Now it may be mentioned that there exist a number of alternative ways of representing both Euclidean and non-Euclidean geometries. Which way is to be preferred depends largely on the scope of our investigations. One of these alternative procedures has often been appealed to in popular writings (by Helmholtz in particular) because it enables us to visualise the sequence of theorems involved, without abandoning thereby our habitual Euclidean representations.
Thus three-dimensional Euclidean space is taken as a starting point. It is then proposed to show how, in this three-dimensional Euclidean space, a two-dimensional non-Euclidean geometry can arise. The fundamental space which we are here postulating being Euclidean, distances in this space must be computed with rigid Euclidean rods. Then it is shown that if we apply tiny Euclidean rods to the surface of a sphere and conduct measurements on this surface, we shall obtain a series of numerical results which are representative of Riemann’s geometry. The chief advantage of this method of presentation is that it allows us to foresee at a rapid glance that Riemann’s geometry must be consistent,[17] since in the present case it reduces to Euclidean geometry on a sphere, which is a particular case of Euclidean geometry in three-dimensional space; and Euclidean geometry is known to be consistent. But aside from this advantage the procedure is to be avoided, for it tends to obscure the philosophical importance of non-Euclidean geometry.
In the first place it is not always possible to follow this method. Consider, for example, the case of Lobatchewski’s geometry. Just as Riemann’s geometry was that of the spherical surface, so Lobatchewski’s geometry turns out to be that of a peculiar saddle-shaped surface called the pseudosphere, as was proved by Beltrami. Hilbert, however, has shown that there cannot exist a surface free from singularities which would represent the total spread of Lobatchewski’s plane geometry; hence here is a first reason for being on our guard against a number of unsuspected difficulties.
But this is not all. If we adopt the method of representing two-dimensional non-Euclidean geometry as the geometry obtained by taking Euclidean measurements on a curved surface, how are we to conceive of three-dimensional Riemannian geometry? Obviously we must start with a four-dimensional Euclidean space in which a three-dimensional spherical surface is embedded. But as a four-dimensional Euclidean space transcends our immediate experience and is in the nature of a mathematical fiction, we should be led to the erroneous conclusion that three-dimensional non-Euclidean geometry must likewise be a purely conceptual construction having nothing in common with measurements which might be conducted with material rods.
We should be led into still greater difficulties if we wished to represent a three-dimensional non-Euclidean space possessing various degrees of non-Euclideanism from place to place. Reasoning by analogy, we should be tempted to say that just as an irregularly curved surface embedded in three-dimensional Euclidean space yielded a two-dimensional geometry of varying degrees of non-Euclideanism, so now all we should have to do would be to conceive of an irregularly curved three-dimensional surface embedded in a four-dimensional Euclidean space. But the analogy would be deceptive; for calculation shows that it would be impossible to represent an arbitrarily curved three-dimensional surface in a four-dimensional Euclidean space. In the general case we should have to situate our variously bumped three-dimensional surface in a six-dimensional Euclidean space. In the same way, a four-dimensional non-Euclidean space of variable curvature could be represented only in a ten-dimensional Euclidean space; and so on.[18]
Now in view of these difficulties, in view of the fact that to represent a non-Euclidean space of three dimensions in terms of Euclidean space, we may be compelled to appeal to a Euclidean space of six dimensions, it is certainly exalting Euclidean space unduly to regard it as The Space. This is especially obvious when we remember that we could have conceived of our three-dimensional non-Euclidean space directly, as a result of measurements with squirming rods, without any reference to Euclidean geometry, and without ever having had to introduce any greater number of dimensions.
At this juncture the beginner is often inclined to argue as follows: “You say that non-Euclidean geometry of two dimensions is the geometry obtained by executing Euclidean measurements on a suitably curved surface. But your concept of curvature is meaningless unless we conceive it as contrasted with some pre-existing standard of straightness. Does not your presentation prove, therefore, that Euclidean space as representative of flatness or straightness is logically antecedent to non-Euclidean space, which connotes curvature?”
This argument is radically incorrect, and arises from too loose an understanding of non-Euclidean geometry. In the first place, to assume that the concept of curvature presupposes the concept of straightness is no more inevitable than to assume that the concept of straightness presupposes that of curvature. For if it is true to say that that which is curved is that which is not straight, it is equally true to say that that which is straight is that which is not curved; so that in the absence of curvature, straightness would in turn be meaningless. The question of deciding, for example, whether a circle or a straight line is the more fundamental is strictly a matter of opinion. The Greeks held circular motion to be the noblest of all motions; and between the designations “noblest” and “most fundamental,” the distinction is exceedingly slight. And even this is not all, for thus far we have been arguing as though Euclidean space were truly flat, and non-Euclidean space truly curved. But we must remember that this curvature of which we speak is not to be construed as representing anything absolute; it arises solely from the particular type of representation we have agreed to select.
Thus, in the presentation of non-Euclidean geometry which we have just been discussing, we started from three-dimensional Euclidean space as a basis and considered curved surfaces embedded in this space. This form of presentation was equivalent, as we know, to stating that Euclidean measurements must be adhered to on the curved surface. But we might have proceeded otherwise. We might have started with three-dimensional non-Euclidean space and considered the geometry of curved surfaces (of a sphere, for example) embedded in this space. This would have been equivalent to applying non-Euclidean rods on our curved surface.
Under suitable conditions the geometry of our curved surface (or sphere) would then become Euclidean. Euclideanism itself would thus be linked with curvature, while non-Euclideanism, which would here be the type of geometry obtained on the plane embedded in our three-dimensional non-Euclidean space, would accordingly be linked with flatness. In short, we must not allow the word “curvature” to mislead us into forming a false impression about non-Euclidean geometry. In many respects Riemann’s choice of the word “curvature” has proved unfortunate and the more scientific appellation non-Euclideanism should be adhered to.[19]