Page:Eddington A. Space Time and Gravitation. 1920.djvu/95

As an example of a mesh-system on a curved surface, we may take the lines of latitude and longitude on a sphere.

For latitude and longitude $(\beta ,\lambda )$$ds^{2}=d\beta ^{2}+\cos ^{2}\beta \,d\lambda ^{2}$.

These expressions form a test, and in fact the only possible test, of the kind of coordinates we are using. It may perhaps seem inconceivable that an observer should for an instant be in doubt whether he was using the mesh-system of Fig. 10 or Fig. 11. He sees at a glance that Fig. 11 is not what he would call a rectangular mesh-system. But in that glance, he makes measures with his eye, that is to say he determines $ds$ for pairs of points, and he notices how these values are related to the number of intervening channels. In fact he is testing which formula for $ds$ will fit. For centuries man was in doubt whether the earth was flat or round—whether he was using plane rectangular coordinates or some kind of spherical coordinates. In some cases an observer adopts his mesh-system blindly and long afterwards discovers by accurate measures that $ds$ does not fit the formula he assumed—that his mesh-system is not exactly of the nature he supposed it was. In other cases he deliberately sets himself to plan out a mesh-system of a particular variety, say rectangular coordinates; he constructs right angles and rules parallel lines; but these constructions are all measurements of the way the $x$-channels and $y$-channels ought to go, and the rules of construction reduce to a formula connecting his measures $ds$ with $x$ and $y$.

The use of special symbols for the coordinates, varying according to the kind of mesh-system used, thus anticipates a knowledge which is really derived from the form of the formulae. In order not to give away the secret prematurely, it will be better to use the symbols $x_{1}$, $x_{2}$ in all cases. The four kinds of coordinates already considered then give respectively the relations,${\begin{aligned}ds^{2}&=d{x_{1}}^{2}+d{x_{2}}^{2}&{\text{(rectangular),}}\\ds^{2}&=d{x_{1}}^{2}+{x_{1}}^{2}d{x_{2}}^{2}&{\text{(polar),}}\\ds^{2}&=d{x_{1}}^{2}-2\kappa \,dx_{1}dx_{2}+d{x_{2}}^{2}&{\text{(oblique),}}\\ds^{2}&=d{x_{1}}^{2}+\cos ^{2}x_{1}d{x_{2}}^{2}&{\text{(latitude and longitude).}}\\\end{aligned}}$ If we have any mesh-system and want to know its nature, we