and then the question is reduced to, finding two numbers of which the sum is 2s, and their product p, or in other words, to divide a number 2s, into two such parts, that their product may be equal to a given number p.

6. A grazier bought as many sheep as cost him £60, and after reserving fifteen out of the number, he sold the remainder for £54, and gained 2s. a head on those he sold : how many did he buy? Ans. 75.

7. A merchant bought cloth for which he paid £33 15s., which he sold again at £2 8s. per piece, and gained by the bargain as much as one piece cost him how many pieces did he buy? Ans. 15.

8. What number is that, which, being divided by the product of its digits, the quotient is 3; and if 18 be added to it, the digits will be inverted? Ans. 24.

9. To find a number, such that if you subtract it from 10, and multiply the remainder by the number itself, the product shall be 21. Ans. 7 or 3.

10. Two persons, A and B, departed from different places at the same time, and travelled towards each other. On meeting, it appeared that A had travelled 18 miles more than B; and that A could have gone B's journey in 153 days, but B would have been 28 days in performing A's journey. How far did each travel?

11. There are two numbers whose difference is 15, and half their product is equal to the cube of the lesser number. What are those numbers ? Ans. 3 and 18.

12. What two numbers are those whose sum, multiplied by the greater, is equal to 77; and whose difference, multiplied by the lesser, is equal to 12?

Ans. 4 and 7, or √2 and

√2. 13. To divide 100 into two such parts, that the sum of their square roots may be 14. Ans. 64 and 36.

14. It is required to divide the number 24 into two such parts, that their product may be equal to 35 times their difference. Ans. 10 and 14.

15. The sum of two numbers is 8, and the sum of their cubes 152. What are the numbers? Ans. 3 and 5.

16. Two merchants each sold the same kind of stuff; the second sold 3 yards more of it than the first, and together they receive 35 dollars. The first said to the second, "I would have received 24 dollars for your stuff;" the other replied, "And I should have received 12 dollars for yours." How many yards did each of them sell?

17. A widow possessed 13,000 dollars, 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 dollars interest; and if she had placed the second out at the same rate as the first, she would have drawn for it 490 dollars interest. What were the two rates of interest?

Ans. 7 and 6 per cent.

CHAPTER VII.

Of Proportions and Progressions.

135. Two quantities of the same kind may be compared together in two ways:

1st. By considering how much one is greater or less than the other, which is shown by their difference; and,

2nd. 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 4 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.

QUEST.-135. In how many ways may two quantities be compared together? What does the first method consider? What the second? What is the first of these methods called? What is the second called? How then do you define the two proportions?

Of Arithmetical Proportion and Progression..

136. 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.

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.

137. When the difference between the first antecedent and consequent is the same as between any two adjacent terms of the proportion, the proportion is called an arith

progression. Hence, a progression by differences, or an arithmetical progression, is a series in which the successive terms continually increase or decrease by a constant number, which is called the common difference of the progression.

QUEST.-136. When are four numbers in arithmetical proportion? What is the first called? What is the second called? What is the third called? What is the fourth called?

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, . . . designate the terms of a progression by differences; it has been agreed to write them thus:

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

138. Let r represent the common difference of the progression

which we will consider increasing.

From the definition of the progression, it evidently follows that

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

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

1=a+(n-1)r.

QUEST.-137. What is an arithmetical progression? What is the number called by which the terms are increased or diminished? What is an increasing progression? What is a decreasing progression ? Which term is only an antecedent? Which only a consequent?