Chapter III: Section A (2)

=137.= The next step in the argument, which assumes two or more dimensions, is concerned with the general analogues of straight lines and planes, _i.e._ with figures--which may be regarded either as relations between positions or as series of positions--uniquely determined by two or by three positions. If this step can be successfully taken, our deduction of the above projective axioms will be complete, and descriptive Geometry will be established as the abstract _à priori_ doctrine of forms of externality.

To prove this contention, consider of what nature the relations can be by which positions are defined. We have seen already that our form is purely relational and infinitely divisible, and that positions (points) are the self-contradictory outcome of the search for something other than relations. What we really mean, therefore, by the relations defining a position, is, when we undo our previous abstraction, the relations of externality by which some thing is related to other things. But how, when we remain in the abstract form, must such relations appear?

=138.= We have to prove that two positions must have a relation independent of any reference to other positions. To prove this, let us recur to what was said, in connection with dimensions, as to the passivity and homogeneity of our form. Since positions are defined only by relations, there must be relations, within the form, between positions. But if there are such relations, there must be a relation which is intrinsic to two positions. For to suppose the contrary, is to attribute an interaction or causal connection, of some kind, between those two positions and other positions--a supposition which the perfect homogeneity of our form renders absurd, since all positions are qualitatively similar, and cannot be changed without losing their identity. We may put this argument thus: since positions are only defined by their relations, such definition could never begin, unless it began with a relation between only two positions. For suppose three positions _A_, _B_, _C_ were necessary, and gave rise to the relation _abc_ between the three. Then there would remain no means of defining the different pairs _BC_, _CA_, _AB_, since the only relation defining them would be one common to all three pairs. Nothing would be gained, in this case, by reference to fresh points, for it follows, from the homogeneity and passivity of the form, that these fresh points could not affect the internal relations of our triad, which relations, if they can give definiteness at all, must give it without the aid of external reference. Two positions must, therefore, if definition is to be possible, have some relation which they by themselves suffice to define. Precisely the same argument applies to three positions, or to four; the argument loses its scope only when we have exhausted the dimensions of the form considered. Thus, in three dimensions, five positions have no fresh relation, not deducible from those already known, for by the definition of dimensions, all the relations involved can be deduced from those of the fourth point to the first three, together with those of the fifth to the first three.

We may give the argument a more concrete, and perhaps a more convincing shape, by considering the matter arranged in our form. If two things are mutually external, they must since they belong to the same world, have some relation of externality; there is, therefore, a relation of externality between two things. But since our form is homogeneous, the same relation of externality may subsist in other parts of the form, _i.e._ while the two things considered alter their relations of externality to other things. The relation of externality between two things is, therefore, independent of other things. Hence, when we return to the abstract language of the form, two positions have a relation determined by those two positions alone, and independent of other positions.

Precisely the same argument applies to the relations of three positions, and in each case the relation must appear in the form as not a mere inference from the positions it relates. For relations, as we have seen, actually constitute a form of externality, and are not mere inferences from terms, which are nowhere to be found in the form[144].

To sum up: Since position is relative, two positions must have _some_ relation to each other; and since our form of externality is homogeneous, this relation can be kept unchanged while the two positions change their relations to other positions. Hence their relation is intrinsic, and independent of other positions. Since the form is a mere complex of relations, the relation in question must, if the form is sensuous or intuitive, be itself sensuous or intuitive, and not a mere inference. In this case, a unique relation must be a unique figure--in spatial terms, the straight line joining the two points.

=139.= With this, our deduction of projective Geometry from the _à priori_ conceptual properties of a form of externality is completed. That such a form, when regarded as an independent thing, is self-contradictory, has been abundantly evident throughout the discussion. But the science of the form has been founded on the opposite way of regarding it: we have held it throughout to be a mere complex of relations, and have deduced its properties exclusively from this view of it. The many difficulties, in applying such an _à priori_ deduction to intuitive space, and in explaining, as logical necessities, properties which appear as sensuous or intuitional data, must be postponed to Chapter IV. For the present, I wish to point out that projective Geometry is wholly _à priori_; that it deals with an object whose properties are logically deduced from its definition, not empirically discovered from data; that its definition, again, is founded on the possibility of experiencing diversity in relation, or multiplicity in unity; and that our whole science, therefore, is logically implied in, and deducible from, the possibility of such experience.

=140.= In metrical Geometry, on the contrary, we shall find a very different result. Although the geometrical conditions which render spatial measurement possible, will be found identical, except for slight differences in the form of statement, with the _à priori_ axioms discussed above, yet the actual measurement--which deals with actually given space, not the mere intellectual construction we have been just discussing--gives results which can only be known empirically and approximately, and can be deduced by no necessity of thought. The Euclidean and non-Euclidean spaces give the various results which are _à priori_ possible; the axioms peculiar to Euclid--which are properly not axioms, but empirical results of measurement--determine, within the errors of observation, which of these _à priori_ possibilities is realized in our actual space. Thus measurement deals throughout with an empirically given matter, not with a creature of the intellect, and its _à priori_ elements are only the conditions presupposed in the possibility of measurement. What these conditions are, we shall see in the second section of this chapter.

Section B.

THE AXIOMS OF METRICAL GEOMETRY.

=141.= We have now reviewed the axioms of projective Geometry, and have seen that they are _à priori_ deductions from the fact that we can experience externality, _i.e._ a coexistent multiplicity of different but interrelated things. But projective Geometry, in spite of its claims, is not the whole science of space, as is sufficiently proved by the fact that it cannot discriminate between Euclidean and non-Euclidean spaces[145]. For this purpose, spatial measurement is required: metrical Geometry, with its quantitative tests, can alone effect the discrimination. For all application of Geometry to physics, also, measurement is required; the law of gravitation, for example, requires the determination of actual distances. For many purposes, in short, projective Geometry is wholly insufficient: thus it is unable to distinguish between different kinds of conics, though their distinction is of fundamental importance in many departments of knowledge.

Metrical Geometry is, then, a necessary part of the science of space, and a part not included in descriptive Geometry. Its _à priori_ element, nevertheless, so far as this is spatial and not arithmetical, is the same as the postulate of projective Geometry, namely, the homogeneity of space, or its equivalent, the relativity of position. We can see, in fact, that the _à priori_ element in both is likely to be the same. For the _à priori_ in metrical Geometry will be whatever is presupposed in the possibility of spatial measurement, _i.e._ of quantitative spatial comparison. But such comparison presupposes simply a known identity of quality, the determination of which is precisely the problem of projective Geometry. Hence the conditions for the possibility of measurement, in so far as they are not arithmetical, will be precisely the same as those for projective Geometry.

=142.= Metrical Geometry, therefore, though distinct from projective Geometry, is not independent of it, but presupposes it, and arises from its combination with the extraneous idea of _quantity_. Nevertheless the mathematical form of the axioms, in metrical Geometry, is slightly different from their form in projective Geometry. The homogeneity of space is replaced by its equivalent, the axiom of Free Mobility. The axiom of the straight line is replaced by the axiom of distance: Two points determine a unique quantity, distance, which is unaltered in any motion of the two points as a single figure. This axiom, indeed, will be found to involve the axiom of the straight line--such a quantity could not exist unless the two points determined a unique curve--but its mathematical form is changed. Another important change is the collapse of the principle of duality: quantity can be applied to the straight line, because it is divisible into similar parts, but cannot be applied to the indivisible point. We thus obtain a reason, which was wanting in descriptive Geometry, for preferring points, as spatial elements, to straight lines or planes[146]. Finally, an entirely new idea is introduced with quantity, namely, the idea of _Motion_. Not that we study motion, or that any of our results have reference to motion, but that they cannot, though in projective Geometry they could, be obtained without at least an ideal motion of our figures through space.

Let us now examine in detail the prerequisites of spatial measurement. We shall find three axioms, without which such measurement would be impossible, but with which it is adequate to decide, empirically and approximately, the Euclidean or non-Euclidean nature of our actual space. We shall find, further, that these three axioms can be deduced from the conception of a form of externality, and owe nothing to the evidence of intuition. They are, therefore, like their equivalents the axioms of projective Geometry, _à priori_, and deducible from the conditions of spatial experience. This experience, accordingly, can never disprove them, since its very existence presupposes them.

I. _The Axiom of Free Mobility._

=143.= Metrical Geometry, to begin with, may be defined as the science which deals with the comparison and relations of spatial magnitudes. The conception of magnitude, therefore, is necessary from the start. Some of Euclid's axioms, accordingly, 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. These axioms, it is said, are purely arithmetical, and do not, like the others, ascribe an adjective to space. As regards their use in arithmetic, this is of course true. But if an arithmetical axiom is to be applied to spatial magnitudes, it must have some spatial import[147], and thus even this class is not, in Geometry, _merely_ arithmetical. Fortunately, the geometrical element is the same in all the axioms of this class--we can see at once, in fact, that it can amount to no more than a definition of spatial magnitude[148]. Again, since the space with which Geometry deals is infinitely divisible, a definition of spatial magnitude reduces itself to a definition of spatial equality, for, as soon as we have this last, we can compare two spatial magnitudes by dividing each into a number of equal units, and counting the number of such units in each[149]. The ratio of the number of units is, of course, the ratio of the two magnitudes.

=144.= We require, then, at the very outset, some criterion of spatial equality: without such a criterion metrical Geometry would become wholly impossible. It might appear, at first sight, as though this need not be an axiom, but might be a mere definition. In part this is true, but not wholly. The part which is merely a definition is given in Euclid's eighth axiom: "Magnitudes which exactly coincide are equal." But this gives a sufficient criterion only when the magnitudes to be compared already occupy the same position. When, as will normally be the case, the two spatial magnitudes are external to one another--as, indeed, must be the case, if they are distinct, and not whole and part--the two magnitudes can only be made to coincide by a motion of one or both of them. In order, therefore, that our definition of spatial magnitude may give unambiguous results, coincidence when superposed, if it can ever occur, must occur always, whatever path be pursued in bringing it about. Hence, if mere motion could alter shapes, our criterion of equality would break down. It follows that the application of the conception of magnitude to figures in space involves the following axiom[150]: _Spatial magnitudes can be moved from place to place without distortion_; or, as it may be put, _Shapes do not in any way depend upon absolute position in space_.

The above axiom is the axiom of Free Mobility[151]. I propose to prove (1) that the denial of this axiom would involve logical and philosophical absurdities, so that it must be classed as wholly _à priori_; (2) that metrical Geometry, if it refused this axiom, would be unable, without a logical absurdity, to establish the notion of spatial magnitude at all. The conclusion will be, that the axiom cannot be proved or disproved by experience, but is an _à priori_ condition of metrical Geometry. As I shall thus be maintaining a position which has been much controverted, especially by Helmholtz and Erdmann, I shall have to enter into the arguments at some length.

=145.= A. _Philosophical Argument._ The denial of the axiom involves absolute position, and an action of mere space, _per se_, on things. For the axiom does not assert that real bodies, as a matter of empirical fact, never change their shape in any way during their passage from place to place: on the contrary, we know that such changes do occur, sometimes in a very noticeable degree, and always to some extent. But such changes are attributed, not to the change of place as such, but to physical causes: changes of temperature, pressure, etc. What our axiom has to deal with is not actual material bodies, but geometrical figures[152], and it asserts that a figure which is possible in any one position in space is possible in every other. Its meaning will become clearer by reference to a case where it does not hold, say the space formed by the surface of an egg. Here, a triangle drawn near the equator cannot be moved without distortion to the point, as it would no longer fit the greater curvature of the new position: a triangle drawn near the point cannot be fitted on to the flatter end, and so on. Thus the method of superposition, such as Euclid employs in Book I. Prop. IV., becomes impossible; figures cannot be freely moved about, indeed, given any figure, we can determine a certain series of possible positions for it on the egg, outside which it becomes impossible. What I assert is, then, that there is a philosophic absurdity in supposing space in general to be of this nature. On the egg we have marked points, such as the two ends; the space formed by its surface is not homogeneous, and if things are moved about in it, it must of itself exercise a distorting effect upon them, quite independently of physical causes; if it did not exercise such an effect, the things could not be moved. Thus such a space would not be homogeneous, but would have marked points, by reference to which bodies would have absolute position, quite independently of any other bodies. Space would no longer be passive, but would exercise a definite effect upon things, and we should have to accommodate ourselves to the notion of marked points in empty space; these points being marked, not by the bodies which occupied them, but by their effects on any bodies which might from time to time occupy them. This want of homogeneity and passivity is, however, absurd; space must, since it is a form of externality, allow only of relative, not of absolute, position, and must be completely homogeneous throughout. To suppose it otherwise, is to give it a thinghood which no form of externality can possibly possess. We must, then, on purely philosophical grounds, admit that a geometrical figure which is possible anywhere is possible everywhere, which is the axiom of Free Mobility.

=146.= B. _Geometrical Argument._ Let us see next what sort of Geometry we could construct without this axiom. The ultimate standard of comparison of spatial magnitudes must, as we saw in introducing the axiom, be equality when superposed; but need we, from this equality, infer equality when separated? It has been urged by Erdmann that, for the more immediate purposes of Geometry, this would be unnecessary[153]. We might construct a new Geometry, he thinks, in which sizes varied with motion on any definite law. Such a view, as I shall show below, involves a logical error as to the nature of magnitude. But before pointing this out, let us discuss the geometrical consequences of assuming its truth. Suppose the length of an infinitesimal arc in some standard position were _ds_; then in any other position _p_ its length would be _ds.f(p)_, where the form of the function _f(p)_ must be supposed known. But how are we to determine the position _p_? For this purpose, we require _p_'s coordinates, _i.e._, some measurement of distance from the origin. But the distance from the origin could only be measured if we assumed our law _f(p)_ to measure it by. For suppose the origin to be _O_, and _Op_ to be a straight line whose length is required. If we have a measuring rod with which we travel along the line and measure successive infinitesimal arcs, the measuring rod will change its size as we move, so that an arc which appears by the measure to be _ds_ will really be _f(s).ds_, where _s_ is the previously traversed distance. If, on the other hand, we move our line _Op_ slowly through the origin, and measure each piece as it passes through, our measure, it is true, will not alter, but now we have no means of discovering the law by which any element has changed its length in coming to the origin. Hence, until we assume our function _f(p)_, we have no means of determining _p_, for we have just seen that distances from the origin can only be estimated by means of the law _f(p)_. It follows that experience can neither prove nor disprove the constancy of shapes throughout motion, since, if shapes were not constant, we should have to _assume_ a law of their variation before measurement became possible, and therefore measurement could not itself reveal that variation to us[154].

Nevertheless, such an arbitrarily assumed law _does_, at first sight, give a mathematically possible Geometry. The fundamental proposition, that two magnitudes which can be superposed in any one position can be superposed in any other, still holds. For two infinitesimal arcs, whose lengths in the standard position are _ds{1}_ and _ds{2}_, would, in any other position _p_, have lengths _f(p).ds{1}_ and _f(p).ds{2}_, so that their ratio would be unaltered. From this constancy of ratio, as we know through Riemann and Helmholtz, the above proposition follows. Hence all that Geometry requires, it would seem, as a basis for measurement, is an axiom that the alteration of shapes during motion follows a definite known law, such as that assumed above.

=147.= There is, however, in such a view, as I remarked above, a logical error as to the nature of magnitude. This error has been already pointed out in dealing with Erdmann[155], and need only be briefly repeated here. A judgment of magnitude is essentially a judgment of comparison: in unmeasured quantity, comparison as to the mere more or less, but in measured magnitude, comparison as to the precise how many times. To speak of differences of magnitude, therefore, in a case where comparison cannot reveal them, is logically absurd. Now in the case contemplated above, two magnitudes, which appear equal in one position, appear equal also when compared in another position. There is no sense, therefore, in supposing the two magnitudes unequal when separated, nor in supposing, consequently, that they have changed their magnitudes in motion. This senselessness of our hypothesis is the logical ground of the mathematical indeterminateness as to the law of variation. Since, then, there is no means of comparing two spatial figures, as regards magnitude, except superposition, the only logically possible axiom, if spatial magnitude is to be self-consistent, is the axiom of Free Mobility in the form first given above.

=148.= Although this axiom is _à priori_, its application to the measurement of actual bodies, as we found in discussing Helmholtz's views, always involves an empirical element[156]. Our axiom, then, only supplies the _à priori_ condition for carrying out an operation which, in the concrete, is empirical--just as arithmetic supplies the _à priori_ condition for a census. As this topic has been discussed at length in Chapter II., I shall say no more about it here.

=149.= There remain, however, a few objections and difficulties to be discussed. First, how do we obtain equality in solids, and in Kant's cases of right and left hands, or of right and left-handed screws, where actual superposition is impossible? Secondly, how can we take congruence as the only possible basis of spatial measurement, when we have before us the case of time, where no such thing as congruence is conceivable? Thirdly, it might be urged that we can immediately estimate spatial equality by the eye, with more or less accuracy, and thus have a measure independent of congruence. Fourthly, how is metrical Geometry possible on non-congruent surfaces, if congruence be the basis of spatial measurement? I will discuss these objections successively.

=150.= (1) How do we measure the equality of solids? These could only be brought into actual congruence if we had a fourth dimension to operate in[157], and from what I have said before of the absolute necessity of this test, it might seem as though we should be left here in utter ignorance. Euclid is silent on the subject, and in all works on Geometry it is assumed as self-evident that two cubes of equal side are equal. This assumption suggests that we are not so badly off as we should have been without congruence, as a test of equality in one or two dimensions; for now we can at least be sure that two cubes have all their sides and all their faces equal. Two such cubes differ, then, in no sensible spatial quality save position, for volume, in this case at any rate, is not a sensible quality. They are, therefore, as far as such qualities are concerned, indiscernible. If their places were interchanged, we might know the change by their colour, or by some other non-geometrical property; but so far as any property of which Geometry can take cognisance is concerned, everything would seem as before. To suppose a difference of volume, then, would be to ascribe an effect to mere position, which we saw to be inadmissible while discussing Free Mobility. Except as regards position, they are geometrically indiscernible, and we may call to our aid the Identity of Indiscernibles to establish their agreement in the one remaining geometrical property of volume. This may seem rather a strange principle to use in Mathematics, and for Geometry their equality is, perhaps, best regarded as a definition; but if we demand a philosophical ground for this definition, it is, I believe, only to be found in the Identity of Indiscernibles. We can, without error, make our _definition_ of three-dimensional equality rest on two-dimensional congruence. For since direct comparison as to volume is impossible, we are at liberty to _define_ two volumes as equal, when all their various lines, surfaces, angles and solid angles are congruent, since there remains, in such a case, no _measurable_ difference between the figures composing the two volumes. Of course, as soon as we have established this one case of equality of volumes, the rest of the theory follows; as appears from the ordinary method of integrating volumes, by dividing them into small cubes.

Thus congruence _helps_ to establish three-dimensional equality, though it cannot directly _prove_ such equality; and the same philosophical principle, of the homogeneity of space, by which congruence was proved, comes to our rescue here. But how about right-handed and left-handed screws? Here we can no longer apply the Identity of Indiscernibles, for the two are very well discernible. But as with solids, so here, Free Mobility can help us much. It can enable us, by ordinary measurement, to show that the internal relations of both screws are the same, and that the difference lies only in their relation to other things in space. Knowing these internal relations, we can calculate, by the Geometry which Free Mobility has rendered possible, all the geometrical properties of these screws--radius, pitch, etc.--and can show them to be severally equal in both. But this is all we require. Mediate comparison is possible, though immediate comparison is not. Both can, for instance, be compared with the cylinder on which both would fit, and thus their equality can be proved. A precisely similar proof holds, of course, for the other cases, right and left hands, spherical triangles, etc. On the whole, these cases confirm my argument; for they show, as Kant intended them to show[158], the essential relativity of space.

=151.= (2) As regards time, no congruence is here conceivable, for to effect congruence requires always--as we saw in the case of solids--one more dimension than belongs to the magnitudes compared. No day can be brought into temporal coincidence with any other day, to show that the two exactly cover each other; we are therefore reduced to the arbitrary assumption that some motion or set of motions, given us in experience, is uniform. Fortunately, we have a large set of motions which all roughly agree; the swing of the pendulum, the rotation and revolution of the earth and the planets, etc. These do not exactly agree, but they lead us to the laws of motion, by which we are able, on our arbitrary hypothesis, to estimate their small departures from uniformity; just as the assumption of Free Mobility enabled us to measure the departures of actual bodies from rigidity. But here, as there, another possibility is mathematically open to us, and can only be excluded by its philosophic absurdity; we might have assumed that the above set of approximately agreeing motions all had velocities which varied approximately as some arbitrarily assumed function of the time, _f(t)_ say, measured from some arbitrary origin. Such an assumption would still keep them as nearly synchronous as before, and would give an equally possible, though more complex, system of Mechanics; instead of the first law of motion, we should have the following: A particle perseveres in its state of rest, or of rectilinear motion with velocity varying as _f(t)_, except in so far as it is compelled to alter that state by the action of external forces. Such a hypothesis _is_ mathematically possible, but, like the similar one for space, it is excluded logically by the comparative nature of the judgment of quantity, and philosophically by the fact that it involves absolute time, as a determining agent in change, whereas time can never, philosophically, be anything but a passive form, abstracted from change. I have introduced this parallel from time, not as directly bearing on the argument, but as a simpler case which may serve to illustrate my reasoning in the more complex case of space. For since time, in mathematics, is one-dimensional, the mathematical difficulties are simpler than in Geometry; and although nothing accurately corresponds to congruence, there is a very similar mixture of mathematical and philosophical necessity, giving, finally, a thoroughly definite axiom as the basis of time-measurement, corresponding to congruence as the basis of space-measurement[159].

=152.= (3) The case of time-measurement suggests the third of the above objections to the absolute necessity of the axiom of Free Mobility. Psycho-physics has shown that we have an approximate power, by means of what may be called the sense of duration, of immediately estimating equal short times. This establishes a rough measure independent of any assumed uniform motion, and in space also, it may be said, we have a similar power of immediate comparison. We can see, by immediate inspection, that the sub-divisions on a foot rule are not grossly inaccurate; and so, it may be said, we both have a measure independent of congruence, and also could discover, by experience, any gross departure from Free Mobility. Against this view, however, there is at the outset a very fundamental psychological objection. It has been urged that all our comparison of spatial magnitudes proceeds by ideal superposition. Thus James says (Psychology, Vol. II. p. 152): "Even where we only feel one sub-division to be vaguely larger or less, the mind must pass rapidly between it and the other sub-division, and receive the immediate sensible shock of the more," and "so far as the sub-divisions of a sense-space are to be _measured_ exactly against each other, objective forms occupying one sub-division must be directly or indirectly superposed upon the other[160]."

Even if we waive this fundamental objection, however, others remain. To begin with, such judgments of equality are only very rough approximations, and cannot be applied to lines of more than a certain length, if only for the reason that such lines cannot well be seen together. Thus this method can only give us any security in our own immediate neighbourhood, and could in no wise warrant such operations as would be required for the construction of maps &c., much less the measurement of astronomical distances. They might just enable us to say that some lines were longer than others, but they would leave Geometry in a position no better than that of the Hedonical Calculus, in which we depend on a purely subjective measure. So inaccurate, in fact, is such a method acknowledged to be, that the foot-rule is as much a need of daily life as of science. Besides, no one would trust such immediate judgments, but for the fact that the stricter test of congruence to some extent confirms them; if we could not apply this test, we should have no ground for trusting them even as much as we do. Thus we should have, here, no real escape from our absolute dependence upon the axiom of Free Mobility.

=153.= (4) One last elucidatory remark is necessary before our proof of this axiom can be considered complete. We spoke above of the Geometry on an egg, where Free Mobility does not hold. What, I may be asked, is there about a thoroughly non-congruent Geometry, more impossible than this Geometry on the egg? The answer is obvious. The Geometry of non-congruent surfaces is _only_ 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 Euclidean 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. Thus the axiom of Free Mobility, as regards three-dimensional space, is necessarily implied and presupposed in the Geometry of non-congruent surfaces; the possibility of the latter, therefore, is a dependent and derivative possibility, and can form no argument against the _à priori_ necessity of congruence as the test of equality.

=154.= It is to be observed that the axiom of Free Mobility, as I have enunciated it, includes also the axiom to which Helmholtz gives the name of Monodromy. This asserts that a body does not alter its dimensions in consequence of a complete revolution through four right angles, but occupies at the end the same position as at the beginning. The supposed mathematical necessity of making a separate axiom of this property of space has been disproved by Sophus Lie (v. Chap. I. § 45); philosophically, it is plainly a particular case of Free Mobility[161], and indeed a particularly obvious case, for a translation really does make some change in a body, namely, a change in position, but a rotation through four right angles may be supposed to have been performed any number of times without appearing in the result, and the absurdity of ascribing to space the power of making bodies grow in the process is palpable; everything that was said above on congruence in general applies with even greater evidence to this special case.

=155.= The axiom of Free Mobility involves, if it is to be true, the homogeneity of space, or the complete relativity of position. For if any shape, which is possible in one part of space, be always 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 defined 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, metrical Geometry would be impossible. This relativity of position is the fundamental postulate of all Geometry, to which each of the necessary metrical axioms leads, and from which, conversely, each of these axioms can be deduced.

=156.= This converse deduction, as regards Free Mobility, is not very difficult, and follows from the argument of Section A[162], which I will briefly recapitulate. 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. Finally Free Mobility follows from homogeneity, for our form would not be homogeneous unless it allowed, in every part, shapes or systems of relations, which it allowed in any other part. Free Mobility, therefore, is a necessary property of every possible form of externality.

=157.= In summing up the argument we have just concluded, we may exhibit it, in consequence of the two preceding paragraphs, in the form of a completed circle. Starting from the conditions of spatial measurement, we found that the comparison, required for measurement, could only be effected by superposition. But we found, further, that the result of such comparison will only be unambiguous, if spatial magnitudes and shapes are unaltered by motion in space, if, in other words, shapes do not depend upon absolute position in space. But this axiom can only be true if space is homogeneous and position merely relative. Conversely, if position is assumed to be merely relative, a change of magnitude in motion--involving as it does, the assertion of absolute position--is impossible, and our test of spatial equality is therefore adequate. But position in any form of externality must be purely relative, since externality cannot be an intrinsic property of anything. Our axiom, therefore, is _à priori_ in a double sense. It is presupposed in all spatial measurement, and it is a necessary property of any form of externality. A similar double apriority, we shall see, appears in our other necessary axioms.

II. _The Axiom of Dimensions[163]._

=158.= We have seen, in discussing the axiom of Free Mobility, that all position is relative, that is, a position exists only by virtue of relations[164]. It follows that, if positions can be defined at all, they must be uniquely and exhaustively defined 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 relations to any fresh known point are deducible from the relations already given. Hence we obtain, as an _à 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 definition of a point constitutes a dimension, and a fraction of a relation is meaningless. The number of relations required must be finite, since an infinite number of dimensions would be practically impossible to determine. If we remember our axiom of Free Mobility, and remember also that space is a continuum, we may state our axiom, for metrical Geometry, in the form given by Helmholtz (v. Chap. I. § 25): "In a space of n dimensions, the position of every point is uniquely determined by the measurement of n continuous independent variables (coordinates).[165]"

=159.= So much, then, is _à priori_ necessary to metrical Geometry. The restriction of the dimensions to three seems, on the contrary, to be wholly the work of experience[166]. This restriction cannot be logically necessary, for as soon as we have formulated any analytical system, it appears wholly arbitrary. Why, we are driven to ask, cannot we add a fourth coordinate to our _x_, _y_, _z_, or give a geometrical meaning to _x^{4}_? In this more special form, we are tempted to regard the axiom of dimensions, like the number of inhabitants of a town, as a purely statistical fact, with no greater necessity than such facts have.

Geometry affords intrinsic evidence of the truth of my division of the axiom of dimensions into an _à priori_ and empirical portion. For while the extension of the number of dimensions to four, or to _n_, alters nothing in plane and solid Geometry, but only adds a new branch which interferes in no way with the old, _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[167].

=160.= Let us, since the point seems of some interest, repeat our proof of the apriority of this axiom from a slightly different point of view. We will begin, this time, from the most abstract conception of space, such as we find in Riemann's dissertation, or in Erdmann's extents. We have here, an ordered manifold, infinitely divisible and allowing of Free Mobility[168]. Free Mobility involves, as we saw, the power of passing continuously from any one point to any other, by any course which may seem pleasant to us; it involves, also, that, in such a course, no changes occur except changes of mere position, _i.e._, positions do not differ from one another in any qualitative way. (This absence of qualitative difference is the distinguishing mark of space as opposed to other manifolds, such as the colour- and tone-systems: in these, every element has a definite qualitative sensational value, whereas in space, the sensational value of a position depends wholly on its spatial relation to our own body, and is thus not intrinsic, but relative.) From the absence of qualitative differences among positions, it follows logically that positions exist only by virtue of other positions; one position differs from another just because they are two, not because of anything intrinsic in either. Position is thus defined simply and solely by relation 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 new position, this new position being defined by the same number of relations. Now, in order that such definition may be at all possible, a finite number of relations must suffice. But every such relation constitutes a dimension. Therefore, if Geometry is to be possible, it is _à priori_ necessary that space should have a finite integral number of dimensions.

=161.= The limitation of the dimensions to three is, as we have seen, empirical; nevertheless, it is not liable to the inaccuracy and uncertainty 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 number--so that _small_ errors are out of the question[169]. Hence the final 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 considerable as to be utterly impossible to overlook. It follows that the certainty of our whole axiom, that the number of dimensions is three, is almost as great as that of the _à priori_ element, since this element leaves to sense a definite disjunction of discrete possibilities.

III. _The Axiom of Distance._

=162.= We have already seen, in discussing projective Geometry, that two points must determine a unique curve, the straight line. In metrical Geometry, the corresponding axiom is, that two points must determine a unique spatial quantity, distance. I propose to prove, in what follows, (1) that if distance, as a quantity completely determined by two points, did not exist, spatial magnitude would not be measurable; (2) that distance can only be determined by two points, if there is an actual curve in space determined by those two points; (3) that the existence of such a curve can be deduced from the conception of a form of externality, and (4) that the application of quantity to such a curve necessarily leads to a certain magnitude, namely distance, uniquely determined by any two points which determine the curve. The conclusion will be, if these propositions can be successfully maintained, that the axiom of distance is _à priori_ in the same double sense as the axiom of Free Mobility, _i.e._ it is presupposed in the possibility of measurement, and it is necessarily true of any possible form of externality.

=163.= (1) The possibility of spatial measurement allows us to infer the existence of a magnitude uniquely determined by any two points. The proof of this depends on the axiom of Free Mobility, or its equivalent, the homogeneity of space. We have seen that these are involved in the possibility of spatial measurement; we may employ them, therefore, in any argument as to the conditions of this possibility.

Now 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 position or localization. But 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, forming a figure congruent with the given pair, can be constructed in any part of space. If this were not possible, we have seen that metrical Geometry could not exist. But both the figures may be regarded as composed of two points and their relation; if the two figures are congruent, therefore, it follows that the relation is quantitatively the same for both figures, since congruence is the test of spatial equality. Hence the two points have a quantitative relation, which is such that they can traverse all space in a combined motion without in any way altering that relation. But in such a general motion, any external relation of the two points, any relation involving other points or figures in space, must be altered[170]. 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[171].

=164.= It might be objected, to the above argument, that it involves a _petitio principii_. For it has been assumed that the two points and their relation form a figure, to which other figures can be congruent. Now if two points have no intrinsic relation, it would seem that they cannot form such a figure. The argument, therefore, apparently assumes what it had to prove. Why, it may be asked, should not three points be required, before we obtain any relation, which Free Mobility allows us to construct afresh in other parts of space?

The answer to this, as to the corresponding question in the first section of this chapter, lies, I think, in the passivity of space, or the mutual independence of its parts. For it follows, from this independence, that any figure, or any assemblage of points, may be discussed without reference to other figures or points. This principle is the basis of infinite divisibility, of the use of quantity in Geometry, and of all possibility of isolating particular figures for discussion. It follows that two points cannot be dependent, as to their relation, on any other points or figures, for if they were so dependent, we should have to suppose some action of such points or figures on the two points considered, which would contradict the mutual independence of different positions. To illustrate by an example: the relation of two given points does not depend on the other points of the straight line on which the given points lie. For only through their relation, _i.e._ through the straight line which they determine, can the other points of the straight line be known to have any peculiar connection with the given pair.

=165.= But why, it may be asked, should there be only one such relation between two points? Why not several? The answer to this lies in the fact that points are wholly constituted by relations, and have no intrinsic nature of their own[172]. 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 qualities 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 only be one relation determined by any two points.

=166.= (2) We have thus established our first proposition--two points have one and only one relation uniquely determined by those two points. This relation we call their distance apart. It remains to consider the conditions of the measurement of distance, _i.e._, how far a unique value for distance involves a curve uniquely determined by the two points.

In the first place, some curve joining the two points is involved in the above notion of a combined motion of the two points, or of two other points forming a figure congruent with the first two. For without some such curve, the two point-pairs cannot be known as congruent, nor can we have any test by which to discover when a point-pair is moving as a single figure[173]. Distance must be measured, therefore, by some line which joins the two points. But need this be a line which the two points completely determine?

=167.= We are accustomed to the definition of the straight line as the _shortest_ distance between two points, which implies that distance might equally well be measured by curved lines. This implication I believe to be false, for the following reasons. When we speak of the length of a curve, we can give a meaning to our words only by supposing the curve divided into infinitesimal rectilinear arcs, whose sum gives the length of an equivalent straight line; thus unless we presuppose the straight line, we have no means of comparing the lengths of different curves, and can therefore never discover the applicability of our definition. It might be thought, perhaps, that some other line, say a circle, might be used as the basis of measurement. But in order to estimate in this way the length of any curve other than a circle, we should have to divide the curve into infinitesimal circular arcs. Now two successive points do not determine a circle, so that an arc of two points would have an indeterminate length. It is true that, if we exclude infinitesimal radii for the measuring circles, the lengths of the infinitesimal arcs would be determinate, even if the circles varied, but that is only because all the small circular arcs through two consecutive points coincide with the straight line through those two points. Thus, even with the help of the arbitrary restriction to a finite radius, all that happens is that we are brought back to the straight line. If, to mend matters, we take three consecutive points of our curve, and reckon distance by the arc of the circle of curvature, the notion of distance loses its fundamental property of being a relation between _two_ points. For two consecutive points of the arc could not then be said to have any corresponding distance apart--three points would be necessary before the notion of distance became applicable. Thus the circle is not a possible basis for measurement, and similar objections apply, of course, with increased force, to any other curve. All this argument is designed to show, in detail, the logical impossibility of measuring distance by any curve not completely defined by the two points whose distance apart is required. If in the above we had taken distance as measured by circles of _given radius_, we should have introduced into its definition a relation to other points besides the two whose distance was to be measured, which we saw to be a logical fallacy. Moreover, how are we to know that all the circles have equal radii, until we have an independent measure of distance?

=168.= A straight line, then, is not the _shortest_ distance, but is simply _the_ distance between two points--so far, this conclusion has stood firm. But suppose we had two or more curves through two points, and that all these curves were congruent _inter se_. We should then say, in accordance with the definition of spatial equality, that the lengths of all these curves were equal. Now it might happen that, although no one of the curves was uniquely determined by the two end-points, yet the common length of all the curves was so determined. In this case, what would hinder us from calling this common length the distance apart, although no unique figure in space corresponded to it? This is the case contemplated by spherical Geometry, where, as on a sphere, antipodes can be joined by an infinite number of geodesics, all of which are of equal length. The difficulty supposed is, therefore, not a purely imaginary one, but one which modern Geometry forces us to face. I shall consequently discuss it at some length.

=169.= To begin with, I must point out that my axiom is not quite equivalent to Euclid's. Euclid's axiom states that two straight lines cannot enclose a space, _i.e._, cannot have more than one common point. Now if every two points, without exception, determine a unique straight line, it follows, of course, that two different straight lines can have only one point in common--so far, the two axioms are equivalent. But it may happen, as in spherical space, that two points _in general_ determine a unique straight line, but fail to do so when they have to each other the special relation of being antipodes. In such a system every pair of straight lines in the same plane meet in two points, which are each other's antipodes; but two points, _in general_, still determine a unique straight line. We are still able, therefore, to obtain distances from unique straight lines, except in limiting cases; and in such cases, we can take any point intermediate between the two antipodes, join it by the _same_ straight line to both antipodes, and measure its distance from those antipodes in the usual way. The sum of these distances then gives a unique value for the distance between the antipodes.

Thus even in spherical space, we are greatly assisted by the axiom of the straight line; all linear measurement is effected by it, and exceptional cases can be treated, through its help, by the usual methods for limits. Spherical space, therefore, is not so adverse as it at first appeared to be to the _à priori_ necessity of the axiom. Nevertheless we have, so far, not attacked the kernel of the objection which spherical space suggested. To this attack it is now our duty to proceed.