Viewing the situation as it now stands, we may say that material rods, visually recognised as rigid, will be taken as norms of measurement in physical space. In other words, it will be assumed that rods which coincide when brought together will continue to remain congruent when transferred, independently of one another, to other regions of space. Of course, the physicist will guard himself as much as possible against local contingent influences, such as variations of pressure and temperature, which might influence the behaviour of his rods. But when all these elementary precautions have been observed, the geometry determined by our rods will automatically become the geometry of space. This physical definition of congruence may be termed practical congruence, as distinguished from theoretical congruence, which is embodied by the mathematical types we have discussed.
With his standard rigid bodies defined in this way by physical objects, the physicist can perform measurements in the space of his frame and in consequence obtains the numerical results of three-dimensional Euclidean geometry. Inasmuch as the more careful he is to guard against such contingent influences as variations of temperature, the more accurately will his numerical results approximate to those of pure Euclidean geometry, he feels justified in stating that rigid objects behave like rigid Euclidean bodies and that the space of our experience is rigorously Euclidean.
We may also recall that his rigid bodies having been defined, the definition of a straight line as the axis of rotation of a revolving solid, two of whose points are fixed, or as the shortest distance between two points in space, follows immediately; and of course the straight line thus defined satisfies Euclid’s parallel postulate, since it is derived from the behaviour of Euclidean solids.
At this stage, a number of popular exponents of non-Euclidean geometry have fallen into a rather unfortunate error. They have argued that material bodies under perfect conditions must necessarily behave like Euclidean solids, for if they behaved like non-Euclidean bodies when displaced they would squirm and change in shape. As it would be inadmissible to credit any such distorting influence to that void which we call empty space, a non-Euclideanism of material bodies would be debarred on first principles. But these men overlook the fact that Riemann’s and Lobatchewski’s geometries do not in any way refer to bodies which squirm and are distorted in any absolute sense as they move about. The non-Euclidean bodies are merely distorted when contrasted with Euclidean bodies taken as standards; but it would be equally true to state that Euclidean bodies likewise would squirm when displaced if we were to contrast them with non-Euclidean bodies taken as standards. In any case, both Euclidean and non-Euclidean bodies behave in a homogeneous way throughout space.[11] By this we mean that wherever they might be situated in empty space, measurements computed with them would yield the same numerical results. As for Euclidean congruence and Euclidean rigidity, it is by no means more representative of real rigidity than are the non-Euclidean varieties. There is therefore no reason to appeal to a distorting effect of empty space in order to account for a possible non-Euclidean behaviour of our material solids when displaced from point to point. Non-Euclideanism may or may not exist in real space, but this is a point for physical measurement and not for philosophy or mathematics to decide.
All we can say is that the principle of sufficient reason compels us to credit empty space with a sameness throughout, and that our measuring rods and material bodies must also behave homogeneously and isotropically, as indeed they do in the three geometries discussed. Only if measurements undertaken with our rods in different parts of space yielded variable non-homogeneous numerical results should we have to assume that space was not really empty and that our rods were subject to local influences.
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