, 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.
Gauss appears to have been the first to undertake space explorations of this sort, when he conducted experiments on light rays transmitted from one mountain top to another. But his observations were too crude and executed over too small an area to detect any trace of non-Euclideanism. Lobatchewski suggested astronomical observations conducted on the course of rays of starlight through interstellar space. For instance, if two light rays emitted from a very distant star and striking the earth at two different points of its orbit appeared to manifest converging directions, we should know that space was Riemannian. If the two rays appeared to diverge from a common point, space would be Lobatchewskian; and, finally, if for very distant stars these two directions appeared identical, space would be truly Euclidean. Yet the most refined astronomical measurements of stellar parallaxes failed to reveal the slightest trace of non-Euclideanism. Hence it was assumed that if any trace of non-Euclideanism was present in real space it was without doubt exceedingly slight, so that for all practical purposes the geometry of space might be regarded as Euclidean. Such were the results obtained by a physical exploration of space.
From all this we see that the physicist, basing his exploration of space on empirical methods, is perfectly justified in stating that its geometry can be determined, that a true definition of congruence can be arrived at, and that the equality of two lengths and of two spatial configurations has a definite significance in nature.
And yet, when we submit all these various examples to a critical analysis, we cannot help but see that this determination of the geometry of space is essentially physical and is, therefore, contingent on the behaviour of material objects and of rays of light. Had the behaviour of material bodies when displaced been regulated by other physical laws, had rays of light followed different courses, the geometry we should have attributed to space might have been entirely different. And we may well wonder what the behaviour of physical objects should have to do with the geometry of space. We shall return to this aspect of the problem later.
Also, it has sometimes been argued that our recognition of shape and size must possess a much deeper significance and cannot be attributed merely to the laws of behaviour of material objects and of rays of light. For instance, it is pointed out that even a child who knows nothing of measurement judges, on simple visual inspection, that a coin (when viewed from a perpendicular direction) is round and an egg oval. He does not feel it necessary to verify this fact by applying a ruler. However, regardless of what opinions we may eventually defend on the subject of a geometry intrinsic to physical space, it can scarcely be held that this last argument of the critic proves his point in the slightest degree.
It may be instructive to consider this illustration in greater detail. In the first place, we must realise that all we are in a position to appreciate when merely viewing an object, is its image on our retina. Indeed, were we to interpose a microscope or a deforming lens between object and eye, the object would suffer no change; but its image cast on our retina would, of course, be modified. As a result, our judgment of its shape and size would vary. Hence, when we decide unhesitatingly that the coin is round, the only inference to be drawn is that the coin’s image on the retina is judged by the brain (or the mind, or whatever we decide to call it) to be circular. We do not wish to intrude on the ground of the eye-specialist or physiologist, who is better equipped than we are to proceed farther in this analysis, but we may point out that the problem which we are now considering is of an entirely different nature from the one whence we started. There we were considering whether shape in empty space was absolute; here we are considering whether an image cast on our retina should impress itself upon our recognition with any definite shape.
The difference amounts to just this: In mathematical space, and even in physical space, absolute measurement seemed to elude us, since in view of the continuity of space it appeared impossible to proceed with an enumeration of points. But in the case of the retina, its surface is no longer homogeneous; it possesses a heterogeneous structure like all tissues, probably a discrete one forming a pattern. In all such cases a definite metrics suggests itself naturally, just as on a net, in the absence of a ruler, we would compare lengths instinctively by counting the holes separating our points.