THE purpose of the present paper is purely logical, and aims at applying principles of general logic to geometrical reasoning. The inquiry which I propose to conduct may be divided into two parts: (1) an analysis of the actual reasoning of Geometry, with a view to discovering those essential axioms, and that fundamental postulate, without which this reasoning would become formally impossible; (2) a deduction, from the fundamental nature of a form of externality, of the principles which must be true of any such form, when treated in abstraction as the subject-matter of a special science. Our conclusion, as might be expected, will be the same in both cases: for the two arguments are fundamentally the same, and form together a completed circle. The element in geometrical reasoning, which both methods reveal as necessary to any Geometry, I shall call a priori.

As I have ventured to use the word a priori in a slightly unconventional sense, I will preface the geometrical inquiry by a few elucidatory remarks of a general nature.

The a 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 a priori in knowledge be primarily the necessary, it must be the necessary on some hypothesis, and the ground of necessity must be included as a 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.

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 sought in 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 a 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 a priori. Rather it is that element, in the subject-matter, which makes possible the branch of experience dealt with by the science in question.¹ The importance of this distinction will appear more clearly as we proceed.

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.

After these general remarks, I will proceed to Geometry. I propose, first, by an analysis of geometrical reasoning, to prove that its possibility depends on three axioms, which I shall call the axiom of congruence or free mobility, the generalised axiom of dimensions, and the generalised axiom of the straight line, and that the truth of these axioms involves the homogeneity of space and the complete relativity of position. So far the argument proceeds by a mere analysis of Geometry. But passing now to the consideration of space, I shall contend. with Kant, that some form of externality, given either by sensation or by intuition (psychological questions are here irrelevant), is a necessary condition of our experience of an external world. But I shall contend that the particular form of externality, which we know as Euclidean space, is not necessary; that, on the contrary, all that is necessary is complete relativity of position. This is necessary, because externality cannot be an intrinsic property of anything. I shall then deduce, from the relativity of position, the three axioms which we found essential to Geometry. This will complete the circle of the double argument. The three axioms will be a priori, in the sense that any form of externality, in beings with our laws of thought and a knowledge of an external world, must conform to them. The remaining axioms required to define Euclidean space will remain, for Geometry, empirical, since no geometrical principle and no possibility of experience of an outer world can prove them to be necessary.

A. Geometrical Argument

Geometry has always possessed, since Euclid, a great advantage – at least for the philosopher – over most of the other sciences, in that it prefaces its reasonings with a definite statement of those axioms which it believes to be fundamental. Until the very end of last century, Euclid's statement, though felt to be unsatisfactory, was believed to be accurate: many attempts were made to prove the axiom of parallels, but none to dispense with it. At first by Gauss, however, and then by a whole school of geometricians, an investigation was made into the effects of a denial of the doubtful axiom. It was found that geometries, whose reasonings, granted the premisses, were equally cogent and equally free from contradictions, could be constructed on a basis opposed to Euclid. Hence arose the idea, chiefly fostered by Helmholtz, that all the axioms were empirical, and that nothing but conformity to experienced space prompted the acceptance even of those axioms which Metageometry retained. Riemann and Helmholtz believed, in fact, that a Geometry could be constructed without the axioms which they retained, but neither they nor subsequent mathematicians have constructed such a Geometry. Three axioms are always retained in non-Euclidian systems, namely, the axiom of Free Mobility, the axiom that space has a finite integral number of dimensions, and the axiom that any two points have an invariant relation, namely, distance, which is unaltered by a combined motion of the two points as one figure.² I will discuss these axioms successively, and endeavour to prove that, without them, spatial magnitude would be unmeaning, and localisation would be impossible.

I. The Axiom of Free Mobility. Some of Euclid’s axioms have been classed as arithmetical, and have been supposed to have nothing particular to do with space. Such are the axioms that equals added to or subtracted from equals give equals, and that things which are equal to the same thing are equal to one another. But if these arithmetical axioms are to be applied to spatial magnitudes, they must have some spatial import, and thus even this class is not, in Geometry, merely arithmetical. The spatial element involved in the geometrical use of all these axioms is, in fact, a definition of spatial magnitude.³ A definition of spatial magnitude, again, since geometrical space is infinitely divisible, reduces itself to a definition of spatial equality. For, given this last, we can always divide two unequal magnitudes into a number of equal parts, and count the number of such parts in each.

We require, then, at the very outset, if Geometry is to be possible at all, some criterion of spatial equality. Now, two spatial magnitudes, unless they are whole and part, are necessarily external to one another, and cannot, therefore, as they stand, be directly compared. For comparison, therefore, some axiom is required. The requisite axiom is given by Euclid in the form, “Magnitudes which exactly coincide are equal.” But this form does not bring out the real point of the difficulty, which we only discover when he uses his axiom (e.g., in Book I, Prop. IV). The two magnitudes have to be brought into coincidence by a motion of one or both of them. Hence, if mere motion could alter shapes, our criterion of spatial equality would break down, and Geometry would become impossible. It follows that the application of the conception of magnitude to Geometry involves the following axiom: Spatial magnitudes can be moved from place to place without distortion; or, as it may be put, shapes do not depend in any way upon absolute position in space.

This is the axiom of Free Mobility, or of Congruence, as it is sometimes called. It has been regarded by Helmholtz and Erdmann as derived empirically from our experience of rigid bodies. But if the above deduction be correct, Helmholtz’s view must be mistaken. For the mere notion of spatial magnitude, and hence of a rigid body, as one whose shape remains unaltered through motion, becomes meaningless if the axiom be denied. The axiom, in fact, renders the experience in question possible, and cannot, therefore, be itself logically dependent on this experience. I do not mean to deny, of course, that we are made aware of Free Mobility, psychologically, by experience of approximately rigid bodies, and that, in a world of fluids, much more acute wits might have been needed to discover it. But in that case, the inhabitants of a fluid world would have to do without any Geometry until they discovered the axiom. The facts, I think, may be stated thus: When our axiom has made the conception of a rigid body possible, we find certain bodies, such as the platinum bar in the exchequer, which approximately answer to this conception. Having eliminated mere motion as a cause of change of shape, we can discover the other causes – change of temperature, etc. – by the ordinary empirical inductive methods. Our actual measurements, therefore, which are always effected by means of such approximately rigid bodies as we can discover, must always be empirical and inaccurate. But Geometry, as the science which deals with the conditions of measurement, and discusses shapes rather than actual bodies, will not be rendered empirical by this fact. Actual motions must be motions of matter, whose rigidity can be only empirically known; but geometrical motions are motions of shapes, and afford the accurate and a priori standard by which alone the empirical inaccuracies become discoverable and acquire a logical meaning.

Having now, I hope, made clear the meaning of the axiom of Free Mobility, I will endeavour to elucidate its necessity by discussing two cases where it does not hold, namely, the Geometry of non-congruent surfaces (e.g., an egg), and the measurement of time.

We have, in ordinary Geometry, many elaborate and satisfactory systems for non-congruent surfaces, e.g., for ellipsoids and hyperboloids. Why, I may be asked, is a thoroughly non-congruent Geometry more impossible than such systems ? The answer is obvious. The whole of the Geometry of such surfaces is only rendered possible by the use of infinitesimals, and in the infinitesimal all surfaces become plane. The fundamental formula, that for the length of an infinitesimal arc, is only obtained on the assumption that such an arc may be treated as a straight line, and that Euclidean Plane Geometry may be applied in the immediate neighbourhood of any point. If we had not our Euclidian measure, which could be moved without distortion, we should have no method of comparing small arcs in different places, and the Geometry of non-congruent surfaces would break down. The possibility of such systems, therefore, pre-supposes the axiom of Free Mobility, and can form no argument against its a priori necessity.

It remains to discuss the case of time-measurement. This case is very peculiar, for although time, like space, is homogeneous in all its parts, motion through time is impossible, because time has only one dimension. Superposition, as contemplated in the above criterion of spatial equality, is only possible for magnitudes of at least one dimension less than that of the manifold in question. In space, we cannot superpose two solids; in time, since we have only one dimension altogether, we cannot superpose two times. No event can be made to recur, without alteration, at another place in the time-series, in order to test the equality of its duration with that of some other event. No day can be superposed upon another day. Hence, it would seem, if the above discussion be correct, time measurement must be impossible. As regards direct time-measurement, this is indeed the case. Time is measured indirectly, by means of space. We are reduced, in fact, to the more or less arbitrary assumption that some motion, or set of motions, given in experience, is uniform. Equal times, on this assumption, are measured by equal spaces traversed. Fortunately, we have a large set of motions which all roughly agree: the swing of the pendulum, the rotation of the earth, the revolutions of the planets, etc. These do not exactly agree, but they- suffice to suggest the laws of motion, by which we are able, on our arbitrary hypothesis, to estimate their small departures from uniformity, just as we were able, by assuming Free Mobility, to measure the departures of actual bodies from rigidity. Philosophically, of course, the assumption is not arbitrary, but mathematically, the only reason we can give for it is, that without it time-measurement would be impossible.

The Axiom of Free Mobility involves, for its truth, the homogeneity of Space, or the complete relativity of position. For if any shape which is possible in one part of space be possible in another, it follows that all parts of space are qualitatively similar, and cannot, therefore, be distinguished by any intrinsic property. Hence, positions in space, if our axiom be true, must be wholly constituted by external relations, i.e., Position is not an intrinsic, but a purely relative, property of things in space. If there could be such a thing as absolute position, in short, all Geometry would be impossible. This is the fundamental postulate of Geometry, to which each of the three necessary axioms leads, and from which, conversely, as we shall see in the second part of the argument, each of these axioms can be deduced.

II. The Axiom of Dimensions. We have just seen, in discussing the axiom of Free Mobility, that all position is relative, i.e., that a position exists only by virtue of relations. It follows that, if positions are definable at all, they must be uniquely and exhaustively described by some finite number of such relations. If Geometry is to be possible, it must happen that, after enough relations have been given to determine a point uniquely, its relation to any fresh known point must be deducible from the relations already given. Hence we obtain, as an a priori condition of Geometry, logically indispensable to its existence, the axiom that Space must have a finite integral number of Dimensions. For every relation required in the localisation of a point constitutes a dimension, and a fraction of a relation is meaningless, while an infinite number of dimensions would be practically impossible to determine.

This axiom, like that of Free Mobility, is accepted by all metageometers. It is thus stated, for example, by Helmholtz: “In a space of n dimensions, the position of a point is uniquely determined by the measurement of n continuous variables (co-ordinates).” (Wissenschaftliche Abhandlungen, vol. ii, p. 614) So much, then, is a priori necessary to Geometry. The restriction of the dimensions to three, on the other hand, appears to have no logical necessity, and may be set down as wholly empirical. The extension of the dimensions to four, or to n, alters nothing in plane and solid Geometry, and presents no new logical difficulties; but some definite number of dimensions is assumed in all Geometries, nor is it possible to conceive of a Geometry which should be free from this assumption.

The limitation of the dimensions to three, though it is empirical, is not liable to the uncertainty and inaccuracy which usually belong, to empirical knowledge. For the alternatives which logic leaves to sense are discrete – if the dimensions are not three, they must be two or four, or some other integer – so that small errors are impossible. Hence the certainty of the axiom of three dimensions, though in part due to experience, is of quite a different order from that of (say) the law of gravitation. In the latter, a small inaccuracy might exist and remain undetected; in the former an error would have to be so large as to be utterly impossible to overlook. Hence the certainty of our whole axiom is almost as great as that of its a priori element.

III. The Axiom of the Straight Line. We have one more axiom to consider, before we have exhausted the logical pre-requisites of spatial measurement. This axiom is required to render possible the measurement of a special, but fundamental, spatial relation, namely distance.

We have seen that the measurement of spatial magnitudes, at bottom, is the same problem as that of localisation, and consists in the determination of spatial relations. I have hitherto spoken of these relations as though their meaning were self-evident, and as though spatial magnitude were a term which any one might use unchallenged. The time has come to examine more minutely into these assumptions.

First of all, what is the relation between two points? Obviously their distance apart. Well and good, but how is this to be measured? If it is to be measurable at all, it must be measured by a curve which joins the two points; and if it is to be unique, it must be measured by a curve which those two points completely determine. But such a curve is a straight line, for a straight line is defined as a curve determined by any two of its points. Hence, if two points are to have to each other a determinate intrinsic relation, without reference to any other point or figure in space, space must allow of curves uniquely determined by any two of their points, i.e., of straight lines.

This is the axiom of the straight line, but we can hardly regard its necessity for Geometry as established by so summary an argument. Many difficulties must be met before it can be regarded as proved. The chief of these is a mathematical difficulty, arising out of spherical Geometry, for a discussion of which I have not time to-night, but must refer to Mind, N.S., No. 17, pp. 16-18. But the first and obvious difficulty is that of turning the above hypothetical conclusion into a categorical one. Why should Geometry be impossible, if the relation between two points were not independent of the rest of space? Why must oar spatial magnitudes be built up out of distances and their relations? Why should not a reference to outside space be always necessary in determining distance?

The answer to this question lies in the axiom of Free Mobility, or its equivalent, the homogeneity of space. To begin with, two points must, if Geometry is to be possible, have some relation to each other, for we have seen that such relations alone constitute localisation. Now, if two points have a relation to each other, this must be an intrinsic relation. For it follows, from the axiom of Free Mobility, that two points, having the same relation to each other, can be constructed in any other part of space; if this were not possible, we have seen that Geometry could not exist. Hence the two points can traverse all space in a combined motion, without altering their relation to each other. But, in such a motion, any external relation of the two points, any relation which involves any other point or figure in space, must be altered;⁴ hence the relation between the two points, being unaltered, must be an intrinsic relation, a relation involving no other point or figure in space; and this intrinsic relation we call distance.

It might be thought, perhaps, that the distance between two points involved a reference to the intermediate points on the straight line joining them, but the above argument shows that this is not the case. For the intermediate points, in a combined motion of the two whose distance we are discussing, do not accompany the two moving points, and yet the distance remains unaltered. In any given position of the two points, the fact that the others are intermediate is deduced from the nature of distance, as measured by a certain curve on which the other points lie, and not vice versa. Only when distance, and its measurement by the straight line, have been established, do the intermediate points on the straight line acquire any peculiar relation to the two whose distance is in question.

Thus the first objection is disposed of. If Geometry is to be possible, any two points must have to each other a relation independent of the rest of space. But why only one ? Why not several? The answer to this lies in the fact that, as we saw in discussing Free Mobility, points must be wholly constituted by relations, and can have no intrinsic nature of their own. A point is defined by its relations to other points, and when once the relations necessary for definition have been given, no fresh relations to the points used in definition are possible, since the point defined has no properties from which such relations could flow. Now one relation to any one other point is as good for definition as more would be, since, however many we had, they would all remain unaltered in a combined motion of both points. Hence there can be only one intrinsic relation between any two points.

Hence two points must have one, and only one, relation independent of the rest of space. But need this relation be measured or defined by a curve which the two points completely determine?

We are used to the definition of the straight line as the shortest distance between two points, and this definition implies that distance might be equally well measured by other curves. This implication I believe to be wholly false. For when we speak of the length of a curve, we can give a meaning to our words only by dividing it into infinitesimal arcs, of which each may be regarded as a straight line, and whose sum gives a straight line equivalent in length to the curve measured. This necessity of the straight line is implied in the mathematical phrase, the rectification of carves. Unless we pre-suppose the straight line, as the measure of the infinitesimal arcs, we can never discover the length of our curve, and hence call never discover the applicability of our definition of the straight line as the shortest distance.

Again, let us suppose that some other curve, say a circle, is taken as oar measure of distance. Then the distance between any two points becomes indeterminate, for through two points we can draw an infinite number of circles, each of which will give a different distance. To overcome this indeterminateness, we should have to define our measuring circles as circles of given radius. But a given radius involves a measure of distance already given, otherwise we cannot know that two radii are equal. If it is said that we could define our circles as congruent inter se, without introducing the notion of a given radius, the choice of our typical circle would be perfectly arbitrary, and could only be effected by introducing, into distance, a relation to some third point, such as the centre of the circle. (For three points are needed to define a circle.) But we have seen that distance must be an intrinsic relation between two points. Hence the circle is impossible as a basis for the measurement of distance. If we seek a last escape from this conclusion, by restricting the direct measurement of distance to infinitesimal arcs, we do, it is true, so long as we exclude circles of infinitesimal radius, get a definite measure for the distance between two consecutive points, and hence for distance in general. But that is only because an infinitesimal arc of a circle is a straight line, and the measurement is effected through the above general method of measuring curves by dividing them into infinitesimal straight lines.

Hence, once more, the circle is not available for our present purpose. Similar objections apply, of course, to any other curve. Hence, for the measurement of the intrinsic relation between two points, we require a curve which those two points uniquely determine, i.e., the straight line.⁵

We can now take a farther step in the argument, and say: Not only must the intrinsic relation between two points be measured by a curve which those two points uniquely determine, but the relation must actually be that curve. If, as we saw in discussing the axiom of Free Mobility, all position is relative, and all spatial magnitudes are spatial relations, then it follows that a curve in space must actually be a relation or complex of relations.⁶ If I had used this doctrine at the start, I might have greatly shortened the above discussion. But it seemed better to avoid a controverted view in the first introduction of the axiom. Now, however, when the axiom has been shown, to the best of my ability, to be necessary to Geometry, we may argue thus: Two points can have no relation but what is given by lines which join them, and therefore, if they have a relation independent of the rest of space, there must be one line joining them which they completely determine. For what sort of thing can a spatial relation between two distinct points be? It must be something spatial, and it must, since points are wholly constituted by their relations, be at least as real and tangible as the points it relates. Only a line joining the points, it would seem, is capable of satisfying these requirements. Hence, once more, an intrinsic relation must involve – nay, must actually be – a unique line joining the two points. That is, linear magnitude, and hence Geometry in general, is logically impossible, unless space allows of curves uniquely determined by any two of their points.

To sum up: If, as the possibility of spatial measurement requires, all position is purely relative, then every point must have, to every other, one, and only one, relation independent of the rest of space. This relation is the distance between the two points. Now a relation between two points can only be defined – can only be, in fact – a line joining them. Hence, a unique relation involves a unique line, i.e., a line determined by any two of its points. Only in a space which allows of such lines is linear magnitude, or localisation in general, a logically possible conception.⁷

The above three axioms, then, are a priori necessary to Geometry. No others can be necessary, since geometries logically as unassailable as Euclid, geometries in which space is perfectly homogeneous, and position is purely relative, have been constructed without any other axioms by the metageometers. The remaining axioms of Euclidean Geometry – the axiom of parallels, the axiom that the number of dimensions is three, and the axiom of the straight line in Euclid’s special form (two straight lines cannot enclose a space) – are not essential to the possibility of a Geometry, i.e., are not deducible from the fact that a science of spatial relations and magnitudes is possible. These, then, must be derived from more special experience.

In summing up the argument we have just concluded, we may give it a more general form, and discuss the conditions of measurement, in any manifold, i.e., the qualities necessary to the manifold, in order that magnitudes in it may be determinable, not only as to the more or less, but as to the precise how much.

Measurement, we may say, is the application of number to continua, or, if we prefer it, the transformation of mere quantity into number of units. Using quantity to denote the vague more or less, and magnitude to denote the precise number of units, the problem of measurement may be defined as the transformation of quantity into magnitude.

Now number is discrete, and springs from the pure intellect, while quantity is continuous, and springs from sense. Hence, measurement, which arises from the effort to synthesise this dualism, involves fundamental and unavoidable contradictions. But disregarding these for the moment, let us discuss the conditions which enable us, in using the notion of magnitude, to avoid any other than the unavoidable contradictions.

Number, to begin with, is a whole, consisting of smaller units, all of these units being qualitatively alike. In order, therefore, that a continuous quantity may be expressible as a number, it must, on the one hand, be itself a whole, and must, on the other hand, be divisible into qualitatively similar parts. In the aspect of a whole, the quantity is intensive; in the aspect of an aggregate of parts, it is extensive. A purely intensive quantity, therefore, is not numerable. On the other hand, a quantity which can be divided at all into parts, must, since it is continuous, be infinitely divisible, otherwise the points at which it could be divided would form natural barriers, and destroy its continuity. But, further, it is not sufficient that there should be a possibility of division into mutually external parts; while the parts, to be perceptible as parts, must be mutually external, they must also, to be knowable as equal parts, be capable of overcoming their mutual externality. For this, as we have seen, superposition, which involves Free Mobility and perfect homogeneity, is necessary; in time, where superposition is impossible, direct measurement is also impossible, although all the other requisites for measurement are satisfied. Hence, infinite divisibility, free mobility, and homogeneity are necessary for the possibility of measurement in any manifold, and these, as we have seen, are equivalent to our three axioms. Hence these axioms are necessary, not only for spatial measurement, but for all measurement. The only manifold given in experience, in which these conditions are satisfied, is space. All other exact measurement – as could be proved, I believe, for every separate case – is effected, as we saw in the measurement of time, by reduction to a spatial correlative. This explains the paramount importance, to exact science, of the mechanical view of nature, which reduces all phenomena to motions in time and space. For number is, of all conceptions, the easiest to operate with, and science seeks everywhere for an opportunity to apply it, but finds this opportunity only by means of spatial equivalents to phenomena.

B. Philosophical Argument

We come now to the second part of our argument, which may be distinguished from the first part as deductive rather than analytic. This argument starts from the conditions of spatial experience, and deduces the above axioms as necessary consequences of these conditions, while our previous argument started from Geometry as its datum, and arrived, through the axioms, at the necessary conditions of their truth. The conditions for the truth of our essential axioms, as we have seen, resolve themselves into the homogeneity. of space, or the complete relativity of position. In our present argument which may be brief – since it is little more than a repetition, in inverse order, of the geometrical argument – we shall see that the relativity of position is deducible from the fact that we experience an external world, and that our three axioms are deducible from the relativity of position.

All experience of an external world, we may say with Kant, depends on the existence, psychologically, of some form of externality. I shall not enter into the arguments for this view, as I have nothing to add to the stock argument of the Transcendental Aesthetic, that to experience anything as external involves, as a logically prior condition, some form of externality in the mind. Whether this form is given in sensation, or in a pure intuition, is to me wholly irrelevant. All that is relevant is the purely logical point, that, if experience excludes solipsism, if perception presents me with an external world, then this externality cannot be logically dependent on the experience in which it is given, but is itself the logical prerequisite of such an experience. Accepting, then, as a mere experienced fact, the existence of an external world, some form of externality is an a priori condition of our knowledge of such a world.

So far with Kant. But Kant extended this a priori necessity to the whole of Euclidean space, and in this, I think, Metageometry proves that he went too far. For all that his argument proves is, that the properties which must belong to any form of externality are a priori. Now Euclidean space, as we have seen, has properties which are not necessary to any form of externality, and are not shared by non-Euclidean spaces. We can see, with very little trouble, that the properties which must belong to any form of externality are precisely those axioms which are shared by Euclid and Metageometry, and which are a priori necessary to any Geometry. I will give, very briefly, the outlines of this deduction.

In the first place, externality is an essentially relative conception – nothing can be external to itself. To be external to something, is to be an other with some relation to that thing. Hence, when we abstract a form of externality from all material content, and study it in isolation, position will appear, of necessity, as purely relative – it can have no intrinsic quality, for our form consists of pure externality, and externality contains no shadow or trace of an intrinsic quality. Hence we derive our fundamental postulate, the relativity of position. From this follows the homogeneity of our form, for any quality in one position, which marked out that position from another, would be necessarily more or less intrinsic, and would contradict the pure relativity. From the homogeneity of our form follows Free Mobility, for our form would not be homogeneous unless it allowed, in every part, shapes, or systems of relations, which it allowed in another. From the relativity of position follows the finite integral number of dimensions. For positions are defined simply and solely by relations to other positions. Any position, therefore, is completely defined when, and only when, enough such relations have been given to enable us to determine its relation to any fresh known position; and every such relation constitutes a dimension.

At this point, it is true, a difficulty arises: why should any number of these relations of externality suffice to determine the relations to a fresh known position? This difficulty is serious, and the only answer I can think of lies in the systematic unity of the world. If the relations between one point and a certain number of others never gave indication of its relation to fresh known points, we should have to suppose a system of relations perpetually reaching out into new worlds, and this, while it would throw doubt on the homogeneity of our form, would leave a scrappy world without systematic unity. It is essential to such unity – a unity which, being equivalent to the law of excluded middle,⁸ is a priori in a far higher degree than any geometrical principle – that the relations of A. to B and of A to C should throw light on the relations of B to C, and our axiom of dimensions is only the special form which this principle takes at the abstract geometrical level.

It is noticeable that the above argument cannot prove the necessity of any particular number of dimensions, and, in particular, cannot prove that there must be more than one dimension. In a space of one dimension, however, as we saw in discussing time, our axiom of Free Mobility would break down. In such a space no figure could be superposed upon any other, and Geometry would be impossible. Hence, also, the measurement of time would be impossible. At the same time, motion would remain possible, unless we adopt the doctrine of the plenum. Nevertheless, the existence of more than one dimension is, I think, an a priori condition of any experience in the least resembling our own. For in a world of only one dimension we could perceive only one point, or one object, throughout our whole lifetime, and the knowledge of a varied world of interrelated objects would be impossible. On this ground, I think, we may regard the existence of two or more dimensions as an essential condition of anything worth the name of experience of an external world.

As regards the axiom of the straight line, our argument here may be brief, since it was already deduced above from the relativity of position. Since position is relative, any two points must have some relation to each other: since our form of externality is homogeneous, this relation can be kept unchanged while the two points move in the form, i.e., change their relations to other points; hence their relation to each other is an intrinsic relation, independent of their relations to other points. Since our form is merely a complex of relations, a relation of externality must appear in the form, i.e., must be immediately presented, and not a mere inference; hence the intrinsic relation of two points must be a unique figure in our form, i.e., in spatial terms, the straight line joining the two points.

All the above essential axioms, therefore, are deducible from the fundamental properties of any form of externality, or, more generally still, from the mere fact that we have experience of an external world, quite apart from the specific nature of such experience. The special axioms of Euclid, on the contrary, can only be derived from an observation of the special nature of our actual spatial experience, and are therefore, for Geometry, empirical.

Finally, it is interesting to observe that our fundamental postulate, the relativity of position, involves, when we endeavour to study space apart from the matter which it relates, the most glaring contradictions. For, as we have seen, it is only possible to define points by their relations, i.e., by lines, while lines can only be defined by the points they relate. This involves either an infinite regress or a vicious circle, the penalty of our attempt to give thinghood to a mere complex of relations. If this contradiction is inherent in the notion of a form of externality taken in abstraction, it affords no argument against our reasoning, but rather evidence that the difficulties dealt with have been fundamental. To remove the contradiction, we should have to abandon the purely geometrical standpoint, and introduce, to begin with, the notion of matter, as that which is localised in space, as that which is related by spatial relations. With the notion of matter new contradictions are introduced, as is shown by the difficulties of atomism. But these contradictions fall outside the limits of the present paper, and could only be dealt with by a detailed criticism of Physics.

*  Bertrand Russell, “The A Priori in Geometry,” Proceedings of the Aristotelian Society 3, no. 2 (1895-96), 97-112

¹ I use “experience” here in the widest possible sense, the sense in which the word is used by Bradley.

² Vide Sophus Lie, Die Grundlagen der Geometrie, Leipziger Berichte, 1890

³ I ought to point out that most modern geometers, as adherents of the “projective school,” regard magnitude as a conception wholly irrelevant to Geometry, and refuse to admit measurement as an aim of geometrical reasoning. This fact, however, does not invalidate my argument, for the following reasons: (1) Spatial measurement remains a requisite for other sciences and for daily life. (2) The reduction of metrical to projective Geometry is effected, except in the case of hyperbolic space, by the use of imaginaries, and has therefore a purely technical, not a philosophical, import (Vide Klein, Vorlesungen über Nicht- Euklidische Geometrie, Göttingen, 1890). (3) Even projective Geometry involves localisation, and is even called, by many of its adherents, the Geometry of position; and localisation, as I have endeavoured to show, is exactly the same problem as that of spatial measurement. (4) Projective Geometry uses precisely those three axioms which I have regarded as a priori necessary for spatial measurement. (Vide Sophus Lie, op. cit.)

⁴ This is subject, in spherical space, to the modification pointed out in Mind, N.S., No. 17, 18

⁵ It is not intended to imply, in the above argument, that two points have no relation but the intrinsic relation of distance. On the contrary, they have also an extrinsic relation, direction, which involves a reference to other straight lines. But direction is meaningless apart from straight lines; it is, in fact, primarily a relation between straight lines, just as distance is a relation between points. The basis of spatial relations therefore remains the simple intrinsic relation of distance, as measured by the straight line. What I had to prove was that two points have a relation independent of external reference, not that they have no relation dependent on external reference.

⁶ Compare James, Psychology, vol. ii, pp. 149, 150

⁷ Spherical Geometry forces us slightly to modify the above axiom, and state it in the form: Any two points must, in general, determine a unique straight line. In this form the axiom is no longer equivalent to Euclid’s

⁸ See Bosanquet’s Logic, book ii, chap. vii