GEO]fflETBICAL AXIOMS. 51 



sions of the space in question. It is further assumed 

 that with the movement of the point A, the magnitudes 

 used as co-ordinates vary continuously. 



Secondly, the definition of a solid body, or rigid 

 system of points, must be made in such a way as to 

 admit of magnitudes being compared by congruence. 

 As we must not, at this stage, assume any special 

 methods for the measurement of magnitudes, our defi- 

 nition can, in the first instance, run only as follows : 

 Between the co-ordinates of any two points belonging 

 to a sojid body, there must be an equation which, how- 

 ever the body is moved, expresses a constant spatial 

 relation (proving at last to be the distance) between 

 the two points, and which is the same for congruent 

 pairs of points, that is to say, such pairs as can be 

 made successively to coincide in space with the same 

 fixed pair of points. 



However indeterminate in appearance, this defini- 

 tion involves most important consequences, because 

 with increase in the number of points, the number of 

 equations increases much more quickly than the number 

 of co-ordinates which they determine. Five points, 

 A, B, C, D, E, give ten different pairs of points 

 AB, AC, AD, AE, 

 BC, BD, BE, 

 CD, CE, 

 DE, 



B 2 



