Page:Eddington A. Space Time and Gravitation. 1920.djvu/121

VI] since the field is plane. Let us now dismiss from our minds all idea of distances in the field or straight lines in the field, and assume that distances on the map merely represent the minimum number of hurdles between two points; straight lines on the map will represent the corresponding routes. This has the advantage that if an earthquake occurs, deforming the field, the map will still be correct. The path of fewest hurdles will still cross the same hurdles as before the earthquake; it will be twisted out of the straight line in the field; but we should gain nothing by taking a straighter course, since that would lead through a region where the hurdles are more crowded. We do not alter the number of hurdles in any path by deforming it.

This can be illustrated by Figs. 14 and 15. Fig. 14 represents the original undistorted field with the hurdles uniformly placed. The straight line $$PQ$$ represents the path of fewest hurdles from $$P$$ to $$Q$$, and its length is proportional to the number of hurdles. Fig. 15 represents the distorted field, with $$PQ$$ distorted into a curve; but $$PQ$$ is still the path of fewest hurdles from $$P$$ to $$Q$$, and the number of hurdles in the path is the same as before. If therefore we map according to hurdle-counts we arrive at Fig. 14 again, just as though no deformation had taken place.

To make any difference in the hurdle-counts, the hurdles must be taken up and replanted. Starting from a given point as centre, let us arrange them so that they gradually thin out towards the boundaries of the field. Now choose a circle with this point as centre;—but first, what is a circle? It has to be defined in terms of hurdle-counts; and clearly it must be a curve such that the minimum number of hurdles between any point on it and the centre is a constant (the radius). With this definition we can defy earthquakes. The number of hurdles in the circumference of such a circle will not bear the same proportion to the number in the radius as in the field of uniform hurdles; owing to the crowding near the centre, the ratio will be less. Thus we have a suitable analogy for a circle whose circumference is less than $$\pi$$ times its diameter.

This analogy enables us to picture the condition of space round a heavy particle, where the ratio of the circumference of a circle to the diameter is less than $$\pi$$. Hurdle-counts will no