in ordinary geometry supposed that the length is equally determinate. But if space is not flat the case is different. Imagine a two-dimensional observer confined to the curved surface of the earth trying to perform this task. As he does not appreciate the third dimension he will not immediately perceive the impossibility; but he will find that the direction which he has transferred to differs according to the route chosen. Or if he went round a complete circuit he would find on arriving back at that the direction he had so carefully tried to preserve on the journey did not agree with that originally drawn[1]. We describe this by saying that in curved space, direction is not integrable; and it is this non-integrability of direction which characterises the gravitational field. In the case considered the length would be preserved throughout the circuit; but it is possible to conceive a more general kind of space in which the length which it was attempted to preserve throughout the circuit, as well as the direction, disagreed on return to the starting point with that originally drawn. In that case length is not integrable; and the non-integrability of length characterises the electromagnetic field. Length associated with direction is called a vector; and the combined gravitational and electric field describe that influence of the world on our measurements by which a vector carried by physical measurement round a closed circuit changes insensibly into a different vector.
The welding together of electricity and gravitation into one geometry is the work of Prof. H. Weyl, first published in 1918[2]. It appears to the writer to carry conviction, although up to the present no experimental test has been proposed. It need scarcely be said that the inconsistency of length for an ordinary circuit would be extremely minute[3], and the ordinary manifestations of the electromagnetic field are the consequential results of
- ↑ It might be thought that if the observer preserved mentally the original direction in three-dimensional space, and obtained the direction at any point in his two-dimensional space by projecting it, there would be no ambiguity. But the three-dimensional space in which a curved two-dimensional space is conceived to exist is quite arbitrary. A two-dimensional observer cannot ascertain by any observation whether he is on a plane or a cylinder, a sphere or any other convex surface of the same total curvature.
- ↑ Appendix, Note 15.
- ↑ I do not think that any numerical estimate has been made.
