[ .8 ] 



theorems for the fine and cofine of multiple arcs, when the mnl- 

 tiplier is any whole pofitive number even or odd, were given by 

 feveral authors — But all the writers on this fubjed that I have 

 feen, except Dr. Waring, have deduced the law of the feries from, 

 obfervation in a few inftances without a general demonftration 



of its truth Dr. Waring has (Curv. algebr. Propr. Theor. 26 



& Cor.) by help of his admirable theorem for finding the fums 

 of the powers of the roots of an equat. given a general de- 

 monftration of the feries for finding ihe chord of the fapple- 

 ment of a multiple arc in terms of the chord of the fupple- 

 * ment of the fimple arc, and confequently a general demon- 

 ftration of the theorem for the cofine of a multiple arc in terms 

 of the cofine of the fimple arc, and alfo of the fine of a multiple 

 arc when the multiplier is an odd number. But in the cafe 

 where the multiplier is an even number no demonftration, as 

 far as I have feen, has ever been given by any author. 

 Dr. Waring's method of demonftration cannot be applied to 

 this cafe — The following demonftration extends to every mul- 

 tiplier whether even or odd. The demonftrations for the fine 

 and cofine of the multiple arc in terms of the cofine of the 

 fimple arc, from whence the other theorems are immediately 

 deducible, are of this kind-^The probable law is deduced from 

 obfervation in a few inftances and then the general truth of 

 that conjedure is proved. Dr. Waring's demonftration, although 

 by a very different procefs, being founded upon the properties 

 of algebraical equations, is alfo of this kind, as it depends 



upon 



