An Essay on the Foundations of Geometry

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