values, and making use of the same means that have been just employed, we shall have the total product of the polynomial raised to any power m.

EVOLUTION.

Evolution consists in extracting the root of any given quantity.

From the theory of exponents, it is evident, that there is no difficulty in extracting the root of a monomial; for, nothing more is necessary, than to divide the exponent of each one of its factors, by the index of the root to be extracted.

But the extracting of the root of a polynomial, requires particular rules, founded on a knowledge of the composition of the several powers of a binomial. Nevertheless, this operation is by no means difficult. The given quantity must, in the first place, be arranged; then we must take the root of the first term, and afterwards proceed according to the same rules that have been laid down for extracting the Square and Cube Roots in arithmetic.

In the mean while, it may be remarked, that, when the quantity to be worked on, is a perfect power of a binomial, the two terms of the root may be readily obtained, by extracting the desired root from the first and last term of the quantity, which is supposed to be properly ordered. This rule is proved in raising a binomial to its different powers,

Let the square root of x2±2ax+ a2 be required, which we have seen above to be an exact power of

xa; if the square root of x2 is taken, we shall have x first term of the root, and, upon taking the square root of a', we have ±a, the second term of the root.

As to the double sign before the second term of the root, it may be remarked, that, when the second term of the square is negative, the second term of the root will also be negative.

What has been just said of the extraction of the square root of a binomial, is applied immediately to solving equations of the second degree, and therefore deserves great attention.

The general rule, just given, can be followed, when an exact root is required; but when the quantity is such, that an exact root cannot be obtained, but only something near it, we must have recourse to the binomial rule of Newton, whose method for raising a quantity to an integral exponent, may be applied in the same way to fractionary exponents, so

-2 a2 x

+

m

n

Let m=1, and n = 2; upon performing the ope

ration, we shall have (x+a=x+

аз

5a1

This series would extend ad infinitum. In order to obtain a proximate root of x+a, we should simply take the first terms of the series, with this caution, that this means is never to be used, but when the terms of the series diminish continually; and it is evident, that the proximate value will be so much the more exact as the series decreases more rapidly, since then the terms that will be thrown away, will be trifling, and may be considered as nothing. It is very perceptible, that, if the terms of a series increased continually, it would not be possible to gain any thing near the value of

that series.

Those series which continually decrease, are called converging ones; and those that continually increase, are called diverging ones.

The Binomial Rule can be readily applied to numbers, to extract any of their roots; in order to do it, the number should be divided into two parts, the first of which should be always greater than the second, and an exact power of the required root; which can always be done by making the second part positive or negative; then treating the two parts of this number, as the two terms of a binomial, the required root will be found.

We shall content ourselves with pointing out this operation, which is rather an object of curiosity than useful for the purpose of this work; as, in order to extract the square, or cube root of a number, the method taught in Arithmetic is more expeditious, and even, if the number is not an exact square, or cube, it is even better for obtaining a proximate root to make use of the decimal parts.

EQUATIONS.

An Equation is the equality of two or more quantities, separated by the sign

which signifies equal to. Equations are of great use in solving problems.

A Problem is a question, the expression of which consists of known and unknown quantities, the latter of which are to be determined from the former.

Analysis is the art of solving problems; in order to effect it, the known must be combined with the unknown quantities, so as to form equations.

An Equation is solved, when the unknown quantity is found by itself in one of its members,* freed from all coefficients and exponents.

Equations are of different degrees. The highest exponent of the unknown quantity, determines the degree of the equation.- -The unknown quanti

ties are usually represented by the latter letters of the alphabet, u, t, x,y,z. Such an equation, as ya, is also of the second degree ;

of the third degree, &c.

-x y z = b is

That a problem may be soluble, the conditions of the problem must be such, that as many equations can be formed, as there are unknown quantities;

* The members of an equation are the two parts of it on the right and left of the sign of equality. The part on the right is called the first member; the other, the second member.

when this is not the case, the problem is said to be indeterminate, that is, it has more solutions than

one.

N. B. It sometimes happens, though there are as many equations as unknown quantities, the question is nevertheless indetermined: this takes place when some of the conditions, though apparently different, are essentially the same, which the method to be pursued in the solution of equations will teach how to distinguish ; in this case, the operation leads to an identical equation, consisting of the same terms on both sides of the sign, which consequently gives no result.

It is evident, that problems must occur in every degree, containing more than one unknown quantity. In order to solve a problem with several unknown quantities, the different equations must be combined together, so as to have but one equation, and but one unknown quantity in this equation.

There are no general rules for forming a problem into one equation; it is by a certain power of the min, quickened by practice, that we are enabled at once to perceive the relation between known and unknown quantities: but, in order to become ac◄ quainted with this relation, and the manner of forming equations more easily, we should always begin by representing the unknown quantities by letters, and working with them as if they were known.

Every operation to be performed with equations, is founded on principles very easy to be comprehended; but it is by practice alone, we can acquire any skill in the art of analysis.