v NON-EUCLIDEAN GEOMETRY 89 



constant curvature making up the texture of space, just as 

 well as the Euclidean plane surface. This intrinsic texture 

 would produce uniform and calculable deformation or 

 crinkling in all bodies immersed in it, and these might 

 conceivably be aware of this deformation as they moved 

 in a non-Euclidean space, just as they are now aware of 

 the direction of their movements. In the Euclidean 

 case the homogeneity of Space is entire in all respects, 

 in the spherical only in some. It is argued (2) that meta- 

 geometry is dependent on Euclidean geometry, because it 

 is reached only through the latter. But it is not clear 

 that it may not be logically independent, even though 

 historically it has developed out of Euclidean geometry, 

 and even though psychologically the latter affords the 

 simplest means of representing spatial images. And it 

 has become clear that both the conception of a manifold 

 and that of a general space admitting of specific 

 determinations is logically prior to that of Euclidean 

 space. 



Theoretically, then, metageometry seems to be able 

 to give a very good account of itself. But it must be 

 confessed that this at present only accentuates its practical 

 failure. It is admitted that Euclidean geometry yields the 

 simplest formulas for calculating spatial relations, and even 

 M. Calinon x hardly ventures to hope that non-Euclidean 

 formulas will be found serviceable. Metageometers 

 mostly confine themselves to supposing imaginary worlds, 

 of which the laws would naturally suggest a non-Euclidean 

 formulation. 2 In short, practically the supremacy of the 

 old geometry remains incontestable, because of its greater 

 simplicity and consequent facility of application. 



II. I pass on to the second question, the light thrown 

 by non-Euclidean geometry on the nature of Space. In 

 this respect incomparably its most important achievement 

 seems to have been to force upon all the distinction 

 between perceptual and conceptual space, or rather spaces. 

 On this point both parties are at one, and we find, e.g., 



1 Rev. Phil, xvii . 12. 

 2 E.g. , M. Poincar6, Rev. de Mlt. iii. 6, pp. 641 ff. 



