Note 173 — Introductory material

(3) But farther, the existence of curves uniquely determined by two points can be deduced from the nature of any form of externality[176]. For we saw, in discussing Free Mobility, that this axiom, together with homogeneity and the relativity of position, can be so deduced, and we saw in the beginning of our discussion on distance, that the existence of a unique relation between two points could be deduced from the homogeneity of space. Since position is relative, we may say, any two points must have some relation to each other: since our form of externality is homogeneous, this relation can be kept unchanged while the two points move in the form, i.e., change their relations to other points; hence their relation to each other is an intrinsic relation, independent of their relations to other points. But since our form is merely a complex of relations, a relation of externality must appear in the form, with the same evidence as anything else in the form; thus if the form be intuitive or sensational, the relation must be immediately presented, and not a mere inference. Hence the intrinsic relation between two points must be a unique figure in our form, i.e. in spatial terms, the straight line joining the two points.