A Treatise of Algebra. 147

cisicd (for zat = —■-a 3 ) we shall, by following

2

the preceding Directions, have x 2 -(- ax -(- a 1 =\/ b* _|_ a* ; whence x will bs found by the common Me*thods. So, if the Equatioa, x 4 — 8 ax 3 4a*—2 6*Xx 2 -}“ 48a 5 + 8 ab z x x = c 4 , 'Was proposed j by follow-ing the fame Method, we should have x z — e^ax —> 6 a z

-b z = '/c* + — 6a z — b z

But now to explain the Reason of these Operationslet x4 -f- ax 3 -j- £x 2 -|- ex = be any Equation of thisKind; wherein the unknown Terms, on the left-handSide, are the fame with those arising from the Square ofsome Trinomial, as x z -f- Ax -j- B - y then this Quantitybeing actually squared, we shall have x 4 -j- 2 Ax 3 -j-2B -j- A 2 xx 2 -)- 2 ABx (-j- B 2 ) — x 4 -)- ax 3 -)- bx z 4- ex(-(- B 2 ). From whence, by equating the Coefficients ofthe like Terms, we have 2 A = a, 2B -f- A 2 = b, and

a b — A 2 \ b

2 AB = c \ therefore A = — , B (=2 — -I — -

2 2. ' 2.

a 1 c ab a 3

— — — and c (— 2 AB) —-—; from .whence the

8 a . 2 8

Reason of the Whole is obvious.

Of the Resolution cf Equations by converging Scries.

The Methods hitherto given, for finding the Roots ofEquations, are all either very troublesome and laborious,or else confined to particular Cafes; but that by con-verging Series, which we are here going to explain, isuniversal, extending to all Kinds of Equations ; and,though not accurately true, gives the Value sought, withlittle trouble, to a very great Degree of Exactness. Whenan Equation is proposed, to be solved by this Method,the Root thereof must first of all be nearly estimated,(which, from the Nature of the Problem, and a few'1 rials, may, in most Cafes, be very easily done) andsome Letter or unknown Quantity (as z) must be assmn-

I. a «d,