38 ORIGIN AND SIGNIFICANCE OF 



of all were the same ; nor are more examples neces- 

 sary to show that geometrical axioms must vary ac- 

 cording to the kind of space inhabited by beings 

 whose powers of reason are quite in conformity with 

 ours. But let us proceed still farther. 



Let us think of reasoning beings existing on the 

 surface of an egg-shaped body. Shortest lines could 

 be drawn between three points of such a surface and 

 a triangle constructed. But if the attempt were made 

 to construct congruent triangles at different parts of 

 the surface, it would be found that two triangles, with 

 three pairs of equal sides, would not have their angles 

 equal. The sum of the angles of a triangle drawn at 

 the sharper pole of the body would depart farther from 

 two right angles than if the triangle were drawn at the 

 blunter pole or at the equator. Hence it appears that 

 not even such a simple figure as a triangle can be 

 moved on such a surface without change of form. It 

 would also be found that if circles of equal radii were 

 constructed at different parts of such a surface (the 

 length of the radii being always measured by shortest 

 lines along the surface) the periphery would be greater 

 at the blunter than at the sharper end. 



We see accordingly that, if a surface admits of the 

 figures lying on it being freely moved without change 

 of any of their lines and angles as measured along it, 

 the property is a special one and does not belong to 



