SCIENCE ABSOLUTE OF SPACE. 45 



(a great circle being called p) this A is mani 

 festly 



z tf 



=27 | (32, VI); 



consequently, any triangle of whose angles the 

 excess is z, is 



-^ 



4*1 



43. Now, in S, the area of a rectilineal A 

 is expressed by means of the sum of its angles. 

 M M N N t If AB increases to infinity; 



( 42) A ABC : (ri.^-u-v) 

 will be constant. But A ABC 

 =BACN ( 32, V), and rt.Z 

 uv=z ( 1) ; and so 

 BACN:^=AABC: (rt. Z- 



9 A 



FIG. 39. 

 tan z : tan z . 



But for y ^=o, we have 

 BD C N , , 



^~ =1 and also 



D Moreover, manifestly ( 30) 

 BDCN : BD C N = r : * * 



consequently, 



BDCN : BACN = tan z : z. 

 But (32) 



BDCN = r.* =* 2 tan z; 

 therefore, BACN=^./ 2 . 



