SPACE AND GEOMETRY 



57 



new perpendiculars to the original perpendicu- 

 lar are for the Euclidean, shear, and asymptotic 

 geometries, respectively. 



VI + s-\ 1, VI 



Let us consider the three lines 

 OA, OB and OC of Figure 9. If s^ 

 is the slope between the first and 

 second lines, and S2 the slope be- 

 tween the second and third, then 53, 

 the slope between the first and 

 third, is given by the three follow- 

 ing equations for the Euclidean, 

 the shear, and the asymptotic geometries, re- 

 spectively : 



FiGUBE 9 



Si + ^2 



1 2 



^1 ~\- Soj S3 



S1 + S2 



1 + Si S2 



I shall not continue further the discussion of 

 these interesting geometries. These few exam- 

 ples will serve to illustrate their general char- 

 acter, and these examples are the ones which we 

 shall find useful in our later discussions. 



