Page:Eddington A. Space Time and Gravitation. 1920.djvu/62

generalised distance between two events, of which the distance in space and the separation in time are particular components. This extension in space and time combined is called the "interval" between the two events; it is the same for all observers, however they resolve it into space and time separately. We may think of the interval as something intrinsic in external nature—an absolute relation of the two events, which postulates no particular observer. Its practical measurement is suggested by analogy with the distance of two points in space.

In two dimensions on a plane, two points $P_{1}$, $P_{2}$ (Fig. 2) can be specified by their rectangular coordinates $(x_{1},y_{1})$ and $(x_{2},y_{2})$, when arbitrary axes have been selected. In the figure, $OX_{1}=x_{1}$, $OY_{1}=y_{1}$, etc. We have ${\begin{aligned}P_{1}{P_{2}}^{2}&=P_{1}M^{2}+M{P_{2}}^{2}\\&=X_{1}{X_{2}}^{2}+Y_{1}{Y_{2}}^{2}\\&=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}{\text{,}}\end{aligned}}$ so that if $s$ is the distance between $P_{1}$ and $P_{2}$ $s^{2}=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}$.

The extension to three dimensions is, as we should expect, $s^{2}=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{2})^{2}$. Introducing the times of the events $t_{1}$, $t_{2}$, we should naturally expect that the interval in the four-dimensional world would be given by $s^{2}=(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})+(t_{2}-t_{1})^{2}$.

An important point arises here. It was, of course, assumed that the same scale was used for measuring $x$ and $y$ and $z$. But how are we to use the same scale for measuring $t$? We cannot use a scale at all; some kind of clock is needed. The most natural connection between the measure of time and length is given by the fact that light travels 300,000 kilometres in 1 second. For the four-dimensional world we shall accordingly regard 1 second as the equivalent of 300,000 kilometres, and measure lengths and times in seconds or kilometres indiscriminately; in other words we make the velocity of light the unit of