the product, is multiplied by the sum of the quantities, what will be the amount?

Let a be one of the quantities and b the other, then,

So that we perceive that the sum of the squares wanting the product, when multiplied by the sum, produces the sum of the cubes. Let us now perform the division, first taking the one factor for a divisor, and then the other; and let us see whether in each case the result of the division will give us the other factor. First, dividing by the multiplier, we have this operation :

The first line of the above operation consists of the divisor, a+b, the dividend, a3+b3, and a place for the quotient as it is found. The divisor and dividend are placed with the letters in the same order, which is a matter of convenience though not one of necessity. Comparing a, the first term of the divisor, with a3, the first of the dividend, we find that as a2xa is = a3, the first term of the quotient must be a2; and because the term of the divisor and that of the dividend have both the sign +, or, which is the same thing, are without any sign, a2 in the quotient must be +.

EXPLANATION.

117.

We next multiply both terms of the divisor by a2, and change the signs, which is the same as subtracting the products, and thus we get the second line —a3—a2b. Comparing the terms of this with those of the dividend we find a3 with the sign +, and also with the sign which destroy each other, and we have remaining a2b+63, which is the third line of the operation; and we again compare its first term with the first term of the divisor, that is, we compare a with -a2b. It is easy to see that a will be got ab times in this term, and that the sign must be or that in order to convert the products into remainders we must make their signs the same as that of the divisor. Performing this multiplication we have +ab+ab2, in which the first terms destroy each other, and there remains ab2+b3 for the fifth line of our operation. Comparing the first term of this with the first of the divisor we perceive that +b2 in the quotient will, if multiplied by a in the divisor, and the sign changed, produce ab2, which extinguishes the first term. Multiplying both terms of the divisor by b2, and changing the signs, we obtain —ab2-b3, which exterminates the whole dividend. Therefore our whole quotient is a2-ab+b2, which is exactly the same as our multiplicand, though the terms are not arranged in exactly the same order.

It will be seen from this operation that the process of dividing algebraically is so simple as to be merely mechanical; for at each step we have only to select such a term for the quotient as shall with the first term of the divisor produce the same combination of letters as that of the dividend, and shall have such a sign as when changed shall be opposite to that of the dividend. It is of no consequence whether any of the other terms are the same or not, because the changing of the signs of those parts obtained by multiplication converts them into remainders; and if the multiplication and change of the signs be rightly per

formed every step of the operation will lead to a true result, whatever be the difference of its appearance.

may

Neither need we conclude that we have committed errors, though the product of a compound divisor and quotient do not amount to the same identical expression as the dividend; because we have already shown that equi-multiples and like parts of any divisor and dividend will all lead to exactly the same quotient.

Before we can thoroughly understand division, and those general relations which are founded upon its principle, or rather in which its principle consists, it is necessary to have recourse to some farther explanations, which can be more conveniently made in a new section; we shall therefore close this one by subjoining the operation for the above example, as divided by the other factor.

a2—ab+b2) a3+b3 (a+b

-a3+a2b—ab2

+a2b-ab2+63

—a2b+ab2—b3

0.

SECTION VII.

NATURE AND MANAGEMENT OF FRACTIONS.

A FRACTION is a quantity viewed in its relation to some other quantity of the same kind which is considered as a whole; and as every case of division may be considered as reducible either exactly or to any degree of nicety that may be required to an expression in which the divisor is one, or, which is exactly the

RELATION TO DIVISION.

119

same in effect, every possible case of division may be conceived as consisting of equi-multiples of 1 and the quotient by the divisor, the simple expression of division by writing the dividend over the divisor, and separating them by a line, is also the

a

general expression of a fraction. Thus is an expression for

b

any fraction, in which the quantity b is understood to mean a whole, or the number 1, and a any quantity whatever, only it must be one of the same kind with b. If a and b were expressed arithmetically it would be necessary to express them both in the same unit, in order that the numbers might express the same relation as the values; and when general expressions are used it is necessary to understand them in this manner.

Perhaps the simplest notion we can have of the nature of a fraction is the arithmetical one, which supposes that the whole is divided into as many equal parts as the under term of the fraction expresses, while the value of the fraction consists in the number of those parts which the upper term expresses. Thus,

19

in the expression the under number 20 shows that some

20

thing considered as a whole is understood to be divided into 20 equal parts; and the upper number 19 shows that the value of this particular fraction is 19 of those parts. From this it follows that the value of the fraction does not depend upon the absolute numbers in which it is expressed, but upon the relation of those numbers to each other; and that each of the two numbers has a distinct operation to perform.

One whole, by whatever number it may be expressed, may be considered as always meaning the very same quantity, unless the contrary is expressly stated; and thus, the larger number which the under term of a fraction expresses, the smaller must be the value of every individual 1 of that number; but the larger the

upper term, the value must always be the greater. A fraction may thus be considered as having a sort of double value, or a value which may at any rate be considered as the result of two operations, a division by the under term to find the value of 1 in the upper number, and a multiplication of the value so found by the upper number.

In an arithmetical point of view, the number of the under term fixes the denomination of the fraction, in the very same way as the denominations of real quantities are fixed by the standards in which they are counted; and for this reason the under number is called the denominator of the fraction. The denominator is thus, as it were, "small change" for the integer number 1, just as shillings are small change for a pound, or yards are small change for a mile. The upper number shows how much of this small change the fraction consists of, and for this reason it is called the numerator, or "the teller of the number" of the fraction. It may be any number, equal to the denominator, or greater, or less; and it may be a number which cannot be exactly expressed in terms of the denominator, at the same time that there is between the two a relation which we can perfectly understand.

Hence the doctrine of fractions is a very general one in mathematical science, as it involves all comparisons in which the whole value of one quantity is compared with the whole value of another. There is something neat in the signs which are used to express the comparisons. ab is relation generally, and says little more than that a and b are quantities of the same kind; is a more definite statement of the relation, for it points out that a is the standard with which b is compared. ab with the compound sign is more definite still, for it points out the difference of the related quantities; but the line

a