geodesic is completely determined by one point and its direction at that point, or by one point and the next consecutive point. Hence a geodesic through the point considered is determined by the ratios of the increments of coordinates, dx₁ dx₂ ... dx_n. Suppose we have two such geodesics, in which the i′th increments are respectively d′x_i and d″x_i. Then all the geodesics given by

dx_i = λ′d′x_i + λ″d″x_i

form a singly infinite series, since they contain one parameter, namely λ′: λ″. Such a series of geodesics, therefore, must form a two-dimensional manifold, with a measure of curvature in the ordinary Gaussian sense. This measure of curvature can be determined from the above formula for the elementary arc, by the help of Gauss's general formula alluded to above. We thus obtain an infinite number of measures of curvature at a point, but from n.(n – 1) 2 of these, the rest can be deduced (Riemann, Gesammelte Werke, p. 262). When all the measures of curvature at a point are constant, and equal to all the measures of curvature at any other point, we get what Riemann calls a manifold of constant curvature. In such a manifold free mobility is possible, and positions do not differ intrinsically from one another. If a be the measure of curvature, the formula for the arc becomes, in this case,

ds² = Σdx² _/ (1 + a 4 Σx²)².

In this case only, as I pointed out above, can the term "measure of curvature" be properly applied to space without reference to a higher dimension, since free mobility is logically indispensable to the existence of quantitative or metrical Geometry.

23. The mathematical result of Riemann's dissertation may be summed up as follows. Assuming it possible to apply magnitude to space, i.e. to determine its elements and figures by means of algebraical quantities, it follows that space can be brought under the conception of a manifold, as a system of quantitatively determinable elements. Owing, however, to the peculiar nature of spatial measurement, the quantitative determination of space demands that magnitudes shall be independent of place—in so far as this is not the case, our measurement will be necessarily inaccurate. If we now assume, as the quantitative relation of distance between two elements, the square root of a quadratic function of the coordinates—a formula subsequently proved by Helmholtz and Lie—then it follows, since magnitudes are to be independent of place, that space must, within the limits of observation, have a constant measure of curvature, or must, in other words, be homogeneous in all its parts. In the infinitesimal, Riemann says (p. 267), observation could not detect a departure from constancy on the part of the measure of curvature; but he makes no attempt to show how Geometry could remain possible under such circumstances, and the only Geometry he has constructed is based entirely on Free Mobility. I shall endeavour to prove, in Chapter III., that any metrical Geometry, which should endeavour to dispense with this axiom, would be logically impossible. At present I will only point out that Riemann, in spite of his desire to prove that all the axioms can be dispensed with, has nevertheless, in his mathematical work, retained three fundamental axioms, namely, Free Mobility, the finite integral number of dimensions, and the axiom that two points have a unique relation, namely distance. These, as we shall see hereafter, are retained, in actual mathematical work, by all metrical Metageometers, even when they believe, like Riemann and Helmholtz, that no axioms are philosophically indispensable.

24. Helmholtz, the historically nearest follower of Riemann, was guided by a similar empirical philosophy, and arrived independently at a very similar method of formulating the axioms. Although Helmholtz published nothing on the subject until after Riemann's death, he had then only just seen Riemann's dissertation (which was published posthumously), and had worked out his results, so far as they were then completed, in entire independence both of Riemann and of Lobatchewsky. Helmholtz is by far the most widely read of all writers on Metageometry, and his writings, almost alone, represent to philosophers the modern mathematical standpoint on this subject. But his importance is much greater, in this domain, as a philosopher than as a mathematician; almost his only original mathematical result, as regards Geometry, is his proof of Riemann's formula for the infinitesimal arc, and even this proof was far from rigid, until Lie reformed it by his method of continuous groups. In this chapter, therefore, only two of his writings need occupy us, namely the two articles in the Wissenschaftliche Abhandlungen, Vol. II., entitled respectively "Ueber die thatsächlichen Grundlagen der Geometrie," 1866 (p. 610 ff.), and "Ueber die Thatsachen, die der Geometrie zum Grunde liegen," 1868 (p. 618 ff.).

25. In the first of these, which is chiefly philosophical, Helmholtz gives hints of his then uncompleted mathematical work, but in the main contents himself with a statement of results. He announces that he will prove Riemann's quadratic formula for the infinitesimal arc; but for this purpose, he says, we have to start with Congruence, since without it spatial measurement is impossible. Nevertheless, he maintains that Congruence is proved by experience. How we could, without the help of measurement, discover lapses from Congruence, is a point which he leaves undiscussed. He then enunciates the four axioms which he considers essential to Geometry, as follows:

(1) As regards continuity and dimensions. In a space of n dimensions, a point is uniquely determined by the measurement of n continuous variables (coordinates).

(2) As regards the existence of moveable rigid bodies. Between the 2n coordinates of any point-pair of a rigid body, there exists an equation which is the same for all congruent point-pairs. By considering a sufficient number of point-pairs, we get more equations than unknown quantities: this gives us a method of determining the form of these equations, so as to make it possible for them all to be satisfied.

(3) As regards free mobility. Every point can pass freely and continuously from one position to another. From (2) and (3) it follows, that if two systems A and B can be brought into congruence in any one position, this is also possible in every other position.

(4) As regards independence of rotation in rigid bodies (Monodromy). If (n – 1) points of a body remain fixed, so that every other point can only describe a certain curve, then that curve is closed.

These axioms, says Helmholtz, suffice to give, with the axiom of three dimensions, the Euclidean and non-Euclidean systems as the only alternatives. That they suffice, mathematically, cannot be denied, but they seem, in some respects, to go too far. In the first place, there is no necessity to make the axiom of Congruence apply to actual rigid bodies—on this subject I have enlarged in Chapter II.[29] Again, Free Mobility, as distinct from Congruence, hardly needs to be specially formulated: what barrier could empty space offer to a point's progress? The axiom is involved in the homogeneity of space, which is the same thing as the axiom of Congruence. Monodromy, also, has been severely criticized; not only is it evident that it might have been included in Congruence, but even from the purely analytical point of view, Sophus Lie has proved it to be superfluous[30]. Thus the axiom of Congruence, rightly formulated, includes Helmholtz's third and fourth axioms and part of his second axiom. All the four, or rather, as much of them as is relevant to Geometry, are consequences, as we shall see hereafter, of the one fundamental principle of the relativity of position.

26. The second article, which is mainly mathematical, supplies the promised proof of the arc-formula, which is Helmholtz's most important contribution to Geometry. Riemann had assumed this formula, as the simplest of a number of alternatives: Helmholtz proved it to be a necessary consequence of his axioms. The present paper begins with a short repetition of the first, including the statement of the axioms, to which, at the end of the paper, two more are added, (5) that space has three dimensions, and (6) that space is infinite. It is supposed in the text, as also in the first paper, that the measure of curvature cannot be negative, and, consequently, that an infinite space must be Euclidean. This error in both papers is corrected in notes, added after the appearance of Beltrami's paper on negative curvature. It is a sample of the slightly unprofessional nature of Helmholtz's mathematical work on this subject, which elicits from Klein the following remarks[31]: "Helmholtz is not a mathematician by profession, but a physicist and physiologist.... From this non-mathematical quality of Helmholtz, it follows naturally that he does not treat the mathematical portion of his work with the thoroughness which one would demand of a mathematician by trade (von Fach)." He tells us himself that it was the physiological study of vision which led him to the question of the axioms, and it is as a physicist that he makes his axioms refer to actual rigid bodies. Accordingly, we find errors in his mathematics, such as the axiom of Monodromy, and the assumption that the measure of curvature must be positive. Nevertheless, the proof of Riemann's arc-formula is extremely able, and has, on the whole, been substantiated by Lie's more thorough investigations.

27. Helmholtz's other writings on Geometry are almost wholly philosophical, and will be discussed at length in Chapter II. For the present, we may pass to the only other important writer of the second period, Beltrami. As his work is purely mathematical, and contains few controverted points, it need not, despite its great importance, detain us long.

The "Saggio di Interpretazione della Geometria non-Euclidea[32]," which is principally confined to two dimensions, interprets Lobatchewsky's results by the characteristic method of the second period. It shows, by a development of the work of Gauss and Minding[33], that all the propositions in plane Geometry, which Lobatchewsky had set forth, hold, within ordinary Euclidean space, on surfaces of constant negative curvature. It is strange, as Klein points out[34], that this interpretation, which was known to Riemann and perhaps even to Gauss, should have remained so long without explicit statement. This is the more strange, as Lobatchewsky's "Géométrie Imaginaire" had appeared in Crelle, Vol. XVII.[35], and Minding's article, from which the interpretation follows at once, had appeared in Crelle, Vol. XIX. Minding had shewn that the Geometry of surfaces of constant negative curvature, in particular as regards geodesic triangles, could be deduced from that of the sphere by giving the radius a purely imaginary value ia[36]. This result, as we have seen, had also been obtained by Lobatchewsky for his Geometry, and yet it took thirty years for the connection to be brought to general notice.

28. In Beltrami's Saggio, straight lines are, of course, replaced by geodesics; his coordinates are obtained through a point-by-point correspondence with an auxiliary plane, in which straight lines correspond to geodesics on the surface. Thus geodesics have linear equations, and are always uniquely determined by two points. Distances on the surface, however, are not equal to distances on the plane; thus while the surface is infinite, the corresponding portion of the plane is contained within a certain finite circle. The distance of two points on the surface is a certain function of the coordinates, not the ordinary function of elementary Geometry. These relations of plane and surface are important in connection with Cayley's theory of distance, which we shall have to consider next. If we were to define distance on the plane as that function of the coordinates which gives the corresponding distance on the surface, we should obtain what Klein calls "a plane with a hyperbolic system of measurement (Massbestimmung)" in which Cayley's theory of distance would hold. It is evident, however, that the ordinary notion of distance has been presupposed in setting up the coordinate system, so that we do not really get alternative Geometries on one and the same plane. The bearing of these remarks will appear more fully when we come to consider Cayley and Klein.

29. The value of Beltrami's Saggio, in his own eyes, lies in the intelligible Euclidean sense which it gives to Lobatchewsky's planimetry: the corresponding system of Solid Geometry, since it has no meaning for Euclidean space, is barely mentioned in this work. In a second paper[37], however, almost contemporaneous with the first, he proceeds to consider the general theory of n-dimensional manifolds of constant negative curvature. This paper is greatly influenced by Riemann's dissertation; it begins with the formula for the linear element, and proves from this first, that Congruence holds for such spaces, and next, that they have, according to Riemann's definition, a constant negative measure of curvature. (It is instructive to observe, that both in this and in the former Essay, great stress is laid on the necessity of the Axiom of Congruence.)

This work has less philosophical interest than the former, since it does little more than repeat, in a general form, the results which the Saggio had obtained for two dimensions—results which sink, when extended to n dimensions, to the level of mere mathematical constructions. Nevertheless, the paper is important, both as a restoration of negative curvature, which had been overlooked by Helmholtz, and as an analytical treatment of Lobatchewsky's results—a treatment which, together with the Saggio, at last restored to them the prominence they deserved.

Third Period.

30. The third period differs radically, alike in its methods and aims, and in the underlying philosophical ideas, from the period which it replaced. Whereas everything, in the second period, turned on measurement, with its apparatus of Congruence, Free Mobility, Rigid Bodies, and the rest, these vanish completely in the third period, which, swinging to the opposite extreme, regards quantity as a perfectly irrelevant category in Geometry, and dispenses with congruence and the method of superposition. The ideas of this period, unfortunately, have found no exponent so philosophical as Riemann or Helmholtz, but have been set forth only by technical mathematicians. Moreover the change of fundamental ideas, which is immense, has not brought about an equally great change in actual procedure; for though spatial quantity is no longer a part of projective Geometry, quantity is still employed, and we still have equations, algebraic transformations, and so on. This is apt to give rise to confusion, especially in the mind of the student, who fails to realise that the quantities used, so far as the propositions are really projective, are mere names for points, and not, as in metrical Geometry, actual spatial magnitudes.

Nevertheless, the fundamental difference between this period and the former must strike any one at once. Whereas Riemann and Helmholtz dealt with metrical ideas, and took, as their foundations, the measure of curvature and the formula for the linear element—both purely metrical—the new method is erected on the formulae for transformation of coordinates required to express a given collineation. It begins by reducing all so-called metrical notions—distance, angle, etc.—to projective forms, and obtains, from this reduction, a methodological unity and simplicity before impossible. This reduction depends, however, except where the space-constant is negative, upon imaginary figures—in Euclid, the circular points at infinity; it is moreover purely symbolic and analytical, and must be regarded as philosophically irrelevant. As the question concerning the import of this reduction is of fundamental importance to our theory of Geometry, and as Cayley, in his Presidential Address to the British Association in 1883, formally challenged philosophers to discuss the use of imaginaries, on which it depends, I will treat this question at some length. But first let us see how, as a matter of mathematics, the reduction is effected.

31. We shall find, throughout this period, that almost every important proposition, though misleading in its obvious interpretation, has nevertheless, when rightly interpreted, a wide philosophical bearing. So it is with the work of Cayley, the pioneer of the projective method.

The projective formula for angles, in Euclidean Geometry, was first obtained by Laguerre, in 1853. This formula had, however, a perfectly Euclidean character, and it was left for Cayley to generalize it so as to include both angles and distances in Euclidean and non-Euclidean systems alike[38].

Cayley was, to the last, a staunch supporter of Euclidean space, though he believed that non-Euclidean Geometries could be applied, within Euclidean space, by a change in the definition of distance[39]. He has thus, in spite of his Euclidean orthodoxy, provided the believers in the possibility of non-Euclidean spaces with one of their most powerful weapons. In his "Sixth Memoir upon Quantics" (1859), he set himself the task of "establishing the notion of distance upon purely descriptive principles." He showed that, with the ordinary notion of distance, it can be rendered projective by reference to the circular points and the line at infinity, and that the same is true of angles[40]. Not content with this, he suggested a new definition of distance, as the inverse sine or cosine of a certain function of the coordinates; with this definition, the properties usually known as metrical become projective properties, having reference to a certain conic, called by Cayley the Absolute. (The circular points are, analytically, a degenerate conic, so that ordinary Geometry forms a particular case of the above.) He proves that, when the Absolute is an imaginary conic, the Geometry so obtained for two dimensions is spherical Geometry. The correspondence with Lobatchewsky, in the case where the Absolute is real, is not worked out: indeed there is, throughout, no evidence of acquaintance with non-Euclidean systems. The importance of the memoir, to Cayley, lies entirely in its proof that metrical is only a branch of descriptive Geometry.

32. The connection of Cayley's Theory of Distance with Metageometry was first pointed out by Klein[41]. Klein showed in detail that, if the Absolute be real, we get Lobatchewsky's (hyperbolic) system; if it be imaginary, we get either spherical Geometry or a new system, analogous to that of Helmholtz, called by Klein elliptic; if the Absolute be an imaginary point-pair, we get parabolic Geometry, and if, in particular, the point-pair be the circular points, we get ordinary Euclid. In elliptic Geometry, two straight lines in the same plane meet in only one point, not two as in Helmholtz's system. The distinction between the two kinds of Geometry is difficult, and will be discussed later.

33. Since these systems are all obtained from a Euclidean plane, by a mere alteration in the definition of distance, Cayley and Klein tend to regard the whole question as one, not of the nature of space, but of the definition of distance. Since this definition, on their view, is perfectly arbitrary, the philosophical problem vanishes—Euclidean space is left in undisputed possession, and the only problem remaining is one of convention and mathematical convenience[42]. This view has been forcibly expressed by Poincaré: "What ought one to think," he says, "of this question: Is the Euclidean Geometry true? The question is nonsense." Geometrical axioms, according to him, are mere conventions: they are "definitions in disguise[43]." Thus Klein blames Beltrami for regarding his auxiliary plane as merely auxiliary, and remarks that, if he had known Cayley's Memoir, he would have seen the relation between the plane and the pseudosphere to be far more intimate than he supposed[44]. A view which removes the problem entirely from the arena of philosophy demands, plainly, a full discussion. To this discussion we will now proceed.

34. The view in question has arisen, it would seem, from a natural confusion as to the nature of the coordinates employed. Those who hold the view have not adequately realised, I believe, that their coordinates are not spatial quantities, as in metrical Geometry, but mere conventional signs, by which different points can be distinctly designated. There is no reason, therefore, until we already have metrical Geometry, for regarding one function of the coordinates as a better expression of distance than another, so long as the fundamental addition-equation[45] is preserved. Hence, if our coordinates are regarded as adequate for all Geometry, an indeterminateness arises in the expression of distance, which can only be avoided by a convention. But projective coordinates—so our argument will contend—though perfectly adequate for all projective properties, and entirely free from any metrical presupposition, are inadequate to express metrical properties, just because they have no metrical presupposition. Thus where metrical properties are in question, Beltrami remains justified as against Klein; the reduction of metrical to projective properties is only apparent, though the independence of these last, as against metrical Geometry, is perfectly real.

35. But what are projective coordinates, and how are they introduced? This question was not touched upon in Cayley's Memoir, and it seemed, therefore, as if a logical error were involved in using coordinates to define distance. For coordinates, in all previous systems, had been deduced from distance; to use any existing coordinate system in defining distance was, accordingly, to incur a vicious circle. Cayley mentions this difficulty in a note, where he only remarks, however, that he had regarded his coordinates as numbers arbitrarily assigned, on some system not further investigated, to different points. The difficulty has been treated at length by Sir R. Ball (Theory of the Content, Trans. R. I. A. 1889), who urges that if the values of our coordinates already involve the usual measure of distance, then to give a new definition, while retaining the usual coordinates, is to incur a contradiction. He says (op. cit. p. 1): "In the study of non-Euclidean Geometry I have often felt a difficulty which has, I know, been shared by others. In that theory it seems as if we try to replace our ordinary notion of distance between two points by the logarithm of a certain anharmonic ratio[46]. But this ratio itself involves the notion of distance measured in the ordinary way. How, then, can we supersede our old notion of distance by the non-Euclidean notion, inasmuch as the very definition of the latter involves the former?"

36. This objection is valid, we must admit, so long as anharmonic ratio is defined in the ordinary metrical manner. It would be valid, for example, against any attempt to found a new definition of distance on Cremona's account of anharmonic ratio[47], in which it appears as a metrical property unaltered by projective transformation. If a logical error is to be avoided, in fact, all reference to spatial magnitude of any kind must be avoided; for all spatial magnitude, as will be shown hereafter[48], is logically dependent on the fundamental magnitude of distance. Anharmonic ratio and coordinates must alike be defined by purely descriptive properties, if the use afterwards made of them is to be free from metrical presuppositions, and therefore from the objections of Sir R. Ball.

Such a definition has been satisfactorily given by Klein[49], who appeals, for the purpose, to v. Staudt's quadrilateral construction[50]. By this construction, which I have reproduced in outline in [Chapter III. Section A, § 112 ff.], we obtain a purely descriptive definition of harmonic and anharmonic ratio, and, given a pair of points, we can obtain the harmonic conjugate to any third point on the same straight line. On this construction, the introduction of projective coordinates is based. Starting with any three points on a straight line, we assign to them arbitrarily the numbers 0, 1, ∞. We then find the harmonic conjugate to the first with respect to 1, ∞, and assign to it the number 2. The object of assigning this number rather than any other, is to obtain the value –1 for the anharmonic ratio of the four numbers corresponding to the four points[51]. We then find the harmonic conjugate to the point 1, with respect to 2, ∞, and assign to it the number 3; and so on. Klein has shown that by this construction, we can obtain any number of points, and can construct a point corresponding to any given number, fractional or negative. Moreover, when two sets of four points have the same anharmonic ratio, descriptively defined[52], the corresponding numbers also have the same anharmonic ratio. By introducing such a numerical system on two straight lines, or on three, we obtain the coordinates of any point in a plane, or in space. By this construction, which is of fundamental importance to projective Geometry, the logical error, upon which Sir R. Ball bases his criticism, is satisfactorily avoided. Our coordinates are introduced by a purely descriptive method, and involve no presupposition whatever as to the measurement of distance.

37. With this coordinate system, then, to define distance as a certain function of the coordinates is not to be guilty of a vicious circle. But it by no means follows that the definition of distance is arbitrary. All reference to distance has been hitherto excluded, to avoid metrical ideas; but when distance is introduced, metrical ideas inevitably reappear, and we have to remember that our coordinates give no information, primâ facie, as to any of these metrical ideas. It is open to us, of course, if we choose, to continue to exclude distance in the ordinary sense, as the quantity of a finite straight line, and to define the word distance in any way we please. But the conception, for which the word has hitherto stood, will then require a new name, and the only result will be a confusion between the apparent meaning of our propositions, to those who retain the associations belonging to the old sense of the word, and the real meaning, resulting from the new sense in which the word is used.

This confusion, I believe, has actually occurred, in the case of those who regard the question between Euclid and Metageometry as one of the definition of distance. Distance is a quantitative relation, and as such presupposes identity of quality. But projective Geometry deals only with quality—for which reason it is called descriptive—and cannot distinguish between two figures which are qualitatively alike. Now the meaning of qualitative likeness, in Geometry, is the possibility of mutual transformation by a collineation[53]. Any two pairs of points on the same straight line, therefore, are qualitatively alike; their only qualitative relation is the straight line, which both pairs have in common; and it is exactly the qualitative identity of the relations of the two pairs, which enables the difference of their relations to be exhaustively dealt with by quantity, as a difference of distance. But where quantity is excluded, any two pairs of points on the same straight line appear as alike, and even any two sets of three: for any three points on a straight line can be projectively transformed into any other three. It is only with four points in a line that we acquire a projective property distinguishing them from other sets of four, and this property is anharmonic ratio, descriptively defined. The projective Geometer, therefore, sees no reason to give a name to the relation between two points, in so far as this relation is anything over and above the unlimited straight line on which they lie; and when he introduces the notion of distance, he defines it, in the only way in which projective principles allow him to define it, as a relation between four points. As he nevertheless wishes the word to give him the power of distinguishing between different pairs of points, he agrees to take two out of the four points as fixed. In this way, the only variables in distance are the two remaining points, and distance appears, therefore, as a function of two variables, namely the coordinates of the two variable points. When we have further defined our function so that distance may be additive, we have a function with many of the properties of distance in the ordinary sense. This function, therefore, the projective Geometer regards as the only proper definition of distance.

We can see, in fact, from the manner in which our projective coordinates were introduced, that some function of these coordinates must express distance in the ordinary sense. For they were introduced serially, so that, as we proceeded from the zero-point towards the infinity-point, our coordinates continually grew. To every point, a definite coordinate corresponded: to the distance between two variable points, therefore, as a function dependent on no other variables, must correspond some definite function of the coordinates, since these are themselves functions of their points. The function discussed above, therefore, must certainly include distance in the ordinary sense.

But the arbitrary and conventional nature of distance, as maintained by Poincaré and Klein, arises from the fact that the two fixed points, required to determine our distance in the projective sense, may be arbitrarily chosen, and although, when our choice is once made, any two points have a definite distance, yet, according as we make that choice, distance will become a different function of the two variable points. The ambiguity thus introduced is unavoidable on projective principles; but are we to conclude, from this, that it is really unavoidable? Must we not rather conclude that projective Geometry cannot adequately deal with distance? If A, B, C, be three different points on a line, there must be some difference between the relation of A to B and of A to C, for otherwise, owing to the qualitative identity of all points, B and C could not be distinguished. But such a difference involves a relation, between A and B, which is independent of other points on the line; for unless we have such a relation, the other points cannot be distinguished as different. Before we can distinguish the two fixed points, therefore, from which the projective definition starts, we must already suppose some relation, between any two points on our line, in which they are independent of other points; and this relation is distance in the ordinary sense[54]. When we have measured this quantitative relation by the ordinary methods of metrical Geometry, we can proceed to decide what base-points must be chosen, on our line, in order that the projective function discussed above may have the same value as ordinary distance. But the choice of these base-points, when we are discussing distance in the ordinary sense, is not arbitrary, and their introduction is only a technical device. Distance, in the ordinary sense, remains a relation between two points, not between four; and it is the failure to perceive that the projective sense differs from, and cannot supersede, the ordinary sense, which has given rise to the views of Klein and Poincaré. The question is not one of convention, but of the irreducible metrical properties of space. To sum up: Quantities, as used in projective Geometry, do not stand for spatial magnitudes, but are conventional symbols for purely qualitative spatial relations. But distance, quâ quantity, presupposes identity of quality, as the condition of quantitative comparison. Distance in the ordinary sense is, in short, that quantitative relation, between two points on a line, by which their difference from other points can be defined. The projective definition, however, being unable to distinguish a collection of less than four points from any other on the same straight line, makes distance depend on two other points besides those whose relation it defines. No name remains, therefore, for distance in the ordinary sense, and many projective Geometers, having abolished the name, believe the thing to be abolished also, and are inclined to deny that two points have a unique relation at all. This confusion, in projective Geometry, shows the importance of a name, and should make us chary of allowing new meanings to obscure one of the fundamental properties of space.

38. It remains to discuss the manner in which non-Euclidean Geometries result from the projective definition of distance, as also the true interpretation to be given to this view of Metageometry. It is to be observed that the projective methods which follow Cayley deal throughout with a Euclidean plane, on which they introduce different measures of distance. Hence arises, in any interpretation of these methods, an apparent subordination of the non-Euclidean spaces, as though these were less self-subsistent than Euclid's. This subordination is not intended in what follows; on the contrary, the correlation with Euclidean space is regarded as valuable, first, because Euclidean space has been longer studied and is more familiar, but secondly, because this correlation proves, when truly interpreted, that the other spaces are self-subsistent. We may confine ourselves chiefly, in discussing this interpretation, to distances measured along a single straight line. But we must be careful to remember that the metrical definition of distance—which, according to the view here advocated, is the only adequate definition—is the same in Euclidean and in non-Euclidean spaces; to argue in its favour is not, therefore, to argue in favour of Euclid.

The projective scheme of coordinates consists of a series of numbers, of which each represents a certain anharmonic ratio and denotes one and only one point, and which increase uniformly with the distance from a fixed origin, until they become infinite on reaching a certain point. Now Cayley showed that, in Euclidean Geometry, distance may be expressed as the limit of the logarithm of the anharmonic ratio of the two points and the (coincident) points at infinity on their straight line; while, if we assumed that the points at infinity were distinct, we obtained the formula for distance in hyperbolic or spherical Geometry, according as these points were real or imaginary. Hence it follows that, with the projective definition of distance, we shall obtain precisely the formulae of hyperbolic, parabolic or spherical Geometry, according as we choose the point, to which the value +∞ is assigned, at a finite, infinite or imaginary distance (in the ordinary sense) from the point to which we assign the value 0. Our straight line remains, all the while, an ordinary Euclidean straight line. But we have seen that the projective definition of distance fits with the true definition only when the two fixed points to which it refers are suitably chosen. Now the ordinary meaning of distance is required in non-Euclidean as in Euclidean Geometries—indeed, it is only in metrical properties that these Geometries differ. Hence our Euclidean straight line, though it may serve to illustrate other Geometries than Euclid's, can only be dealt with correctly by Euclid. Where we give a different definition of distance from Euclid's, we are still in the domain of purely projective properties, and derive no information as to the metrical properties of our straight line. But the importance, to Metageometry, of this new interpretation, lies in the fact that, having independently established the metrical formulae of non-Euclidean spaces, we find, as in Beltrami's Saggio, that these spaces can be related, by a homographic correspondence, with the points of Euclidean space; and that this can be effected in such a manner as to give, for the distance between two points of our non-Euclidean space, the hyperbolic or spherical measure of distance for the corresponding points of Euclidean space.

39. On the whole, then, a modification of Sir R. Ball's view, which is practically a generalized statement of Beltrami's method, seems the most tenable. He imagines what, with Grassmann, he calls a Content, i.e. a perfectly general three-dimensional manifold, and then correlates its elements, one by one, with points in Euclidean space. Thus every element of the Content acquires, as its coordinates, the ordinary Euclidean coordinates of the corresponding point in Euclidean space. By means of this correlation, our calculations, though they refer to the Content, are carried on, as in Beltrami's Saggio, in ordinary Euclidean space. Thus the confusion disappears, but with it, the supposed Euclidean interpretation also disappears. Sir R. Ball's Content, if it is to be a space at all, must be a space radically different from Euclid's[55]; to speak, as Klein does, of ordinary planes with hyperbolic or elliptic measures of distance, is either to incur a contradiction, or to forego any metrical meaning of distance. Instead of ordinary planes, we have surfaces like Beltrami's, of constant measure of curvature; instead of Euclid's space, we have hyperbolic or spherical space. At the same time, it remains true that we can, by Klein's method, give a Euclidean meaning to every symbolic proposition in non-Euclidean Geometry. For by substituting, for distance, the logarithm above alluded to, we obtain, from the non-Euclidean result, a result which follows from the ordinary Euclidean axioms. This correspondence removes, once for all, the possibility of a lurking contradiction in Metageometry, since, to a proposition in the one, corresponds one and only one proposition in the other, and contradictory results in one system, therefore, would correspond to contradictory results in the other. Hence Metageometry cannot lead to contradictions, unless Euclidean Geometry, at the same moment, leads to corresponding contradictions. Thus the Euclidean plane with hyperbolic or elliptic measure of distance, though either contradictory or not metrical as an independent notion, has, as a help in the interpretation of non-Euclidean results, a very high degree of utility.

40. We have still to discuss Klein's third kind of non-Euclidean Geometry, which he calls elliptic. The difference between this and spherical Geometry is difficult to grasp, but it may be illustrated by a simpler example. A plane, as every one knows, can be wrapped, without stretching, on a cylinder, and straight lines in the plane become, by this operation, geodesics on the cylinder. The Geometries of the plane and the cylinder, therefore, have much in common. But since the generating circle of the cylinder, which is one of its geodesics, is finite, only a portion of the plane is used up in wrapping it once round the cylinder. Hence, if we endeavour to establish a point-to-point correspondence between the plane and the cylinder, we shall find an infinite series of points on the plane for a single point on the cylinder. Thus it happens that geodesics, though on the plane they have only one point in common, may on the cylinder have an infinite number of intersections. Somewhat similar to this is the relation between the spherical and elliptic Geometries. To any one point in elliptic space, two points correspond in spherical space. Thus geodesics, which in spherical space may have two points in common, can never, in elliptic space, have more than one intersection.

But Klein's method can only prove that elliptic Geometry holds of the ordinary Euclidean plane with elliptic measure of distance. Klein has made great endeavours to enforce the distinction between the spherical and elliptic Geometries[56], but it is not immediately evident that the latter, as distinct from the former, is valid.

In the first place, Klein's elliptic Geometry, which arises as one of the alternative metrical systems on a Euclidean plane or in a Euclidean space, does not by itself suffice, if the above discussion has been correct, to prove the possibility of an elliptic space, i.e. of a space having a point-to-point correspondence with the Euclidean space, and having as the ordinary distance between two of its points the elliptic definition of the distance between corresponding points of the Euclidean space. To prove this possibility, we must adopt the direct method of Newcomb (Crelle's Journal, Vol. 83). Now in the first place Newcomb has not proved that his postulates are self-consistent; he has only failed to prove that they are contradictory[57]. This would leave elliptic space in the same position in which Lobatchewsky and Bolyai left hyperbolic space. But further there seems to be, at first sight, in two-dimensional elliptic space, a positive contradiction. To explain this, however, some account of the peculiarities of the elliptic plane will be necessary.

The elliptic plane, regarded as a figure in three-dimensional elliptic space, is what is called a double surface[58], i.e. as Newcomb says (loc. cit. p. 298): "The two sides of a complete plane are not distinct, as in a Euclidean surface.... If ... a being should travel to distance 2D, he would, on his return, find himself on the opposite surface to that on which he started, and would have to repeat his journey in order to return to his original position without leaving the surface." Now if we imagine a two-dimensional elliptic space, the distinction between the sides of a plane becomes unmeaning, since it only acquires significance by reference to the third dimension. Nevertheless, some such distinction would be forced upon us. Suppose, for example, that we took a small circle provided with an arrow, as in the figure, and moved this circle once round the universe. Then the sense of the arrow would be reversed. We should thus be forced, either to regard the new position as distinct from the former, which transforms our plane into a spherical plane, or to attribute the reversal of the arrow to the action of a motion which restores our circle to its original place. It is to be observed that nothing short of moving round the universe would suffice to reverse the sense of the arrow. This reversal seems like an action of empty space, which would force us to regard the points which, from a three-dimensional point of view, are coincident though opposite, as really distinct, and so reduce the elliptic to the spherical plane. But motion, not space, really causes the change, and the elliptic plane is therefore not proved to be impossible. The question is not, however, of any great philosophic importance.

41. In connection with the reduction of metrical to projective Geometry, we have one more topic for discussion. This is the geometrical use of imaginaries, by means of which, except in the case of hyperbolic space, the reduction is effected. I have already contended, on other grounds, that this reduction, in spite of its immense technical importance, and in spite of the complete logical freedom of projective Geometry from metrical ideas, is purely technical, and is not philosophically valid. The same conclusion will appear, if we take up Cayley's challenge at the British Association, in his Presidential Address of 1883.

In this address, Professor Cayley devoted most of his time to non-Euclidean systems. Non-Euclidean spaces, he declared, seemed to him mistaken à priori[59]; but non-Euclidean Geometries, here as in his mathematical works, were accepted as flowing from a change in the definition of distance. This view has been already discussed, and need not, therefore, be further criticised here. What I wish to speak about, is the question with which Cayley himself opened his address, namely, the geometrical use and meaning of imaginary quantities. From the manner in which he spoke of this question, it becomes imperative to treat it somewhat at length. For he said (pp. 8–9):

"... The notion which is the really fundamental one (and I cannot too strongly emphasize the assertion) underlying and pervading the whole notion of modern analysis and Geometry, [is] that of imaginary magnitude in analysis, and of imaginary space (or space as the locus in quo of imaginary points and figures) in Geometry: I use in each case the word imaginary as including real.... Say even the conclusion were that the notion belongs to mere technical mathematics, or has reference to nonentities in regard to which no science is possible, still it seems to me that (as a subject of philosophical discussion) the notion ought not to be thus ignored; it should at least be shown that there is a right to ignore it."

42. This right it is now my purpose to demonstrate. But for fear non-mathematicians should miss the point of Cayley's remark (which has sometimes been erroneously supposed to refer to non-Euclidean spaces), I may as well explain, at the outset, that this question is radically distinct from, and only indirectly connected with, the validity or import of Metageometry. An imaginary quantity is one which involves √–1 : its most general form is a + √–1 b where a and b are real; Cayley uses the word imaginary so as to include real, in order to cover the special case where b = 0. It will be convenient, in what follows, to exclude this wider meaning, and assume that b is not zero. An imaginary point is one whose coordinates involve √–1, i.e. whose coordinates are imaginary quantities. An imaginary curve is one whose points are imaginary—or, in some special uses, one whose equation contains imaginary coefficients. The mathematical subtleties to which this notion leads need not be here discussed; the reader who is interested in them will find an excellent elementary account of their geometrical uses in Klein's Nicht-Euklid, II. pp. 38–46. But for our present purpose, we may confine ourselves to imaginary points. If these are found to have a merely technical import, and to be destitute of any philosophical meaning, then the same will hold of any collection of imaginary points, i.e. of any imaginary curve or surface.

That the notion of imaginary points is of supreme importance in Geometry, will be seen by any one who reflects that the circular points are imaginary, and that the reduction of metrical to projective Geometry, which is one of Cayley's greatest achievements, depends on these points. But to discuss adequately their philosophical import is difficult to me, since I am unacquainted with any satisfactory philosophy of imaginaries in pure Algebra. I will therefore adopt the most favourable hypothesis, and assume that no objection can be successfully urged against this use. Even on this hypothesis, I think, no case can be made out for imaginary points in Geometry.

In the first place, we must exclude, from the imaginary points considered, those whose coordinates are only imaginary with certain special systems of coordinates. For example, if one of a point's coordinates be the tangent from it to a sphere, this coordinate will be imaginary for any point inside the sphere, and yet the point is perfectly real. A point, then, is only to be called imaginary, when, whatever real system of coordinates we adopt, one or more of the quantities expressing these coordinates remains imaginary. For this purpose, it is mathematically sufficient to suppose our coordinates Cartesian—a point whose Cartesian coordinates are imaginary, is a true imaginary point in the above sense.

To discuss the meaning of such a point, it is necessary to consider briefly the fundamental nature of the correspondence between a point and its coordinates. Assuming that elementary Geometry has proved—what I think it does satisfactorily prove—that spatial relations are susceptible of quantitative measurement, then a given point will have, with a suitable system of coordinates, in a space of n dimensions, n quantitative relations to the fixed spatial figure forming the axes of coordinates, and these n quantitative relations will, under certain reservations, be unique—i.e., no other point will have the same quantities assigned to it. (With many possible coordinate systems, this latter condition is not realized: but for that very reason they are inconvenient, and employed only in special problems.) Thus given a coordinate system, and given any set of quantities, these quantities, if they determine a point at all, determine it uniquely. But, by a natural extension of the method, the above reservation is dropped, and it is assumed that to every set of quantities some point must correspond. For this assumption there seems to me no vestige of evidence. As well might a postman assume that, because every house in a street is uniquely determined by its number, therefore there must be a house for every imaginable number. We must know, in fact, that a given set of quantities can be the coordinates of some point in space, before it is legitimate to give any spatial significance to these quantities: and this knowledge, obviously, cannot be derived from operations with coordinates alone, on pain of a vicious circle. We must, to return to the above analogy, know the number of houses in Piccadilly, before we know whether a given number has a corresponding house or not; and arithmetic alone, however subtly employed, will never give us this information.

Thus the distinction which is important is, not the distinction between real and imaginary quantities, but between quantities to which points correspond and quantities to which no points correspond. We can conventionally agree to denote real points by imaginary coordinates, as in the Gaussian method of denoting by the single quantity (a + √–1 b) the point whose ordinary coordinates are a, b. But this does not touch Cayley's meaning. Cayley means that it is of great utility in mathematics to regard, as points with a real existence in space, the assumed spatial correlates of quantities which, with the coordinate system employed, have no correlates in every-day space; and that this utility is supposed, by many mathematicians, to indicate the validity of so fruitful an assumption. To fix our ideas, let us consider Cartesian axes in three-dimensional Euclidean space. Then it appears, by inspection, that a point may be situated at any distance to right or left of any of the three coordinate planes; taking this distance as a coordinate, therefore, it appears that real points correspond to all quantities from -∞ to +∞. The same appears for the other two coordinates; and since elementary Geometry proves their variations mutually independent, we know that one and only one real point corresponds to any three real quantities. But we also know, from the exhaustive method pursued, that all space is covered by the range of these three variable quantities: a fresh set of quantities, therefore, such as is introduced by the use of imaginaries, possesses no spatial correlate, and can be supposed to possess one only by a convenient fiction.

43. The fact that the fiction is convenient, however, may be thought to indicate that it is more than a fiction. But this presumption, I think, can be easily explained away. For all the fruitful uses of imaginaries, in Geometry, are those which begin and end with real quantities, and use imaginaries only for the intermediate steps. Now in all such cases, we have a real spatial interpretation at the beginning and end of our argument, where alone the spatial interpretation is important: in the intermediate links, we are dealing in a purely algebraical manner with purely algebraical quantities, and may perform any operations which are algebraically permissible. If the quantities with which we end are capable of spatial interpretation, then, and only then, our result may be regarded as geometrical. To use geometrical language, in any other case, is only a convenient help to the imagination. To speak, for example, of projective properties which refer to the circular points, is a mere memoria technica for purely algebraical properties; the circular points are not to be found in space, but only in the auxiliary quantities by which geometrical equations are transformed. That no contradictions arise from the geometrical interpretation of imaginaries, is not wonderful: for they are interpreted solely by the rules of Algebra, which we may admit as valid in their application to imaginaries. The perception of space being wholly absent, Algebra rules supreme, and no inconsistency can arise. Wherever, for a moment, we allow our ordinary spatial notions to intrude, the grossest absurdities do arise—every one can see that a circle, being a closed curve, cannot get to infinity. The metaphysician, who should invent anything so preposterous as the circular points, would be hooted from the field. But the mathematician may steal the horse with impunity.

Finally, then, only a knowledge of space, not a knowledge of Algebra, can assure us that any given set of quantities will have a spatial correlate, and in the absence of such a correlate, operations with these quantities have no geometrical import. This is the case with imaginaries in Cayley's sense, and their use in Geometry, great as are its technical advantages, and rigid as is its technical validity, is wholly destitute of philosophical importance.

44. We have now, I think, discussed most of the questions concerning the scope and validity of the projective method. We have seen that it is independent of all metrical presuppositions, and that its use of coordinates does not involve the assumption that spatial magnitudes are measured or expressed by them. We have seen that it is able to deal, by its own methods alone, with the question of the qualitative likeness of geometrical figures, which is logically prior to any comparison as to quantity, since quantity presupposes qualitative likeness. We have seen also that, so far as its legitimate use extends, it applies equally to all homogeneous spaces, and that its criterion of an independently possible space—the determination of a straight line by two points[60]—is not subject to the qualifications and limitations which belong, as we have seen in the case of the cylinder, to the metrical criterion of constant curvature. But we have also seen that, when projective Geometry endeavours to grapple with spatial magnitude, and bring distance and the measurement of angles beneath its sway, its success, though technically valid and important, is philosophically an apparent success only. Metrical Geometry, therefore, if quantity is to be applied to space at all, remains a separate, though logically subsequent branch of Mathematics.

45. It only remains to say a few words about Sophus Lie. As a mathematician, as the inventor of a new and immensely powerful method of analysis, he cannot be too highly praised. Geometry is only one of the numerous subjects to which his theory of continuous groups applies, but its application to Geometry has made a revolution in method, and has rendered possible, in such problems as Helmholtz's, a treatment infinitely more precise and exhaustive than any which was possible before.

The general definition of a group is as follows: If we have any number of independent variables x₁ x₂...x_n, and any series of transformations of these into new variables—the transformations being defined by equations of specified forms, with parameters varying from one transformation to another—then the series of transformations form a group, if the successive application of any two is equivalent to a single member of the original series of transformations. The group is continuous, when we can pass, by infinitesimal gradations within the group, from any one of the transformations to any other.

Now, in Geometry, the result of two successive motions or collineations of a figure can always be obtained by a single motion or collineation, and any motion or collineation can be built up of a series of infinitesimal motions or collineations. Moreover the analytical expression of either is a certain transformation of the coordinates of all the points of the figure[61]. Hence the transformations determining a motion or a collineation are such as to form a continuous group. But the question of the projective equivalence of two figures, to which all projective Geometry is reducible, must always be dealt with by a collineation; and the question of the equality of two figures, to which all metrical Geometry is reducible, must always be decided by a motion such as to cause superposition; hence the whole subject of Geometry may be regarded as a theory of the continuous groups which define all possible collineations and motions.

Now Sophus Lie has developed, at great length, the purely analytical theory of groups; he has therefore, by this method of formulating the problem, a very powerful weapon ready for the attack. In two papers "On the foundations of Geometry[62]," undertaken at Klein's urgent request, he takes premisses which roughly correspond to those of Helmholtz, omitting Monodromy, and applies the theory of groups to the deduction of their consequences[63]. Helmholtz's work, he says, can hardly be looked upon as proving its conclusions, and indeed the more searching analysis of the group-theory reveals several possibilities unknown to Helmholtz. Nevertheless, as a pioneer, devoid of Lie's machinery, Helmholtz deserves, I think, more praise than Lie is willing to give him[64].

Lie's method is perfectly exhaustive; omitting the premiss of Monodromy, the others show that a body has six degrees of freedom, i.e. that the group giving all possible motions of a body will have six independent members; if we keep one point fixed, the number of independent members is reduced to three. He then, from his general theory, enumerates all the groups which satisfy this condition. In order that such a group should give possible motions, it is necessary, by Helmholtz's second axiom, that it should leave invariant some function of the coordinates of any two points. This eliminates several of the groups previously enumerated, each of which he discusses in turn. He is thus led to the following results:

I. In two dimensions, if free mobility is to hold universally, there are no groups satisfying Helmholtz's first three axioms, except those which give the ordinary Euclidean and non-Euclidean motions; but if it is to hold only within a certain region, there is also a possible group in which the curve described by any point in a rotation is not closed, but an equiangular spiral. To exclude this possibility, Helmholtz's axiom of Monodromy is required.

II. In three dimensions, the results go still more against Helmholtz. Assuming free mobility only within a certain region, we have to distinguish two cases: Either free mobility holds, within that region, absolutely without exception, i.e. when one point is held fast, every other point within the region can move freely over a surface: in this case the axiom of Monodromy is unnecessary, and the first three axioms suffice to define our group as that of Euclidean and non-Euclidean motions. Or free mobility, within the specified region, holds only of every point of general position, while the points of a certain line, when one point is fixed, are only able to move on that line, not on a surface: when this is the case, other groups are possible, and can only be excluded by Helmholtz's fourth axiom.

Having now stated the purely mathematical results of Lie's investigations, we may return to philosophical considerations, by which Helmholtz's work was mainly motived. It becomes obvious, not only that exceptions within a certain region, but also that limitation to a certain region, of the axiom of Free Mobility, are philosophically quite impossible and inconceivable. How can a certain line, or a certain surface, form an impassable barrier in space, or have any mobility different in kind from that of all other lines or surfaces? The notion cannot, in philosophy, be permitted for a moment, since it destroys that most fundamental of all the axioms, the homogeneity of space. We not only may, therefore, but must take Helmholtz's axiom of Free Mobility in its very strictest sense; the axiom of Monodromy thus becomes mathematically, as well as philosophically, superfluous. This is, from a philosophical standpoint, the most important of Lie's results.

46. I have now come to the end of my history of Metageometry. It has not been my aim to give an exhaustive account of even the important works on the subject—in the third period, especially, the names of Poincaré, Pasch, Cremona, Veronese, and others who might be mentioned, would have cried shame upon me, had I had any such object. But I have tried to set forth, as clearly as I could, the principles at work in the various periods, the motives and results of successive theories. We have seen how the philosophical motive, at first predominant, has been gradually extruded by the purely mathematical and technical spirit of most recent Geometers. At first, to discredit the Transcendental Aesthetic seemed, to Metageometers, as important as to advance their science; but from the works of Cayley, Klein or Lie, no reader could gather that Kant had ever lived. We have also seen, however, that as the interest in philosophy waned, the interest for philosophy increased: as the mathematical results shook themselves free from philosophical controversies, they assumed gradually a stable form, from which further development, we may reasonably hope, will take the form of growth, rather than transformation. The same gradual development out of philosophy might, I believe, be traced in the infancy of most branches of mathematics; when philosophical motives cease to operate, this is, in general, a sign that the stage of uncertainty as to premisses is past, so that the future belongs entirely to mathematical technique. When this stable stage has been attained, it is time for Philosophy to borrow of Science, accepting its final premisses as those imposed by a real necessity of fact or logic.

47. Now in discussing the systems of Metageometry, we have found two kinds, radically distinct and subject to different axioms. The historically prior kind, which deals with metrical ideas, discusses, to begin with, the conditions of Free Mobility, which is essential to all measurement of space. It finds the analytical expression of these conditions in the existence of a space-constant, or constant measure of curvature, which is equivalent to the homogeneity of space. This is its first axiom.

Its second axiom states that space has a finite integral number of dimensions, i.e. in metrical terms, that the position of a point, relative to any other figure in space, is uniquely determined by a finite number of spatial magnitudes, called coordinates.

The third axiom of metrical Geometry may be called, to distinguish it from the corresponding projective axiom, the axiom of distance. There exists one relation, it says, between any two points, which can be preserved unaltered in a combined motion of both points, and which, in any motion of a system as one rigid body, is always unaltered. This relation we call distance.

The above statement of the three essential axioms of metrical Geometry is taken from Helmholtz as amended by Lie. Lie's own statement of the axioms, as quoted above, has been too much influenced by projective methods to give a historically correct rendering of the spirit of the second period; Helmholtz's statement, on the other hand, requires, as Lie has shewn, very considerable modifications. The above compromise may, therefore, I hope be taken as accepting Lie's corrections while retaining Helmholtz's spirit.

48. But metrical Geometry, though it is historically prior, is logically subsequent to projective Geometry. For projective Geometry deals directly with that qualitative likeness, which the judgment of quantitative comparison requires as its basis. Now the above three axioms of metrical Geometry, as we shall see in Chapter III. Section B, do not presuppose measurement, but are, on the contrary, the conditions presupposed by measurement. Without these axioms, which are common to all three spaces, measurement would be impossible; with them, so I shall contend, measurement is able, though only empirically, to decide approximately which of the three spaces is valid of our actual world. But if these three axioms themselves express, not results, but conditions, of measurement, must they not be equivalent to the statement of that qualitative likeness on which quantitative comparison depends? And if so, must we not expect to find the same axioms, though perhaps under a different form, in projective Geometry?

49. This expectation will not be disappointed. The above three axioms, as we shall see hereafter, are one and all philosophically equivalent to the homogeneity of space, and this in turn is equivalent to the axioms of projective Geometry. The axioms of projective Geometry, in fact, may be roughly stated thus:

I. Space is continuous and infinitely divisible; the zero of extension, resulting from infinite division, is called a Point. All points are qualitatively similar, and distinguished by the mere fact that they lie outside one another.

II. Any two points determine a unique figure, the straight line; two straight lines, like two points, are qualitatively similar, and distinguished by the mere fact that they are mutually external.

III. Three points not in one straight line determine a unique figure, the plane, and four points not in one plane determine a figure of three dimensions. This process may, so far as can be seen à priori, be continued, without in any way interfering with the possibility of projective Geometry, to five or to n points. But projective Geometry requires, as an axiom, that the process should stop with some positive integral number of points, after which, any fresh point is contained in the figure determined by those already given. If the process stops with (n + 1) points, our space is said to have n dimensions.

These three axioms, it will be seen, are the equivalents of the three axioms of metrical Geometry[65], expressed without reference to quantity. We shall find them to be deducible, as before, from the homogeneity of space, or, more generally still, from the possibility of experiencing externality. They will therefore appear as à priori, as essential to the existence of any Geometry and to experience of an external world as such.

50. That some logical necessity is involved in these axioms might, I think, be inferred as probable, from their historical development alone. For the systems of Metageometry have not, in general, been set up as more likely to fit facts than the system of Euclid; with the exception of Zöllner, for example, I know of no one who has regarded the fourth dimension as required to explain phenomena. As regards the space-constant again, though a small space-constant is regarded as empirically possible, it is not usually regarded as probable; and the finite space-constants, with which Metageometry is equally conversant, are not usually thought even possible, as explanations of empirical fact[66]. Thus the motive has been throughout not one of fact, but one of logic. Does not this give a strong presumption, that those axioms which are retained, are retained because they are logically indispensable? If this be so, the axioms common to Euclid and Metageometry will be à priori, while those peculiar to Euclid will be empirical. After a criticism of some differing theories of Geometry, I shall proceed, in Chapters III. and IV., to the proof and consequences of this thesis, which will form the remainder of the present work.

FOOTNOTES:

[5] V. Mémoires de l'Académie royale des Sciences de l'lnstitut de France, T. XII. 1833, for a full statement of his results, with references to former writings.

[6] This bolder method, it appears, had been suggested, nearly a century earlier, by an Italian, Saccheri. His work, which seems to have remained completely unknown until Beltrami rediscovered it in 1889, is called "Euclides ab omni naevo vindicatus, etc." Mediolani, 1733. (See Veronese, Grundzüge der Geometrie, German translation, Leipzig, 1894, p. 636.) His results included spherical as well as hyperbolic space; but they alarmed him to such an extent that he devoted the last half of his book to disproving them.

[7] Klein's first account of elliptic Geometry, as a result of Cayley's projective theory of distance, appeared in two articles entitled "Ueber die sogenannte Nicht-Euklidische Geometrie, I, II," Math. Annalen 4, 6 (1871–2). It was afterwards independently discovered by Newcomb, in an article entitled "Elementary Theorems relating to the geometry of a space of three dimensions, and of uniform positive curvature in the fourth dimension," Crelle's Journal für die reine und angewandte Mathematik, Vol. 83 (1877). For an account of the mathematical controversies concerning elliptic Geometry, see Klein's "Vorlesungen über Nicht-Euklidische Geometrie," Göttingen 1893, I. p. 284 ff. A bibliography of the relevant literature up to the year 1878 was given by Halsted in the American Journal of Mathematics, Vols. 1, 2.

[8] Veronese (op. cit. p. 638) denies the priority of Gauss in the invention of a non-Euclidean system, though he admits him to have been the first to regard the axiom of parallels as indemonstrable. His grounds for the former assertion seem scarcely adequate: on the evidence against it, see Klein, Nicht-Euklid, I. pp. 171–174.

[9] V. Briefwechsel mit Schumacher, Bd. II. p. 268.

[10] f. Helmholtz, Wiss. Abh. II. p. 611.

[11] Crelle's Journal, 1837.

[12] Theorie der Parallellinien, Berlin, 1840. Republished, Berlin, 1887. Translated by Halsted, Austin, Texas, U.S.A. 4th edition, 1892.

[13] Frischauf, Absolute Geometrie, nach Johann Bolyai, Leipzig, 1872. Halsted, The Science Absolute of Space, translated from the Latin, 4th edition, Austin, Texas, U.S.A. 1896.

[14] Both Lobatchewsky and Bolyai, as Veronese remarks, start rather from the point-pair than from distance. See Frischauf, Absolute Geometrie, Anhang.

[15] Compare Stallo, Concepts of Modern Physics, p. 248.

[16] Gesammelte Werke, pp. 255–268.

[17] On the history of this word, see Stallo, Concepts of Modern Physics, p. 258. It was used by Kant, and adapted by Herbart to almost the same meaning as it bears in Riemann. Herbart, however, also uses the word Reihenform to express a similar idea. See Psychologie als Wissenschaft, I. § 100 and II. § 139, where Riemann's analogy with colours is also suggested.

[18] Compare Erdmann's "Grössenbegriff vom Raum."

[19] Compare Veronese, op. cit. p. 642: "Riemann ist in seiner Definition des Begriffs Grösse dunkel." See also Veronese's whole following criticism.

[20] Vorträge und Reden, Vol. II. p. 18.

[21] Cf. Klein, Nicht-Euklid, I. p. 160.

[22] Since we are considering the curvature at a point, we are only concerned with the first infinitesimal elements of the geodesics that start from such a point.

[23] Disquisitiones generales circa superficies curvas, Werke, Bd. IV. SS. 219–258, 1827.

[24] Nevertheless, the Geometries of different surfaces of equal curvature are liable to important differences. For example, the cylinder is a surface of zero curvature, but since its lines of curvature in one direction are finite, its Geometry coincides with that of the plane only for lengths smaller than the circumference of its generating circle (see Veronese, op. cit. p. 644). Two geodesics on a cylinder may meet in many points. For surfaces of zero curvature on which this is not possible, the identity with the plane may be allowed to stand. Otherwise, the identity extends only to the properties of figures not exceeding a certain size.

[25] For we may consider two different parts of the same surface as corresponding parts of different surfaces; the above proposition then shows that a figure can be reproduced in one part when it has been drawn in another, if the measures of curvature correspond in the two parts.

[26] Crelle, Vols, XIX., XX., 1839–40.

[27] In this formula, u, v may be the lengths of lines, or the angles between lines, drawn on the surface, and having thus no necessary reference to a third dimension.

[28] In what follows, I have given rather Klein's exposition of Riemann, than Riemann's own account. The former is much clearer and fuller, and not substantially different in any way. V. Klein, Nicht-Euklid, I. pp. 206 ff.

[29] See [§§ 69–73.]

[30] Grundlagen der Geometrie, I. and II., Leipziger Berichte, 1890; v. end of present chapter, [§ 45.]

[31] Nicht-Euklid, I. pp. 258–9.

[32] Giornale di Matematiche, Vol. VI., 1868. Translated into French by J. Hoüel in the "Annales Scientifiques de l'École Normale Supérieure," Vol. VI. 1869.

[33] Crelle's Journal, Vols. XIX. XX., 1839–40.

[34] Nicht-Euklid, I. p. 190.

[35] This article is more trigonometrical and analytical than the German book, and therefore makes the above interpretation peculiarly evident.

[36] Such surfaces are by no means particularly remote. One of them, for example, is formed by the revolution of the common Tractrix

x = a sin φ, y = a ₍log tan φ 2 + cos φ₎.

[37] "Teoria fondamentale degli spazii di curvatura costanta," Annali di Matematica, II. Vol. 2, 1868–9. Also translated by J. Hoüel, loc. cit.

[38] See Klein, Nicht-Euklid, I. p. 47 ff., and the references there given.

[39] See quotation below, from his British Association Address.

[40] Compare the opening sentence, due to Cayley, of Salmon's Higher Plane Curves.

[41] V. Nicht-Euklid, I. Chaps. I. and II.

[42] See p. 9 of Cayley's address to the Brit. Ass. 1883. Also a quotation from Klein in Erdmann's Axiome der Geometrie, p. 124 note.

[43] Nature, Vol. XLV. p. 407.

[44] Nicht-Euklid, I. p. 200.

[45] I.e. the equation AB + BC = AC, for three points in one straight line.

[46] The formula substituted by Klein for Cayley's inverse sine or cosine. The two are equivalent, but Klein's is mathematically much the more convenient.

[47] Elements of Projective Geometry, Second Edition, Oxford, 1893, Chap. IX.

[48] Chap. III. Section B.

[49] See Nicht-Euklid, I. p. 338 ff.

[50] See his Geometrie der Lage, § 8, Harmonische Gebilde.

[51] The anharmonic ratio of four numbers, p, q, r, s, is defined as

(p - q).(r - s) _/ (p - r).(q - s).

[52] I.e. as transformable into each other by a collineation. See [Chap. III. Sec. A, § 110.]

[53] See [Chap. III. Sec. A.]

[54] It follows from this, that the reduction of metrical to projective properties, even when, as in hyperbolic Geometry, the Absolute is real, is only apparent, and has a merely technical validity.

[55] Sir R. Ball does not regard his non-Euclidean content as a possible space (v. op. cit. p. 151). In this important point I disagree with his interpretation, holding such a content to be a space as possible, à priori, as Euclid's, and perhaps actually true within the margin due to errors of observation.

[56] See Nicht-Euklid, I. p. 97 ff. and p. 292 ff.

[57] Newcomb says (loc. cit. p. 293): "The system here set forth is founded on the following three postulates.

"1. I assume that space is triply extended, unbounded, without properties dependent either on position or direction, and possessing such planeness in its smallest parts that both the postulates of the Euclidean Geometry, and our common conceptions of the relations of the parts of space are true for every indefinitely small region in space.

"2. I assume that this space is affected with such curvature that a right line shall always return into itself at the end of a finite and real distance 2D without losing, in any part of its course, that symmetry with respect to space on all sides of it which constitutes the fundamental property of our conception of it.

"3. I assume that if two right lines emanate from the same point, making the indefinitely small angle a with each other, their distance apart at the distance r from the point of intersection will be given by the equation

s = 2aD π sin rπ 2D .

The right line thus has this property in common with the Euclidean right line that two such lines intersect only in a single point. It may be that the number of points in which two such lines can intersect admit of being determined from the laws of curvature, but not being able so to determine it, I assume as a postulate the fundamental property of the Euclidean right line."

It is plain that in the absence of the determination spoken of, the possibility of elliptic space is not established. It may be possible, for example, to prove that, in a space where there is a maximum to distance, there must be an infinite number of straight lines joining two points of maximum distance. In this event, elliptic space would become impossible.

[58] For an elucidation of this term, see Klein, Nicht-Euklid, I. p. 99 ff.

[59] Cf. p. 9 of Report: "My own view is that Euclid's twelfth axiom, in Playfair's form of it, does not need demonstration, but is part of our notion of space, of the physical space of our experience, but which is the representation lying at the bottom of all external experience."

[60] The exception to this axiom, in spherical space, presupposes metrical Geometry, and does not destroy the validity of the axiom for projective Geometry. See [Chap. III. Sec. B, § 171.]

[61] Mathematicians of Lie's school have a habit, at first somewhat confusing, of speaking of motions of space instead of motions of bodies, as though space as a whole could move. All that is meant is, of course, the equivalent motion of the coordinate axes, i.e. a change of axes in the usual elementary sense.

[62] "Ueber die Grundlagen der Geometrie," Leipziger Berichte, 1890. The problem of these two papers is really metrical, since it is concerned, not with collineations in general, but with motions. The problem, however, is dealt with by the projective method, motions being regarded as collineations which leave the Absolute unchanged. It seemed impossible, therefore, to discuss Lie's work, until some account had been given of the projective method.

[63] Lie's premisses, to be accurate, are the following:

Let

x₁ = f (x, y, z, a₁, a₂...)
x₂ = φ (x, y, z, a₁, a₂...)
x₃ = ψ (x, y, z, a₁, a₂...)

give an infinite family of real transformations of space, as to which we make the following hypotheses:

A. The functions f, φ, ψ, are analytical functions of

x, y, z, a₁, a₂....

B. Two points x₁y₁z₁, x₂y₂z₂ possess an invariant, i.e.

Ω(x₁, y₁, z₁, x₂, y₂, z₂) = Ω(x₁′, y₁′, z₁′, x₂′, y₂′, z₂′)

where x₁′..., x₂′..., are the transformed coordinates of the two points.

C. Free Mobility: i.e., any point can be moved into any other position; when one point is fixed, any other point of general position can take up ∞² positions; when two points are fixed, any other of general position can take up ∞¹ positions; when three, no motion is possible—these limitations being results of the equations given by the invariant Ω.

[64] On this point, cf. Klein, Höhere Geometrie, Göttingen, 1893, II. pp. 225–244, especially pp. 230–1.

[65] Axiom II. of the metrical triad corresponds to Axiom III. of the projective, and vice versâ.

[66] Cf. Helmholtz, Wiss. Abh. Vol. II. p. 640, note: "Die Bearbeiter der Nicht-Euklidischen Geometrie (haben) deren objective Wahrheit nie behauptet."

CHAPTER II.
CRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL THEORIES OF GEOMETRY.

51. We have now traced the mathematical development of the theory of geometrical axioms, from the first revolt against Euclid to the present day. We may hope, therefore, to have at our command the technical knowledge required for the philosophy of the subject. The importance of Geometry, in the theories of knowledge which have arisen in the past, can scarcely be exaggerated. In Descartes, we find the whole theory of method dominated by analytical Geometry, of whose fruitfulness he was justly proud. In Spinoza, the paramount influence of Geometry is too obvious to require comment. Among mathematicians, Newton's belief in absolute space was long supreme, and is still responsible for the current formulation of the laws of motion. Against this belief on the one hand, and against Leibnitz's theory of space on the other, and not, as Caird has pointed out[67], against Hume's empiricism, was directed that keystone of the Critical Philosophy, the Kantian doctrine of space. Thus Geometry has been, throughout, of supreme importance in the theory of knowledge.

But in a criticism of representative modern theories of Geometry, which is designed to be, not a history of the subject, but an introduction to, and defence of, the views of the author, it will not be necessary to discuss any more ancient theory than that of Kant. Kant's views on this subject, true or false, have so dominated subsequent thought, that whether they were accepted or rejected, they seemed equally potent in forming the opinions, and the manner of exposition, of almost all later writers.

Kant.

52. It is not my purpose, in this chapter, to add to the voluminous literature of Kantian criticism, but only to discuss the bearing of Metageometry on the argument of the Transcendental Aesthetic, and the aspect under which this argument must be viewed in a discussion of Geometry[68]. On this point several misunderstandings seem to me to have had wide prevalence, both among friends and foes, and these misunderstandings I shall endeavour, if I can, to remove.

In the first place, what does Kant's doctrine mean for Geometry? Obviously not the aspect of the doctrine which has been attacked by psychologists, the "Kantian machine-shop" as James calls it—at any rate, if this can be clearly separated from the logical aspect. The question whether space is given in sensation, or whether, as Kant maintained, it is given by an intuition to which no external matter corresponds, may for the present be disregarded. If, indeed, we held the view which seems crudely to sum up the standpoint of the Critique, the view that all certain knowledge is self-knowledge, then we should be committed, if we had decided that Geometry was apodeictic, to the view that space is subjective. But even then, the psychological question could only arise when the epistemological question had been solved, and could not, therefore, be taken into account in our first investigation. The question before us is precisely the question whether, or how far, Geometry is apodeictic, and for the moment we have only to investigate this question, without fear of psychological consequences.

53. Now on this question, as on almost all questions in the Aesthetic or the Analytic, Kant's argument is twofold. On the one hand, he says, Geometry is known to have apodeictic certainty: therefore space must be à priori and subjective. On the other hand, it follows, from grounds independent of Geometry, that space is subjective and à priori; therefore Geometry must have apodeictic certainty. These two arguments are not clearly distinguished in the Aesthetic, but a little analysis, I think, will disentangle them. Thus in the first edition, the first two arguments deduce, from non-geometrical grounds, the apriority of space; the third deduces the apodeictic certainty of Geometry, and maintains, conversely, that no other view can account for this certainty[69]; the last two arguments only maintain that space is an intuition, not a concept. In the second edition, the double argument is clearer, the apriority of space being proved independently of Geometry in the metaphysical deduction, and deduced from the certainty of Geometry, as the only possible explanation of this, in the transcendental deduction. In the Prolegomena, the latter argument alone is used, but in the Critique both are employed.

54. Now it must be admitted, I think, that Metageometry has destroyed the legitimacy of the argument from Geometry to space; we can no longer affirm, on purely geometrical grounds, the apodeictic certainty of Euclid. But unless Metageometry has done more than this—unless it has proved, what I believe it alone cannot prove, that Euclid has not apodeictic certainty—then Kant's other line of argument retains what force it may ever have had. The actual space we know, it may say, is admittedly Euclidean, and is proved, without any reference to Geometry, to be à priori; hence Euclid has apodeictic certainty, and non-Euclid stands condemned. To this it is no answer to urge, with the Metageometers, that non-Euclidean systems are logically self-consistent; for Kant is careful to argue that geometrical reasoning, by virtue of our intuition of space, is synthetic, and cannot, though à priori, be upheld by the principle of contradiction alone[70]. Unless non-Euclideans can prove, what they have certainly failed to prove up to the present, that we can frame an intuition of non-Euclidean spaces, Kant's position cannot be upset by Metageometry alone, but must also be attacked, if it is to be successfully attacked, on its purely philosophical side.

55. For such an attack, two roads lie open: either we may disprove the first two arguments of the Aesthetic, or we may criticize, from the standpoint of general logic, the Kantian doctrine of synthetic à priori judgments and their connection with subjectivity. Both these attacks, I believe, could be conducted with some success; but if we are to disprove the apodeictic certainty of Geometry, one or other is essential, and both, I believe, will be found only partially successful. It will be my aim to prove, in discussing these two lines of attack, (1) that the distinction of synthetic and analytic judgments is untenable, and further, that the principle of contradiction can only give fruitful results on the assumption that experience in general, or, in a particular science, some special branch of experience, is to be formally possible; (2) that the first two arguments of the Transcendental Aesthetic suffice to prove, not Euclidean space, but some form of externality—which may be sensational or intuitional, but not merely conceptual—a necessary prerequisite of experience of an external world. In the third and fourth chapters, I shall contend, as a result of these conclusions, that those axioms, which Euclid and Metageometry have in common, coincide with those properties of any form of externality which are deducible, by the principle of contradiction, from the possibility of experience of an external world. These properties, then, may be said, though not quite in the Kantian sense, to be à priori properties of space, and as to these, I think, a modified Kantian position may be maintained. But the question of the subjective or objective nature of space may be left wholly out of account during the course of this discussion, which will gain by dealing exclusively with logical, as opposed to psychological points of view.

56. (1) Kant's logical position. The doctrine of synthetic and analytic judgments—at any rate if this is taken as the corner-stone of Epistemology—has been so completely rejected by most modern logicians[71], that it would demand little attention here, but for the fact that an enthusiastic French Kantian, M. Renouvier, has recently appealed to it, with perfect confidence, on the very question of Geometry[72]. And it must be owned, with M. Renouvier, that if such judgments existed, in the Kantian sense, non-Euclidean Geometry, which makes no appeal to intuition, could have nothing to say against them. M. Renouvier's contention, therefore, forces us briefly to review the arguments against Kant's doctrine, and briefly to discuss what logical canon is to replace it.

Every judgment—so modern logic contends—is both synthetic and analytic; it combines parts into a whole, and analyses a whole into parts[73]. If this be so, the distinction of analysis and synthesis, whatever may be its importance in pure Logic, can have no value in Epistemology. But such a doctrine, it must be observed, allows full scope to the principle of contradiction: this criterion, since all judgments, in one aspect at least, are analytic, is applicable to all judgments alike. On the other hand, the whole which is analysed must be supposed already given, before the parts can be mutually contradictory: for only by connection in a given whole can two parts or adjectives be incompatible. Thus the principle of contradiction remains barren until we already have some judgments, and even some inference: for the parts may be regarded, to some extent, as an inference from the whole, or vice versâ. When once the arch of knowledge is constructed, the parts support one another, and the principle of contradiction is the keystone: but until the arch is built, the keystone remains suspended, unsupported and unsupporting, in the empty air. In other words, knowledge once existent can be analysed, but knowledge which should have to win every inch of the way against a critical scepticism, could never begin, and could never attain that circular condition in which alone it can stand.

But Kant's doctrine, if true, is designed to restrain a critical scepticism even where it might be effective. Certain fundamental propositions, he says, are not deducible from logic, i.e. their contradictories are not self-contradictory; they combine a subject and predicate which cannot, in any purely logical way, be shewn to have any connection, and yet these judgments have apodeictic certainty. But concerning such judgments, Kant is generally careful not to rely upon the mere subjective conviction that they are undeniable: he proves, with every precaution, that without them experience would be impossible. Experience consists in the combination of terms which formal logic leaves apart, and presupposes, therefore, certain judgments by which a framework is made for bringing such terms together. Without these judgments—so Kant contends—all synthesis and all experience would be impossible. If, therefore, the detail of the Kantian reasoning be sound, his results may be obtained by the principle of contradiction plus the possibility of experience, as well as by his distinction of synthetic and analytic judgments.

Logic, at the present day, arrogates to itself at once a wider and a narrower sphere than Kant allowed to it. Wider, because it believes itself capable of condemning any false principle or postulate; narrower, because it believes that its law of contradiction, without a given whole or a given hypothesis, is powerless, and that two terms, per se, though they may be different, cannot be contradictories, but acquire this relation only by combination in a whole about which something is known, or by connection with a postulate which, for some reason, must be preserved. Thus no judgment, per se, is either analytic or synthetic, for the severance of a judgment from its context robs it of its vitality, and makes it not truly a judgment at all. But in its proper context it is neither purely synthetic nor purely analytic; for while it is the further determination of a given whole, and thus in so far analytic, it also involves the emergence of new relations within this whole, and is so far synthetic.

57. We may retain, however, a distinction roughly corresponding to the Kantian à priori and à posteriori, though less rigid, and more liable to change with the degree of organisation of knowledge. Kant usually endeavoured to prove, as observed above, that his synthetic à priori propositions were necessary prerequisites of experience; now although we cannot retain the term synthetic, we can retain the term à priori, for those assumptions, or those postulates, from which alone the possibility of experience follows. Whatever can be deduced from these postulates, without the aid of the matter of experience, will also, of course, be à priori. From the standpoint of general logic, the laws of thought and the categories, with the indispensable conditions of their applicability, will be alone à priori; but from the standpoint of any special science, we may call à priori whatever renders possible the experience which forms the subject-matter of our science. In Geometry, to particularize, we may call à priori whatever renders possible experience of externality as such.

It is to be observed that this use of the term is at once more rationalistic and less precise than that of Kant. Kant would seem to have supposed himself immediately aware, by inspection, that some knowledge was apodeictic, and its subject-matter, therefore, à priori: but he did not always deduce its apriority from any further principle. Here, however, it is to be shown, before admitting apriority, that the falsehood of the judgment in question would not be effected by a mere change in the matter of experience, but only by a change which should render some branch of experience formally impossible, i.e. inaccessible to our methods of cognition. The above use is also less precise, for it varies according to the specialization of the experience we are assuming possible, and with every progress of knowledge some new connection is perceived, two previously isolated judgments are brought into logical relation, and the à priori may thus, at any moment, enlarge its sphere, as more is found deducible from fundamental postulates.

58. (2) Kant's arguments for the apriority of space. Having now discussed the logical canon to be used as regards the à priori, we may proceed to test Kant's arguments as regards space. The argument from Geometry, as remarked above, is upset by Metageometry, at least so far as those properties are concerned, which belong to Euclid but not to non-Euclidean spaces; as regards the common properties of both kinds of space, we cannot decide on their apriority till we have discussed the consequences of denying them, which will be done in Chapter III. As regards the two arguments which prove that space is an intuition, not a concept, they would call for much discussion in a special criticism of Kant, but here they may be passed by with the obvious comment that infinite homogeneous Euclidean space is a concept, not an intuition—a concept invented to explain an intuition, it is true, but still a pure concept[74]. And it is this pure concept which, in all discussions of Geometry, is primarily to be dealt with; the intuition need only be referred to where it throws light on the functions or the nature of the concept. The second of Kant's arguments, that we can imagine empty space, though not the absence of space, is false if it means a space without matter anywhere, and irrelevant if it merely means a space between matters and regarded as empty[75]. The only argument of importance, then, is the first argument. But I must insist, at the outset, that our problem is purely logical, and that all psychological implications must be excluded to the utmost possible extent. Moreover, as will be proved in Chapter IV., the proper function of space is to distinguish between different presented things, not between the Self and the object of sensation or perception. The argument then becomes the following: consciousness of a world of mutually external things demands, in presentations, a cognitive but non-inferential element leading to the discrimination of the objects presented. This element must be non-inferential, for from whatever number or combination of presentations, which did not of themselves demand diversity in their objects, I could never be led to infer the mutual externality of their objects. Kant says: "In order that sensations may be ascribed to something external to me ... and similarly in order that I may be able to present them as outside and beside one another, ... the presentation of space must be already present." But this goes rather too far: in the first place, the question should be only as to the mutual externality of presented things, not as to their externality to the Self[76]; and in the second place, things will appear mutually external if I have the presentation of any form of externality, whether Euclidean or non-Euclidean. Whatever may be true of the psychological scope of this argument—whose validity is here irrelevant—the logical scope extends, not to Euclidean space, but only to any form of externality which could exist intuitively, and permit knowledge, in beings with our laws of thought, of a world of diverse but interrelated things.

Moreover externality, to render the scope of the argument wholly logical, must not be left with a sensational or intuitional meaning, though it must be supposed given in sensation or intuition. It must mean, in this argument, the fact of Otherness[77], the fact of being different from some other thing: it must involve the distinction between different things, and must be that element, in a cognitive state, which leads us to discriminate constituent parts in its object. So much, then, would appear to result from Kant's argument, that experience of diverse but interrelated things demands, as a necessary prerequisite, some sensational or intuitional element, in perception, by which we are led to attribute complexity to objects of perception[78]; that this element, in its isolation may be called a form of externality; and that those properties of this form, if any such be found, which can be deduced from its mere function of rendering experience of interrelated diversity possible, are to be regarded as à priori. What these properties are, and how the various lines of argument here suggested converge to a single result, we shall see in Chapters III. and IV.

59. In the philosophers who followed Kant, Metaphysics, for the most part, so predominated over Epistemology, that little was added to the theory of Geometry. What was added, came indirectly from the one philosopher who stood out against the purely ontological speculations of his time, namely Herbart. Herbart's actual views on Geometry, which are to be found chiefly in the first section of his Synechologie, are not of any great value, and have borne no great fruit in the development of the subject. But his psychological theory of space, his construction of extension out of series of points, his comparison of space with the tone and colour-series, his general preference for the discrete above the continuous, and finally his belief in the great importance of classifying space with other forms of series (Reihenformen[79]), gave rise to many of Riemann's epoch-making speculations, and encouraged the attempt to explain the nature of space by its analytical and quantitative aspect alone[80]. Through his influence on Riemann, he acquired, indirectly, a great importance in geometrical philosophy. To Riemann's dissertation, which we have already discussed in its mathematical aspect, we must now return, considering, this time, only its philosophical views.

Riemann.

60. The aim of Riemann's dissertation, as we saw in Chapter I., was to define space as a species of manifold, i.e. as a particular kind of collection of magnitudes. It was thus assumed, to begin with, that spatial figures could be regarded as magnitudes, and the axioms which emerged, accordingly, determined only the particular place of these among the many algebraically possible varieties of magnitudes. The resulting formulation of the axioms—while, from the mathematical standpoint of metrical Geometry, it was almost wholly laudable—must, from the standpoint of philosophy, be regarded, in my opinion, as a petitio principii. For when we have arrived at regarding spatial figures as magnitudes, we have already traversed the most difficult part of the ground. The axioms of metrical Geometry—and it is metrical Geometry, exclusively, which is considered in Riemann's Essay—will appear, in Chapter III., to be divisible into two classes. Of these, the first class—which contains the axioms common to Euclid and Metageometry, the only axioms seriously discussed by Riemann—are not the results of measurement, nor of any conception of magnitude, but are conditions to be fulfilled before measurement becomes possible. The second class only—those which express the difference between Euclidean and non-Euclidean spaces—can be deduced as results of measurement or of conceptions of magnitude. As regards the first class, on the contrary, we shall see that the relativity of position—by which space is distinguished from all other known manifolds, except time—leads logically to the necessity of three of the most distinctive axioms of Geometry, and yet this relativity cannot be called a deduction from conceptions of magnitude. In analytical Geometry, owing to the fact that coordinate systems start from points, and hence build up lines and surfaces, it is easy to suppose that points can be given independently of lines and of each other, and thus the relativity of position is lost sight of. The error thus suggested by mathematics was probably reinforced by Herbart's theory of space, which, by its serial character, as we have seen, appeared to him to facilitate a construction out of successive points, and to which Riemann acknowledges his indebtedness both in his Dissertation and elsewhere. The same error reappears in Helmholtz, in whom it is probably due wholly to the methods of analytical Geometry. It is a striking fact that, throughout the writings of these two men, there is not, so far as I know, one allusion to the relativity of position, that property of space from which, as our next chapter will shew, the richest quarry of consequences can be extracted. This is not a result of any conception of magnitude, but follows from the nature of our space-intuition; yet no one, surely, could call it empirical, since it is bound up in the very possibility of locating things there as opposed to here.

61. Indeed we can see, from a purely logical consideration of the judgment of quantity, that Riemann's manner of approaching the problem can never, by legitimate methods, attain to a philosophically sound formulation of the axioms. For quantity is a result of comparison of two qualitatively similar objects, and the judgment of quantity neglects altogether the qualitative aspect of the objects compared. Hence a knowledge of the essential properties of space can never be obtained from judgments of quantity, which neglect these properties, while they yet presuppose them. As well might one hope to learn the nature of man from a census. Moreover, the judgment of quantity is the result of comparison, and therefore presupposes the possibility of comparison. To know whether, or by what means, comparison is possible, we must know the qualities of the things compared and of the medium in which comparison is effected; while to know that quantitative comparison is possible, we must know that there is a qualitative identity between the things compared, which again involves a previous qualitative knowledge. When spatial figures have once been reduced to quantity, their quality has already been neglected, as known and similar to the quality of other figures. To hope, therefore, for the qualities of space, from a comparison of its expression as pure quantity with other pure quantities, is an error natural to an analytical geometer, but an error, none the less, from which there is no return to the qualitative basis of spatial quantity.

62. We must entirely dissent, therefore, from the disjunction which underlies Riemann's philosophy of space. Either the axioms must be consequences of general conceptions of magnitude, he thinks, or else they can only be proved by experience (p. 255). Whatever can be derived from general conceptions of magnitude, we may retort, cannot be an à priori adjective of space: for all the necessary adjectives of space are presupposed in any judgment of spatial quantity, and cannot, therefore, be consequences of such a judgment. Riemann's disjunction, accordingly, since one of its alternatives is obviously impossible, really begs the question. In formulating the axioms of metrical Geometry, our question should be: What axioms, i.e. what adjectives of space, must be presupposed, in order that quantitative comparison of the parts of space may be possible at all? And only when we have determined these conditions, which are à priori necessary to any quantitative science of space, does the second question arise: what inferences can we draw, as to space, from the observed results of this quantitative science, i.e. of this measurement of spatial figures? The conditions of measurement themselves, though not results of any conception of magnitude, will be à priori, if it can be shown that, without them, experience of externality would be impossible.

After this initial protest against Riemann's general philosophical position, let us proceed to examine, in detail, his use of the notion of a manifold.

63. In the first place there is, if I am not mistaken, considerable obscurity in the definition of a manifold, of which an almost verbal rendering was given in Chapter I. What is meant, to begin with, by a general conception capable of various determinations? Does not this property belong to all conceptions? It affords, certainly, a basis for counting, but if continuous quantity is to arise, we must, surely, have some less discrete formulation. It might afford a basis, for example, for the distinction of points in projective Geometry, but projective Geometry has nothing to do with quantity. Something more fluid and flexible than a conception, one would think, is necessary as the basis of continua. Then, again, what is meant by a quantum of a manifold? In space, the answer is obvious: what is meant is a piece of volume. But how about Riemann's other continuous manifold, colour? Does a quantum of colour mean a single line in the spectrum, or a band of finite thickness? In either case, what are the magnitudes to be compared? And how is superposition necessary, or even possible? A colour is fixed by its position in the spectrum: two lines in the same spectrum cannot be superposed, and two lines in different spectra need not be—their positions in their respective spectra suffice, or even, roughly, their immediate sense-quality. The fact is, Riemann had space in his mind from the start, and many of the properties, which he enunciates as belonging to all manifolds, belong, as a matter of fact, only to space. It is far from clear what the magnitudes are which the various determinations make possible. Do these magnitudes measure the elements of the manifold, or the relations between elements? This is surely a very fundamental point, but it is one which Riemann never touches on. In the former case, the superposition which he speaks of becomes unnecessary, since the magnitude is inherent in the element considered. We do not require superposition to measure quantities corresponding to different tones or colours; these can be discovered by analysis of single tones or colours. With space, on the other hand, if we seek for elements, we can find none except points, and no analysis of a point will find magnitudes inherent in it—such magnitudes are a fiction of coordinate Geometry. The magnitudes which space deals with, as we shall see in Chapter III., are relations between points, and it is for this reason that superposition is essential to space-measurement. There is no inherent quality in a single point, as there is in a single colour, by which it can be quantitatively distinguished from another. Thus the conception of a manifold, as defined by Riemann, either does not include colours, or does not involve superposition as the only means of measurement. From this dilemma there is no escape.

64. But if "measurement consists in a superposition of the magnitudes compared" (p. 256), does it not follow immediately that measurement is logically possible only where such superposition leaves the magnitudes unchanged? And therefore that measurement, as above defined, involves, as an à priori condition, that magnitudes are unchanged by motion? This consequence is not drawn by Riemann; indeed he proceeds immediately (pp. 256–7) to consider what he calls a general portion of the doctrine of magnitude (Grössenlehre), independent of measurement. But how is any doctrine of magnitude possible, in which the magnitudes cannot be measured? The reason of the confusion is, that Riemann's definition of measurement is applicable to no single manifold except space, since it depends on the noteworthy property that what we measure in Geometry is not points, but relations between points, and the latter, though not the former, may of course be unaltered by motion. Let us try, in illustration, to apply Riemann's definition of measurement to colours. We must remember that motion, in dealing with the colour manifold, means—not motion in space but—motion in the colour manifold itself. Now since every point of the colour manifold is completely determined by three magnitudes, which are given in fact, and cannot be arbitrarily chosen, it is plain that measurement by superposition—involving, as it does, motion, and therefore change in these determining magnitudes—is totally out of the question. The superposition of one colour on another, as a means of measurement, is sheer nonsense. And yet measurement is possible in the colour-manifold, by means of Helmholtz's law of mixture (Mischungsgesetz); but the measurement is of every separate element, not of the relations between elements, and is thus radically different from space-measurement[81]. The elements are not, like points in space, qualitatively alike, and distinguished by the mere fact of their mutual externality. What we have, in colours, is three fundamental qualitatively distinct elements, out of certain proportions of which we can build up all the other elements of the manifold—each of the resulting elements having the same combination of qualitative diversity and similarity as the three original elements. But in space, what could we make of such a procedure? Given three points, how are we to combine them in certain proportions? The phrase is meaningless. If some one makes the obvious retort, that we have to combine lines, not points, my rejoinder is equally obvious. To begin with, lines are not elements. Metaphysically, space has no elements, being, as the sequel will show, mere relations between non-spatial elements. Mathematically, this fact exhibits itself in the self-contradictory notion of the point, or zero magnitude in space, as the limit in our vain search for spatial elements. But even if we allow the line to pass as the spatial element, what does the combination of three lines in definite proportions give us? It gives us, simply, the coordinates of a point. Here again we see a great difference between the colour and space-manifolds. In colours, the combination of magnitudes gives a new magnitude of the same kind; in space, it defines, not a magnitude at all, but a would-be element of a different kind from the defining magnitudes. In the tone-manifold, we should find still different conditions. Here, no one of the measuring magnitudes can vanish without the tone vanishing too, and all three are so bound up together, in the single resulting sensation, that none can exist without a finite quantity of the others. They are all qualitatively different, both from each other, and from any possible tone, being constituents of it, as mass and velocity are constituents of momentum. All these different conditions require to be examined, before a manifold can be completely defined; and until we have conducted such an examination in detail, we cannot pronounce as to the à priori or empirical nature of the laws of the manifold. As regards space, I have attempted such an examination in the third and fourth chapters of this Essay.

65. I do not wish to deny, however, the great value of the conception of space as a manifold. On the contrary, this conception seems to have become essential to any treatment of the question. I only wish to urge that the purely algebraical treatment of any manifold, important as it may be in deducing fresh consequences from known premisses, tends rather to conceal than to make clear the basis of the premisses themselves, and is therefore misleading in a philosophical investigation. For mathematics, where quantity reigns supreme, Riemann's conception has proved itself abundantly fruitful; for philosophy, on the contrary, where quantity appears rather as a cloak to conceal the qualities it abstracts from, the conception seems to me more productive of error and confusion than of sound doctrine.

We are thus brought back to the point from which we started, namely, the falsity of Riemann's initial disjunction, and the consequent fallacy in his proof of the empirical nature of the axioms. His philosophy is chiefly vitiated, to my mind, by this fallacy, and by the uncritical assumption that a metrical coordinate system can be set up independently of any axioms as to space-measurement[82]. Riemann has failed to observe, what I have endeavoured to prove in the next chapter, that, unless space had a strictly constant measure of curvature, Geometry would become impossible; also that the absence of constant measure of curvature involves absolute position, which is an absurdity. Hence he is led to the conclusion that all geometrical axioms are empirical, and may not hold in the infinitesimal, where observation is impossible. Thus he says (p. 267): "Now the empirical conceptions, on which spatial measurements are based, the conceptions of the rigid body and the light-ray, appear to lose their validity in the infinitesimal: it is therefore quite conceivable that the relations of spatial magnitudes in the infinitesimal do not correspond to the presuppositions of Geometry, and this would, in fact, have to be assumed, as soon as it would enable us to explain the phenomena more simply." From this conclusion I must entirely dissent. In very large spaces, there might be a departure from Euclid; for they depend upon the axiom of parallels, which is not contained in the axiom of Free Mobility; but in the infinitesimal, departures from Euclid could only be due to the absence of Free Mobility, which, as I hope my third chapter will show, is once for all impossible.

Helmholtz.

66. Helmholtz, like Riemann, was important both in the mathematics and in the philosophy of Geometry. From the mathematical point of view, his work has been already considered in Chapter I.; the consideration of his philosophy, which must occupy us here, will be a more serious task. Like Riemann, he endeavoured to prove that all the axioms are empirical, and like Riemann, he based his proof chiefly on Metageometry. He had an additional resource, however, in the physiology of the senses, which first led him to reject the Transcendental Aesthetic, and enabled him to attack Kant from the psychological as well as the mathematical side[83].

The principal topics, for a criticism of Helmholtz, are three: First, his criterion of the à priori; second, his discussion with Land as to the "imaginability" of non-Euclidean spaces; third—and this is by far the most important of the three—his theory of the dependence of Geometry on Mechanics. Let us discuss these three points successively.

67. Helmholtz's criterion of apriority is difficult to discover, as he never, to my knowledge, gives a precise statement of it. From his discussion of physical and transcendental Geometry[84], however, it would appear that he regards as empirical whatever applies to empirical matter. For he there maintains, that even if space were an à priori form, yet any Geometry, which aimed at an application to Physics, would, since the actual places of bodies are not known à priori, be necessarily empirical[85]. It seems the more probable that he regards this as a possible criterion, as it is adopted, in several passages, by his disciple Erdmann[86], and so strange a test could hardly be accepted by a philosopher, unless he had found it in his master. I have called this a strange test, because it seems to me completely to ignore the work of the Critical Philosophy. For if there is one thing which, one might have hoped, had been made sufficiently clear by Kant's Critique, it is this, that knowledge which is à priori, being itself the condition of possible experience, applies—and in Kant's view, applies only—to empirical matter. Helmholtz and Erdmann, therefore, in setting up this test without discussion, simply ignore the existence of Kant and the possibility of a transcendental argument. Helmholtz assumes always that empirical knowledge must be wholly empirical, that there can be no à priori conditions of the experience in question, that experience will always be possible, and may give any kind of result. Thus in discussing "physical" Geometry, he assumes that the possibility of empirical measurement involves no à priori axioms, and that no à priori element can be contained in the process. This assumption, as we shall see in Chapter III., is quite unwarrantable: certain properties of space, in fact, are involved in the possibility of measuring matter. In spite of the fact, therefore, that we apply measurement to empirical matter, and that our results are therefore empirical, there may well be an à priori element in measurement, which is presupposed in its possibility. Such a criterion, therefore, must pronounce everything empirical, but must itself be pronounced worthless.

Another and a better criterion, it is true, is also to be found in Helmholtz, and has also been adopted by Erdmann. Whatever might, by a different experience, have been rendered different—so this criterion contends—must itself be dependent on experience, and so empirical. This criterion seems perfectly sound, but Helmholtz's use of it is usually vitiated by his neglecting to prove the possibility of the different experience in question. He says, for example, that if our experience showed us only bodies which changed their shapes in motion, we should not arrive at the axiom of Congruence, which he pronounces accordingly to be empirical. But I shall endeavour to prove, in Chapter III., that without the axiom of Congruence, experience of spatial magnitude would be impossible. If my proof be correct, it follows that no experience can ever reveal spatial magnitudes which contradict this axiom—a possibility which Helmholtz nowhere discusses, in setting up his hypothetical experience. Thus this second criterion, though perfectly sound, requires always an accompanying transcendental argument, as to the conditions of possible experience. But this accompaniment is seldom to be found in Helmholtz.

68. One of the few cases, in which Helmholtz has attempted such an accompaniment, occurs in connection with our second point, the imaginability of non-Euclidean spaces. The argument on this point was elicited by Helmholtz's Kantian opponents, who maintained that the merely logical possibility of these spaces was irrelevant, since the basis of Geometry was not logic, but intuition. The axioms, they said, are synthetic propositions, and their contraries are, therefore, not self-contradictory; they are nevertheless apodeictic propositions, since no other intuition than the Euclidean is possible to us[87]. I have already criticized this line of argument in the beginning of the present chapter. Helmholtz's criticism, however, was different: admitting the internal consistency of the argument, he denied one of its premisses. We can imagine non-Euclidean spaces, he said, though their unfamiliarity makes this difficult. From this view it followed, of course, that Kant's argument, even if it were formally valid, could not prove the apriority of Euclidean space in particular, but only of that general space which included Euclid and non-Euclid alike[88].

Although I agree with Helmholtz in thinking the distinction between Euclidean and non-Euclidean spaces empirical, I cannot think his argument on the "imaginability" of the latter a very happy one. The validity of any proof must turn, obviously, on the definition of imaginability. The definition which Helmholtz gives in his answer to Land is as follows: Imaginability requires "die vollständige Vorstellbarkeit derjenigen Sinneseindrücke, welche das betreffende Object in uns nach den bekannten Gesetzen unserer Sinnesorgane unter allen denkbaren Bedingungen der Beobachtung erregen, und wodurch es sich von anderen ähnlichen Objecten unterscheiden würde" (Wiss. Abh. II. p. 644). This definition is not very clear, owing to the ambiguity of the word "Vorstellbarkeit." The following definition seems less ambiguous: "Wenn die Reihe der Sinneseindrücke vollständig und eindeutig angegeben werden kann, muss man m. E. die Sache für anschaulich vorstellbar erklären" (Vorträge und Reden, II. p. 234). This makes clear, what also appears from his manner of proof, that he regards things as imaginable which can be described in conceptual terms. Such, as Land remarks (Mind, Vol. II. p. 45), "is not the sense required for argumentation in this case." That Land's criticism is just, is shown by Helmholtz's proof for non-Euclidean spaces, for it consists only in an analogy to the volume inside a sphere, which is mathematically obtained thus: We take the symbols representing magnitudes in "pseudo-spherical" (hyperbolic) space, and give them a new Euclidean meaning; thus all our symbolic propositions become capable of two interpretations, one for pseudo-spherical space, and one for the volume inside a sphere. It is, however, sufficiently obvious that this procedure, though it enables us to describe our new space, does not enable us to imagine it, in the sense of calling up images of the way things would look in it. We really derive, from this analogy, no more knowledge than a man born blind may derive, as to light, from an analogy with heat. The dictum "Nihil est in intellectu quod non fuerit ante in sensu," would unquestionably be true, if for intellect we were to substitute imagination; it is vain, therefore, if our actual space be Euclidean, to hope for a power of imagining a non-Euclidean space. What Helmholtz might, I believe with perfect truth, have urged against Land, is that the image we actually have of space is not sufficiently accurate to exclude, in the actual space we know, all possibility of a slight departure from the Euclidean type. But in maintaining that we cannot imagine, though we can conceive and describe, a space different from that we actually have, Land is, in my opinion, unquestionably in the right. For a pure Kantian, who maintains, with Land, that none of the axioms can be proved, this question is of great importance. But if, as I have maintained, some of the axioms are susceptible of a transcendental proof, while the others can be verified empirically, the question is freed from psychological implications, and the imaginability or non-imaginability of metageometrical spaces becomes unimportant.

69. We come now to the third and most important question, the relation of Geometry to Mechanics. There are three senses in which Helmholtz's appeal to rigid bodies may be taken: the first, I think, is the sense in which he originally intended it; the second seems to be the sense which he adopted in his defence against Land; while the third is admitted by Land, and will be admitted in the following argument. These three senses are as follows:

(1) It may be asserted that the actual meaning of the axiom of Free Mobility lies in the assertion of empirical rigid bodies, and that the two propositions are equivalent to one another. This is certainly false.

(2) The axiom of Free Mobility, it may be said, is logically distinguishable from the assertion of rigid bodies, and may even be not empirical; but it is barren, even for pure Geometry, without the aid of measures, which must themselves be empirical rigid bodies. This sense is more plausible than the first, but I believe we can show that, in this sense also, the proposition is false.

(3) For pure Geometry and the abstract study of space, it may be said, Free Mobility, as applied to an abstract geometrical matter, gives a sufficient possibility of quantitative comparison; but the moment we extend our results to mixed mathematics, and apply them to empirically given matter, we require also, as measures, empirically given rigid bodies, or bodies, at least, whose departures from rigidity are empirically known. In this sense, I admit, the proposition is correct[89].

In discussing these three meanings, I shall not confine myself strictly to the text of Helmholtz or Land: if I endeavoured to do so, I should be met by the difficulty that neither of them defines the à priori, and that each is too much inclined, in my opinion, to test it by psychological criteria. I shall, therefore, take the three meanings in turn, without laying stress on their historical adequacy to the views of Land or Helmholtz.

70. (1) Congruence may be taken to mean—as Helmholtz would certainly seem to desire—that we find actual bodies, in our mechanical experience, to preserve their shapes with approximate constancy, and that we infer, from this experience, the homogeneity of space. This view, in my opinion, radically misconceives the nature of measurement, and of the axioms involved in it. For what is meant by the non-rigidity of a body? We mean, simply, that it has changed its shape. But this involves the possibility of comparison with its former shape, in other words, of measurement. In order, therefore, that there may be any question of rigidity or non-rigidity, the measurement of spatial magnitudes must be already possible. It follows that measurement cannot, without a vicious circle, be itself derived from experience of rigid bodies. Geometrical measurement, in fact, is the comparison of spatial magnitudes, and such comparison involves, as will be proved at length in Chapter III., the homogeneity of space. This is, therefore, the logical prerequisite of all experience of rigid bodies, and cannot be the result of such experience. Without the homogeneity of space, the very notion of rigidity or non-rigidity could not exist, since these mean, respectively, the constancy or inconstancy of spatial magnitude in pieces of matter, and both alike, therefore, presuppose the possibility of spatial measurement. From the homogeneity of space, we learn that a body, when it moves, will not change its shape without some physical cause; that it actually does not change its shape, is never asserted, and is indeed known to be false. As soon as measurement is possible, actual changes of shape can be estimated, and their empirical causes can be sought. But if space were not homogeneous, measurement would be impossible, constant shape would be a meaningless phrase, and rigidity could never be experienced. Congruence asserts, in short, that a body can, so far as mere space is concerned, move without change of shape; rigidity asserts that it actually does so move—a very different proposition, involving obviously, as its logical prius, the former geometrical proposition.

This argument may be summed up by the following disjunction: If bodies change their shapes in motion—and to some extent, since no body is perfectly rigid, they must all do so—then one of two cases must occur. Either the changes of shape, as bodies move from place to place, follow no geometrical law, are not, for instance, functions of the amount or direction of motion; in which case the law of causation requires that they should not be effects of the change of place, but of some simultaneous non-geometrical change, such as temperature. Or the changes are regular, and the shape S becomes, in a new position p, Sf(p). In this case, the law of concomitant variations leads us to attribute the change of shape to the mere motion, and shape thus becomes a function of absolute position. But this is absurd, for position means merely a relation or set of relations; it is impossible, therefore, that mere position should be able to effect changes in a body. Position is one term in a relation, not a thing per se; it cannot, therefore, act on a thing, nor exist by itself, apart from the other terms of the relation. Thus Helmholtz's view, that Congruence depends on the existence of rigid bodies, must, since it involves absolute position, be condemned as a logical fallacy. Congruence, in fact, as I shall prove more fully in Chapter III., is an à priori deduction from the relativity of position.

71. (2) The above argument seems to me to answer satisfactorily Helmholtz's contention in the precise form which he first gave it. The axiom of Congruence, we must agree, is logically distinguishable from the existence of rigid bodies. Nevertheless some reference to matter is logically involved in Geometry[90], but whether this reference makes Geometry empirical, or does not, rather, show an à priori element in dynamics, is a further question.

The reference to matter is necessitated by the homogeneity of empty space. For so long as we leave matter out of account, one position is perfectly indistinguishable from another, and a science of the relations of positions is impossible. Indeed, before spatial relations can arise at all, the homogeneity of empty space must be destroyed, and this destruction must be effected by matter. The blank page is useless to the geometer until he defaces its homogeneity by lines in ink or pencil. No spatial figures, in short, are conceivable, without a reference to a not purely spatial matter. Again, if Congruence is ever to be used, there must be motion: but a purely geometrical point, being defined solely by its spatial attributes, cannot be supposed to move without a contradiction in terms. What moves, therefore, must be matter. Hence, in order that motion may afford a test of equality, we must have some matter which is known to be unaffected throughout the motion, that is, we must have some rigid bodies. And the difficulty is, that these bodies must not only undergo no change due solely to the nature of space, but must, further, be unchanged by their changing relation to other bodies. And here we have a requisite which can no longer be fulfilled à priori: which, indeed, we know to be, in strictness, untrue. For the forces acting on a body depend upon its spatial relations to other bodies, and changing forces are liable to produce changing configuration. Hence, it would seem, actual measurement must be purely empirical, and must depend on the degree of rigidity to be obtained, during the process of measurement, in the bodies with which we are conversant.

This conclusion, I believe, is valid of all actual measurement. But the possibility of such empirical and approximate rigidity, I must insist, depends on the à priori law that mere motion, apart from the action of other matter, cannot effect a change of shape. For without this law, the effect of other matter would not be discoverable; the laws of motion would be absurd, and Physics would be impossible. Consider the second law, for example: How could we measure the change of motion, if motion itself produced a change in our measures? Or consider the law of gravitation: How could we establish the inverse square, unless we were able, independently of Dynamics, to measure distances? The whole science of Dynamics, in short, is fundamentally dependent on Geometry, and but for the independent possibility of measuring spatial magnitudes, none of the magnitudes of Dynamics could be measured. Time, force, and mass are alike measured by spatial correlates: these correlates are given, for time, by the first law, for force and mass, by the second and third. It is true, then, that an empirical element appears unavoidably in all actual measurement, inasmuch as we can only know empirically that a given piece of matter preserves its shape throughout the necessary change of dynamical relations to other matter involved in motion; but it is further true that, for Geometry—which regards matter simply as supplying the necessary breach in the homogeneity of space, and the necessary term for spatial relations, not as the bearer of forces which change the configuration of other material systems—for Geometry, which deals with this abstract and merely kinematical matter, rigidity is à priori, in so far as the only changes with which it is cognizant—changes of mere position, namely—are incapable of affecting the shapes of the imaginary and abstract bodies with which it deals. To use a scholastic distinction, we may say that matter is the causa essendi of space, but Geometry is the causa cognoscendi of Physics. Without a Geometry independent of Physics, Physics itself, which necessarily assumes the results of Geometry, could never arise; but when Geometry is used in Physics, it loses some of its à priori certainty, and acquires the empirical and approximate character which belongs to all accounts of actual phenomena.

72. (3) This argument leads us to Land's distinction of physical and geometrical rigidity. The distinction may be expressed—and I think it is better expressed—by distinguishing between the conceptions of matter proper to Dynamics and to Geometry respectively. In Dynamics, we are concerned with matter as subject to and as causing motion, as affected by and as exerting force. We are therefore concerned with the changes of spatial configuration to which material systems are liable: the description and explanation of these changes is the proper subject-matter of all Dynamics. But in order that such a science may exist, it is obviously necessary that spatial configuration should be already measurable. If this were not the case, motion, acceleration and force would remain perfectly indeterminate. Geometry, therefore, must already exist before Dynamics becomes possible: to make Geometry dependent for its possibility on the laws of motion or any of their consequences, is a gross ὕστερον πρότερον. Nevertheless, as we have seen, some sort of matter is essential to Geometry. But this geometrical matter is a more abstract and wholly different matter from that of Dynamics. In order to study space by itself, we reduce the properties of matter to a bare minimum: we avoid entirely the category of causation, so essential to Dynamics, and retain nothing, in our matter, but its spatial adjectives[91]. The kind of rigidity affirmed of this abstract matter—a kind which suffices for the theory of our science, though not for its application to the objects of daily life—is purely geometrical, and asserts no more than this: That since our matter is devoid, ex hypothesi, of causal properties, there remains nothing, in mere empty space, which is capable of changing the configuration of any geometrical system. A change of absolute position, it asserts, is nothing; therefore the only real change involved in motion is a change of relation to other matter; but such other matter, for the purposes of our science, is regarded as destitute of causal powers; hence no change can occur, in the configuration of our system, by the mere effect of motion through empty space. The necessity of such a principle may be shown by a simple reductio ad absurdum, as follows. A motion of translation of the universe as a whole, with constant direction and velocity, is dynamically negligeable; indeed it is, philosophically, no motion at all, for it involves no change in the condition or mutual relations of the things in the universe. But if our geometrical rigidity were denied, the change in the parameter of space might cause all bodies to change their shapes owing to the mere change of absolute position, which is obviously absurd.

To make quite plain the function of rigid bodies in Geometry, let us suppose a liquid geometer in a liquid world. We cannot suppose the liquid perfectly homogeneous and undifferentiated, in the first place because such a liquid would be indistinguishable from empty space, in the second place because our geometer's body—unless he be a disembodied spirit—will itself constitute a differentiation for him. We may therefore assume