Suppose, he says, a north and south pole, _N_ and _S_, arbitrarily fixed, and an equator _EW_. Suppose a being, _B_, capable of impressions only from things on the surface of the sphere, to move in a meridian _NBS_. Let _B_ start from some point _a_, and finally, after describing a great circle, return to the same point _a_. If _a_ is known only by the quality of the impression it makes on _B_, _B_ may imagine he has not reached the same point _a_, but another similar point _a′_, bearing a relation to _a_ similar to that of the octave in singing: he might even not arrange his impressions spatially at all. In order that this may occur, we require the further assumption, that every difference in the above-mentioned feelings (as he describes the meridian) may be presented as a spatial distance between two places. Even now, _B_ may think he is describing a Euclidean straight line, containing similar points at certain intervals. Allowing, however, that he realizes the identity of _a_ with his initial position, he will now seem, by motion in a straight line, to have returned to the point from which he started, for his motion cannot, without the third dimension, seem to him other than rectilinear. Up to this point, there seems little ground for objection, except, perhaps, to the idea of a straight line with periodical similar points--if _B_ were as philosophical as, in these discussions, we usually suppose him to be, he would probably object to this interpretation of his experiences, on the ground that it regards empty space as something independent of the objects in it. It is worth pointing out, also, that _B_ would not need to describe the whole circle, in order suddenly to find himself home again with his old friends. Accurate measurements of small triangles would suffice to determine his space-constant, and show him the length of a great circle (or straight line, as he would call it). We must admit, also, that so hypothetical a being as _B_ might form no space-intuition at all, but as he is introduced solely for the purposes of the analogy, it is convenient to allow him all possible qualifications for his post. But these points do not touch the kernel of the argument, which lies in the statement that such a straight line, returning into itself after a finite time, would appear to _B_ as an "unendurable contradiction," and thus force him, for logical though not for sensational purposes, into the assumption of a third dimension. This assertion seems to me quite unwarranted: the whole of Metageometry is a solid array in disproof of it. Helmholtz's argument is, it must be remembered, only an analogy, and the contradiction would exist _only_ for a Euclidean. A complete _three_-dimensional Geometry has, we have seen in Chapter I., been developed on the assumption that straight lines are of finite length. A _constant_ value for the measure of curvature, as our discussion of Riemann showed, involves