GEOMETRICAL AXIOMS. 57 



upon a supposition as to the behaviour of certain phy- 

 sical things, either the human body or other instru- 

 ments employed. The supposition may be in the 

 highest degree probable and in closest harmony with 

 all other physical relations known to us, but yet it 

 passes beyond the scope of pure space-intuition. 



It is in fact possible to imagine conditions for 

 bodies apparently solid such that the measurements in 

 Euclid's space become what they would be in spherical 

 or pseudospherical space. Let me first remind the 

 reader that if all the linear dimensions of other bodies, 

 and our own, at the same time were diminished or in- 

 creased in like proportion, as for instance to half or 

 double their size, we should with our means of space- 

 perception be utterly unaware of the change. This 

 would also be the case if the distension or contraction 

 were different in different directions, provided that 

 our own body changed in the same manner, and further 

 that a body in rotating assumed at every moment, 

 without suffering or exerting mechanical resistance, 

 the amount of dilatation in its different dimensions 

 corresponding to its position at the time. Think of 

 the image of the world in a convex mirror. The 

 common silvered globes set up in gardens give the 

 essential features, only distorted by some optical ir- 

 regularities. A well-made convex mirror of moderate 

 aperture represents the objects in front of it as ap- 



