GEOMETEICAL AXIOMS. 43 



batchewsky, professor of mathematics at Kasan, 1 and 

 it was proved that this system could be carried out as 

 consistently as Euclid's. It agrees exactly with the 

 geometry of the pseudospherical surfaces worked out 

 recently by Beltrami. 



Thus we see that in the geometry of two dimen- 

 sions a surface is marked out as a plane, or a sphere, or 

 a pseudospherical surface, by the assumption that any 

 figure may be moved about in all directions without 

 change of dimensions. The axiom, that there is only 

 one shortest line between any two points, distinguishes 

 the plane and the pseudospherical surface from the 

 sphere, and the axiom of parallels marks off the plane 

 from the pseudosphere. These three axioms are in 

 fact necessary and sufficient, to define as a plane the 

 surface to which Euclid's planimetry has reference, as 

 distinguished from all other modes of space in two 

 dimensions. 



The difference between plane and spherical geome- 

 try has been long evident, but the meaning of the 

 axiom of parallels could not be understood till Gauss 

 had developed the notion of surfaces flexible without 

 dilatation, and consequently that of the possibly in- 

 finite continuation of pseudospherical surfaces. In- 

 habiting, as we do, a space of three dimensions and 

 endowed with organs of sense for their perception, we 

 1 Principien der Geometric, Kasan, 1829-30. 



