THE SMITHSONIAN INSTITUTION. 423 



The proposition may also be easily demonstrated. To express it in 



T 



general terms, let a =: — , r and s being any positive whole numbers. 



s 



Then: a+irz: - +-• 



a s s ' 



Suppose s > r and =z r -\- t, then — 



1 + 1 = _!:_ 4- ^ + ^ = ._JL_ -I- 1 + L 



s r r -\- t r r -{- t r 



— 1 4- y^ + r^ + ^2 



= 1 4- (r + 0^ — rt 



r {r -\- t) • 



=: 1 _|.^+^_ '^ 





= 1 + 1 + -- 



r + i 



But since r -f- (5 is greater than r, the expression in the last brackets 



must be positive, and therefore — + — greater than 2. But — + — 



s r s r 



is only a general form for the expression a -j- — > consequently a -\-— 

 is always greater than 2. 



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