[ ^9 ] 



upon his theorem for the fums of the powers of the roots of 

 an equation, of which he has given the fame kind of demon - 

 ftration — Previous to the demonftrationjs of thefe theorems I 

 have given a demonflration of the theorems for exprefling the 

 fine and cofine of multiple arcs in terms compounded of the 

 fine and cofine — Thefe theorems alfo have been given by many 

 authors, and the only general demonftrations of them have been 

 deduced from the hyperbola and the confideration of impoffible 

 quantities — However ufeful impoffible quantities may be ill 

 difcovering mathematical truths they ought never to be ufed 

 in flridl demonflration, and it mufl feem a very circuitous mode 

 to apply the properties of the hyperbola to demonftrate thofe 

 of the circle — Thefe demonftrations are from the properties of 

 the circle and the theorems for combinations. 



The theorems hitherto mentioned are more particularly 

 applicable to the conflrudlion of trig, tables and the refolution 

 of certain equations — In confcquence of the great advances that 

 have beeii made in phyfical aftronomy fince the time of 

 Sir Ifaac Newton, it has been found necefTary for facilitating 

 the calculation of particular fluents to exprefs the powers of 

 the fine and cofine in terms of the fines and cofines of mul- 

 tiple arcs, and theorems for this purpofe have been given by 

 feveral authors. They have all however either deduced the 

 general law from obfervation without demonflration, or gene- 

 rally demonflrated it by help of impoffible logarithms — The 



demonftrations 



