206 A NEW and UNIVERSA L- SO LUTION 



For, in the formula e — % X -> let sr, the firft approx- 



n — v 



imate value already found, be fubftituted for *, and let i denote 



the value of * that will refult from the fubftitution ; or, which 



is the fame thing, let i = s X ^—^- — : And further, let i be 



n — 7T 



fubftituted for e in the equation fin (n — v) = * fin y X cof n, 

 and let *■' denote the correfponding value of v : then will ic be a 

 fecond approximation to the arch u. 



In like manner, t being fubftituted for v in the formula 



e — z X ^— , will give a new value of e y denoted by e" : 



and, by means of the equation fin (n — v) z= e X fin v X cof V, 

 this new value of e will give a third approximation to the arch v, 

 denoted by %' . And it is manifeft, that the feries of arches, 

 r, ?r', t", &c. approximating to the value of v-, may be continued 

 indefinitely. 



6. I now fay, that the arche& t, x', w", &c. which conftitute 

 the feries of approximations to the value of e, are alternately 

 too fmall and too great : that is, the firft, third, fifth, &c. terms 

 in the feries are all lefs ; but the fecond, fourth, fixth, &c. terms 

 in the feries are all greater, than the exacl value of v. 



For, if, in the equation fin (n — v) — e X fin v X cof v, we 

 write fin n cof v — cof n fin v, for fin (n — v), and divide both fides 

 by fin v X cof v, we fhall get 



fin n cof v 

 fin v cof v ' 

 in this formula e vanifhes when n — v : and, fuppofing the 



arch v to decreafe, it is manifeft that the pofitive part, -= , will 



* fin v* 



cof // 

 increafe, and that the negative part, — **—, will decreafe : there- 

 fore e will increafe when v decreaies ; and the lefs the arch v is, 

 the greater will be the value of e. This, it is evident, muft alfo 

 be true, when taken inverfely j that is, the greater e is, the lefs 

 will be the arch v. 



Let 



