(Arith. 21, 66.) Thus, to multiply a by b is to compound with the quantity represented by a arother quantity, in the same manner as the quantity represented by b is with unity.

We have already explained in articles 2. and 7. the signs used to indicate multiplication; and the product of a by b is expressed by ab, or by a . b, or lastly by a b.

We have often occasion to express several successive multiplications, as that of a by b, and that of the product ab by e, also that of this last product by d, and so on. In this case, it is evident, that the last result is a number having for factors the numbers a, b, c, d, (Arith. 22); and to give a general expression of this method we indicate the product by writing the factors composing it in order, one after the other, without any sign between them ; we have accordingly the expression a b c d.

Reciprocally every expression, such as a b c d formed of several letters written in order one after the other, designates always the product of the numbers represented by these letters.

I have already availed myself of this method, in which the numerical coefficients are also included, since they are evidently factors of the quantity proposed. Indeed 15 a b c d, designating the quantity abcd taken fifteen times, expresses likewise the product of the five factors 15, a, b, c, d.

It follows from this, that in order to indicate the multiplication of several simple quantities, such as 4 abc, 5 def, 3 m n, it is necessary to write the quantities in order, one after the other, without any sign between them, and it becomes

4abc5def3mn;

but since, as is shown in arithmetic, (art. 82.) the order of the factors of a product may be changed at pleasure without altering the value of this product, we may avail ourselves of this principle, to bring together the numerical factors, the multiplication of which is performed by the rules of arithmetic; to express then this product, as indicated in the order 4. 5. 3abcdefmn, we multiply together the numbers 4, 5, 3, which give simply

* As the use of algebraic symbols abridges very much the demonstration of this proposition, I have thought it proper to suggest here a method by these symbols.

If we write the product a b c d e f as follows, abc X de xf, and

23. The expression of the product may be much abridged when it contains equal factors. Instead of writing several times in order, the letter which represents one of the factors, it need be written only once with a number annexed, showing how many times it ought to have been written as a factor; but as this number indicates successive multiplications, it ought to be carefully distinguished from a coefficient, which indicates only additions. For this reason, it is placed on the right of the letter and a little higher up, while a coefficient is always placed on the left and on the same line.

Agreeably to this method, the product of a by a, which would be indicated, according to article 21., by a a, becomes a2. The 2 raised, denotes that the number, designed by the letter a, is twice a factor in the expression to which it belongs. It ought not to be confounded with 2 a, which is only an abbreviation of a + a. To render evident the error, which would arise from mistaking one for the other, it is sufficient to substitute numbers instead of the letters. If we have for example a = 5, 2 a would become 2.5= 10, and a2 = a × a = 5,5 = 25.

Extending this method we should denote a product in which a is three times a factor by writing a3 instead of a a a; also a a5 represents a product in which a is five times a factor, and is equivalent to a a ɑ ɑ ɑ.

24. The products formed in this manner by the successive multiplications of a quantity, are called in general powers of that quantity.

The quantity itself, as a, is called the first power.

The quantity multiplied by itself, as a a, or a2, is the second power. It is called also the square.

The quantity multiplied by itself twice in succession, as a a a, or a3, is the third power, and is called also the cube.*

change the order of the factors of the product to e d instead of de, (Arith. 22,) it becomes abc Xed xf, or abc edf. It is evident that we may, by analyzing the product differently, produce any change which we wish in the order of the factors of the product in question.

The denominations square and cube refer to geometrical considerations. They interrupt the uniformity in the nomenclature of products formed by equal factors, and are very improper in algebra, But they are frequently used for the sake of conciseness.

In general, any power whatever is designated by the number of equal factors from which it is formed; c5 or a a a a a 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 a5, 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 a, 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 a2b3 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 at b3 b5 c c2 d,

a6 68 c3 d,

a6 instead of a2 a4

68 instead of b3 b5

c3 instead of cc or of cl 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 products a2 63 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.*

The coefficients expressed in numbers are not considered 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 found 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 multiplicand 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 +.

and is obtained by multiplying each part of the multiplicand by the multiplier, and adding together the two particular products a c and b c. 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 a c + bc, then by multiplying a + b by the second term d of the multiplier, we have a d + bd, and the sum of the two results gives

for the whole.

a c + b c + ad + b d

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 bc; we ought then to subtract be from a c, which gives, as above,

a c -bc.

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