TRACT 33.

HISTORY OF ALGEBRA.

277

lie next adds a particular mode of resolving all sorts ofequations, that have rational roots, on the principle ofthe roots boin<i divisors of the last or absolute term, as beforementioned by Peletarius; and then gives the method ofapproximating to other roots that are not rational, much inthe same way as Stevinus.

Having found one root of an equation, by any of theformer methods, by means of it he depresses the equationone degree lower, then finds another root, and so on tillthey are all found ; for he shows that every algebraic equa-tion, admits of as many solutions or roots, as there are unitsin the index of the highest power, which roots may beeither positive or negative, or imaginary, or, as he callsthem, greater than nothing, or less than nothing, or in-volved ; so the roots of the equation 1 3 equ. 73 — 6 , are2, 1, and — 3 ; and the roots of the equation l^E) equ.40 — 3 are 1 ,

1,

- 1 + ✓ - 2 ,

- 1 - •/- 2 .

In depressing an equation to lower degrees, he does notuse the method of resolution of Harriot, but that whichis derived from the general relation of the roots and coeffi-cients of the terms, which he here fully and universally _states, viz, that the coefficient of the 2 d term is equal to thesum of all the roots ; that of the 3d term equal to the sum ofall the products of the roots, taken two by two; that of the4th term, the sum of the products, taken three bv three ;and so on, to the fast or absolute term, which is the con-tinual product of all the roots ; a property which was beforestated by Vieta, as to the equations that have all their rootspositive ; and here extended by Girard to all sorts of rootswhatever : but how either Vieta or he came by this property,nowhere appears that I know of. From this general property,among other deductions, Girard shows how to find thesums of the powers of the roots of an equation ; thus, letA, b, c, d, &c, be the 1st, 2d, 3d, 4th, Sec, coefficient,