[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. [Illustration]