=162.= 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. =163.= (1) The possibility of spatial measurement allows us to infer the existence of a magnitude uniquely determined by any two points. The proof of this depends on the axiom of Free Mobility, or its equivalent, the homogeneity of space. We have seen that these are involved in the possibility of spatial measurement; we may employ them, therefore, in any argument as to the conditions of this possibility. Now to begin with, two points must, if Geometry is to be possible, have _some_ relation to each other, for we have seen that such relations alone constitute position or localization. But if two points have a relation to each other, this must be an intrinsic relation. For it follows, from the axiom of Free Mobility, that two points, forming a figure congruent with the given pair, can be constructed in any part of space. If this were not possible, we have seen that metrical Geometry could not exist. But both the figures may be regarded as composed of two points and their relation; if the two figures are congruent, therefore, it follows that the relation is quantitatively the same for both figures, since congruence is the test of spatial equality. Hence the two points have a quantitative relation, which is such that they can traverse all space in a combined motion without in any way altering that relation. But in such a general motion, any external relation of the two points, any relation involving other points or figures in space, must be altered[170]. Hence the relation between the two points, being unaltered, must be an intrinsic relation, a relation involving no other point or figure in space; and this intrinsic relation we call distance[171].