Page:The Meaning of Relativity - Albert Einstein (1922).djvu/20

that in applying such transformations the equations of a straight line become equations of a straight line. Reversing equations (3a) by multiplying both sides by $b_{\nu \beta }$ and summing for all the $\nu$'s, we obtain

$\sum b_{\nu \beta }\Delta x_{\nu }'=\sum \limits _{\nu \alpha }b_{\nu \alpha }b_{\nu \beta }\Delta x_{\alpha }=\sum \limits _{\alpha }\delta _{\alpha \beta }\Delta x_{\alpha }=\Delta x_{\beta }.$

(5)

The same coefficients, $b$, also determine the inverse substitution of $\Delta x_{\nu }$. Geometrically, $b_{\nu \alpha }$ is the cosine of the angle between the $x_{\nu }'$ axis and the $x_{\alpha }$ axis.

To sum up, we can say that in the Euclidean geometry there are (in a given space of reference) preferred systems of co-ordinates, the Cartesian systems, which transform into each other by linear orthogonal transformations. The distance $s$ between two points of our space of reference, measured by a measuring rod, is expressed in such co-ordinates in a particularly simple manner. The whole of geometry may be founded upon this conception of distance. In the present treatment, geometry is related to actual things (rigid bodies), and its theorems are statements concerning the behaviour of these things, which may prove to be true or false.

One is ordinarily accustomed to study geometry divorced from any relation between its concepts and experience. There are advantages in isolating that which is purely logical and independent of what is, in principle, incomplete empiricism. This is satisfactory to the pure mathematician. He is satisfied if he can deduce his theorems from axioms correctly, that is, without errors of logic. The question as to whether Euclidean