=173.= (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. =174.= (4) Finally, we have to prove that the existence of such a curve necessarily leads, when quantity is applied to the relation between two points, to a unique magnitude, which those two points completely determine. With this, we shall be brought back to distance, from which we started, and shall complete the circle of our argument. We saw, in section A § 119, that the figure formed by two points is projectively indistinguishable from that formed by any two other points in the same straight line; the figure, in both cases, is, from the projective standpoint, simply the straight line on which the two points lie. The difference of relation, in the two cases, is not qualitative, since projective Geometry cannot deal with it; nevertheless, there is some difference of relation. For instance, if one point be kept fixed, while the other moves, there is obviously some change of relation. This change, since all parts of the straight line are qualitatively alike, must be a change of quantity. If two points, therefore, determine a unique figure, there must exist, for the distinction between the various other points of this figure, a unique quantitative relation between the two determining points, and therefore, since these points are arbitrary, between only two points. This relation is _distance_, with which our argument began, and to which it at least returns.