Note 123 — Introductory material

From the exclusively mathematical standpoint, as we have seen, projective Geometry discusses only what figures can be obtained from each other by projective transformations, i.e. by the operations of projection and section. These operations, in all their forms, presuppose the point, straight line, and plane[131], whose necessity for projective Geometry, from the purely mathematical point of view, is thus self-evident from the start. But philosophically, projective Geometry has, as we saw, a wider aim. This wider aim, which gives, to the investigation of projectively equivalent figures, its chief importance, consists in the determination of qualitative spatial similarity, in the determination, that is, of all the figures which, when any one figure is given, can be distinguished from the given figure, so long as quantity is excluded, only by the mere fact that they are external to it.