SPACE AND GEOMETRY 45 



and the rules for using words that we call gram- 

 mar. Chess has its pieces and its moves, arith- 

 metic its numbers and its operations upon num- 

 bers; geometry has its figures and its methods 

 of comparing figures. As the rules of chess ap- 

 ply, not to all games, but to a single game, so 

 the rules of Euclid apply, not to all geometries, 

 but only to one geometry. 



Now the method which Euclid used for com- 

 paring one figure with another was only a slight 

 idealization of the method of cutting the figures 

 out of paper and moving them about to see how 

 nearly they fit one another. Any such method of 

 transposing a figure may evidently be divided 

 into two movements, one of sliding without 

 turning, and one of turning without sliding, 

 and we shall see that these two types of move- 

 ment are closely connected respectively with the 

 parallel postulate and the circular postulate of 

 Euclid. 



We now proceed more circumspectly than 

 Euclid did, and we have gone further in idealiz- 

 ing the physical process of moving a figure cut 

 out of wood or paper. We do not wish to be 

 limited by the particular properties or by the 

 imperfections of such substances, nor do we 

 wish to be influenced by our intuition. Instead 



