A Treatise of Algebra.

125

SECT. XII.

Of the Resolution of Equations of severalDimensions.

B EFORE I proceed to explain the Methods of re-solving cubic, biquadratic, and other higher Equa-tions, it will be requisite, in order to render that Subjectmore clear and intelligible, to premise something con-cerning the Origin and Composition of Equations.

Mr. Harriot has (hewn how Equations are derivedfrom the continual Multiplication of binomial Factorsinto each other : His Method is this. Let there be anyNumber of binomial Factors, * — a, x — b, x —**— if, &c. in which let the Value of x be such that someone of those Factors may be equal to nothing ; then, ifthey be multiply’d continually together, their Product,it is manifest, will also be equal to nothing, that isx —axx— bxx — ex* — d, &c. — 0 : In which Equation,x may, it is manifest, be equal to any one of the Quan-tities a, b, c, if, &c. since, any of these being substitut-ed instead of x, the whole Expression vanishes. Henceit appears, that an Equation may have as many Rootsas it consists of Dimensions, or as will express the Num-ber of Factors whereof it is .supposed to be produced.Thus the quadratic Equation x — a X x — b-=. o, or x x

_^x-j-airro, has two Roots, a and b-, the cubic

Equation x — a x x — b X x—*•= o, or x* -f- — b\ x z

ab

-p ac ? x — abc — o, three, s, b and c, and the biquadra-

be

tic Equation x— <r*x x — hi — c x x — d ■=. o, or **