SCIENCE ABSOLUTE OF SPACE. 21 



Also (by 21) 

 EC:DC=OEC:ODC(inF) = OAC:OBC (18); 



\ / \ o / &quot; 



and so is also 



0AC:0BC = l:sin CAB; 



whence the theorem is evident for any triangle. 

 26. In any spherical triangle, the sines 

 of the sides are as the sines of the angles 

 opposite. 



For take ZABC=rt.Z, and 

 CED 1 to the radius OA of the 

 sphere. We shall have CED \_ 

 AOB, and (since also BOC _l_ 

 BOA), CDJ_OB. But in the 

 is. triangles CEO, CDO (by 25) 

 OEC:OOC:ODC = sin COE : 1 - sin COD = sin 

 AC : 1 : sin BC; meanwhile also ( 25) OEC : 

 ODC=sin CDE sin CED. Therefore, sin 

 AC : sin BC=sin CDE : sin CED; but CDE= 

 rt.Z = CBA, and CED = CAB. Consequently 



sin AC : sin BC=1 : sin A. 

 Spherical trigonometry , flowing from this, 

 is thus established independently of Axiom 

 XL 



27. If AC and BD are J_ AB, and CAB is 

 carried along the straight AB; we shall have, 

 designating by CD the path of the point C, 

 CD : AB=sin u : sin v. 



