Add a+26-3c-10 to 3b-4a+5c+10 and 56-c.
Add a+b and a-b together.

Add Bab-10 to c-d-a and −4c + 2a —36—7.

Add 3a2+b2-c to 2ab-3a2+bc-b.

Add a+bc-b2 to ab2-abc+62.

Add 9a-8b+10x-6d-7c+ 50 to 2x-3a-5c+4b+6d

-10.

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

From 8x2y+6 5√xy + 2x√xy 7x2+2x-18+ 3b Take-2xy +2 7/xy+3-2xy 9x2-12

+5b+ x2

From 3a + b + c −d — 10, take c + 2a — d.
From 3a+b+c — -d- 10, take b-10 + 3a.
From 2ab+b2 — 4c + bc-

b, take 3a2- ·c + b2. From a3+36°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-4b12c + 12x, take 2a 8a+ 46-5c.

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 effect as to add or unite an equal positive one. So that, 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.

MULTIPLICATION.

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.

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

That this rule for the signs is true, may be thus shown.

1. When a is to be multiplied by+c; the meaning is, that a is to be taken as many times as there are units in c; and since the sum of any number of positive terms is positive, it follows that +ax+c makes+ac.

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 +e, or+c by-a: this is the same thing as taking · a as many times as there are units in +c; and as the sum of any number of negative terms is negative, it follows that—ax +c, or + a X-c make or produce ac.

3. When a is to be multiplied by -c: here—a is to be subtracted as often as there are units in c: but subtracting negatives is the same thing as adding affirmatives, by the demonstration of the rule for subtraction; consequently the product is c times a, or+ac.

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

Other demonstrations upon the principles of proportion, or by means of geometrical diagrams, bave also been given; but the above may suífice.

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.

CASE III.

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.

a2+2ab+b2 12x2-7xy-10y 6x3-3x2y-9xy+6y'

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 the actual operation, 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) × (a — b) × 3ab, or a + b. a—b. 3ab.