But it is best to set down both the co-efficients and the powers of the letters at once, in one line, without the intermediate lines in the above example, as in the example here below.

2. Let a · be involved to the 6th power.

The terms with the co-efficients will be

- 6a5x + 15a2x2

3 Required the 4th power of a — x.

Ans. a 4a3x+6a2x2

6ax+x*.

And thus any other powers may be set down at once, in the same manner; which is the best way.

EVOLUTION.

EVOLUTION is the reverse of Involution, being the method of finding the square root, cube root, &c, of any given quantity, whether simple or compound.

CASE I. To find the Roots of Simple Quantities.

EXTRACT the root of the co-efficient, for the numeral part; and divide the index of the letter or letters, by the index of the power, and it will give the root of the literal part; then annex this to the former, for the whole root sought".

*

Any even root of an affirmative quantity, may be either + or: thus the square root of +a2 is either +a, ora; because + ax +a+a2, and + a2 also.

But an odd root of any quantity will have the same sign as the quantity itself: thus the cube root of a3 is+a and the cube root of a3 isa; for +ax+a×+ a = + a3, and -a X -ax

Any even root of a negative quantity is impossible; for neither +ax+a, nor — a X - a can produce — a2.

--

Any root of a product, is equal to the like root of each of the factors multiplied together. And for the root of a fraction, take the root of the numerator, and the root of the denominator.

EXAMPLES.

To find the Square Root of a Compound Quantity.

THIS is performed like as in numbers, thus:

1. Range the quantities according to the dimensions of one of the letters, and set the root of the first term in the quotient.

2. Subtract the square of the root, thus found, from the first term, and bring down the next two terms to the remainder for a dividend; and take double the root for a divisor.

3. Divide the dividend by the divisor, and annex the result both to the quotient and to the divisor.

4. Multiply the divisor thus increased, by the term last set in the quotient, and subtract the product from the dividend.

And so on, always the same, as in common arithmetic.

EXAMPLES.

1. Extract the square root of a1 — 4a3b+6a2 b2 —4ab3 †ba. a4a3b6a2b2

-

2ab+62 the root.

Note. In the multiplication of compound quantities, it is the best way to set them down in order, according to the powers and the letters of the alphabet. And in multiplying them, begin at the left-hand side, and multiply from the left hand towards the right, in the manner that we write, which is contrary to the way of multiplying numbers. But in setting down the several products, as they arise, in the second and following lines, range them under the like terms in the lincs above, when there are such like quantities; which is the easiest way for adding them up together.

In many cases, the multiplication of compound quantities is only to be performed by setting them down one after another, each within or under a vinculum, with a sign of multiplication between them. As (a + b) x (a - b) x 3ab, ora+b.a- - b. 3ab.

4. Multiply x2
5. Multiply a3 +
6. Multiply a +
7. Multiply 3x2

Ans. 20a c.

Ans. 9a2b-662.
Ans. 9a2-462.
Ans. x3+y3.
Ans. aa—ba.

3. Multiply 3a + 26 by 3a 26. • xy + y2 by x + y. a2b + ab2 +b3 by a-b. ab +b2 by a2 ab + b2. -2xy + 5 by x2 + 2xy 8. Multiply Sa2 2ax +5x2 by 3a2. 9. Multiply 3x3 + 2x2y2 + 3y3 by 2x3 19. Multiply a + ab + b2 by a — 26.

6.

4ax 7x2.

-

DIVISION.

DIVISION in Algebra, like that in numbers, is the converse of multiplication; and it is performed like that of numbers also, by beginning at the left-hand side, and dividing all the parts of the dividend by the divisor, when they can be so divided; or else by setting them down like a fraction, the dividend over the divisor, and then abbreviating the fraction as much as can be done. This will naturally divide into the following particular cases.

CASE

CASE I.
>

When the Divisor and Dividend are both Simple Quantities;

SET the terms both down as in division of numbers, either the divisor before the dividend, or below it, like the denominator of a fraction. Then abbreviate these terms as much as can be done, by cancelling or striking out all the letters that are common to them both, and also dividing the one co-efficient by the other, or abbreviating them after the manner of a fraction, by dividing them by their common

measure.

Note. Like signs in the two factors make + in the quotient; and unlike signs make; the same as in multiplication*.

EXAMPLES.

1. To divide 6ab by 3a.

6ab
3a

Here Gab3a, or 3a) 6ab ( or 26.

Because the divisor multiplied by the quotient, must produce the dividend. Therefore,

1. When both the terms are +, the quotient must be +; because in the divisor in the quotient, produces + in the dividend. 2. When the terms are both, the quotient is also because in the divisor X + in the quotient, produces in the dividend. 3. When one term is + and the other, the quotient must be —; because in the divisor X - in the quotient produces - in the divi dend, or in the divisor in the quotient gives in the dividend. viz. that like signs give +, and unlike

So that the rule is general; signs give, in the quotient.

VOL. I.

Bb

CASE

SURDS.

SURDS are such quantities as have not exact values in numbers; and are usually expressed by fractional indices, or by means of the radical sign✔. Thus, 32, or 3, denotes the

square root of 3;

the square of 2;

and 25 or √22, or 4, the cube root of

where the numerator shows the power to which the quantity is to be raised, and the denominator its root.

PROBLEM I.

To Reduce a Rational Quantity to the Form of a Surd.

RAISE the given quantity to the power denoted by the index of the surd; then over or before this new quantity set the radical sign, and it will be of the form required.

EXAMPLES.

1. To reduce 4 to the form of the square root.
First, 44 X 4 = 16; then ✔ 16 is the answer.

2. To reduce 3a2 to the form of the cube root.
First, 3a2 × 3a2 × Sa2 = (3a2)3 = 27a6 ;

then 27a or (27a) is the answer. 3. Reduce 6 to the form of the cube root.

Ans. (216) or 216.

4. Reduce fab to the form of the square root.

Ans.✔ 1a2b2.
Ans. (16).

5. Reduce 2 to the form of the 4th root.
6. Reduce as to the form of the 5th root.
7. Reduce a + x to the form of the square root.
8. Reduce a to the form of the cube root.

PROBLEM II.

To Reduce Quantities to a Common Index.

1. REDUCE the indices of the given quantities to a common denominator, and involve each of them to the power denoted by its numerator; then I set over the common denominator will form the common index.

Or,