In general, any power whatever is designated by the number of equal factors from which it is formed; a' or a aa aa is the fifth power of a.

I take the number 3 to illustrate these denominations, and I have

The number which denotes the power of any quantity is called the exponent of this quantity.

When the exponent is equal to unity it is not written; thus a is the same as a1.

It is evident then, that to find the power of any number, it is necessary to multiply this number by itself as many times less one, as there are units in the exponent of the power.

25. As the exponent denotes the number of equal factors, which form the expression of which it is a part, and as the product of two quantities must have each of these quantities as factors; it follows that the expression as in which a is five times a factor, multiplied by a3, in which a is three times a factor, ought to give a product in which a is eight times a factor, and consequently expressed by a3, and that in general the product of two powers of the same number ought to have for an exponent the sum of those of the multiplicand and multiplier.

26. It follows from this, that when two simple quantities have common letters, we may abridge the expression of the product of these quantities by adding together the exponents of such letters of the multiplicand and multiplier.

For example, the expression of the product of the quantities a2 b3 c and a4 b5 c2 d, which would be a2 b3 c a4 b5 c2 d, by the foregoing rule, art. 21, is abridged by collecting together the factors designated by the same letter, and

becomes

by writing

a2 a b3 b c c2 d,

as bs c3 d,

as instead of a2 a4

b8 instead of b3 b5

e3 instead of c c2 or of c1 c2.

27. As we distinguish powers by the number of equal factors from which they are formed, so also we denote any products by the number of simple factors or firsts which produce them; and I shall give to these expressions the name of degrees. The product a2 b3 c, for example, will be of the sixth degree, because it contains six simple factors, viz; 2 factors a, 3 factors b, and 1 factor c. It is evident that the factors a, b, and c, here regarded as firsts, are not so, except with respect to algebra, which does not permit us to decompose them; they may, notwithstanding, represent compound numbers, but we here speak of them only with respect to their general import.*

As the coefficients expressed in letters are considered only in estimating the degree of algebraic quantities, we have regard only to the letters.

It is evident, (21, 25) that when we multiply two simple quantities the one by the other, the number which marks the degree of the product is the sum of those which mark the degree of each of the simple quantities.

28. The multiplication of compound quantities consists in that of simple quantities, each term of the multiplicand and multiplier being considered by itself; as in arithmetic we perform the operation upon each figure of the numbers which we propose to multiply. (Arith. 33.) The particular products added together make up the whole product. But algebra presents a circumstance which is not fourd in numbers. These have no negative terms or parts to be subtracted, the units, tens, hundreds, &c. of which they consist, are always considered as added together, and it is very evident, that the whole product must be composed of the sum of the products of each part of the muitiplicand by each part of the multiplier.

*We apply the term dimensions, generally to what I have here called degrees, in conformity to the analogy already pointed out in the note to page 29. This example sufficiently proves the absurdity of the ancient nomenclature, borrowed from the circumstance, that the products of 2 and 3 factors, measure respectively the areas of the surfaces and the bulks of bodies, the former of which have two and the latter three dimensions; but beyond this limit the correspondence between the algebraic expressions and geometrical figures fails, as extension can have only three dimensions.

The same is true of literal expressions when all the terms are connected together by the sign +

The product of multiplied by

is

a+b

ac + bc,

and is obtained by multiplying each part of the multiplicand by the multiplier, and adding together the two particular products ac and bc. The operation is the same when the multiplicand contains more than two parts.

If the multiplier is composed of several terms, it is manifest that the product is made up of the sum of the products of the multiplicand by each term of the multiplier.

for by multiplying first a+b by c, we obtain ac+bc, then by multiplying a+b by the second term d of the multiplier, we have ad+bd, and the sum of the two results gives a c + b c + ad+bd for the whole.

29. When the multiplicand contains parts to be subtracted, the products of these parts by the multiplier must be taken from the others, or in other words, have the sign — prefixed to them. For example,

for each time that we take the entire quantity a, which was to have been diminished by b before the multiplication, we take the quantity b too much; the product a c therefore, in which the whole of a is taken as many times as is denoted by the number c, exceeds the product sought by the quantity b, taken as many times as is denoted by the number c, that is by the product be; we ought then to subtract b c from a c, which gives, as above,

ac -bc.

The same reasoning will apply to each of the parts of the multiplicand, that are to be subtracted, whatever may be the number, and whatever may be that of the terms of the multiplier, pro

vided they all have the sign+. Recollecting that the terms which have no sign are considered as having the sign +, we see by the examples, that the terms of the multiplicand affected by the sign + give a product affected by the sign +, while those which have the sign- give one having the sign. It follows from this, that when the multiplier has the sign+, the product has the same sign as the corresponding part of the multiplicand.

30. The contrary takes place when the multiplier contains parts to be subtracted; the products arising from these parts must be put down with a sign, contrary to that which they would have had by the above rule. This may be shown by the following example.

for the product of the multiplicand, by the first term of the multiplier, will be by the last example ac-bc; but by taking the whole of c for the multiplier instead of c diminished by d, we take the quantity ab so many times too much as is denoted by the number d; so that the product a cbc exceeds that sought by the product of ab by d. Now this last is, by what has been said, a d-bd, and in order to subtract it from the first it is necessary to change the signs (20). We have then

ac-be-ad+bd for the result required.

31. Agreeably to the above examples, we conclude, that the multiplication of polynomials is performed by multiplying successively, according to the rules given for simple quantities (21-26), all the terms of the multiplicand by each term of the multiplier, and by observing that each particular product must have the same sign, as the corresponding part of the multiplicand, when the multiplier has the sign+, and the contrary sign when the individual multiplier has the sign

If we develop the different cases of this last rule, we shall find, 1. That a term having the sign +, multiplied by a term having the sign+, gives a product having the sign +;

2. That a term having the sign, multiplied by a term having the sign+, gives a product which has the sign-;

3. That a term having the sign +, multiplied by a term having the sign, gives a product which has the sign —;

4. That a term having the sign, multiplied by a term having the sign, gives a product which has the sign +.

It is evident from this table, that when the multiplicand and multiplier have the same sign, the product has the sign +, and that when they have different signs, the product has the sign—.

To facilitate the practice of the multiplication of polynomials, I have subjoined a recapitulation of the rules to be observed. 1. To determine the sign of each particular product according to the rule just given; this is the rule for the signs.

2. To form the coefficients by taking the product of those of each multiplicand and multiplier (22); this is the rule for the coeffi cients.

3. To write in order, one after the other, the different letters comtained in each multiplicand and multiplier (21); this is the rule for the letters.

4. To give to the letters, common to the multiplicand and multiplier, an exponent equal to the sum of the exponents of these letters in the multiplicand and multiplier (25); this is the rule for the exponents.

32. The example below will illustrate all these rules. Multiplicand Multiplier

Several products.

5a42a3b+4 a 2 b2

a3-4a3b+2b3

Result reduced 5a* —22ab+12a3 b2 —6a*b3—4a3b*+8a2b3. The first line of the several products contains those of all the terms of the multiplicand by the first term a3 of the multiplier; this term being considered as having the sign +, the products which it gives have the same signs as the corresponding terms of the multiplicand (31).

The first term 5 a of the multiplicand having the sign plus, we do not write that of the first term of the product, which would be+; the coefficient 5 of a being multiplied by the coefficient 1 of a3, gives 5 for the coefficient of this product; the sum of the two exponents of the letter a is 4+ 3, or 7, the first term of the product then is 5 a7.

The second term 2 a3 b of the multiplicand having the sign -, the product has the sign minus; the coefficient 2 of ab mul