become more fimple and general, and are more easily deduced.

In this general method, all the terms of any equation are brought to one fide, and the equation is expreffed by making them equal to o. Therefore, if a root of the equation be inferted instead of (x) the unknown quantity, the positive terms will be equal to the negative, and the whole muft be equal to o.

DEV. When any equation is put into this form, the term in which (x) the unknown quantity is of the highest power, is called the First, that in which the index of x is lefs by 1, is the Second, and fo on, till the laft into which the unknown quantity does not enter, and which is called the Abfolute Term.

Prop. I. If any number of equations be multiplied together, an equation will be produced, of which the dimenfion* is equal to

the

* The term dimenfion, in this treatise, is ufed in fenfes fomewhat different, but fo as not to create any ambiguity. In this chapter it means either the order of an equation, or the number denoting that order,

the fum of the dimenfions of the equations

multiplied.

If any number of fimple equations be multiplied together, as xao, x—b=0, x-co, &c. it is obvious, that the product will be an equation of a dimension containing as many units as there are fimple equations. In like manner, if higher equations are multiplied together, as a cubic and a quadratic, one of the fifth order is produced; and fo on.

Conversely. An equation of any dimenfion is confidered as compounded either of fimple equations, or of others, fuch that the fum of their dimenfions is equal to the dimenfion of the given one. By the refolution of equations thefe inferior equations are difcovered, and by inveftigating the nent fimple equations, the roots of any higher equation are found.

compo

Cor. 1. Any equation admits of as many folutions, or has as many roots, as there are fimple equations which compofe it, that is, as there are units in the dimension of it.

Cor. which was formerly defined to be the highest exponent of the unknown quantity in any term of the equation.

Cor. 2. And converfely, no equation can have more roots than the units in its dimenfion.

Cor. 3. Imaginary or impoffible roots must enter an equation by pairs; for they arife from quadratics, in which both the roots are fuch.

Hence alfo, an equation of an even dimenfion may have all its roots, or any even number of them impoffible, but an equation of an odd dimenfion must at least have one poffible root.

Cor. 4. The roots are either pofitive or negative, according as the roots of the fimple equations, from which they are produced, are positive or negative.

Cor. 5. When one root of an equation is discovered, one of the fimple equations is found, from which the given one is compounded. The given equation, therefore, being divided by this fimple equation, will give an equation of a dimension lower by 1. Thus, any equation may be depreffed as many degrees as there are roots found by any method whatever.

Prop.

Prop. II. To explain the general properties of the signs and coefficients of the terms of an equation.

Let x=a=o, xb=0, x=0, xd o, &c. be fimple equations, of which the roots are any positive quantities +a, +b, +c, +d, &c. and let x+mo, x+n=0, &c. be fimple equations, of which the roots are any negative quantities —m, —n, &c. and let any number of thefe equations be multiplied together, as in the following table.

From this table it is plain,

1. That in a complete equation the num ber of terms is always greater by unit than the dimension of the equation.

2. The coefficient of the firft term is 1. The coefficient of the second term is the fum of all the roots a, b, c, m, &c. with their figns changed.

The coefficient of the third term is the fum of all the products that can be made by multiplying any two of the roots together.

The coefficient of the fourth term is the fum of all the products which can be made by multiplying together any three of the roots with their figns changed; and fo of others.

The laft term is the product of all the roots, with their signs changed.

3. From induction it appears, that in any equation (the terms being regularly arranged as in the preceding example) there are as many positive roots as there are changes in the figns of the terms from + to, and

A a