33. From the manner of proceeding in multiplication, it is evident that if all the terms of the multiplicand are of the same degree (27), and those of the multiplier are also of the same degree, all the terms of the product will be of a degree denoted by the sum of the numbers, which mark the degree of the terms of each of the factors. In the first example, the multiplicand is of the fourth degree, the multiplier of the third; and the product is of the seventh. In the second example, the multiplicand is of the sixth degree, the multiplier of the third; and the product is of the ninth. Expressions of the kind just referred to, the terms of which are all of the same degree, are called homogeneous expressions. The above remark, with respect to their products, may serve to prevent occasional errors, which one may commit by forgetting some of the factors in the several parts of the multiplication. 34. Algebraic operations performed upon literal quantities, as they permit us to see how the several parts of the quantities concur to form the results, often make known some general properties of numbers independent of every system of notation. The multiplications that follow, lead to conclusions of the greatest importance, and of frequent use in the subsequent parts of this work. a+b a2 + ab +ab+b2 a2 +2ab+b2 It appears from the first of these products, that the quantity a+b, multiplied by a b, gives a b2; whence it is evident that, if we multiply the sum of two numbers by their difference, the product will be the difference of the squares of these numbers. If we take, for example, the sum 11 of the numbers 7 and 4, and multiply it by the difference 3 of these numbers, the product 3 X 11, or 33, will be equal to the difference between 49, the square of 7, and 16, the square of 4. By the second example, in which a + b is twice a factor, we learn; that the second power, or the square of a quantity composed of two parts a and b contains the square of the first part, plus double the product of the first part by the second, plus the square of the second. The third example, in which we have multiplied the second power of ab by the first, shows; that, the third power or cube of a quantity composed of two parts contains the cube of the first, plus three times the square of the first multiplied by the second, plus three times the first multiplied by the square of the second plus the cube of the second. 35. As we have often occasion to decompose a quantity into its factors, and as the algebraic operations are dispensed with, when it can be done, in order to exhibit the formation of the quantities to be considered, as distinctly as possible, it is necessary to fix upon some signs proper to indicate multiplication. between complex quantities. We use indeed the marks of a parenthesis to comprehend the factors of a product. The expression 3 a2 b2 + b1) (4 a b2 — a c2 + d3) (b2 — c2), for example, indicates the product of the compound quantities (5 a 5 a4 3a2 b2 + b2, 4 ab3 a c2 + d3, and b2 — c2. Bars were used formerly by some authors placed over the factors thus, 5a4 3a2 b2 + b4 X 4 ab2 ac2 + d3 × b2 · c2; but as these may happen to be too long or too short, they are liable to more uncertainty than the marks of a parenthesis, which can never admit of any doubt with respect to the quantity belonging to each factor. They have accordingly been preferred. Of the Division of Algebraic Quantities. 36. ALGEBRAIC division, like division in arithmetic, is to be regarded as an operation designed to discover one of the factors of a given product, when the other is known. According to this definition, the quotient multiplied by the divisor must produce anew the dividend. By applying what is here said to simple quantities we shall see by art. 21, that the dividend is formed from the factors of the divisor and those of the quotient; whence, by suppressing in the dividend all the factors which compose the divisor, the result will be the quotient sought. Let there be, for example, the simple quantity 72 a3 b3 c2 d to be divided by the simple quantity 9 a b c2; according to the rule above given, we must suppress in the first of these quantities the factors of the second, which are respectively 9, a3, b, and c2. It is necessary then, in order that the division may be performed, that these factors should be in the dividend. Taking them in order, we see in the first place that the coefficient 9 of the divisor, ought to be a factor of the coefficient 72 of the dividend, or that 9 ought to divide 72 without a remainder. This is in fact the case, since 72 9 × 8. By suppressing then the factor 9, there will remain the factor 8 for the coefficient of the quotient. It follows moreover, from the rules of multiplication (25), that the exponent 5 of the letter a in the dividend, is the sum of the exponents belonging to the divisors and quotient; this last exponent therefore will be the difference between the two others, or 53 2. Thus the letter a has in the quotient the exponent 2. For the same reason, the letter b has in the quotient an exponent equal to 31, or 2. The factor ca being common to the dividend and divisor is to be suppressed, and we have for the quotient required. 8 a2 b2 d The same will apply to every other case; we conclude then, that, in order to effect the division of simple quantities, the course to be pursued is, To divide the coefficient of the dividend by that of the divisor; To suppress in the dividend the letters which are common to it and the divisor, when they have the same exponent; and when the exponent is not the same, to subtract the exponent of the divisor from that of the dividend, the remainder being the exponent to be affixed to the letter in the quotient; To write in the quotient the letters of the dividend which are not in the divisor. 37. If we apply the rule now given for obtaining the exponent of the letters of the quotient, to a letter which has the same exponent in the dividend and divisor, we shall find zero to be the exponent which it ought to have in the quotient; a3 divided by as, for example, gives a°. To understand what is the import of such an expression, it is necessary to go back to its origin and to consider, that if we represent the quotient arising from the division of a quantity by itself, it ought to answer to unity, which expresses how many times any quantity is contained in itself. It follows from this, that the expression a' is a symbol equivalent to unity, and may consequently be represented by 1. We may then omit writing the letters which have zero for their exponent, since each of them signifies nothing but unity. Thus a3 b c2 divided by a2 b c2, gives a1 b° co, which becomes a, as is very evident by suppressing the common factors of the dividend and divisor. We see by this, that the proposition, every quantity which has zero for its exponent, is equal to 1, is nothing, properly speaking, but the explanation of a conclusion to which we are brought by the common manner of writing the powers of quantities by exponents. In order that the division may be performed, it is necessary, 1. that the divisor should have no letter which is not found in the dividend; 2. that the exponent of any letter in the divisor should not exceed that of the same letter in the dividend; 3. that the coefficient of the divisor should exactly divide that of the dividend. 38. When these conditions do not exist, the division can only be indicated in the manner pointed out in the 2d article. Still we should endeavour to simplify the fraction by suppressing such factors, as are common to the dividend and divisor, if there are any such; for (Arith. 57) it is manifest, that the theory of arithmetical fractions rests upon principles which are independent of every particular value of their terms, and which would apply to fractions represented by letters, as well as to those which are represented by numbers. According to these principles, we in the first place suppress the numerical factors common to the dividend and divisor, and then the letters which are common to the dividend and divisor, and which have the same exponent in each. When the exponent is not the same in each, we subtract the less from the greater, and affix the remainder, as the exponent to the letter, which is written only in that term of the fraction which has the highest exponent. The following example will illustrate this rule. Let 48 a3 b3 c2 d be divided by 64 a3 b3 ca e; the quotient can only be indicated in the form of a fraction 48 a3 b5 c2 d 64 a3 b3 c4 e But the coefficients 48 and 64 being divisible by 16, by suppressing this common factor, the coefficient of the numerator becomes 3, and that of the denominator 4. The letter a having the same exponent 3 in the two terms of the fraction, it follows that a3 is a factor common to the dividend and divisor, and may consequently be suppressed. To find the number of factors b common to the two terms of the fraction, we must divide the higher b by the lower b3, according to the rule above given, and the quotient ba shows, that b3 = b3 × b3. Suppressing then the common factor b3, there will remain in the numerator the factor b2. With respect to the letter c, the higher factor being c✩ of the denominator, if we divide it by c2 we shall decompose it into c2 × c2; and by suppressing the factor c2 common to the two terms, this letter disappears from the numerator, but will remain in the denominator with the exponent 2. Finally, the letters d and e will remain in their respective places, since in the state in which they are, they indicate no factor common to both. By these several operations the proposed fraction is reduced to 3 b2 d ; 4 c2 e and it is the most simple expression of the quotient, except we give numerical values to the letters; in which case it might be further reduced by cancelling the common factors as before. 39: It ought to be remarked, that, if all the factors of the dividend enter into the divisor, which besides contains others peculiar to it, it is necessary after suppressing the former to put unity in the place of the dividend, as the numerator of the fraction. In this case indeed we may suppress all the terms of the numerator, or, in other words, divide the two terms of the fraction by the numerator; but this being divided by itself must give unity for the quotient, which becomes the new numerator. Suppose for example the fraction |