they receive $35. The first said to the second, "I would have

received $24 for your stuff."

The other replied, "And I should have received $12 for yours." How many yards did each of them sell?

Ans. 1st merchant, 15 or 5; 2d merchant, 18 or 8.

17. A widow possessed $13,000, which she divided into two parts, and placed them at interest in such a manner that the incomes from them were equal. If she had put out the first portion at the same rate as the second, she would have drawn for this part $360 interest; and if she had placed the second out at the same rate as the first, she would have drawn for it $490 interest. What were the two rates of interest?

18. Find three numbers such that the difference between the third and second shall exceed the difference between the second and first by 6, that the sum of the numbers shall be 33, and the sum of their squares 467. Ans. 5, 9, and 19.

19. What number is that which being divided by the product of its two digits the quotient will be 3, and if 18 be added to it the resulting number will be expressed by the digits inverted?

Ans. 24.

20. What two numbers are those which are to each other as m to n, and the sum of whose squares is b?

21. What two numbers are those which are to each other as m to n, and the difference of whose squares is b?

22. Required to find three numbers such that the product of the first and second shall be equal to 2, the product of the

first and third equal to 4, and the sum of the squares of the second and third equal to 20. Ans. 1, 2, and 4.

23. It is required to find three numbers whose sum shall be 38, the sum of their squares 634, and the difference between the second and first greater by 7 than the difference between the third and second. Ans. 3, 15, and 20.

24. Required to find three numbers such that the product of the first and second shall be equal to a, the product of the first and third equal to b, and the sum of the squares of the second and third equal to c.

25. What two numbers are those whose sum multiplied by the greater gives 144, and whose difference multiplied by the less gives 14?

Ans. 9 and 7.

CHAPTER IX.

PROPORTIONS AND PROGRESSIONS.

176. Two quantities of the same kind may be compared, the one with the other, in two ways, first, by considering how much one is greater or less than the other, which is shown by their difference; and, second, by considering how many times one is greater or less than the other, which is shown by their quotient.

Thus, in comparing the numbers 3 and 12 together with respect to their difference, we find that 12 exceeds 3 by 9; and in comparing them together with respect to their quotient, we find that 12 contains 3 four times, or that 12 is four times as great as 3.

The first of these methods of comparison is called arithmetical proportion; and the second, geometrical proportion. Hence

Arithmetical proportion considers the relation of quantities with respect to their difference; and geometrical proportion, the relation of quantities with respect to their quotient.

ARITHMETICAL PROPORTION AND PROGRESSION.

177. If we have four numbers, 2, 4, 8, and 10, of which the difference between the first and second is equal to the difference between the third and fourth, these numbers are said to be in arithmetical proportion. The first term, 2, is called an antecedent; and the second term, 4, with which it is compared, a consequent. The number 8 is also called an antecedent; and the number 10, with which it is compared, a consequent.

258

When the difference between the first and second is equal to the difference between the third and fourth, the four numbers are said to be in proportion. Thus, the numbers

2, 4, 8, 10,

are in arithmetical proportion.

178. When the difference between the first antecedent and consequent is the same as between any two consecutive terms of the proportion, the proportion is called an arithmetical progression. Hence a progression by differences, or an arithmetical progression, is a series in which the successive terms are continually increased or decreased by a constant number, which is called the common difference of the progression.

A series is a succession of terms, each of which is derived from one or more of the preceding ones by a fixed law, called the law of the series.

In the two series,

1, 4, 7, 10, 13, 16, 19, 22, 25,

60, 56, 52, 48, 44, 40, 36, 32, 28,

the first is called an increasing progression, of which the common difference is 3; and the second, a decreasing progression, of which the common difference is 4.

In general, let a, b, c, d, e, f,

denote the terms of a progression by differences. It has been agreed to write them thus:

a.b.c.d.e.f.g. h.i.k.....

d

This series is read, a is to b, as b is to c, as c is to d, as is to e," etc. This is a series of continued equi-differences, in which each term is at the same time an antecedent and a consequent, with the exception of the first term, which is only an antecedent, and the last, which is only a consequent.

179. Let d denote the common difference of the progression,

a.b.c.e.f.g. h, etc.,

which we will consider increasing.

From the definition of the progression, it evidently follows that

b=a+d, c=b+d=a+2d, e=c+d=a+3d;

and, in general, any term of the series is equal to the first term, plus as many times the common difference as there are preceding terms.

Thus, let be any term, and n the number which marks the place of it. The expression for this general term is

Hence, for finding the last term, we have the following rule:

Multiply the common difference by the number of terms

less one.

To the product add the first term. The sum will be the

serves to find any term whatever, without determining all those which precede it.

Exercises.

1. If we make n 1, we have la; that is, the series will have but one term.

2. If we make n = 2, we have la+d; that is, the series will have two terms, and the second term is equal to the first, plus the common difference.