=172.= I have now shown, I hope conclusively, that spherical space affords no objection to the apriority of my axiom. Any two points have one relation, their distance, which is independent of the rest of space, and this relation requires, as its measure, a curve uniquely determined by those two points. I might have taken the bull by the horns, and said: Two points _can_ have no relation but what is given by lines which join them, and therefore, if they have a relation independent of the rest of space, there must be one line joining them which they completely determine. Thus James says[175]: "Just as, in the field of quantity, the relation between two numbers is another number, so in the field of space the relations are facts of the same order with the facts they relate.... When we speak of the relation of direction of two points towards each other, we mean simply the sensation of the line that joins the two points together. _The line is the relation...._ The relation of position between the top and bottom points of a vertical line is that line, and nothing else." If I had been willing to use this doctrine at the beginning, I might have avoided all discussion. A unique relation between two points _must_ in this case, involve a unique line between them. But it seemed better to avoid a doctrine not universally accepted, the more so as I was approaching the question from the logical, not the psychological, side. After disposing of the objections, however, it is interesting to find this confirmation of the above theory from so different a standpoint. Indeed, I believe James's doctrine could be proved to be a logical necessity, as well as a psychological fact. For what sort of thing can a spatial relation between two distinct points be? It must be something spatial, and it must, since points are wholly constituted by their relations, be something at least as real and tangible as the points it relates. There seems nothing which can satisfy these requirements, except a line joining them. Hence, once more, a unique relation must involve a unique line. That is, linear magnitude is logically impossible, unless space allows of curves uniquely determined by any two of their points.