[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.