56 



THE ANATOMY OF SCIENCE 



,^__ 



part in the new geometry, and indeed the word- 

 ing of the theorems is usually identical, except 

 for an occasional difference in sign (while the- 

 orems in the geometry of shear rotation repre- 

 sent a sort of compromise between the other 



two). Thus in the as- 

 ymptotic geometry the 

 square on the hypote- 

 nuse of a right-angled 

 triangle is equal to the 

 difference between the 

 squares on the other 

 two sides. 



So in Figure 8, if 

 we have two parallel 

 lines OA and QR, and 

 the line OQ perpen- 

 dicular to these, we 

 may take a new line 

 OB and draw the perpendicular to it in the 

 three geometries. We obtain OR in Euclidean, 

 OP in the asymptotic, while OQ still remains 

 perpendicular in the shear geometry. Now if s 

 represents what we may call the slope between 

 the lines OA and OB (in Euclidean geometry 

 we call it the tangent of the angle between OA 

 and OB), the ratios of the lengths of the three 



Figure 8 



