SCIENCE ABSOLUTE OF SPACE. 33 



Moreover, - - =OBC, and hence 

 dx 



1 ,0 



- =^- sin u, whence (integrating) 



du 2~ 



ver sin u , 



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) 



FD ver sin u A -nvp, , 



, and so ^=-b D./&amp;gt;. 



p 2/K 



But (21)/=7r.FGD; therefore 



^rr.FD.FDG. But (21) 

 M FD.FDG=FC.FC; consequently 



Now let BJ=CJ=r/ (30) 

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

 areao2r (inF) =*i* ! (Y ^-Y&quot; 1 f . 



Also (IV} 

 area o2jX=^ 2 (Y 2 2+Y~ 2 ) ; 

 therefore, area O2^ (in F) =area O2j^, and so 

 the surface 7. of a segment of a sphere is 

 equal to the surface of the circle described 

 with the chord FC as a radius. 



