NON-EUCLIDEAN GEOMETRY 161 



a combination of them could be expected to appear 

 even in the humblest capacity in the ultimate world- 

 structure. Meanwhile the first step through the gate- 

 way takes us to the geometry obeyed by these lengths 

 — very nearly Euclidean, but actually non-Euclidean and, 

 as we have seen, a distinctive type of non-Euclidean 

 geometry in which the ten principal coefficients of cur- 

 vature vanish. We have shown in this chapter that 

 the limitation is not arbitrary; it is a necessary property 

 of lengths expressed in terms of the extension of a ma- 

 terial standard, though it might have been surprising if 

 it had occurred in lengths defined otherwise. Must we 

 stop to notice the interjection that if we had meant 

 something different by length we should have found a 

 different geometry? Certainly we should; and if we 

 had meant something different by electric force we should 

 have found equations different from Maxwell's equations. 

 Not only empirically but also by theoretical reasoning, 

 we reach the geometry which we do because our lengths 

 mean what they do. 



I have too long delayed dealing with the criticism of 

 the pure mathematician who is under the impression 

 that geometry is a subject that belongs entirely to him. 

 Each branch of experimental knowledge tends to have 

 associated with it a specialised body of mathematical 

 investigations. The pure mathematician, at first called in 

 as servant, presently likes to assert himself as master; 

 the connexus of mathematical propositions becomes for 

 him the main subject, and he does not ask permission 

 from Nature when he wishes to vary or generalise the 

 original premises. Thus he can arrive at a geometry 

 unhampered by any restriction from actual space meas- 

 ures; a potential theory unhampered by any question 

 as to how gravitational and electrical potentials really 



