SUBTRACTION.

SET down in one line the first quantities from which the subtraction is to be made; and underneath them place all the other quantities composing the subtrahend: ranging the like quantities under each other as in Addition.

Then change all the signs (+ and -) of the lower line, or conceive them to be changed; after which, collect all the terms together as in the cases of Addition.*

*This rule is founded on the consideration, that addition and subtraction are opposite to each other in their nature and operation, as are the signs + and -, by which they are expressed and represented. So that, since to unite a negative quantity with a positive one of the same kind, has the effect of diminishing it, or subducting an equal positive one from it, therefore to subtract a positive (which is the opposite of uniting or adding) is to add the equal negative quantity In like manner, to subtract a negative quantity, is the same in eflect as to add or unite an equal positive one. So that, by changing the sign of a quantity from + to, or from-to +, changes its nature from a subductive quantity to an additive one; and any quantity is in effect subtracted, by barely changing its sign.

5xy-30 7x3-2(a+b)
7xy-50 2x2-4(a+b)

3xy3 +20α √(xy+10) 4x2 y2+12a (xy+10)

From a+b, take a − b.
From 4a+16, take b+a.
From 4a-4b, take 3a+5b.

From 8a-12x, take 4a-3x.

From 2x-4a-2b+5, take 8-5b+a+6x.
From 3a+b+c-d-10, take c+2a-d.
From 3a+b+c-d-10, take b-19+3a.
From 2ab+63-4c+bc-b, take 3a2-c+b2.
From a3+3b2 c+ab3 - abc, take b2 +ab2 -abc.

From 12x+6a-4b+40, take 46 - 3a +4x+6d-10.
From 2x-3a+4b+6c-50, take 9a+x+6b-6c-40.
From 6a-46-12c+12x, take 2x-8a+46-5c.

MULTIPLICATION.
PLICATION

This consists of several cases, according as the factors are simple or compound quantities.

CASE 1.

When both the Factors are Simple Quantities.

FIRST multiply the co-efficients of the two terms together, then to the product annex all the letters in those terms, which will give the whole product required.

Like signs. in the factors, produce +, and unlike in the products.

EXAMPLES.

* That this rule for the signs is true, may be thus shown. 1. Whena is to be multiplied by the meaning is, that taken as many times as there are units in c; and since the sum of any positive terms is positive, it follows that a x + c makes+ac.

a is to be number of

2. When

When one of the Factors is a Compound Quantity.

MULTIPLY every term of the multiplicand, or compound quantity, separately, by the multiplier, as in the former case; placing the products one after another, with the proper signs; and the result will be the whole product required.

2. When two quantities are to be multiplied together, the result will be exactly the same, in whatever order they are placed; for a times c is the same as c times a, and therefore, when a is to be multiplied by + c, or + c by this is the same thing as taking as the sum of any number of negative terms is negative, it follows that - aX+ -a as many times as there are units in +e; and c, or + ax c make or produce 3. Whena is to be multiplied by as there are units in c: but subtracting negatives is the same thing as adding -c: herea is to be subtracted as often affirmatives, by the demonstration of the rule for subtraction; consequently the product is c times a, or + ac.

-ac.

Otherwise. Since a-a0, therefore (aa) Xc is also 0, because 0 multiplied by any quantity, is still but 0; and since the first term of the product, or a cis- ac, by the second case; therefore the last term of the product, ora xc, must be ac, to make the sum =0, or ac + ac0: that is, - o = +ac.

EXAMPLES.

When both the Factors are Compound Quantities;

MULTIPLY every term of the multiplier by every term of the multiplicand, separately; setting down the products one after or under another, with their proper signs; and add the several lines of products all together for the whole product required.

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 lines 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, or ath.a-b. 3ab.

EXAMPLES FOR PRACTICE.

1. Multiply 10ac by 2a.

2. Multiply 3a2-2b by 3b.

3. Multiply 3a+26 by 3a-2b.

4. Multiply 2-xy+y2 by x+y

5. Multiply a+ab+ab+be by a-b.

6. Multiply a2+ab‍+b2 by a2 — ab+ b2.
7. Multiply 3x2-xy+5 by x2+2xy-6.

Ans. 20a2c.

Ans. 9a2b-6b2.

Ans. 9a24b2.
Ans. x2+y3.
Ans. a-ba.

8. Multiply 3a2-2ax+5x2 by 3a3-4ax-7x3. 9. Multiply 3x3+2x2 y2+3y3 by 2x-3x2 y2 +3y3. 10. Multiply a2+ub+b2 by a -2b.

DIVISION.

DIVISION in Algebra, like that in numbers, is the converse of multiplication; and 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