An Essay on the Foundations of Geometry

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.