Page:Eddington A. Space Time and Gravitation. 1920.djvu/97

If a being limited to a two-dimensional world finds that his measures agree with the first formula, he can make them agree with the second or third formulae by drawing the meshes differently. But to obtain the fourth formula he must be translated to a different world altogether.

We thus see that there are different kinds of two-dimensional space, betrayed by different metrical properties. They are naturally visualised as different surfaces in Euclidean space of three dimensions. This picture is helpful in some ways, but perhaps misleading in others. The metrical relations on a plane sheet of paper are not altered when the paper is rolled into a cylinder—the measures being, of course, confined to the two-dimensional world represented by the paper, and not allowed to take a short cut through space. The formulae apply equally well to a plane surface or a cylindrical surface; and in so far as our picture draws a distinction between a plane and a cylinder, it is misleading. But they do not apply to a sphere, because a plane sheet of paper cannot be wrapped round a sphere. A genuinely two-dimensional being could not be cognisant of the difference between a cylinder^[1] and a plane; but a sphere would appear as a different kind of space, and he would recognise the difference by measurement.

Of course there are many kinds of mesh-systems, and many kinds of two-dimensional spaces, besides those illustrated in the four examples. Clearly it is not going to be a simple matter to discriminate the different kinds of spaces by the values of the $g$'s. There is no characteristic, visible to cursory inspection, which suggests why the first three formulae should all belong to the same kind of space, and the fourth to a different one. Mathematical investigation has discovered what is the common link between the first three formulae. The $g_{11}$, $g_{12}$, $g_{22}$ satisfy in all three cases a certain differential equation^[2]; and whenever this differential equation is satisfied, the same kind of space occurs.

No doubt it seems a very clumsy way of approaching these intrinsic differences of kinds of space to introduce potentials

1. One should perhaps rather say a roll, to avoid any question of joining the two edges.
2. Appendix, Note 4.