Thus -£. X 4 = −£. + −£. + −£. + −£. =

Hence also the repetition of the operations denoted by four times is equivalent to the performance of those indicated by ; or ×4=§.

If the terms of the numerator and denominator have any common measure, they may be divided by it (Prop. 27.)

Prop. 32. To prove the Rule for division of a fraction by an integer.

As in division of a concrete quantity, the number of the units is divided by the divisor, so in division of a fraction, considered as a concrete quantity, the number of the units being denoted by the numerator, and their kind by the denominator, we must divide the numerator by the divisor. Thus 14 is equal to ; for the fourth part of 12 seventeenths is evidently 3 seventeenths, just on the same principle as the fourth part of 12 shillings is 3 shillings.

But if the numerator be not exactly divisible by the divisor, the quotient may be obtained by multiplying the denominator by the divisor.

For

5

7 X 2

units 5 units (7 × 2) =

units

}

2.

a

a

units=a units (b × c) = (a units b) c

units

c.

Or exhibited algebraically:

If the fraction be considered as abstract, then we seek the symbol of some operations, which, repeated a number of times equal to the divisor, shall produce the same result as those indicated by the fractional dividend. Now since the operations indicated by are équivalent to the repetition of those indicated by four times, therefore 14 = 1.

Prop. 33.-To prove the Rule for finding the value of a

compound fraction.

Let the fraction be of. In order to ascertain what is the meaning, and equivalence of this expression, we consider that is the symbol of two operations, to be performed upon the quantity with which it is joined (Prop. 26.) Thus if it be required to find the value of £ or of £1, the result would be obtained by dividing £1 by 3 and multiplying the quotient by 2, or vice versâ. So if it be required to find the value of of £, the result will be obtained by dividing £ by 3 and multiplying the quotient by 2 X 5 2, and will therefore be £or- £. Hence it appears that the re3X7 sult of the successive operations upon the unit £1 (or anything else), of

10

21

taking first of it and then of this, is the same as the result of taking of it; that is of and represent equivalent operations and are therefore equal;

10

Or, of

=

21

3 7 3 X 7

Similarly it may also be shewn that the value of such expressions as of of of &c. however many fractions there may be, is obtained by multiplying the numerators for a new numerator, and the denominators for a new denominator. For the last two fractions may be reduced to one, as above, then this and the next, and so on.

Prop. 34.-To explain the meaning of the Multiplication

by a fraction, and to deduce a Rule for finding the product.

The operation of Multiplication is that by which is found the result of the repetition of a number or a quantity, as often as there are units in the multiplier. Hence it is plain that the multiplier must be an abstract number, and the multiplicand either abstract or concrete. But the meaning of the result is somewhat different in these two cases; for in the former the result is an abstract number, and as an abstract number is the symbol of an operation, viz. that of repetition, the meaning of the result is that the operation represented by it is equivalent to the repetition of the operation denoted by the multiplicand as often as there are units in the multiplier. In the case of the multiplicand being concrete the meaning of the result is, that the quantity which it represents is equal to as many quantities, equal to the multiplicand, as there are units in the multiplier. We have now to see whether these ideas of multiplication will hold good, when the multiplier is a fraction. Now it is plain, that in the strict sense of the expressions "as often," "as many times," "as many," we cannot speak of repeating an operation "as often," or taking "as many" quantities, as there are units in a fraction. But if we adopt the extended notion of "a time," viz. that it may mean such a repetition of a subdivision of a quantity, as takes place in forming a part of it, then the forming a part of a quantity may be called multiplication by that which expresses the part, viz, the fraction. This operation (of forming a part) is strictly one of multiplication and division combined, but every multiplication might be performed by these combined operations (as for instance to multiply by 5 we might multiply by 10 and divide by 2), that is by such an operation as that which has been called "multiplication by a fraction;" so that the strict sense of multiplication includes the sense of multiplication by a fraction, viz. the repetition of a part of a quantity so as to form a "multiple-part." Therefore in assigning this meaning to the term "multiplication," we introduce no idea contrary to the general one, and shall not by

so doing be led into any absurdity or contradiction. We are then at liberty to call the operation of taking a part of a quantity "multiplication by a fraction," and to express it by the same sign, and we may interpret the meaning of a product of a quantity by a fraction to be the fractional part of the quantity, which is expressed by the multiplier.

In a similar manner it may be shewn that the result of the multiplication of a quantity by several fractions in succession is the same as the result of multiplication by one fraction, which is compounded of them all.

Hitherto it has been supposed that the multiplicand is concrete, but if it be abstract, and therefore represent one or two operations, according as it is a whole number or a fraction, what meaning are we to assign to its "multiplication by a fraction?" and what is to be the interpretation of the product?

It is plain that we cannot conceive any idea of taking a fractional part of an operation; therefore the meaning of the product cannot be the same as in the former case. To determine the meaning, which is to be assigned, let us revert to the idea which we have conceived of the product of two abstract numbers. This is, that it represents an operation which is equivalent to the repetition of that indicated by the multiplicand, as often as there are units in the multiplier. But this repetition is also equivalent to the successive operations indicated by the multiplicand and the multiplier. For multiplication by two numbers in succession is equivalent to the repetition of the multiplication by one of them, as often as there are units in another.

Therefore the operation indicated by the product of two abstract numbers is equivalent to the successive operations indicated by the two. This view of the term product evidently is applicable to the case of fractions, as well as of whole numbers, since every abstract number represents an operation, and may represent two, as every fraction does; therefore we may with equal justice talk of the product of the one kind of number, as of the other. Adopting then this view of the case, we shall see that the product of an abstract number, fractional or not, by a fraction, is the indication of the operation denoted by the fraction of the number. For the successive

multiplication by two fractions has been shewn in the previous part of this Prop. to be equivalent to multiplication by the fraction compounded of them; therefore the operation indicated by the product of two fractions is the same as is indicated by the fraction compounded of them. In other words the product of two fractions is equal to the fraction of a fraction.

Hence, generally it appears that the Rule for multiplication by fractions is the same as that for finding the fraction of the multiplicand, or, if this be fractional, the same as for finding the fraction of a fraction.

-

Prop. 35. To explain the meaning of Division by a frac

tion, and to deduce a Rule for forming the quotient.

Division is the operation by which (according to our first notion of it) is found some number or quantity, which repeated a given number of times, equal to the divisor, becomes equal to a given number or quantity, the dividend. But since the number of units in the quantity sought is the same as the number of times the abstract units in the dividend contain those in the divisor, therefore the operation of division is defined to be that, by which is ascertained how often one number is contained in another. In order to explain the meaning of Division by a fraction, it will be necessary to consider both these notions of the operation.

Now every integer is expressible in the form of a fraction, and every multiplication by an integer might be performed by the repetition of equal parts of the multiplicand, therefore division by an integer might be expressed, as division by a fraction, and the quantity sought in the division would be defined to be such, that the parts of it denoted by the divisor would be equal to the dividend. It appears then that if division by a fraction be defined to be the operation by which is found the number or quantity, of which the part or parts denoted by the divisor are equal to the dividend, no new idea of division is introduced, but one, which is included in the idea of division by an integer. Such then is defined to be the operation of division by a fraction, according to the first notion of division.

If it be considered that the object of division is to find how often the abstract units in the dividend contain those in the divisor, we need no extension of ideas to explain the case of division by a fraction. For, by the number of times which one abstract number contains another is meant the number of times which the operation represented by the one must be repeated in order to produce the effect of that represented by the other; and this number of times is the symbol of the operations of multiplication and division which must be performed upon the result of the one to produce that of the other, so that the object of division might be defined to be the discovering what operations of multiplication and division must be performed upon the result of the operation denoted by the divisor

in order to produce the result of that denoted by the dividend. There is nothing in this definition to exclude a fraction as a divisor.

We have now to deduce a Rule for the formation of the quotient.

1st. Let the dividend be concrete, and represented by the straight line A B; let the divisor be or .

Then we seek a line, of which or shall be equal to A B. Divide A B into three equal parts, then each of these parts is in the one case of the line sought, and in the other is of the same. Therefore the line sought is equal to 8 or 2 of these parts; but each of these parts is equal to A B, therefore quotient is equal to § A B or A B. Hence to divide a concrete quantity by a fraction, we must invert the divisor and multiply by it the dividend.

2nd. Let the dividend be abstract, then we seek the symbol of some operation, such that by performing upon its result the operation denoted by the divisor, a result will be obtained equal to that from performing the operation indicated by the dividend. Now since of 34 of an unit = of an unit, therefore is the symbol of the operations, upon whose result, if the operations denoted by be performed, the same result is obtained as by performing the operations denoted by &; hence 34 is the quotient of &÷; but = X; therefore the rule for obtaining the quotient is the same as before, "invert the divisor and multiply by it the dividend."

3rd. Let it be required to find how often a fraction is contained in a given abstract number; i.e. to find what operation must be performed upon the result of the operations denoted by the divisor to produce the same result as the operations denoted by the dividend. Here, since of of an unit = of an unit, therefore we must perform the operations denoted by upon the result of those denoted by in order to produce the result of those denoted by ; hence denotes the "number of times" that contains; and is obtained from these two fractions as before.

Prop. 36.-To explain the meaning of a complex fraction, and to deduce a Rule for its reduction to a simple fraction.

A complex fraction is one in which the numerator and denominator are, one or both, fractions or fractional expressions. If they are fractional expressions, these may be reduced to simple fractions, therefore it will be necessary only to consider the case of their being simple fractions.

Whatever meaning may be assigned to this class of fractions must be applicable to all cases of them: now every ordinary fraction may be ex