[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.]
[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.