88 HUMANISM v 



geometry, and as logically on a par with spherical and 

 pseudo-spherical geometry. It is a species of a genus, 

 and the differentia which constitutes it is the famous 

 postulate of Euclid/ which Euclid postulated because he 

 could not prove it, and which the failures of all his 

 successors have only brought into clearer light as an 

 indispensable presupposition. The non-Euclideans, on 

 the other hand, have shown that it does not require proof, 

 because it embodies the definition of the sort of space 

 dealt with by ordinary geometry ; and that in both of its 

 equivalent forms, whether as the axiom of parallels or of 

 the equality of the angles of a triangle to two right angles, 

 it forms a special case intermediate between that of spherical 

 and that of pseudo-spherical space. In spherical space 

 nothing analogous to the Euclidean parallels is to be 

 found ; in pseudo-spherical space, on the other hand, 

 not one, but two parallels may be drawn through any 

 point. So while spherical triangles always have their 

 angles greater than two right angles, the pseudo-spherical 

 triangles always have them less than two right angles. 

 Moreover, the Euclidean case can always be reached by 

 supposing the parameter of the non-Euclidean spaces 

 infinitely large. So much for the possibility of a general 

 geometry, including the Euclidean amongst others. 



It has also, I think, been shown that the non-Euclidean 

 geometries would form coherent and consistent systems, 

 like the Euclidean, in which an indefinite number of 

 propositions might be shown to follow from their initial 

 definitions. They are, that is to say, thoroughly thinkable 

 and free from contradiction, and intellectually on a level 

 with the Euclidean conception of space. They are 

 thinkable, but (as yet) no more ; and this explains their 

 defence against the two objections upon which their more 

 unprejudiced opponents incline to lay most stress. It is 

 objected (i) that there is, e.g., no such thing as a spherical 

 space, only a spherical surface. True ; but there is nothing 

 to prevent us from conceiving the peculiar properties of a 

 spherical surface as pervading every portion of the space 

 it bounds. We can conceive a spherical surface of a 



