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 divison, 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, in 11 hours he will go 11 times as much, or of a league multiplied by 11, or 5, that is 18 leagues and 1 mile.

In how many hours will the courier of the preceding 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 180 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 18, or 1. In general, 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 19 for a quotient, which is the case; for in reducing to its most simple terms, we get 1.

18

180

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 138.

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 55 of a league, ought to contain 3 hours, the time required in going 5 leagues, as often as contains 5.

The four numbers 5, 5, 3, 11, are then in proportion; and in reality if we divide by 5, we get, a result equivalent to . It will now be easy to recognize 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.

The first term of a relation is called the antecedent, and the second the consequent. In a proportion 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.

៖

113. To ascertain that there is a proportion between the four numbers 13, 18, 130, and 180, we must see if the fractions 1 and 180 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 180 be equal, it follows that, by reducing them to the same denominator, the numerator of the one will become equal to that of the other, and that, consequently, 18 multiplied by 130 will give the same product as 180 by 13. This is actually the case, and the reasoning by which it is shown, being independent of the particular values of the numbers, proves, that, if four numbers be in proportion, the product of the first and last, or of the two extremes, is equal to the product of the second and third, or of the two means.

We see at the same time, that, if the four given numbers were not in proportion, they would not have the abovementioned property; for the fraction, which expresses the first ratio, not being equivalent to that which expresses the second, the numerator of the one will not be equal to that of the other, when they are reduced to a common denominator.

114. The first consequence, naturally drawn from what has been said, is, that the order of the terms of a proportion may be changed, provided they be so placed, that the product of the extremes shall be equal to that of the means. In the proportion 13:18:130: 180, the following arrangements may be made;

for in each one of these, the product of the extremes is formed of the same factors, and the product of the means of the same factors. The second arrangement, in which the means have chang

ed places with each other, is one of those that most frequently occur.*

115. This change shows that we may either multiply or divide the two antecedents, or the two consequents, by the same number, without destroying the proportion. For this change makes the two antecedents to constitute the first relation, and the two consequents, the second. If, for instance, 55:21::165: 65, changing the places of the means we should have,

55: 165:21: 63;

we might now divide the terms, which form the first relation, by 5, (111) which would give 11: 33:: 21: 63, changing again the places of the means, we should have 11:21 :: 33:63, a proportion which is true in itself, and which does not differ from the given proportion, except in having had its two antecedents divided by 5.

116. Since the product of the extremes is equal to that of the means, one product may be taken for the other, and, as in dividing the product of the extremes, by one extreme, we must necessarily find the other as the quotient, consequently, in dividing by one extreme the product of the means, we shall find the other extreme. For the same reason, if we divide the product of the extremes by one of the means, we shall find the other mean.

.

*It may be observed, that the proportion 13: 130:18: 180 might have been at once presented under this form, according to the solution of the question in article 109; for the value of a yard of cloth may be ascertained in two ways, namely, by dividing the price of the piece of 13 yards by 13, or by dividing the price of 18 yards by 18; it follows then that the price of the first must contain 13 as many times as the price of the second contains 18; we shall then have 13: 130 :: 18:180. We may reason in the same manner with respect to the 2nd question in the article above referred to, as well as with respect to all others of the like kind, and thence derive proportions; but the method adopted in article 109 seemed preferable, because it leads us to compare together numbers of the same denomination, whilst by the others we compare prices, which are sums of money, with yards, which are measures of length; and this cannot be done without reducing them both to abstract numbers.