GEOMETRICAL AXIOMS. 60 



APPENDIX. 



THE elements of the geometry of spherical space are most 

 easily obtained by putting for space of four dimensions the 

 equation for the sphere 



and for the distance ds between the points (x, y, z, t) and 

 [(x+dx) (y+dy) (z+dz) (t + dt)] the value 



...... (2.) 



It is easily found by means of the methods used for three 

 dimensions that the shortest lines are given by equations of 

 the form 



,3 . 



in which a, b, c,f, as well as a, f3, y, <j>, are constants. 



The length of the shortest arc, s, between the points 

 (a;, y, z, t), and (, 17, , r) follows, as in the sphere, from the 

 equation 



cos J*!*l ......... (4.) 



One of the co-ordinates may be eliminated from the values 

 given in 2 to 4, by means of equation 1, and the expressions 

 then apply to space of three dimensions. 



If we take the distances from the points 



=,,==0 



