44 SCIENCE ABSOLUTE OF SPACE. 



and so also will AABC and AABL, and hence 

 manifestly 



BCL+BLC+CBL=st. Z. 



However (by 31), the angle-sum of any tri- \M\ 

 angle, in S, is <st.Z. 

 Therefore L, falls on C. 



42. Let u be the supplement of the angle- 

 sum of the AABC, but v of ADEF; then is 

 :z;. 



For if p be the area of each 

 of the triangles ACG, GCH, 

 HCB, DFK, KFE; and 

 AABC=m./, and ADEF= 

 A~G H~*B n.p; and 5 the angle-sum of 

 FIG. 38. any triangle equivalent top; 



manifestly 



st. ^-um.s (m l)st. Z =st. Z /^(st. Z. s} ; 

 and u=m(st.^ 5); and in like manner V 



Therefore AABC : ADEF=m : n=u\v. 



It is evidently also easily extended to the 

 case of the incommensurability of the triangles 

 ABC, DEF. 



In the same way is demonstrated that tri- 

 angles on a sphere are as the excesses of the 

 sums of their angles above a st.<. 



If two angles of the spherical A are right, 

 the third z will be the said excess. But 



