=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.