Chapter III: Section A (3)

=170.= It will be remembered that, in our _à priori_ proof that two points must have one definite relation, we held it impossible for those two points to have, to the rest of space, any relation which would be unaltered by motion. Now in spherical space, in the particular case where the two points are antipodes, they _have_ a relation, unaltered by motion, to the rest of space--the relation, namely, that their distance is half the circumference of the universe. In our former discussion, we assumed that any relation to outside space must be a relation of position--and a relation of position must be altered by motion. But with a finite space, in which we have absolute magnitude, another relation becomes possible, namely, a relation of magnitude. Antipodal points, accordingly, like coincident points, no longer determine a unique straight line. And it is instructive to observe that there is, in consequence, an ambiguity in the expression for distance, like the ordinary ambiguity in angular measurement. If 1/k^{2} be the space constant, and _d_ be one value for the distance between two points, 2πkn ± d, where _n_ is any integer, is an equally good value. Distance is, in short, a periodic function like angle. Thus such a state of things rather confirms than destroys my contention, that distance depends on a curve uniquely determined by two points. For as soon as we drop this unique determination, we see ambiguities creeping into our expression for distance. Distance still has a set of _discrete_ values, corresponding to the fact that, given one point, the straight line is uniquely determined for all other points but one, the antipodal point. It is tempting to go on, and say: If through _every_ pair of points there were an infinite number of the curves used in measuring distance, distance would be able, for the same pair of points, to take, not only a discrete series, but an infinite _continuous_ series of values.

=171.= This, however, is mere speculation. I come now to the _pièce de résistance_ of my argument. The ambiguity in spherical space arose, as we saw, from a relation of _magnitude_ to the rest of space--such a relation being unaltered by a motion of the two points, and therefore falling outside our introductory reasoning. But what is this relation of magnitude? Simply a relation of the _distance_ between the two points to a _distance_ given in the nature of the space in question. It follows that such a relation _presupposes_ a measure of distance, and need not, therefore, be contemplated in any argument which deals with the _à priori_ requisites for the possibility of definite distances[174].

=172.= I have now shown, I hope conclusively, that spherical space affords no objection to the apriority of my axiom. Any two points have one relation, their distance, which is independent of the rest of space, and this relation requires, as its measure, a curve uniquely determined by those two points. I might have taken the bull by the horns, and said: 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. Thus James says[175]:

"Just as, in the field of quantity, the relation between two numbers is another number, so in the field of space the relations are facts of the same order with the facts they relate.... When we speak of the relation of direction of two points towards each other, we mean simply the sensation of the line that joins the two points together. _The line is the relation...._ The relation of position between the top and bottom points of a vertical line is that line, and nothing else."

If I had been willing to use this doctrine at the beginning, I might have avoided all discussion. A unique relation between two points _must_ in this case, involve a unique line between them. But it seemed better to avoid a doctrine not universally accepted, the more so as I was approaching the question from the logical, not the psychological, side. After disposing of the objections, however, it is interesting to find this confirmation of the above theory from so different a standpoint. Indeed, I believe James's doctrine could be proved to be a logical necessity, as well as a psychological fact. 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 something at least as real and tangible as the points it relates. There seems nothing which can satisfy these requirements, except a line joining them. Hence, once more, a unique relation must involve a unique line. That is, linear magnitude is logically impossible, unless space allows of curves uniquely determined by any two of their points.

=173.= (3) But farther, the existence of curves uniquely determined by two points can be deduced from the nature of any form of externality[176]. For we saw, in discussing Free Mobility, that this axiom, together with homogeneity and the relativity of position, can be so deduced, and we saw in the beginning of our discussion on distance, that the existence of a unique relation between two points could be deduced from the homogeneity of space. Since position is relative, we may say, 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. But since our form _is_ merely a complex of relations, a relation of externality must appear in the form, with the same evidence as anything else in the form; thus if the form be intuitive or sensational, the relation must be immediately presented, and not a mere inference. Hence the intrinsic relation between two points must be a unique figure in our form, _i.e._ in spatial terms, the straight line joining the two points.

=174.= (4) Finally, we have to prove that the existence of such a curve necessarily leads, when quantity is applied to the relation between two points, to a unique magnitude, which those two points completely determine. With this, we shall be brought back to distance, from which we started, and shall complete the circle of our argument.

We saw, in section A § 119, that the figure formed by two points is projectively indistinguishable from that formed by any two other points in the same straight line; the figure, in both cases, is, from the projective standpoint, simply the straight line on which the two points lie. The difference of relation, in the two cases, is not qualitative, since projective Geometry cannot deal with it; nevertheless, there is some difference of relation. For instance, if one point be kept fixed, while the other moves, there is obviously some change of relation. This change, since all parts of the straight line are qualitatively alike, must be a change of quantity. If two points, therefore, determine a unique figure, there must exist, for the distinction between the various other points of this figure, a unique quantitative relation between the two determining points, and therefore, since these points are arbitrary, between only two points. This relation is _distance_, with which our argument began, and to which it at least returns.

=175.= To sum up: If points are defined simply by relations to other points, _i.e._, if all position is relative, _every point must have to every other point 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 by a line joining them--nay further, it may be contended that a relation can only _be_ 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 admits of such a line is linear magnitude a logically possible conception. But when once we have established the possibility, _in general_, of drawing such lines, and therefore of measuring linear magnitudes, we may find that a certain magnitude has a peculiar relation to the constitution of space. The straight line may turn out to be of finite length, and in this case its length will give a certain peculiar magnitude, the space-constant. Two antipodal points, that is, points which bisect the entire straight line, will then have a relation of magnitude which, though unaltered by motion, is rendered peculiar by a certain constant relation to the rest of space. This peculiarity presupposes a measure of linear magnitude in general, and cannot, therefore, upset the apriority of the axiom of the straight line. But it destroys, for points having the peculiar antipodal relation to each other, the argument which proved that the relation between two points could not, since it was unchanged by motion, have reference to the rest of space. Thus it is intelligible that, for such special points, the axiom breaks down, and an infinite number of straight lines are possible between them; but unless we had started with assuming the general validity of the axiom, we could never have reached a position in which antipodal points could have been known to be peculiar, or, indeed, a position which would have enabled us to give any quantitative definition whatever of particular points.

Distance and the straight line, as relations uniquely determined by two points, are thus _à priori_ necessary to metrical Geometry. But further, they are properties which must belong to any form of externality. Since their necessity for Geometry was deduced from homogeneity and the relativity of position, and since these are necessary properties of any form of externality, the same argument proves both conclusions. We thus obtain, as in the case of Free Mobility, a double apriority: The axiom of Distance, and its implication, the axiom of the Straight Line, are, on the one hand, presupposed in the possibility of spatial magnitude, and cannot, therefore, be contradicted by any experience resulting from the measurement of space; while they are consequences, on the other hand, of the necessary properties of any form of externality which is to render possible experience of an external world.

=176.= In connection with the straight line, it will be convenient to discuss the conditions of a metrical coordinate system. The projective coordinate system, as we have seen, aims only at a convenient nomenclature for different points, and can be set up without introducing the notion of spatial quantity. But a metrical coordinate system does much more than this. It defines every point quantitatively, by its quantitative spatial relations to a certain coordinate figure. Only when the system of coordinates is thus metrical, _i.e._, when every coordinate represents some spatial magnitude, which is itself a relation of the point defined to some other point or figure--can operations with coordinates lead to a metrical result. When, as in projective Geometry, the coordinates are not spatial magnitudes, no amount of transformation can give a metrical result. I wish to prove, here, that a metrical coordinate system necessarily involves the straight line, and cannot, without a logical fallacy, be set up on any other basis. The projective system of coordinates, as we saw, is entirely based on the straight line; but the metrical system is more important, since its quantities embody actual information as to spatial magnitudes, which, in projective Geometry, is not the case.

In the first place, a point's metrical coordinates constitute a complete quantitative definition of it; now a point can only be defined, as we have seen, by its relations to other points, and these relations can only be defined by means of the straight line. Consequently, any metrical system of coordinates must involve the straight line, as the basis of its definitions of points.

This _à priori_ argument, however, though I believe it to be quite sound, is not likely to carry conviction to any one persuaded of the opposite. Let us, therefore, examine metrical coordinate systems in detail, and show, in each case, their dependence on the straight line.

We have already seen that the notion of distance is impossible without the straight line. We cannot, therefore, define our coordinates in any of the ordinary ways, as the distances from three planes, lines, points, spheres, or what not. Polar coordinates are impossible, since,--waiving the straightness of the radius vector--the length of the radius vector becomes unmeaning. Triangular coordinates involve not only angles, which must in the limit be rectilinear, but straight lines, or at any rate some well-defined curves. Now curves can only be metrically defined in two ways: _Either_ by relation to the straight line, as, _e.g._, by the curvature at any point, _or_ by purely analytical equations, which presuppose an intelligible system of metrical coordinates. What methods remain for assigning these arbitrary values to different points? Nay, how are we to get any estimate of the difference--to avoid the more special notion of distance--between two points? The very notion of a point has become illusory. When we have a coordinate system, we may define a point by its three coordinates; in the absence of such a system, we may define the notion of point _in general_ as the intersection of three surfaces or of two curves. Here we take surfaces and curves as notions which intuition makes plain, but if we wish them to give us a precise numerical definition of _particular_ points, we must specify the kind of surface or curve to be used. Now this, as we have seen, is only possible when we presuppose either the straight line, or a coordinate system. It follows that every coordinate system presupposes the straight line, and is logically impossible without it.

=177.= The above three axioms, we have seen, are _à priori_ necessary to metrical Geometry. No others can be necessary, since metrical systems, logically as unassailable as Euclid's, and dealing with spaces equally homogeneous and equally relational, have been constructed by the metageometers, without the help of any other axioms. The remaining axioms of Euclidean Geometry--the axiom of parallels, the axiom that the number of dimensions is three, and Euclid's form of the axiom of the straight line (two straight lines cannot enclose a space)--are not essential to the possibility of metrical Geometry, _i.e._, are not deducible from the fact that a science of spatial magnitudes is possible. They are rather to be regarded as empirical laws, obtained, like the empirical laws of other sciences, by actual investigation of the given subject-matter--in this instance, experienced space.

=178.= In summing up the distinctive argument of this Section, we may give it a more general form, and discuss the conditions of measurement in any continuous manifold, _i.e._, the qualities necessary to the manifold, in order that quantities 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 a 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--a purely extensive quantity, if any such could be imagined, would not be a single quantity at all, since it would have to consist of wholly unsynthesized particulars. A measurable quantity, therefore, is a whole divisible into similar parts. But a continuous quantity, if divisible at all, must be _infinitely_ divisible. For otherwise the points at which it could be divided would form natural barriers, and so 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, we require superposition, which involves Free Mobility and homogeneity--the absence of Free Mobility in time, where all other requisites of measurement are fulfilled, renders direct measurement of time impossible. Hence infinite divisibility, free mobility, and homogeneity are necessary for the possibility of measurement in _any_ continuous manifold, and these, as we have seen, are equivalent to our three axioms. These axioms are necessary, therefore, 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 case 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[177].

=179.= We have now seen in what the _à priori_ element of Geometry consists. This _à priori_ element may be defined as the axioms common to Euclidean and non-Euclidean spaces, as the axioms deducible from the conception of a form of externality, or--in metrical Geometry--as the axioms required for the possibility of measurement. It remains to discuss, in a final chapter, some questions of a more general philosophic nature, in which we shall have to desert the firm ground of mathematics and enter on speculations which I put forward very tentatively, and with little faith in their ultimate validity. The chief questions for this final chapter will be two: (1) How is such _à priori_ and purely logical necessity possible, as applied to an actually given subject-matter like space? (2) How can we remove the contradictions which have haunted us in this chapter, arising out of the relativity, infinite divisibility, and unbounded extension of space? These two questions are forced upon us by the present chapter, but as they open some of the fundamental problems of philosophy, it would be rash to expect a conclusive or wholly satisfactory answer. A few hints and suggestions may be hoped for, but a complete solution could only be obtained from a complete philosophy, of which the prospects are far too slender to encourage a confident frame of mind.

FOOTNOTES:

[116] See infra, Axiom of Distance, in Sec. B. of this Chapter.

[117] Thus on a cylinder, two geodesics, _e.g._ a generator and a helix, may have any number of intersections--a very important difference from the plane.

[118] Cf. Cremona, Projective Geometry (Clarendon Press, 2nd ed. 1893) p. 50: "Most of the propositions in Euclid's Elements are metrical, and it is not easy to find among them an example of a purely descriptive theorem."

[119] Op. cit. p. 226.

[120] Some ground for this choice will appear when we come to metrical Geometry.

[121] The straight line _σa_ denotes the straight line common to the planes _σ_ and _a_, the point _σa_ denotes the point common to the plane _σ_ and the straight line _a_, and similarly for the rest of the notation.

[122] Cremona (op. cit. Chap. IX. p. 50) defines anharmonic ratio as a metrical property which is unaltered by projection. This, however, destroys the logical independence of projective Geometry, which can only be maintained by a purely descriptive definition.

[123] There is no corresponding property of _three_ points on a line, because they can be projectively transformed into any other three points on the same line. See § 120.

[124] Due to v. Staudt's "Geometrie der Lage."

[125] See Cremona, op. cit. Chapter VIII.

[126] The corresponding definitions, for the two-dimensional manifold of lines through a point, follow by the principle of duality.

[127] It is important to observe that this definition of the Point introduces metrical ideas. Without metrical ideas, we saw, nothing appears to give the Point precedence of the straight line, or indeed to distinguish it conceptually from the straight line. A reference to quantity is therefore inevitable in defining the Point, if the definition is to be geometrical. A non-metrical definition would have to be also non-geometrical. See Chap. IV. §§ 196-199.

[128] §§ 163-175.

[129] On this axiom, however, compare § 131.

[130] For the proof of this proposition, see Chap. III. Sec. B, Axiom of Dimensions.

[131] The straight line and plane, in all discussions of general Geometry, are not necessarily Euclidean. They are simply figures determined, in general, by two and by three points respectively; whether they conform to the axiom of parallels and to Euclid's form of the axiom of the straight line, is not to be considered in the general definition.

[132] That projective Geometry must have existential import, I shall attempt to prove in Chapter IV.

[133] Logic, Book I. Chapter II.

[134] Cf. Bradley's Logic, p. 63. It will be seen that the sense in which I have spoken of space as a principle of differentiation is not the sense of a "principle of individuation" which Bradley objects to.

[135] Chap. IV. §§ 186-191.

[136] Chap. IV. § 201 ff.

[137] It is important to observe, however, that this way of regarding spatial relations is metrical; from the projective standpoint, the relation between two points is the whole unbounded straight line on which they lie, and need not be regarded as divisible into parts or as built up of points.

[138] §§ 207, 208. Cf. Hegel, Naturphilosophie, § 254.

[139] See Chap. IV. §§ 196-199.

[140] See a forthcoming article on "The relations of number and quantity" by the present writer in _Mind_, July, 1897.

[141] Logic, Vol. II. Chap. VII. p. 211.

[142] Real, as opposed to logical, diversity is throughout intended. Diverse aspects may coexist in a thing at one time and place, but two diverse real things cannot so coexist.

[143] On the insufficiency of time alone, see Chapter IV. § 191.

[144] Geometrically, the axiom of the plane is, not that three points determine a figure at all, which follows from the axiom of the straight line, but that the straight line joining two casual points of the plane lies wholly in the plane. This axiom requires a projective method of constructing the plane, _i.e._ of finding all the triads of points which determine the same projective figure as the given triad. The required construction will be obtained if we can find any projective figure determined by three points, and any projective method of reaching other points which determine the same figure.

Let _O_, _P_, _Q_ be the three points whose projective relation is required. Then we have given us the three straight lines _PQ_, _QO_, _OP_. Metrically, the relation between these points is made up of the area, and the magnitude of the sides and angles, of the triangle _OPQ_, just as the relation between two points is distance. But projectively, the figure is unchanged when _P_ and _Q_ travel along _OP_ and _OQ_, or when _OP_ and _OQ_ turn about _O_ in such a way as still to meet _PQ_. This is a result of the general principle of projective equivalence enunciated above (§§ 108, 109). Hence the projective relation between _O_, _P_, _Q_ is the same as that between _O_, _p_, _q_ or _O_, _P′_, _Q′_; that is, _p_, _q_ and _P′_, _Q′_ lie in the plane _OPQ_. In this way, any number of points on the plane may be obtained, and by repeating the construction with fresh triads, every point of the plane can be reached. We have to prove that, when the plane is so constructed, the straight line joining any two points of the plane lies wholly in the plane.

It is evident, from the manner of construction, that any point of _PQ_, _OP_, _OQ_, _OP′_ or _OQ′_ lies in the plane. If we can prove that any point of _pq_ lies in the plane, we shall have proved all that is required, since _pq_ may be transformed, by successive repetitions of the same construction, into any straight line joining two points of the plane. But we have seen that the same plane is determined by _O_, _p_, _q_ and by _O_, _P_, _Q_. The straight lines _PQ_, _pq_ have, therefore, the same relation to the plane. But _PQ_ lies wholly in the plane; therefore _pq_ also lies wholly in the plane. Hence our axiom is proved.

[145] A detailed proof has been given above, Chap. I. 3rd period. It is to be observed that any reference to infinitely distant elements involves metrical ideas.

[146] Cf. Section A, §§ 115-117.

[147] Contrast Erdmann, op. cit. p. 138.

[148] Cf. Erdmann, op. cit. p. 164.

[149] Strictly speaking, this method is only applicable where the two magnitudes are commensurable. But if we take infinite divisibility rigidly, the units can theoretically be taken so small as to obtain any required degree of approximation. The difficulty is the universal one of applying to continua the essentially discrete conception of number.

[150] Cf. Erdmann, op. cit. p. 50.

[151] Also called the axiom of congruence. I have taken congruence to be the _definition_ of spatial equality by superposition, and shall therefore generally speak of the _axiom_ as Free Mobility.

[152] For the sense in which these figures are to be regarded as material, see criticism of Helmholtz, Chapter II. §§ 69 ff.

[153] Op. cit. p. 60.

[154] The view of Helmholtz and Erdmann, that mechanical experience suffices here, though geometrical experience fails us, has been discussed above, Chapter II. §§ 73, 82.

[155] Chapter II. § 81.

[156] Chapter II. § 72.

[157] Contrast Delbœuf, L'ancienne et les nouvelles géométries, II. Rev. Phil. 1894, Vol. xxxvii. p. 354.

[158] Prolegomena, § 13. See Vaihinger's Commentar, II. pp. 518-532 esp. pp. 521-2. The above was Kant's whole purpose in 1768, but only part of his purpose in the Prolegomena, where the intuitive nature of space was also to be proved.

[159] On the subject of time measurement, cf. Bosanquet's Logic, Vol. i. pp. 178-183. Since time, in the above account, is measured by motion, its measurement presupposes that of spatial magnitudes.

[160] Cf. Stumpf. Ursprung der Raumvorstellung, p. 68.

[161] As is Helmholtz's other axiom, that the possibility of superposition is independent of the course pursued in bringing it about.

[162] Cf. §§ 129, 130.

[163] This deduction is practically the same as that in Sec. A, but I have stated it here with more special reference to space and to metrical Geometry.

[164] The question: "Relations to what?" is a question involving many difficulties. It will be touched on later in this chapter, and answered, as far as possible, in the fourth chapter. For the present, in spite of the glaring circle involved, I shall take the relations as relations to other positions.

[165] Wiss. Abh. Vol. II. p. 614.

[166] Cp. Grassmann, Ausdehnungslehre von 1844, 2nd ed. p. XXIII.

[167] Delbœuf, it is true, speaks of Geometries with _m_/_n_ dimensions, but gives no reference (Rev. Phil. T. xxxvi. p. 450).

[168] In criticizing Erdmann, it will be remembered, we saw that Free Mobility is a necessary property of his extents, though he does not regard it as such.

[169] Cf. Riemann, Hypothesen welche der Geometrie zu Grunde liegen, Gesammelte Werke, p. 266; also Erdmann, op. cit. p. 154.

[170] This is subject, in spherical space, to the modification pointed out below, in dealing with the exception to the axiom of the straight line. See §§ 168-171.

[171] In speaking of distance at once as a quantity and as an intrinsic relation, I am anxious to guard against an apparent inconsistency. I have spoken of the judgment of quantity, throughout, as one of comparison; how, then, can a quantity be intrinsic? The reply is that, although measurement and the judgment of quantity express the result of comparison, yet the terms compared must exist before the comparison; in this case, the terms compared in measuring distances, _i.e._ in comparing them _inter se_, are intrinsic relations between points. Thus, although the _measurement_ of distance involves a reference to other distances, and its expression as a magnitude requires such a reference, yet its existence does not depend on any external reference, but exclusively on the two points whose distance it is.

[172] See the end of the argument on Free Mobility, § 155 ff.

[173] In Frischauf's "Absolute Geometrie nach Johann Bolyai," Anhang, there is a series of definitions, starting from the sphere, as the locus of congruent point-pairs when one point of the pair is fixed, and hence obtaining the circle and the straight line. From the above it follows, that the sphere so defined already involves a curve between the points of the point-pair, by which various point-pairs can be known as congruent; and it will appear, as we proceed, that this curve must be a straight line. Frischauf's definition by means of the sphere involves, therefore, a vicious circle, since the sphere presupposes the straight line, as the test of congruent point-pairs.

[174] Nor in any argument which, like those of projective Geometry, avoids the notion of magnitude or distance altogether. It follows that the propositions of projective Geometry apply, without reserve, to spherical space, since the exception to the axiom of the straight line arises only on metrical ground.

[175] Psychology, Vol. II. pp. 149-150.

[176] This step in the argument has been put very briefly, since it is a mere repetition of the corresponding argument in Section A, and is inserted here only for the sake of logical completeness. See § 137 ff.

[177] Cf. Hannequin, Essai critique sur l'hypothèse des atomes, Paris, 1895, passim.