SCIENCE ABSOLUTE OF SPACE. 33 



Moreover, -=OBC, and hence 



ax . 



j /2 



-j-==- sin u, whence (integrating) 



au 2- 



_ver sin u , 2 

 ~2* 



The F may be conceived on which P falls 

 (passing through the middle F of the seg- 

 ment) ; through AF and AC the planes FEM, 

 CEM are placed, perpendicular to F and cut- 

 ting F along FEG and CE; and consider the 

 L form CD (f rom C 1 to FEG), and the L form 

 CF; (20) CEF=^, and (21) 



gg= ver sin * and so ^=FD./. 



P 47T 



But (21)/=^.FGD; therefore 



^=r:.FD.FDG. But (21) 

 FD.FDG=FC.FC; consequently 

 ^=7r.FC.FC=areaoFC, in F. 



Now let BJ=CJ=r/ (30) 

 2r=*(Y Y- 1 ), and so (21) 

 area O2r (in F) =^ 2 (Y- Y- 1 ; 2 . 

 Also (IV) 



area 



therefore, area O2^ (in F) =area Q2y 1 and so 

 the surface z of a segment of a sphere is 

 equal to the surface of the circle described 

 with the chord FC as a radius. 



