The evolution of scientific thought from Newton to Einstein

by A. D'Abro

For this reason Euclidean geometry has a uniqueness about it which is denied to the non-Euclidean varieties. These latter constitute a class of geometries; and the precise type of the geometry we may happen to be discussing is determined by the intensity of curvature of the sphere or pseudosphere to which it pertains. We also see why it is so difficult to determine by empirical methods whether the space of the universe is truly Euclidean or not. It is because both Riemannian and Lobatchewskian geometry merge by insensible gradations into Euclidean geometry.

The analysis we have given refers to non-Euclidean geometries as derived from the geometry of surfaces; that is, to the numerical results obtained when we use rigid Euclidean rods to effect measurements on curved surfaces. It must be noted, however, that the surfaces we have discussed are of a very special kind. Both spheres and pseudospheres are known as surfaces of constant curvature. There is no need to go into the mathematical definition of what is meant by this; we will limit ourselves to stating the principal characteristic of such surfaces.

Consider, for example, a net made of inextensible threads (inextensible or undeformable being used in the Euclidean sense). If this net can be applied with perfect contact to any portion of a plane, it can be slid over the entire plane without ever losing its perfect contact. The same is true of the sphere and pseudosphere. Thus, a net which could be fitted with perfect contact to any part of a sphere could be slid over any portion of the surface without ever losing its perfect contact.

The only surfaces which possess this property are precisely the sphere, the plane and the pseudosphere (and those derived therefrom without stretching); and these surfaces are the surfaces of constant positive curvature, of constant zero curvature, and of constant negative curvature, respectively. So far, then, as the shape of nets applicable to the surface is concerned, there is no means of distinguishing one part of the surface from any other part, since the net can be slid over the surface, preserving its dimensions and never losing its perfect contact with the surface. In other words, the surface appears to be isotropic and homogeneous when measurements with rigid Euclidean rods are conducted over it, or when the nets we slide over it are Euclideanly inextensible, This is the property called free mobility.

Such would no longer be the case were we to consider surfaces of variable curvature, e.g., the surface of an ellipsoid or of a trumpet. A net which could be applied with perfect contact to one part of the surface would lose its perfect contact when slid over the surface, unless we deformed it by stretching it or causing it to shrink.

It is because all three types of geometry we have mentioned (Euclidean, Riemannian and Lobatchewskian) hold in exactly the same way for all parts of the space in which we are operating, that the surfaces which portray them in two dimensions are necessarily of the constant-curvature type.

We have yet to show the connection between non-Euclidean geometry of two dimensions defined as the geometry obtained with Euclidean rods on a surface of constant curvature, and non-Euclidean geometry obtained with squirming Euclidean rods, i.e., rigid non-Euclidean ones. For this purpose consider a sphere resting on a plane. We may call the south pole that point of the sphere which stands in contact with the plane; and we shall assume that a point-source of light is located at the north pole. If the sphere is transparent to rays of light all figures traced on the surface of the sphere will cast shadows on the plane. But the same will be true of our little Euclidean rods which we place alongside these figures on the sphere for purposes of measurement.

It is easy to see that a circle, for instance, traced on the spherical surface will, generally speaking, have for its shadow an ellipse on the plane. As this circle is displaced on the sphere towards the north pole, its shadow will grow larger and larger, tending finally, when any point of the circle coincides with the north pole, to become a parabola extending to infinity. But inasmuch as the shadows of the little rods with which we measure lengths on the sphere vary in exactly the same way as do the shadows of the figures measured, we shall obtain on the plane exactly the same Riemannian results that endure on the sphere.

In short, if we assume that on the plane our standard measuring rods are given by the shadows of the standard measuring rods used on the sphere, we shall obtain Riemannian geometry on the plane. It is scarcely necessary to state that with our Euclidean ideas of congruence it would be impossible for us to accept the statement that these successive squirming shadows of a Euclidean length displaced on the surface of the sphere were all congruent to one another; but this is not the point at issue. If we were perfectly flat, two-dimensional beings, living on the plane, and if not only all the flat bodies sliding over the plane behaved as did the shadows in our previous example, but also our measuring rods and our own living bodies behaved likewise, we should have no other alternative than to assume that these shadows represented the displacements of bodies that moved while remaining rigid or without change. We should accordingly regard the geometry of our two-dimensional space as Riemannian.

It is impossible to show more clearly that the geometry we attribute to space is nothing but an expression of the properties of our bodies and measuring rods when displaced, since here we have a plane which is Euclidean or Riemannian in its geometry according to the behaviour of the bodies that glide over its surface. And so we see that two alternative methods of interpreting non-Euclidean geometry are open to us. Either we may regard our rods as Euclidean and applied to a curved surface, or else we may attribute non-Euclideanism to the behaviour of our rods on a flat surface.[20]