GEOMETRICAL AXIOMS. f>3 



At the same time it must be noted that as a small 

 elastic flat disk, say of india-rubber, can only be fitted 

 to a slightly curved spherical surface with relative con- 

 traction of its border and distension of its centre, so 

 our bodies, developed in Euclid's flat space, could not 

 pass into curved space without undergoing similar 

 distensions and contractions of their parts, their co- 

 herence being of course maintained only in as far as 

 their elasticity permitted their bending without break- 

 ing. The kind of distension must be the same as in 

 passing from a small body imagined at the centre of 

 Beltrami's sphere to its pseudospherical or spherical 

 representation. For such passage to appear possible, 

 it will always have to be assumed that the body is 

 sufficiently elastic and small in comparison with the 

 real or imaginary radius of curvature of the curved 

 space into which it is to pass. 



These remarks will suffice to show the way in 

 which we can infer from the known laws of our sen- 

 sible perceptions the series of sensible impressions 

 which a spherical or pseudospherical world would give 

 us, if it existed. In doing so, we nowhere meet with 

 inconsistency or impossibility any more than in the 

 calculation of its metrical proportions. "We can re- 

 present to ourselves the look of a pseudospherical 

 world in all directions just as we can develop the con* 

 ception of it. Therefore it cannot be allowed that the 



