GEOMETRICAL AXIOMS. 53 



its movements are limited to rotations round the 

 straight line connecting them. If we turn it com- 

 pletely round once, it again occupies exactly the po- 

 sition it had at first. This fact, that rotation in one 

 direction always brings a solid body back into its ori- 

 ginal position, needs special mention. A system of 

 geometry is possible without it. This is most easily 

 seen in the geometry of a plane. Suppose that with 

 every rotation of a plane figure its linear dimensions in- 

 creased in proportion to the angle of rotation, the figure 

 after one whole rotation through 360 degrees would no 

 longer coincide with itself as it was originally. But 

 any second figure that was congruent with the first in 

 its original position might be made to coincide with it 

 in its second position by being also turned through 

 360 degrees. A consistent system of geometry would 

 be possible upon this supposition, , which does not come 

 under Riemann's formula. 



On the other hand I have shown that the three 

 assumptions taken together form a sufficient basis for 

 the starting-point of Riemann's investigation,, and 

 thence for all his further- results relating to the dis- 

 tinction of different spaces according to their measure 

 of curvature. 



It still remained to be seen whether the laws of 

 motion, as dependent on moving forces, could also 'be 

 consistently transferred to spherical or pseudospherical 



