22 SCIENCE ABSOLUTE OF SPACE. 



For take DElCA; 

 in the triangles ADE, 

 ADB (by 25) 

 OED : OAD : OAB= 



sin u : 1 : sin v. 



G F 



FIG. 19. In revolving BACD 



about AC, B describes OAB, and D describes 

 OED; and designate here by sOCD the path 

 of the said CD. Moreover, let there be any [123 

 polygon BFG. . . inscribed in OAB. 



Passing through all the sides BF, FG, &c., 

 planes J_ to OAB we form also a polygonal fig- 

 ure of the same number of sides in sOCD, and 

 we may demonstrate, as in 23, that CD : AB 

 =DH : BF=HK : FG, &c., and so 



DH+HK &c. : BF+FG &c. : =CD : AB. 



If each of the sides BF, FG . . . approaches 

 the limit zero, manifestly 



BF+FG+ . . . =0 AB and 

 DH+HK+...=OED. 



Therefore also OED : OAB=CD : AB. But 

 we had OED : OAB=sin u : sin v. Conse- 

 quently 



CD : AB=sin u : sin v. 



If AC goes away from BD to infinity, CD : 

 AB, and so also sin u : sin v remains constant; 

 but ^=rt. Z (1), and if DM II BN, v=z; 

 whence CD : AB=1 : sin z. 



