158 GRAVITATION— THE EXPLANATION 



a non-Euclidean geometry of two dimensions. And so 

 if you concentrate your attention on the earth's surface 

 so hard that you forget that there is an inside or an 

 outside to it, you will say that it is a two-dimensional 

 manifold with non-Euclidean geometry; but if you 

 recollect that there is three-dimensional space all round 

 which affords shorter ways of getting from point to 

 point, you can fly back to Euclid after all. You will then 

 "explain away" the non-Euclidean geometry by saying 

 that what you at first took for distances were not the 

 proper distances. This seems to be the easiest way of 

 seeing how a non-Euclidean geometry can arise — 

 through mislaying a dimension — but we must not infer 

 that non-Euclidean geometry is impossible unless it arises 

 from this cause. 



In our four-dimensional world pervaded by gravitation 

 the distances obey a non-Euclidean geometry. Is this 

 because we are concentrating attention wholly on its 

 four dimensions and have missed the short cuts through 

 regions beyond? By the aid of six extra dimensions we 

 can return to Euclidean geometry; in that case our usual 

 distances from point to point in the world are not the 

 "true" distances, the latter taking shorter routes through 

 an eighth or ninth dimension. To bend the world in a 

 super-world of ten dimensions so as to provide these 

 short cuts does, I think, help us to form an idea 

 of the properties of its non-Euclidean geometry; at any 

 rate the picture suggests a useful vocabulary for de- 

 scribing those properties. But we are not likely to accept 

 these extra dimensions as a literal fact unless we regard 

 non-Euclidean geometry as a thing which at all costs 

 must be explained away. 



Of the two alternatives — a curved manifold in a 

 Euclidean space of ten dimensions or a manifold with 



