Note 175 — Introductory material

To sum up: If points are defined simply by relations to other points, i.e., if all position is relative, every point must have to every other point one, and only one, relation independent of the rest of space. This relation is the distance between the two points. Now a relation between two points can only be defined by a line joining them—nay further, it may be contended that a relation can only be a line joining them. Hence a unique relation involves a unique line, i.e., a line determined by any two of its points. Only in a space which admits of such a line is linear magnitude a logically possible conception. But when once we have established the possibility, in general, of drawing such lines, and therefore of measuring linear magnitudes, we may find that a certain magnitude has a peculiar relation to the constitution of space. The straight line may turn out to be of finite length, and in this case its length will give a certain peculiar magnitude, the space-constant. Two antipodal points, that is, points which bisect the entire straight line, will then have a relation of magnitude which, though unaltered by motion, is rendered peculiar by a certain constant relation to the rest of space. This peculiarity presupposes a measure of linear magnitude in general, and cannot, therefore, upset the apriority of the axiom of the straight line. But it destroys, for points having the peculiar antipodal relation to each other, the argument which proved that the relation between two points could not, since it was unchanged by motion, have reference to the rest of space. Thus it is intelligible that, for such special points, the axiom breaks down, and an infinite number of straight lines are possible between them; but unless we had started with assuming the general validity of the axiom, we could never have reached a position in which antipodal points could have been known to be peculiar, or, indeed, a position which would have enabled us to give any quantitative definition whatever of particular points.