APPENDIX III. 



LIGHT PROM NON-EUCLIDEAN SPACES ON THE 

 TEACHING OP ELEMENTARY GEOMETRY. 



BY G. B. HALSTED. 



As foreshadowed by Bolyai and Riemann, 

 founded by Cayley, extended and interpreted 

 for hyperbolic, parabolic, elliptic spaces by 

 Klein, recast and applied to mechanics by Sir 

 Robert Ball, protective metrics may be looked 

 upon as characteristic of what is highest and 

 most peculiarly modern in all the bewildering 

 range of mathematical achievement. 



Mathematicians hold that number is wholly 

 a creation of the human intellect, while on the 

 contrary our space has an empirical element. 

 Of possible geometries we can not say a priori 

 which shall be that of our actual space, the 

 space in which we move. Of course an ad 

 vance so important, not only for mathemat 

 ics but for philosophy, has had some me.taphy- 

 sical opponents, and as long ago as 1878 I 

 mentioned in my Bibliography of Hyper- 



