SCIENCE ABSOLUTE OF SPACE. 45 



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

 festly 



=|- 1 (32, VI); ' 



consequently, any triangle of whose angles the 

 excess is z, is 



A 



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

 is expressed by means of the sum of its angles. 

 M N' If AB increases to infinity; 



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

 7 will be constant. But A ABC 

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

 ; uv=z ( 1); and so 

 : BACN : ^=AABC : (rt. Z- 



D ' Moreover, manifestly ( 30) 

 FIG. 39. BDCN : BD'C'N'-/: r' ~- 



tan z : tan z' . 



But for y'=o, we have 



BD'C'N' , tan*' , 



BAC'N' = ' a ~^~ 



consequently, 



BDCN : BACN=tan z : z. 

 But (32) 



BDCN=r.*'=*' 2 tan ^y 

 therefore, 



