70 ORIGIN AND SIGNIFICANCE OF GEOMETRICAL AXIOMS. 



from which equation 1 gives 7=R, then, 



fs n \_<r 

 sm (j) M 



in which o-=\/a; 2 + 2/ 2 + 2;2 



or, s =fi . arc sin (jA=R . arc tang (jj- (5.) 



In this, SQ is the distance of the point x, y, z, measured 

 from the centre of the co-ordinates. 



If now we suppose the point x, y, z, of spherical space, 

 to be projected in a point of plane space whose co-ordinates 

 are respectively 



_Rx _Ry^ ^fe 

 t t t 



then in the plane space the equations 3, which belong to 

 the straightest lines of spherical space, are equations of the 

 straight line. Hence the shortest lines of spherical space 

 are represented in the system of *-, JT, , by straight lines. 

 For very small values of x, y, z, t=R, and 



Immediately about the centre of the co-ordinates, the 

 measurements of both spaces coincide. On the other hand, 

 we have for the distances from the centre 



Sn=.K . arc 



tang ( ) - - - (6.) 



In this, r may be infinite ; but every point of plane space 

 must be the projection of two points of the sphere, one for 

 which S Q < -| J??r, and one for which S Q > J RTT. The 

 extension in the direction of r is then 



