Ex. Multiply 7s. 10 d. by 7985.

By the table, 7986 = 11 × 11 × 11 × 6.

Sub.

7 10 arising from the multiplier 1, above.

£3152 8 229

I need scarcely tell you, that examples of this kind may always be worked by common reduction; that is, you may reduce the compound quantity to the lowest denomination before multiplying, and then convert the product into the higher denomination.

*In multiplying by 10, you merely annex 0 to the number multiplied; so that when anything below 10 is carried, you have only to annex it to the next quantity in the multiplicand.

(41.) DIVISION OF COMPOUND QUANTITIES.

RULE. Divide the greatest denomination first, put down the quotient; reduce the remainder to the next lower denomination; carry it, thus reduced, to the term of that denomination, in the dividend, and divide as before.

NOTE 1. When the divisor is a composite number, produced by factors, none of which are greater than 12, use those factors, instead of the number itself, and work by short division.

2. When the divisor is a large number which cannot be decomposed into suitable factors, you may regard the question as one of simple reduction: reduce the compound quantity to the lowest denomination in it, and then divide; the quotient will be a quantity in that lowest denomination, which, by reduction, may then be brought into the higher denomination.

£. s. d.

8)34 16 8

3) 4 7 1

1

£1

901

Ex. 1. Divide £34 16s. 8d. by 24. Here the divisor is a composite number, formed by the factors 8 and 3. Dividing first by the 8, we say, 8 in 34, 4 times and 2 over: this 2 being pounds, we reduce it to shillings, carrying the result, 40s., to the 16, and say, 8 in 56, 7 times; then 8 in 8, once. Dividing now by the 3, we say, 3 in 4, once, and 1 over; so that 20s. is carried to the 7, and we say, 3 in 27, 9 times; 3 in 1, no times, and 1 over: this 1d. being 4 farthings, we say, 3 in 4, once; so that the quotient is £1 98. 01d., the remainder being neglected, as we have no coin below the farthing.

(42.) Now I have a remark to make upon this operation, to which you must attend. I have supposed that we have

* See page 59.

been working the above example together, and have imagined ourselves as saying, “8 in 34, 4 times;" "8 in 56, 7 times;" and so on, as arithmeticians would say: but I must tell you, that we have been using incorrect language; and I am the more anxious to draw your attention to this, in order to show you how necessary it is in reading books of this kind, and, indeed, in reading any book at all, that you should think for yourself, and not receive, without thinking, everything that you may find in a printed book. When you had said, as above, "8 in 34, 4 times," and had put down the 4; suppose somebody had asked you what that 4 stood for, you would have answered, 4 pounds; and you would have been right: but how can 8 be contained in £34, £4 times? 4 pounds times is an expression which has no meaning. You see, therefore, that the form of language employed above is faulty: we ought to have said, "the eighth part of £34 is £4, and £2 over;" "the eighth part of 56s. is 7s.," and so on; or, which is the same thing, "E34 divided by 8 gives £4 for quotient, and £2, or 40s., for remainder;""568., divided by 8, gives 7s. for quotient, and no remainder;" and so on. You thus see that the only thing that requires correction, in what is done above, is the form of words used in describing the work; but, as the result obtained is always correct, as to the figures, the faulty language, if thought the more convenient, may be allowed to pass; though it is right that you should know what the objection to it is, and how it may be corrected. There is another thing too, to which your notice has been already called. You have seen that multiplication is a short way of finding the result of addition; and that it is nothing more than this: the multiplier always denotes the number of things, each equal to the multiplicand, that are to be added together, and the product gives their sum: the multiplier, therefore, can never be a commodity, as a sum of money, or a weight of goods; nor yet any measure of length or space; it is simply what is called an abstract number, denoting how many repetitions or times some other abstract number, or concrete quantity, is to be taken.*

* Among those who have advanced further into the practical application of arithmetic, there may be some who may think that this view of multiplication is in opposition to what goes by that name in books on mensuration, surveying, &c., where feet are apparently multiplied by feet, yards by yards, &c. The fact is, however, that although concrete quantities are in such subjects said to be multiplied together, the phrase

Now, just as multiplication is an abridgment of addition, so it has been said that division is an abridgment of repeated subtraction; so it is: but it is also more than this: we could not have any operation in multiplication which could not be performed by addition, though there are plenty of operations in division which could not be performed by subtraction: how, for instance, could the example worked above be done by subtraction? A sum of money, which is a concrete quantity, a real commodity, is to be divided by 24, an abstract number, not 24 things; it would be nonsense to speak of taking an abstract number from a concrete quantity,-from real substantial things: when we divide a concrete quantity by 24, we merely seek that smaller concrete quantity which is the 24th part of the greater; or that smaller quantity, which repeated 24 times, makes up the greater. Division of a concrete quantity replaces subtraction, only when the divisor is a concrete quantity of the same kind also. of money may, for instance, be divided by a smaller sum of

A sum

ology is adopted solely for brevity, and to enable writers on those topics to express the rules of operation in a form easy of recollection, and free from that prolixity of language which the strictly correct form of expression would seem to require. All that is meant is, that we are to proceed in applying the rules of mensuration, &c., as if feet could be multiplied by feet, yards by yards, &c., or as if, instead of these concrete quantities, they were merely abstract numbers. The direction for finding the surface of a rectangle is briefly expressed by saying, " multiply the length by the breadth, the product will be the area in square feet, &c." The meaning is, that we are to multiply these measures together as if they were not measures, but abstract numbers; and then to consider the product as if it were not an abstract number, but so many square feet, &c.; this is all that is to be understood by the expression "feet multiplied by feet produces square feet;" and the same of all expressions of a like kind. Strange to say, however, there are books on arithmetic,—and books, too, of very recent date,—the authors of which, teaching the subject as they themselves have learnt it, that is, merely as a sort of mechanical jugglery with the nine digits-I say, there are modern books on arithmetic, the authors of which, finding the expression "feet multiplied by feet" tolerated, proceed to induct their deluded pupils into the mystery of multiplying cwts. by tons, money by money, and so on! What meaning they attach to their results no one knows; indeed, meaning, or any accounting for their processes by an appeal to reason or common sense, is what never enters the heads of these writers; they would, no doubt, just as readily multiply a house by a house, or one man's name by another's. There is perhaps no class of educational books which has done so much injury to the youthful mind as books on arithmetic. What a benefit would it be to the young if about fivesixths of existing works on arithmetic were collected in one vast pile, and burnt in Smithfield for scientific heterodoxy!

money in this kind of division, we merely seek how many times the smaller sum is contained in the greater; the quotient must evidently, therefore, be an abstract number. It is

only when dividend and divisor are of the same kind, both concrete quantities of one sort, or both abstract numbers, that the operation of division can replace that of successive sub

traction.

(43.) I have directed your attention to these particulars, in order that you may clearly see the true character of your operations in the multiplication and division of compound quantities, and to impress upon you, that a multiplier is always an abstract number, while a divisor may be either an abstract number, or a concrete quantity of the same kind as the dividend; and, moreover, to prepare you for the rule for division in this latter case: this rule is as follows:

(44.) To divide a Compound Quantity by Another of the Same Kind.

RULE. Reduce the two quantities to the lowest denomination to be found in either, and then perform the division: the quotient will express the number of times the smaller quan- tity is contained in the greater.

Ex. Divide £18 5s. by £2 7s. 8d. Here the lowest denomination is pence; we have, therefore, to

reduce both dividend and divisor to this denomination, and then to divide the pence contained in the dividend by the pence contained in the divisor, as in the margin. We thus find that the smaller sum is contained in the greater 7 times and a part of a time, expressed by the fraction 379, which part is the 572nd

5729

22

£. s.

d.

£.

S.

2

7 8

18

5

20

20

47

365

12

12

572

) 4380(7879

4004

376 rem.

part of the number 376. If the sum to be divided were diminished by 376 pence, then the other sum would be contained in it exactly 7 times.