GEOMETRICAL AXIOMS. 37 



between its ends. Thus the notion of the geodetic or 

 straightest line is not quite identical with that of the 

 shortest line. If the two given points are the ends of 

 a diameter of the sphere, every plane passing through 

 this diameter cuts semicircles, on the surface of the 

 sphere, all of which are shortest lines between the 

 ends ; in which case there is an equal number of 

 equal shortest lines between the given points. Ac- 

 cordingly, the axiom of there being only one shortest 

 line between two points would not hold without a 

 certain exception for the dwellers on a sphere. 



Of parallel lines the sphere -dwellers would know 

 nothing. They would maintain that any two straightest 

 lines, sufficiently produced, must finally cut not in one 

 only but in two points. The sum of the angles of a 

 triangle would be always greater than two right angles, 

 increasing as the surface of the triangle grew greater. 

 They could thus have no conception of geometrical 

 similarity between greater and smaller figures of the 

 same kind, for with them a greater triangle must have 

 different angles from a smaller one. Their space 

 would be unlimited, but would be found to be finite or 

 at least represented as such. 



It is clear, then, that such beings must set up a 

 very different system of geometrical axioms from that 

 of the inhabitants of a plane, or from ours with our 

 space of three dimensions, though the logical power* 



