2o6 A NEW and UNI FERSA L- SO LU TI N 



For, in the formula e — eX -> let t, the firft approx- 



imate value already found, be fubftituted for v, and let s denote 

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



is the fame thing, let e =: s X ^.^^-^ E^ ; And further, let / be 



n — T 



fubftituted for e in the equation fm {n — t) zz ^ fm v X cof v, 



and let t denote the correfponding value of v : then will it be a 



fecond approximation to the arch v. 



Int like manner, t' being fubftituted for v in the formula 



fin (« — y) . . „ 



e — £ X — , will give a new value of e, denoted by e : 



and, by means of the equation fin (« — c) := ^ X fin k X cof V, 

 this new value of ^ will give a third approximation to the arch ►, 

 denoted by -tt". And it is manifeft, that the feries of arches, 

 V, 5r', ^", &c. approximating to the value of v, may be continued 

 indefinitely. 



6. I now fay, that the archer v, t', ^r", &c. which conftitute 

 the feries of approximations to the value of v, 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. terras 

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



For, if, ijn the equation fin (« — v) z^ e X fin v X cof f, we 

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

 by fin n X cof v, we fhall get 



fin n cof V 



fin V cof V ' 

 in this formula e vaniflies when « zr v : and, fuppofing the 



c- 



arch V to decreafe, it is manifeft that the pofitive part, -^ — . will 



lin v' 



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

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

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

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

 will be the arch ^. 



Let 



