=170.= It will be remembered that, in our _à priori_ proof that two points must have one definite relation, we held it impossible for those two points to have, to the rest of space, any relation which would be unaltered by motion. Now in spherical space, in the particular case where the two points are antipodes, they _have_ a relation, unaltered by motion, to the rest of space--the relation, namely, that their distance is half the circumference of the universe. In our former discussion, we assumed that any relation to outside space must be a relation of position--and a relation of position must be altered by motion. But with a finite space, in which we have absolute magnitude, another relation becomes possible, namely, a relation of magnitude. Antipodal points, accordingly, like coincident points, no longer determine a unique straight line. And it is instructive to observe that there is, in consequence, an ambiguity in the expression for distance, like the ordinary ambiguity in angular measurement. If 1/k^{2} be the space constant, and _d_ be one value for the distance between two points, 2πkn ± d, where _n_ is any integer, is an equally good value. Distance is, in short, a periodic function like angle. Thus such a state of things rather confirms than destroys my contention, that distance depends on a curve uniquely determined by two points. For as soon as we drop this unique determination, we see ambiguities creeping into our expression for distance. Distance still has a set of _discrete_ values, corresponding to the fact that, given one point, the straight line is uniquely determined for all other points but one, the antipodal point. It is tempting to go on, and say: If through _every_ pair of points there were an infinite number of the curves used in measuring distance, distance would be able, for the same pair of points, to take, not only a discrete series, but an infinite _continuous_ series of values. =171.= This, however, is mere speculation. I come now to the _pièce de résistance_ of my argument. The ambiguity in spherical space arose, as we saw, from a relation of _magnitude_ to the rest of space--such a relation being unaltered by a motion of the two points, and therefore falling outside our introductory reasoning. But what is this relation of magnitude? Simply a relation of the _distance_ between the two points to a _distance_ given in the nature of the space in question. It follows that such a relation _presupposes_ a measure of distance, and need not, therefore, be contemplated in any argument which deals with the _à priori_ requisites for the possibility of definite distances[174].