Page:The Meaning of Relativity - Albert Einstein (1922).djvu/81

concept of this distance is physically significant because the distance can be measured directly by means of a rigid measuring rod. By a suitable choice of Cartesian co-ordinates this distance may be expressed by the formula $ds^{2}=dx_{1}^{2}+dx_{2}^{2}$. We may base upon this quantity the concepts of the straight line as the geodesic ($\delta \int ds=0$), the interval, the circle, and the angle, upon which the Euclidean plane geometry is built. A geometry may be developed upon another continuously curved surface, if we observe that an infinitesimally small portion of the surface may be regarded as plane, to within relatively infinitesimal quantities. There are Cartesian co-ordinates, $X_{1},X_{2}$, upon such a small portion of the surface, and the distance between two points, measured by a measuring rod, is given by

$ds^{2}=dX_{1}^{2}+dX_{2}^{2}.$

If we introduce arbitrary curvilinear co-ordinates, $x_{1},x_{2}$, on the surface, then $dX_{1},dX_{2}$, may be expressed linearly in terms of $dx_{1},dx_{2}$. Then everywhere upon the surface we have

$ds^{2}=g_{11}dx_{1}^{2}+2g_{12}dx_{1}dx_{2}+g_{22}dx_{2}^{2}$

where $g_{11},g_{12},g_{22}$ are determined by the nature of the surface and the choice of co-ordinates; if these quantities are known, then it is also known how networks of rigid rods may be laid upon the surface. In other words, the geometry of surfaces may be based upon this expression for $ds^{2}$ exactly as plane geometry is based upon the corresponding expression.

There are analogous relations in the four-dimensional