this way through all the denominations to the last, the remainder of which, if there be one, must have its quotient represented in the form of a fraction by placing the divisor under it. The sum of the several quotients, thus obtained, will be the whole quotient required. When the divisor is large and can be resolved into two or more simple factors, we may divide first by one of these factors, and then that quotient by another, and so on, and the last quotient will be the same as that which would have been obtained by using a whole divisor in a single operation. Taking the result of the example in the corresponding case of multiplication, we proceed thus,

By dividing £83 18s. 6d. by 2, we obtain one half of this sum, which, being divided by 9, must give one 9th of one half, or one 18th of the whole. The first operation may be considered as separating the dividend into two equal parts, and the second as distributing each of these into nine equal parts, the number of parts therefore will be 18, and being equal, one of them must be one 18th of the whole.

But when the divisor cannot be thus resolved, the operation must be performed by dividing by the whole at once. If the

quotient, which we are seeking, were known, by adding it to, or subtracting from it, the dividend a certain number of times increasing or diminishing the divisor at the same time by as many units, we might change the question into one, whose divisor would admit of being resolved into factors, which would give the same quotient; we should thus preserve the anology which exists between the multiplication and division of compound numbers. But this cannot be done, as it supposes that to be known, which is the object of the operation.

Multiplication and division, where compound numbers are concerned, mutually prove each other, as in the case of simple numbers. This may be seen by comparing the examples, which are given at length to illustrate these rules.

Ans. 9mls. 4fur. 39pls. 2ft. 8in.

Divide 375mls. 2fur. 7pls. 2yds. 1ft. 2in. by 39.

If 9 yards of cloth cost £4 3s. 74d. what is it per yard? Ans. 9s. 3d. 2q.

If

a hogshead of wine cost £33 12s. what is it per gallon? Ans. 10s. 8d.

If a dozen silver spoons weigh 3lb. 2oz. 13pwt. 12grs. what is the weight of each spoon? Ans. 3oz. 4pwt. 11grs.

If a person's income be £150 a year, what is it per day?

A capital of £223 16s. 8d.

what is the value of a share?

Ans. 8s. 2 d. nearly.

being divided into 96 shares, Ans. £2 6s. 71⁄2d.

Proportion.

108. We have shown in the preceding part of this work, the different methods necessary for performing on all numbers, whether

whole or fractional, or consisting of different denominations, the four fundamental operations of arithmetic, namely, addition, subtraction, multiplication, and division; and all questions relative to numbers ought to be regarded as solved, when, by an attentive examination of the manner in which they are stated, they can be reduced to some one of these operations. Consequently, we might here terminate all that is to be said on arithmetic, for what remains belongs, properly speaking, to the province of algebra. We shall, nevertheless, for the sake of exercising the learner, now resolve some questions which will prepare him for algebraic analysis, and make him acquainted with ́ a very important theory, that of ratios and proportions, which is ordinarily comprehended in arithmetic.

109. A piece of cloth 13 yards long was sold for 130 dollars; what will be the price of a piece of the same cloth 18 yards long?

It is plain, that if we knew the price of one yard of the cloth that was sold, we might repeat this price 18 times, and the result would be the price of the piece 18 yards long. Now since 13 yards cost 130 dollars, one yard must have cost the thirteenth part of 130 dollars, or 130; performing the division, we find for the result 10 dollars, and multiplying this number by 18 we have 180 dollars for the answer; which is the true cost of the piece 18 yards long.

A courier, who travels always at the same rate, having gone 5 leagues in 3 hours, how many will he go in 11 hours?'

Reasoning as in the last example, we see, that the courier goes in one hour of 5 leagues, or, and consequently, ir 11 hours he will go 11 times as much, or of a league multiplied by 11, or 55, that is 18 leagues and 1 mile.

In how many hours will the courier of the precedeng question go 22 leagues?

We see, that if we knew the time he would occupy in going one league, we should have only to repeat this number 22 times, and the result would be the number of hours required. Now the courier, requiring 3 hours to go 5 leagues, will require only

of the time, of an hour, to go one league; this number, multiplied by 22, gives or 13 hours and, that is, 13 hours and 12 minutes.

110. We have discovered the unknown quantities by an analysis of each of the preceding statements, but the known numbers and those required depend upon each other in a manner, that it would be well to examine.

To do this, let us resume the first question, in which it was required to find the price of 18 yards of cloth, of which 13 cost 130 dollars.

It is plain, that the price of this piece would be double, if the number of yards it contained were double that of the first; that if the number of yards were triple, the price would be triple also, and so on; also that for the half or two thirds of the piece we should have to pay only one half or two thirds of the whole price.

According to what is here said, which all those, who understand the meaning of the terms, will readily admit, we see that if there be two pieces of the same cloth, the price of the second ought to contain that of the first, as many times as the length of the second contains the length of the first; and this circumstance is stated in saying, that the prices are in proportion to the lengths or have the same relation to each other as the lengths.

This example will serve to establish the meaning of several terms which frequently occur.

111. The relation of the lengths is the number, whether whole or fractional, which denotes how many times one of the lengths contains the other. If the first piece had 4 yards and the second 8, the relation, or ratio, of the former to the latter would be 2, because 8 contains 4 twice. In the above example, the first piece had 13 yards and the second 18; the ratio of the former to the latter is then, or 15. In genaral, the relation or ratio of two numbers is the quotient arising from dividing one by the other.

As the prices have the same relation to each other, that the lengths have, 180 divided by 130 must give for a quotient, which is the case; for in reducing to its most simple terms, we get 1.

8

8 0

The four numbers, 13, 18, 130, 180, written in this order, are then such that the second contains the first as many times as the fourth contains the third, and thus they form what is called a proportion.

We see also, that a proportion is the combination of two equal

ratios.

We may observe, in this connexion, that a relation is not changed by multiplying each of its terms by the same number; and this is plain, because a relation, being nothing but the quotient of a division, may always be expressed in a fractional form. Thus the relation is the same as 1.

The same considerations apply also to the second example. The courier, who went 5 leagues in 3 hours, would go twice as far in double that time, three times as far in triple that time; thus 11 hours, the time spent by the courier in going 18 leagues and, or of a league, ought to contain 3 hours, the time required in going 5 leagues, as often as 55 contains 5.

The four numbers 5, 55, 3, 11, are then in proportion; and in reality if we divide 55 by 5, we get 55, a result equivalent to . It will now be easy to recognise all the cases, where there may be a proportion between the four numbers.

112. To denote that there is a proportion between the numbers 13, 18, 130, and 180, they are written thus,

13 18:130: 180,

which is read 13 is to 18 as 130 is to 180; that is, 13 is the same part of 18 that 130 is of 180, or that 13 is contained in 18 as many times as 130 is in 180, or lastly that the relation of 18 to 13 is the same as that of 180 to 130.

e first term of a relation is called the antecedent, and the

second the consequent. In a proporion there are two antecedents and two consequents, viz. the antecedent of the first relation and that of the second; the consequent of the first relation and that of the second. In the proportion 13: 18: 130: 180, the antecedents are 13, 130; the consequents 18 and 180.

We shall in future take the consequent for the numerator, and the antecedent for the denominator of the fraction which expresses the relation.

18

113. To ascertain that there is a proportion between the four numbers 13, 18, 130, and 180, we must see if the fractions and be equal, and to do this, we reduce the second to its most simple terms; but this verification may also be made by considering, that if, as is supposed by the nature of proportion, the two fractions and be equal, it follows that, by reducing them to the same denominator, the numerator of the one will be