Note 162 — Introductory material

We have already seen, in discussing projective Geometry, that two points must determine a unique curve, the straight line. In metrical Geometry, the corresponding axiom is, that two points must determine a unique spatial quantity, distance. I propose to prove, in what follows, (1) that if distance, as a quantity completely determined by two points, did not exist, spatial magnitude would not be measurable; (2) that distance can only be determined by two points, if there is an actual curve in space determined by those two points; (3) that the existence of such a curve can be deduced from the conception of a form of externality, and (4) that the application of quantity to such a curve necessarily leads to a certain magnitude, namely distance, uniquely determined by any two points which determine the curve. The conclusion will be, if these propositions can be successfully maintained, that the axiom of distance is à priori in the same double sense as the axiom of Free Mobility, i.e. it is presupposed in the possibility of measurement, and it is necessarily true of any possible form of externality.