GEOMETEICAL AXIOMS. 59 



world as we can look into theirs, without overstepping 

 the boundary, they must declare it to be a picture in a 

 spherical mirror, and would speak of us just as we 

 speak of them ; and if two inhabitants of the different 

 worlds could communicate with one another, neither, 

 so far as I can see, would be able to convince the other 

 that he had the true, the other the distorted, relations. 

 Indeed I cannot see that such a question would have 

 any meaning at all, so long as mechanical considerations 

 are not mixed up with it. 



Now Beltrami's representation of pseudospherical 

 space in a sphere of Euclid's space, is quite similar, ex- 

 cept that the background is not a plane as in the 

 convex mirror, but the surface of a sphere, and that 

 the proportion in which the images as they approach 

 the spherical surface contract, has a different mathe- 

 matical expression. 1 If we imagine then, conversely, 

 that in the sphere, for the interior of which Euclid's 

 axioms hold good, moving bodies contract as they 

 depart from the centre like the images in a convex 

 mirror, and in such a way that their representatives 

 in pseudospherical space retain their dimensions 

 unchanged, observers whose bodies were regularly 

 subjected to the same change would obtain the 

 same results from the geometrical measurements 

 they could make as if they lived in pseudospherical 

 space. 



1 Compare the Appendix at the end of this Lecture. 



