Page:The Meaning of Relativity - Albert Einstein (1922).djvu/21

geometry is true or not does not concern him. But for our purpose it is necessary to associate the fundamental concepts of geometry with natural objects; without such an association geometry is worthless for the physicist. The physicist is concerned with the question as to whether the theorems of geometry are true or not. That Euclidean geometry, from this point of view, affirms something more than the mere deductions derived logically from definitions may be seen from the following simple consideration.

between $n$ points of space there are ${\frac {n(n-1)}{2}}$ distances, $s_{\mu \nu }$; between these and the $3n$ co-ordinates we have the relations

$s_{\mu \nu }^{2}=(x_{1(\mu )}-x_{1(\nu )})^{2}+(x_{2(\mu )}-x_{2(\nu )})^{2}+...$

From these ${\frac {n(n-1)}{2}}$ equations the $3n$ co-ordinates may be eliminated, and from this elimination at least ${\frac {n(n-1)}{2}}-3n$ equations in the $s_{\mu \nu }$ will result.^[1] Since the $s_{\mu \nu }$ are measurable quantities, and by definition are independent of each other, these relations between the $s_{\mu \nu }$ are not necessary a priori.

From the foregoing it is evident that the equations of transformation (3), (4) have a fundamental significance in Euclidean geometry, in that they govern the transformation from one Cartesian system of co-ordinates to another. The Cartesian systems of co-ordinates are characterized

1. In reality there are ${\frac {n(n-1)}{2}}-3n+6$