From these three examples may be collected the following rule for the multiplication of algebraic quantities: Multiply each term of the multiplicand by each term of the multiplier; when the two terms have both + or both-before them, put + before their product; when one has + and the other-, put-before their product. In using the first terms, which have no sign, apply the rule as if they had the sign +.

68. For example, (a + b)(a + b) gives aa + ab + ab + bb. But ab + ab is 2ab; hence the square of a + b is aa + 2ab + bb. Again (a- b)(a-b) gives aa-ab-ab + bb. But two subtractions of ab are equivalent to subtracting 2ab; hence the square of a- b is aa-2ab + bb. Again, (a + b)(a-b) gives aa + ab-ab -bb. But the addition and subtraction of ab makes no change; hence the product of a + b and a- b is aa-bb.

Again, the square of a + b + c + d or (a + b + c + d)(a + b + c + d) will be found to be aa + 2ab + 2ac + 2ad + bb + 2bc + 2bd + cc + 2cd + dd; or the rule for squaring such a quantity is: Square the first term, and multiply all that come after by twice that term; do the same with the second, and so on to the end.

SECTION IV.
DIVISION.

69. Suppose I ask whether 156 can be divided into a number of parts each of which is 13, or how many thirteens 156 contains; I propose a question, the solution of which is called DIVISION. In this case, 156 is called the dividend, 13 the divisor, and the number of parts required is the quotient; and when I find the quotient, I am said to divide 156 by 13.

70. The simplest method of doing this is to subtract 13 from 156, and then to subtract 13 from the remainder, and so on; or, in common language, to tell off 156 by thirteens. A similar process has already occurred in the exercises on subtraction, Art. (46). Do this, and mark one for every subtraction that is made, to remind you that each subtraction takes 13 once from 156, which operations will stand as follows:

Begin by subtracting 13 from 156, which leaves 143. Subtract 13 from 143, which leaves 130; and so on. At last 13 only remains, from which when 13 is subtracted, there remains nothing. Upon counting the number of times which you have subtracted 13, you find that this number is 12; or 156 contains twelve thirteens, or contains 13 twelve times.

This method is the most simple possible, and might be done with pebbles. Of these you would first count 156. You would then take 13 from the heap, and put them into one heap by themselves. You would then take another 13 from the heap, and place them in another heap by themselves; and so on until there were none left. You would then count the number of heaps, which you would find to be 12.

71. Division is the opposite of multiplication. In multiplication you have a number of heaps, with the same number of pebbles in each, and you want to know how many pebbles there are in all. In division you know how many there are in all, and how many there are to be in each heap, and you want to know how many heaps there are.