An essay on the foundations of geometry

by Bertrand Russell

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.

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TO

JOHN McTAGGART ELLIS McTAGGART

TO WHOSE DISCOURSE AND FRIENDSHIP IS OWING

THE EXISTENCE OF THIS BOOK.

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[TABLE OF CONTENTS.]

	INTRODUCTION.
	OUR PROBLEM DEFINED BY ITS RELATIONS TO LOGIC,PSYCHOLOGY AND MATHEMATICS.
		PAGE
[1.]	The problem first received a modern form through Kant, who connected the à priori with the subjective	1
[2.]	A mental state is subjective, for Psychology, when its immediate cause does not lie in the outer world	2
[3.]	A piece of knowledge is à priori, for Epistemology, when without it knowledge would be impossible	2
[4.]	The subjective and the à priori belong respectively to Psychology and to Epistemology. The latter alone will be investigated in this essay	3
[5.]	My test of the à priori will be purely logical: what knowledge is necessary for experience?	3
[6.]	But since the necessary is hypothetical, we must include, in the à priori, the ground of necessity	4
[7.]	This may be the essential postulate of our science, or the element, in the subject-matter, which is necessary to experience;	4
[8.]	Which, however, are both at bottom the same ground	5
[9.]	Forecast of the work	5
	CHAPTER I.
	A SHORT HISTORY OF METAGEOMETRY.
[10.]	Metageometry began by rejecting the axiom of parallels	7
[11.]	Its history may be divided into three periods: the synthetic, the metrical and the projective	7
[12.]	The first period was inaugurated by Gauss,	10
[13.]	Whose suggestions were developed independently by Lobatchewsky	10
[14.]	And Bolyai	11
[15.]	The purpose of all three was to show that the axiom of parallels could not be deduced from the others, since its denial did not lead to contradictions	12
[16.]	The second period had a more philosophical aim, and was inspired chiefly by Gauss and Herbart	13
[17.]	The first work of this period, that of Riemann, invented two new conceptions:	14
[18.]	The first, that of a manifold, is a class-conception, containing space as a species,	14
[19.]	And defined as such that its determinations form a collection of magnitudes	15
[20.]	The second, the measure of curvature of a manifold, grew out of curvature in curves and surfaces	16
[21.]	By means of Gauss's analytical formula for the curvature of surfaces,	19
[22.]	Which enables us to define a constant measure of curvature of a three-dimensional space without reference to a fourth dimension	20
[23.]	The main result of Riemann's mathematical work was to show that, if magnitudes are independent of place, the measure of curvature of space must be constant	21
[24.]	Helmholtz, who was more of a philosopher than a mathematician,	22
[25.]	Gave a new but incorrect formulation of the essential axioms,	23
[26.]	And deduced the quadratic formula for the infinitesimal arc, which Riemann had assumed	24
[27.]	Beltrami gave Lobatchewsky's planimetry a Euclidean interpretation,	25
[28.]	Which is analogous to Cayley's theory of distance;	26
[29.]	And dealt with n-dimensional spaces of constant negative curvature	27
[30.]	The third period abandons the metrical methods of the second, and extrudes the notion of spatial quantity	27
[31.]	Cayley reduced metrical properties to projective properties, relative to a certain conic or quadric, the Absolute;	28
[32.]	And Klein showed that the Euclidean or non-Euclidean systems result, according to the nature of the Absolute;	29
[33.]	Hence Euclidean space appeared to give rise to all the kinds of Geometry, and the question, which is true, appeared reduced to one of convention	30
[34.]	But this view is due to a confusion as to the nature of the coordinates employed	30
[35.]	Projective coordinates have been regarded as dependent on distance, and thus really metrical	31
[36.]	But this is not the case, since anharmonic ratio can be projectively defined	32
[37.]	Projective coordinates, being purely descriptive, can give no information as to metrical properties, and the reduction of metrical to projective properties is purely technical	33
[38.]	The true connection of Cayley's measure of distance with non-Euclidean Geometry is that suggested by Beltrami's Saggio, and worked out by Sir R. Ball,	36
[39.]	Which provides a Euclidean equivalent for every non-Euclidean proposition, and so removes the possibility of contradictions in Metageometry	38
[40.]	Klein's elliptic Geometry has not been proved to have a corresponding variety of space	39
[41.]	The geometrical use of imaginaries, of which Cayley demanded a philosophical discussion,	41
[42.]	Has a merely technical validity,	42
[43.]	And is capable of giving geometrical results only when it begins and ends with real points and figures	45
[44.]	We have now seen that projective Geometry is logically prior to metrical Geometry, but cannot supersede it	46
[45.]	Sophus Lie has applied projective methods to Helmholtz's formulation of the axioms, and has shown the axiom of Monodromy to be superfluous	46
[46.]	Metageometry has gradually grown independent of philosophy, but has grown continually more interesting to philosophy	50
[47.]	Metrical Geometry has three indispensable axioms,	50
[48.]	Which we shall find to be not results, but conditions, of measurement,	51
[49.]	And which are nearly equivalent to the three axioms of projective Geometry	52
[50.]	Both sets of axioms are necessitated, not by facts, but by logic	52
	CHAPTER II.
	CRITICAL ACCOUNT OF SOME PREVIOUS PHILOSOPHICAL THEORIES OF GEOMETRY.
[51.]	A criticism of representative modern theories need not begin before Kant	54
[52.]	Kant's doctrine must be taken, in an argument about Geometry, on its purely logical side	55
[53.]	Kant contends that since Geometry is apodeictic, space must be à priori and subjective, while since space is à priori and subjective, Geometry must be apodeictic	55
[54.]	Metageometry has upset the first line of argument, not the second	56
[55.]	The second may be attacked by criticizing either the distinction of synthetic and analytic judgments, or the first two arguments of the metaphysical deduction of space	57
[56.]	Modern Logic regards every judgment as both synthetic and analytic,	57
[57.]	But leaves the à priori, as that which is presupposed in the possibility of experience	59
[58.]	Kant's first two arguments as to space suffice to prove some form of externality, but not necessarily Euclidean space, a necessary condition of experience	60
[59.]	Among the successors of Kant, Herbart alone advanced the theory of Geometry, by influencing Riemann	62
[60.]	Riemann regarded space as a particular kind of manifold, i.e. wholly quantitatively	63
[61.]	He therefore unduly neglected the qualitative adjectives of space	64
[62.]	His philosophy rests on a vicious disjunction	65
[63.]	His definition of a manifold is obscure,	66
[64.]	And his definition of measurement applies only to space	67
[65.]	Though mathematically invaluable, his view of space as a manifold is philosophically misleading	69
[66.]	Helmholtz attacked Kant both on the mathematical and on the psychological side;	70
[67.]	But his criterion of apriority is changeable and often invalid;	71
[68.]	His proof that non-Euclidean spaces are imaginable is inconclusive;	72
[69.]	And his assertion of the dependence of measurement on rigid bodies, which may be taken in three senses,	74
[70.]	Is wholly false if it means that the axiom of Congruence actually asserts the existence of rigid bodies,	75
[71.]	Is untrue if it means that the necessary reference of geometrical propositions to matter renders pure Geometry empirical,	76
[72.]	And is inadequate to his conclusion if it means, what is true, that actual measurement involves approximately rigid bodies	78
[73.]	Geometry deals with an abstract matter, whose physical properties are disregarded; and Physics must presuppose Geometry	80
[74.]	Erdmann accepted the conclusions of Riemann and Helmholtz,	81
[75.]	And regarded the axioms as necessarily successive steps in classifying space as a species of manifold	82
[76.]	His deduction involves four fallacious assumptions, namely:	82
[77.]	That conceptions must be abstracted from a series of instances;	83
[78.]	That all definition is classification;	83
[79.]	That conceptions of magnitude can be applied to space as a whole;	84
[80.]	And that if conceptions of magnitude could be so applied, all the adjectives of space would result from their application	86
[81.]	Erdmann regards Geometry alone as incapable of deciding on the truth of the axiom of Congruence,	86
[82.]	Which he affirms to be empirically proved by Mechanics.	88
[83.]	The variety and inadequacy of Erdmann's tests of apriority	89
[84.]	Invalidate his final conclusions on the theory of Geometry	90
[85.]	Lotze has discussed two questions in the theory of Geometry:	93
[86.]	(1) He regards the possibility of non-Euclidean spaces as suggested by the subjectivity of space,	93
[87.]	And rejects it owing to a mathematical misunderstanding,	96
[88.]	Having missed the most important sense of their possibility,	96
[89.]	Which is that they fulfil the logical conditions to which any form of externality must conform	97
[90.]	(2) He attacks the mathematical procedure of Metageometry	98
[91.]	The attack begins with a question-begging definition of parallels	99
[92.]	Lotze maintains that all apparent departures from Euclid could be physically explained, a view which really makes Euclid empirical	99
[93.]	His criticism of Helmholtz's analogies rests wholly on mathematical mistakes	101
[94.]	His proof that space must have three dimensions rests on neglect of different orders of infinity	104
[95.]	He attacks non-Euclidean spaces on the mistaken ground that they are not homogeneous	107
[96.]	Lotze's objections fall under four heads	108
[97.]	Two other semi-philosophical objections may be urged,	109
[98.]	One of which, the absence of similarity, has been made the basis of attack by Delbœuf,	110
[99.]	But does not form a valid ground of objection	111
[100.]	Recent French speculation on the foundations of Geometry has suggested few new views	112
[101.]	All homogeneous spaces are à priori possible, and the decision between them is empirical	114
	CHAPTER III.
	Section A. the axioms of projective geometry.
[102.]	Projective Geometry does not deal with magnitude, and applies to all spaces alike	117
[103.]	It will be found wholly à priori	117
[104.]	Its axioms have not yet been formulated philosophically	118
[105.]	Coordinates, in projective Geometry, are not spatial magnitudes, but convenient names for points	118
[106.]	The possibility of distinguishing various points is an axiom	119
[107.]	The qualitative relations between points, dealt with by projective Geometry, are presupposed by the quantitative treatment	119
[108.]	The only qualitative relation between two points is the straight line, and all straight lines are qualitatively similar	120
[109.]	Hence follows, by extension, the principle of projective transformation	121
[110.]	By which figures qualitatively indistinguishable from a given figure are obtained	122
[111.]	Anharmonic ratio may and must be descriptively defined	122
[112.]	The quadrilateral construction is essential to the projective definition of points,	123
[113.]	And can be projectively defined,	124
[114.]	By the general principle of projective transformation	126
[115.]	The principle of duality is the mathematical form of a philosophical circle,	127
[116.]	Which is an inevitable consequence of the relativity of space, and makes any definition of the point contradictory	128
[117.]	We define the point as that which is spatial, but contains no space, whence other definitions follow	128
[118.]	What is meant by qualitative equivalence in Geometry?	129
[119.]	Two pairs of points on one straight line, or two pairs of straight lines through one point, are qualitatively equivalent	129
[120.]	This explains why four collinear points are needed, to give an intrinsic relation by which the fourth can be descriptively defined when the first three are given	130
[121.]	Any two projectively related figures are qualitatively equivalent, i.e. differ in no non-quantitative conceptual property	131
[122.]	Three axioms are used by projective Geometry,	132
[123.]	And are required for qualitative spatial comparison,	132
[124.]	Which involves the homogeneity, relativity and passivity of space	133
[125.]	The conception of a form of externality,	134
[126.]	Being a creature of the intellect, can be dealt with by pure mathematics	134
[127.]	The resulting doctrine of extension will be, for the moment, hypothetical	135
[128.]	But is rendered assertorical by the necessity, for experience, of some form of externality	136
[129.]	Any such form must be relational	136
[130.]	And homogeneous	137
[131.]	And the relations constituting it must appear infinitely divisible	137
[132.]	It must have a finite integral number of dimensions,	139
[133.]	Owing to its passivity and homogeneity	140
[134.]	And to the systematic unity of the world	140
[135.]	A one-dimensional form alone would not suffice for experience	141
[136.]	Since its elements would be immovably fixed in a series	142
[137.]	Two positions have a relation independent of other positions,	143
[138.]	Since positions are wholly defined by mutually independent relations	143
[139.]	Hence projective Geometry is wholly à priori,	146
[140.]	Though metrical Geometry contains an empirical element	146
	Section B. the axioms of metrical geometry.
[141.]	Metrical Geometry is distinct from projective, but has the same fundamental postulate	147
[142.]	It introduces the new idea of motion, and has three à priori axioms	148
	I. The Axiom of Free Mobility.
[143.]	Measurement requires a criterion of spatial equality	149
[144.]	Which is given by superposition, and involves the axiom of Free Mobility	150
[145.]	The denial of this axiom involves an action of empty space on things	151
[146.]	There is a mathematically possible alternative to the axiom,	152
[147.]	Which, however, is logically and philosophically untenable	153
[148.]	Though Free Mobility is à priori, actual measurement is empirical	154
[149.]	Some objections remain to be answered, concerning—	154
[150.]	(1) The comparison of volumes and of Kant's symmetrical objects	154
[151.]	(2) The measurement of time, where congruence is impossible	156
[152.]	(3) The immediate perception of spatial magnitude; and	157
[153.]	(4) The Geometry of non-congruent surfaces	158
[154.]	Free Mobility includes Helmholtz's Monodromy	159
[155.]	Free Mobility involves the relativity of space	159
[156.]	From which, reciprocally, it can be deduced	160
[157.]	Our axiom is therefore à priori in a double sense	160
	II. The Axiom of Dimensions.
[158.]	Space must have a finite integral number of dimensions	161
[159.]	But the restriction to three is empirical	162
[160.]	The general axiom follows from the relativity of position	162
[161.]	The limitation to three dimensions, unlike most empirical knowledge, is accurate and certain	163
	III. The Axiom of Distance.
[162.]	The axiom of distance corresponds, here, to that of the straight line in projective Geometry	164
[163.]	The possibility of spatial measurement involves a magnitude uniquely determined by two points,	164
[164.]	Since two points must have some relation, and the passivity of space proves this to be independent of external reference	165
[165.]	There can be only one such relation	166
[166.]	This must be measured by a curve joining the two points,	166
[167.]	And the curve must be uniquely determined by the two points	167
[168.]	Spherical Geometry contains an exception to this axiom,	168
[169.]	Which, however, is not quite equivalent to Euclid's	168
[170.]	The exception is due to the fact that two points, in spherical space, may have an external relation unaltered by motion,	169
[171.]	Which, however, being a relation of linear magnitude, presupposes the possibility of linear magnitude	170
[172.]	A relation between two points must be a line joining them	170
[173.]	Conversely, the existence of a unique line between two points can be deduced from the nature of a form of externality,	171
[174.]	And necessarily leads to distance, when quantity is applied to it	172
[175.]	Hence the axiom of distance, also, is à priori in a double sense	172
[176.]	No metrical coordinate system can be set up without the straight line	174
[177.]	No axioms besides the above three are necessary to metrical Geometry	175
[178.]	But these three are necessary to the direct measurement of any continuum	176
[179.]	Two philosophical questions remain for a final chapter	177
	CHAPTER IV.
	PHILOSOPHICAL CONSEQUENCES.
[180.]	What is the relation to experience of a form of externality in general?	178
[181.]	This form is the class-conception, containing every possible intuition of externality; and some such intuition is necessary to experience	178
[182.]	What relation does this view bear to Kant's?	179
[183.]	It is less psychological, since it does not discuss whether space is given in sensation,	180
[184.]	And maintains that not only space, but any form of externality which renders experience possible, must be given in sense-perception	181
[185.]	Externality should mean, not externality to the Self, but the mutual externality of presented things	181
[186.]	Would this be unknowable without a given form of externality?	182
[187.]	Bradley has proved that space and time preclude the existence of mere particulars,	182
[188.]	And that knowledge requires the This to be neither simple nor self-subsistent	183
[189.]	To prove that experience requires a form of externality, I assume that all knowledge requires the recognition of identity in difference	184
[190.]	Such recognition involves time	184
[191.]	And some other form giving simultaneous diversity	185
[192.]	The above argument has not deduced sense-perception from the categories, but has shown the former, unless it contains a certain element, to be unintelligible to the latter	186
[193.]	How to account for the realization of this element, is a question for metaphysics	187
[194.]	What are we to do with the contradictions in space?	188
[195.]	Three contradictions will be discussed in what follows	188
[196.]	(1) The antinomy of the Point proves the relativity of space,	189
[197.]	And shows that Geometry must have some reference to matter,	190
[198.]	By which means it is made to refer to spatial order, not to empty space	191
[199.]	The causal properties of matter are irrelevant to Geometry, which must regard it as composed of unextended atoms, by which points are replaced	191
[200.]	(2) The circle in defining straight lines and planes is overcome by the same reference to matter	192
[201.]	(3) The antinomy that space is relational and yet more than relational,	193
[202.]	Seems to depend on the confusion of empty space with spatial order	193
[203.]	Kant regarded empty space as the subject-matter of Geometry,	194
[204.]	But the arguments of the Aesthetic are inconclusive on this point,	195
[205.]	And are upset by the mathematical antinomies, which prove that spatial order should be the subject-matter of Geometry	196
[206.]	The apparent thinghood of space is a psychological illusion, due to the fact that spatial relations are immediately given	196
[207.]	The apparent divisibility of spatial relations is either an illusion, arising out of empty space, or the expression of the possibility of quantitatively different spatial relations	197
[208.]	Externality is not a relation, but an aspect of relations. Spatial order, owing to its reference to matter, is a real relation	198
[209.]	Conclusion	199

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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.