Whence it is very natural to infer, that in an equation of 

 the fifth degree, the roots will have the form, 



*= \/ p + 1/ 9 + y'r-j- 



x = a tf p + a- \/ q + <r ^ r -f- a v ^/ j 



ROOT. 



(i) «■ 



* + a* 



*' + 



^* 



n (n — l) (n — 2) 



■' b- + &c 



x = a' ;// 



? + a *V'»' + a \Z f 



.»■ = a' v'/ + a 1 £/ 7 + a 1/ r + a \/s 



5 r > 5 5 



x = a' ^/ p -f a \ / q + a r y r + a' ^/ s 



where a, a', a, a', are the four imaginary roots of iy 1. 



This generalization of the forms of the roots of equations, 

 though generally cttributed to Euler, is, we believe, due 

 to Waring j but neither of thefe able analylts were able to 

 derive from it the folutionof any general equation above the 

 fourth degree. On the (ubjeft of equations, -viz. of finding 

 - their roots, we would refer the reader to Waring's " Medi- 



If b be negative, the odd powers of b will alfo be nega- 

 tive ; that is, 



(*-*)• = 



(2) a' 



a— ' b + 



n{n — 1) in - 2) 



n (n — n) 



b' - 



Otherwife 



1 



let 



b- + &c. 



reprefent the index, and put — = 



a 



tationes Algebraicx," Lea's " Refolution of the higher Q ; alfo, let A, B, C, D, &c. reprefent the firft, fecond, 

 Equations in Algebra," La Grange's treatife on the re- third, &c. terms of the feries, with their proper figns ; 

 folution " Des Equations Numeriques," and the Introduc- then will 



equations JNumenq 

 tion to Barlow's " Mathematical Tables," from which 

 we haie extracted the following general fynopfis of refolu- 

 tion. See alfo Bonnycallle's Algebra. 



Genera! Synopfis of the direct and approximate Methods of 

 afcertaining the Roots of Numbers and Equations. 



ExtraBion of the Roots of Numbers. 

 I. Let n reprefent the index of the root, 



x the number of which the root is required, 

 /■ an approximate value of £/ .v ; 



which may be found by trial, or otherwife, as near as con- 

 venient, and cither in excefs or defect : then will, 



2 r (x — r") 



I. y x = r + 



very nearly. 



- i)r" + (n- 1) 



And ufing this new approximate value in the fame way, 

 another value may be found (till nearer ; and fo on, to any 

 degree of accuracy required. 



This general formula rcfolves itfelf into the following 

 particular one:, ; 



A 



m 



(3) « T + 



CQ 



+ ^^DQ + fa., 



4 n 



which is by far the moll convenient form in the cafe of 

 fractional or negative indices. 



Roots of Equations. 

 Quadratic Equations. 



III. Let x 1 + ax — b = o, or ** + ax = b, reprefent 

 any quadratic equation, a and b being either politive or ne- 

 gative : then will 



— a 



2 



+ 



\/(t ♦ •)• 



But if a and b, independent of the fign by which they are 

 preceded, be always fuppofed politive ; then this general 

 formula rcfolves itfelf into the four following particular 

 ones', viz. 



1 . x* 4/ "X 



o, where . 



" a 

 2 



By the Binomial Theorem. 



[I. Let (a + b) reprefent any binomial, and n the index 

 of the root or power, which may be either pofitive or ne- 

 gative, integral or fractional ; then will 



(« + *)"=» 



x' — a x — b ~ o, where .1= — + « / f— +6), 

 *' — ax + I = o, where x = — 4j .. /( ' b \ 



#' -|- a j. • + b = o, where .r = — + 



2 ~ 



By Sines and Tangents. 



I. x '- + px = q. 

 2 



Put s / q = tan. z ; then 



P 



f + s / q x tan. i s, or 



1 — \' q X cof. \ z. 



v/(t - »> 



IV. 



