254 Wisconsin Academy of Sciences, Arts and Letters. 



Etc, to exponent infinite. 



i 7t i^ represents the mass of the whole sphere. The values, 

 then, for 



Cos. :& = |. 

 Cos.2 ^ = |. 

 Cos.^ 3^ = |: etc., to COS. with exponent infinite. 



Sin^^ = 1— 008^5 = |. 



Sin^^ = (1— cos^^)'^ = 1—2 cos-^ + cos^S- =j |:f 



Sin'''^ = (I— cos^^)^ = 1—4 cos-5 + 6 008*5-4 cos'=5 + cos^S- = fr|:|, 



Etc. for higher powers. 



The truth contained in the celebrated proposition of the 

 square of the hypothenuse was doubtless well known long 

 before the time of Pythagoras. Pythagcras in joy from the 

 simplicity and the exactness of his demonstration of the 

 celebrated proposition gave exclamation to the Greek word. 

 Eureka. The memoirs of Legendre, Laplace and Ivory on 

 attraction do not contain Eureka demonstrations. It re- 

 quires inventive genius as well as mathematical ability to 

 make such demonstrations. For borrowed knowledge on 

 the subject of my writing to Sir Isaac Newton, I am in- 

 debted more by far than to all the rest combined. 



