SPACE AND GEOMETRY 43 



"flat" geometry of Euclid, although it is by no 

 means implied that they find applicability only 

 upon curved surfaces. As that absolute magni- 

 tude to which I referred becomes smaller and 

 smaller, the geometries approach the geometry 

 of Euclid as a limit. Moreover, the geometry of 

 any given region approaches more nearly to the 

 Euclidean as the region considered becomes 

 smaller, just as in our allegory the South Sea 

 islanders found their elementary geometry suffi- 

 cient for the home. 



Are these geometries true and is EucKd's ge- 

 ometry false .? This is a question which no longer 

 conveys any meaning to our minds. Is chess 

 true? Provided that a geometry contains within 

 itself no inconsistencies or absurdities, then we 

 regard it as true just in so far as it is interest- 

 ing or useful. Certainly the laws of navigation 

 are true and the only two-dimensional geome- 

 try that they fit is a non-Euclidean geometry. 

 This is one of the so-called geometries of posi- 

 tive curvature, and all of the geometries of this 

 class have attained enormous importance owing 

 to recent theories of gravitation, the fringes of 

 which we shall touch in the fourth chapter. The 

 geometry of so-called negative curvature, typi- 

 fied by the original non-EucHdean geometry of 



