Page:Eddington A. Space Time and Gravitation. 1920.djvu/63

velocity. It is not essential to do this, but it greatly simplifies the discussion.

Secondly, the formulae here given for $s^{2}$ are the characteristic formulae of Euclidean geometry. So far as three-dimensional space is concerned the applicability of Euclidean geometry is very closely confirmed by experiment. But space-time is not Euclidean; it does, however, conform (at least approximately) to a very simple modification of Euclidean geometry indicated by the corrected formula $s^{2}=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}-(t_{2}-t_{1})^{2}$. There is only a sign altered; but that minus sign is the secret of the differences of the manifestations of time and space in nature.

This change of sign is often found puzzling at the start. We could not define $s$ by the expression originally proposed (with the positive sign), because the expression does not define anything objective. Using the space and time of one observer, one value is obtained; for another observer, another value is obtained. But if $s$ is defined by the expression now given, it is found that the same result is obtained by all observers^[1]. The quantity $s$ is thus something which concerns solely the two events chosen; we give it a name—the interval between the two events. In ordinary space the distance between two points is the corresponding property, which concerns only the two points and not the extraneous coordinate system of location which is used. Hence interval, as here defined, is the analogue of distance; and the analogy is strengthened by the evident resemblance of the formula for $s$ in both cases. Moreover, when the difference of time vanishes, the interval reduces to the distance. But the discrepancy of sign introduces certain important differences. These differences are summed up in the statement that the geometry of space is Euclidean, but the geometry of space-time is semi-Euclidean or "hyperbolic." The association of a geometry with any continuum always implies the existence of some uniquely measurable quantity like interval or distance; in ordinary space, geometry without the idea of distance would be meaningless.

1. Appendix, Note 2.