NON-EUCLIDEAN GEOMETRY 159 



non-Euclidean geometry and no extra dimensions — 

 which is right? I would rather not attempt a direct 

 answer, because I fear I should get lost in a fog of 

 metaphysics. But I may say at once that I do not take 

 the ten dimensions seriously; whereas I take the non- 

 Euclidean geometry of the world very seriously, and 

 I do not regard it as a thing which needs explaining 

 away. The view, which some of us were taught at 

 school, that the truth of Euclid's axioms can be seen in- 

 tuitively, is universally rejected nowadays. We can no 

 more settle the laws of space by intuition than we can 

 settle the laws of heredity. If intuition is ruled out, the 

 appeal must be to experiment — genuine open-minded ex- 

 periment unfettered by any preconception as to what the 

 verdict ought to be. We must not afterwards go back 

 on the experiments because they make out space to be 

 very slightly non-Euclidean. It is quite true that a way 

 out could be found. By inventing extra dimensions we 

 can make the non-Euclidean geometry of the world 

 depend on a Euclidean geometry of ten dimensions; had 

 the world proved to be Euclidean we could, I believe, 

 have made its geometry depend on a non-Euclidean 

 geometry of ten dimensions. No one would treat the 

 latter suggestion seriously, and no reason can be given 

 for treating the former more seriously. 



I do not think that the six extra dimensions have any 

 stalwart defenders; but we. often meet with attempts to 

 reimpose Euclidean geometry on the world in another 

 way. The proposal, which is made quite unblushingly, 

 is that since our measured lengths do not obey Euclidean 

 geometry we must apply corrections to them — cook them 

 — till they do. A closely related view often advocated 

 is that space is neither Euclidean nor non-Euclidean; 

 it is all a matter of convention and we are free to 



