At any rate, the early non-Euclidean geometers, realising that space as a result of measurement might turn out to be non-Euclidean, busied themselves with devising means of settling the question once and for all. Now, as a result of their measurements with material rods there was no doubt that space was very approximately Euclidean. But here it must be realised that the two non-Euclidean geometries as opposed to the Euclidean variety are not unique. We may conceive of various intensities of non-Euclideanism of both types, merging by insensible gradations into Euclidean geometry. It was therefore still an open question whether space, in spite of its apparent Euclidean characteristics, might not betray a slight trace of non-Euclideanism. A simple illustration will make this point clearer. We saw that in Euclidean geometry the ratio of the length of a circumference to its diameter was always the same number, . In Riemann’s geometry this number was always smaller than , and decreased progressively from the value to the value zero as the diameter of the circle increased. But there was nothing definite about this rate of decrease; it might be very rapid, just as it might be exceedingly slight. We must conceive, therefore, of varying intensities of non-Euclideanism, or of departure from Euclideanism. Hence, if the non-Euclideanism of real space were exceedingly slight, it might require measurements extending over a circumference of gigantic proportions, reaching as far as the stars in order to detect it; and measurements conducted in restricted areas could not be considered conclusive. The only means of disclosing slight traces of non-Euclideanism would therefore be obtained by having recourse to measurements conducted over cosmic distances. Of course, in an attempt of this sort, measurements with material rods were out of the question and it was necessary to appeal to other methods of exploration. These were obtained by taking advantage of the propagation of light rays in empty space. It was argued that the principle of sufficient reason precluded light rays from deviating to the right or to the left from their course along the straightest path through empty space; this belief was also in accord with the important physical principle of Least Action, as deduced from the laws of mechanics.[12] Accordingly, light rays would follow geodesics in empty space; and, as we have seen, a knowledge of the geodesics or straight lines reveals as much about the geometry of space as congruence itself. [Pg 50]