Note 41 — Introductory material

In connection with the reduction of metrical to projective Geometry, we have one more topic for discussion. This is the geometrical use of imaginaries, by means of which, except in the case of hyperbolic space, the reduction is effected. I have already contended, on other grounds, that this reduction, in spite of its immense technical importance, and in spite of the complete logical freedom of projective Geometry from metrical ideas, is purely technical, and is not philosophically valid. The same conclusion will appear, if we take up Cayley's challenge at the British Association, in his Presidential Address of 1883.