Note 141 — Introductory material

We have now reviewed the axioms of projective Geometry, and have seen that they are à priori deductions from the fact that we can experience externality, i.e. a coexistent multiplicity of different but interrelated things. But projective Geometry, in spite of its claims, is not the whole science of space, as is sufficiently proved by the fact that it cannot discriminate between Euclidean and non-Euclidean spaces[145]. For this purpose, spatial measurement is required: metrical Geometry, with its quantitative tests, can alone effect the discrimination. For all application of Geometry to physics, also, measurement is required; the law of gravitation, for example, requires the determination of actual distances. For many purposes, in short, projective Geometry is wholly insufficient: thus it is unable to distinguish between different kinds of conics, though their distinction is of fundamental importance in many departments of knowledge.