+15a—6a2+b2+5cd—2de +5xy—x2—2x+y2

SUBTRACTION.

RULE. Place the quantities one under the other, and change all the figns of the fubtrahend; that is, where there is an affirmative fign, place a negative one; and vice verfa. Then add the quantities together, as in addition.

Subtraction, as well as each of the other four rules in algebra, is proved in the fame manner as in common arithmetic *.

The reafon of this rule is evident from hence, that if a decrement or a negative quantity be taken away from an affirmative quantity, the remainder will be the fame as if an increment or an affirmative quantity of equal quantity be added to the original quantity. For every negative quantity always decreases the value of any quantity with which it is joined. Thus, if -b be taken from a-b, there will remain a, and if +ỏ be added to s—b, the fum is likewise a.

MULTIPLICATION.

RULE. Multiply each term of the multiplier into every term of the multiplicand: that is, the coefficients into the coefficients, and the letters into the letters; and to each prefix its fign; viz. + to like figns, and — to unlike signs.

Examples.
Multiply 3d

Multiply 7a
By 36

Multiply 5d2

Pro. +84aa Sac-12cc P.—as+a2b+2a2b2—ac2—2ab3 +2c2b2

The foregoing examples may be proved by divifion, as in common arithmetic *.

This rule depends upon the fame principle as multiplication in common arithmetic. And that two quantities having like figns, should give a product with the fign+, and two quantities of unlike figns the fign -, may be proved from hence: viz. 1. If an affirmative quantity be multiplied by an affirmative quantity, the product must of course be an affirmative quantity.-z. If a negative quantity be multiplied by an affirmative one, the negative quantity must be taken as often as there are units in the affirmative one, and the fum of any number of negative quantities will be negative. And if an affirmative quantity be multiplied by a negative one, the affirmative quantity must be fubtracted as often as there are units in the negative one; and the sum of any number of negatives will be negative.-3. Again, if a negative quantity be multiplied by a negative quantity, the multiplicand is to be fubtracted, as often as there are units in the multiplier: but, to fubtract a negative quantity is the fame thing as to add an equal affirmative one: therefore, the product will be affirmative. From hence the general rule, that like figns produce +, and unlike figns —.

DIVISION.

DIVISION.

RULE. If the quantities be fimple, divide the coefficient of the dividend, by the coefficient of the divifor, and place the answer in the quotient; annexing thereto thofe letters in the dividend, which are not found in the divifor: obferving that like figns produce +, and unlike figns -.

2. But if the quantities be compound, divide the first term of the dividend by the firft term of the divifor, and place the refult in the quotient. Then multiply the whole divifor thereby, and fubtract the product from the dividend, and to the remainder bring down the next term in the dividend. And repeat the operation as in common arithmetic. But the terms in the dividend fhould be ranged in a proper order, that is, according to the dimenfions of fome letter; the quantities reprefented by a being generally placed firft; thofe by b or ab next, as follows;

Examples.

9ac)36abed(4bd

36abcd

4a2b2)-24a2b3d(-6bd
24a2b3d

4a-6c) 16abb-24cbb (4bb
16abb-24cbb

3a-b)3a3-12a2-ba2+10ab—2b2 (a2-4a+26

When the divifor will not exactly divide the dividend, as is often the cafe, the dividend is to be placed over the divifor, and a line drawn between them, like a fraction, throwing out fuch letters as are found in both the divifor and dividend. Thus, If ab+c3 was to be divided by a b-c2, it would stand ab+c3 +63 + When the power of a quantity is to

thus,

ab-c3

=

be divided by any other power of the fame quantity, it is done by fubtracting the exponent of the divifor, from that of the dividend:

BEFORE the ftudent proceed to equations, it is necessary that he know how to manage fractional quantities; and to raise a quantity to any given power; and, on the contrary, to extract the root of any quantity; to manage furd quantities, &c.

The rules for managing Algebraic Fractions are exactly the fame as thofe for Vulgar Fractions in arithmetic, and therefore need not be repeated; as few perfons would attempt Algebra, till they were fufficiently fkilled in common arithmetic. An example or two may, however, be of fervice. '.

* To prove the reason of this rule, that like figns give +, and unlike figns it is only neceffary that the divifor be multiplied by the quotient, and the product will be equal to the dividend.

VOL. 11.

c

EXAMBLE

EXAMPLE I. Reduce the mixed quantity a, to a frac

quantity.

Here dividing bd-b by d, the quantity will be

b, and the remainder -62, and therefore the mixed quantity

tions of the fame value, having a common denominator.

Here

adf cbf ebd

bdf bdfbdf are the fractions required.

Involution; or, to find any given Power of any given Quantity.

RULE. Multiply the quantity into itself as often as the index contains units, except one, and the laft product will be the required power; or, which is more convenient, multiply the index of the quantity by the index of the power.

Thus, let it be required to raise the quantities +c and be to the third power, or the cube,