Page:Eddington A. Space Time and Gravitation. 1920.djvu/204

a point $A$ and a sufficient number of other points, and also between $B$ and the same points, can we calculate what will be the interval between $A$ and $B$? In ordinary geometry this would be possible; but, since in the present case we know nothing of the relation signified by the word interval, it is impossible to predict any law a priori. But we have found in our previous work that there is such a rule, expressed by the formula $ds^{2}=g_{11}\,{dx_{1}}^{2}+g_{22}\,{dx_{2}}^{2}+\ldots +2g_{12}\,dx_{1}\,dx_{2}+\ldots$ This means that, having assigned our identification numbers $(x_{1},x_{2},x_{3},x_{4})$ to the point-events, we have only to measure ten different intervals to enable us to determine the ten coefficients, $g_{11}$, etc., which in a small region may be considered to be constants; then all other intervals in this region can be predicted from the formula. For any other region we must make fresh measures, and determine the coefficients for a new formula.

I think it is unlikely that the individual interval-relations of point-events follow any such definite rule. A microscopic examination would probably show them as quite arbitrary, the relations of so-called intermediate points being not necessarily intermediate. Perhaps even the primitive interval is not quantitative, but simply $1$ for certain pairs of point-events and $0$ for others. The formula given is just an average summary which suffices for our coarse methods of investigation, and holds true only statistically. Just as statistical averages of one community may differ from those of another, so may this statistical formula for one region of the world differ from that of another. This is the starting point of the infinite variety of nature.

Perhaps an example may make this clearer. Compare the point-events to persons, and the intervals to the degree of acquaintance between them. There is no means of forecasting the degree of acquaintance between $A$ and $B$ from a knowledge of the familiarity of both with $C$, $D$, $E$, etc. But a statistician may compute in any community a kind of average rule. In most cases if $A$ and $B$ both know $C$, it slightly increases the probability of their knowing one another. A community in which this correlation was very high would be described as cliquish. There may be differences among communities in this respect, corresponding to their degree of cliquishness; and so