GEOMETRICAL AXIOMS. 49 



portrayed in the interior of a sphere in Euclid's homa- 

 loid space, that every straightest line. or flattest surface 

 of the pseudospherical space is represented by a 

 straight line or a plane, respectively, in the sphere. 

 The surface itself of the sphere corresponds to the 

 infinitely distant points of the pseudospherical space ; 

 and the different parts of this space, as represented in 

 the sphere, become smaller, the nearer they lie to the 

 spherical surface, diminishing more rapidly in the direc- 

 tion of the radii than in that perpendicular to them. 

 Straight lines in the sphere, which only intersect 

 beyond its surface, correspond to straightest lines of 

 the pseudospherical space which never intersect. 



Thus it appeared that space, considered as a region 

 of measurable quantities, does not at all correspond 

 with the most general conception of an aggregate of 

 three dimensions, but involves also special conditions, 

 depending on the perfectly free mobility of solid 

 bodies without change of form to all parts of it and 

 with all possible changes of direction ; and, further, on 

 the special value of the measure of curvature which 

 for our actual space equals, or at least is not distin- 

 guishable from, zero. This latter definition is given 

 in the axioms of straight lines and parallels. 



Whilst Kiemann entered upon this new field from 

 the side of the most general and fundamental questions 

 of analytical geometry, I myself arrived at similar 



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