Page:The Meaning of Relativity - Albert Einstein (1922).djvu/18

we obtain a line which has all the properties of the straight lines of the Euclidean geometry. In particular, it easily follows that by laying off $n$ times the interval $s$ upon a straight line, an interval of length $n\cdot s$ is obtained. A length, therefore, means the result of a measurement carried out along a straight line by means of a unit measuring rod. It has a significance which is as independent of the system of co-ordinates as that of a straight line, as will appear in the sequel.

We come now to a train of thought which plays an analogous rôle in the theories of special and general relativity. We ask the question: besides the Cartesian co-ordinates which we have used are there other equivalent co-ordinates? An interval has a physical meaning which is independent of the choice of co-ordinates; and so has the spherical surface which we obtain as the locus of the end points of all equal intervals that we lay off from an arbitrary point of our space of reference. If $x_{\nu }$ as well as $x_{\nu }'$ ($\nu$ from 1 to 3) are Cartesian co-ordinates of our space of reference, then the spherical surface will be expressed in our two systems of co-ordinates by the equations

$\sum \Delta x_{\nu }^{2}={\text{const.}}$

(2)

$\sum \Delta x_{\nu }'^{2}={\text{const.}}$

(2a)

How must the $x_{\nu }'$ be expressed in terms of the $x_{\nu }$ in order that equations (2) and (2a) may be equivalent to each other? Regarding the $x_{\nu }'$ expressed as functions of the $x_{\nu }$, we can write, by Taylor's theorem, for small values of the $\Delta x_{\nu }$,