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

*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

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: 63, 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.

We can then find any one term of a proportion, when we know the other three, for the term sought must be either one of the extremes or one of the means.

The question of article (109) may be resolved by one of these rules. Thus, when we have perceived that the prices of the two pieces are in the proportion of the number of yards contained in each, we write the proportion in this manner,

13:18:: 130: x,

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..

putting the letter x instead of the required price of 18 yards; and we find the price, which is one of the extremes, by multiplying together the two means, 18 and 130, which makes 2340, and dividing this product by the known extreme, 13, we obtain, for the result, 180.

The operation, by which, when any three terms of a proportion are given, we find the fourth, is called the Rule of Three. Writers on arithmetic have distinguished it into several kinds, but this is unnecessary, when the nature of proportion and the enunciation of the question are well understood; as a few examples will sufficiently show.

117. A person having travelled 217,5 miles in 9 days; it is asked, how long he will be in travelling 423,9 miles, he being supposed to travel at the same rate.

In this question the unknown quantity is the number of days, which ought to contain the 9 days spent in going 217,5 miles, as many times as 423,9 contains 217,5; we thus get the following proportion;

days

217,5: 423,99 x, and we find for x, 17,54 nearly.

118. All the difficulty in these questions consists in the manner of stating the proportion. The following rules will be sufficient to guide the learner in all cases.

Among the four numbers which constitute a proportion, there are two of the same kind, and two others also of the same kind, but different from the first two. In the preceding example, two of the terms are miles, and the other two, days.

First, then, it is necessary to distinguish the two terms of each kind, and when this is done, we shall necessarily have the quotient of the greatest term of the second kind by the smallest of the same kind, equal to the quotient of the greatest term of the first kind by the smallest of the same kind, which will give us this proportion;

the smaller term of the first kind

is

to the larger of the same kind

as

the smaller term of the second kind

is

to the larger of this kind.

In the preceding example this rule immediately gives

217,5: 423,9 :: 9:x,

for the unknown term ought to be greater than 9, since a greater number of days will be necessary to complete a longer journey.

119. If it were required to find how many days it would take 27 men to perform a piece of work, which 15 men, working at the same rate, would do in 18 days; we see that the days should be less in proportion as the number of men is greater, and reciprocally. There is still a proportion in this case, but the order of the terms is inverted; for, if the number of workmen in the second set were triple of that in the first, they would require only one third of the time. The first number of days then would contain the second as many times, as the second number of workmen would contain the first. This order of the terms being the reverse of that assigned to them by the enunciation of the question, we say, that the number of workmen is in the inverse ratio of the number of days. If we compare the two first and the two last, in the order in which they present themselves, the ratio of the former will be 3, or 3, and that of the latter, which is the same as the preceding with the terms inverted.

It is evident, indeed, that we invert a ratio by inverting the terms of the fraction, which expresses it, since we make the antecedent take the place of the consequent, and the consequent that of the antecedent. or 2: 3 is the inverse of or 3 : 2.

The mode of proceeding in such cases may be rendered very simple; for we have only to take the numbers denoting the two sets of workmen, for the quantities of the first kind, and the numbers denoting the days, for those of the second, and to place the one and the other in the order of their magnitude; proceding thus we have the following proportion,

15: 27 :: x: 18,

from which we immediately find x equal to 10.

Recapitulating the remarks already given, we have the following rule; (make the number which is of the same kind with the answer the third term, and the two remaining ones the first and second, putting the greater or the less first, according as the third is greater or less than the term sought; then the fourth term will be found by multiplying together the second and third, and dividing the product by the first

120. 1st. A man placed 3575 dollars at interest at the rate of 5 per cent yearly; it is asked what will be the interest of this sum at the end of one year?

The expression, 5 per cent. interest, means, that for a sum of one hundred dollars, 5 dollars is allowed at the end of a year; if then we take the two principals for the quantities of the first kind, and the interest of those for the second, we shall have,

100: 3575:5: x,

a proportion which may be reduced to 20 : 3575 :: 1: x, according to the observation in article 115; then dividing the two terms of the first relation by 5, we shall have 4 : 715 :: 1 : x, whence x is equal to 715, or $178,75 cts.

We may also resolve this question by considering that 5 is of 100, and that consequently we shall obtain the interest of any sum put out at this rate by taking the twentieth part of this sum. Now of $3575 is $178,75; a result which agrees with the one

before found.

2d. A merchant gives his note for $800,00 payable in a year; the note is sold to a broker, who advances the money for it 8 months before the time of payment; how much ought the broker to give?

As the broker advances, from his own funds, a sum which is not to be replaced till the expiration of 8 months, it is proper that he should be allowed interest for his money during this time.

Let the interest for a year be 6 per cent. the interest for 8 months will be, or, of 6, or 4; a sum then of 100 dollars, lent for 8 months, must be entitled to 4 dollars interest; that is,