Note 168 — Introductory material

A straight line, then, is not the shortest distance, but is simply the distance between two points—so far, this conclusion has stood firm. But suppose we had two or more curves through two points, and that all these curves were congruent inter se. We should then say, in accordance with the definition of spatial equality, that the lengths of all these curves were equal. Now it might happen that, although no one of the curves was uniquely determined by the two end-points, yet the common length of all the curves was so determined. In this case, what would hinder us from calling this common length the distance apart, although no unique figure in space corresponded to it? This is the case contemplated by spherical Geometry, where, as on a sphere, antipodes can be joined by an infinite number of geodesics, all of which are of equal length. The difficulty supposed is, therefore, not a purely imaginary one, but one which modern Geometry forces us to face. I shall consequently discuss it at some length.