I 152 ] 



ftration of the tlieorem for exprefling the cofine of a multiple arc 

 in terms of the cofine of the fimple arc. No author before 

 Dr. Waring has given a general demonftration of this latter 

 theorem, and confequentiy all demonftrations of Cotes's property 

 by the circle alone previous to his, cannot be general fo far as that 

 theorem is concerned, and it will be found that in another circum- 

 ftance not lefs important they are all defedlive. Dr. Waring in 

 his letter to Dr. Powell has from his theorem for the chords of 

 the fupplement of a multiple arc fhewn the truth of Cotes's pro- 

 perty in particular inftances, and in his " Propr. Algebr. Curv. 

 Prob. 32," has given the heads of a general folution. But it ap- 

 pears one of the fteps there omitted is the only difficult part of 

 the demonftration after conceding the theorem for the cofine of a 

 multiple arc. 



The demonftration here given is general and probably as dired 

 and fimple as the propofition will admit of. The proof of the 

 lemma which it was neceft'ary to premife is much the moft 

 difficult part of the whole, and it is in that ftep of the de- 

 monftration where the Lemma is applied that all demonftrations 

 heretofore have been defedive and only applicable to particular 

 inftances. 



Lemma. 



