connect by a line each price that is less than the mean rate with one or more that is greater, and each price greater than the mean rate with one or more that is less.

12. Place the difference between the mean rate and that of each of the simples opposite the price, with which they are connected.

13. Then, if only one difference stands against any price, it expresses the quantity of that price; but if there be several, their sum will express the quantity.

14. A merchant has several sorts of tea, some at 10s., some at 11s., some at 13s. and some at 24s. per lb.; what proportions of each must be taken to make a composition worth 12s. per

lb.?

15. How much wine, at 5s. per gallon and 3s. per gallon, must be mixed together, that the compound may be worth 4s. per gallon? A. 1 gallon of each.

16. How much corn, at 42 cents, 60 cents, 67 cents, and 78 cents, per bushel, must be mixed together, that the compound may be worth 64 cents per bushel? A. 14bu. at 42c.; 3 at 60; 4 at 67; 22 at 78.

17. A grocer would mix different quantities of sugar, viz.20, one at 23, and one at 26 cents per lb.; what quantity of each sort must be taken to make a mixture worth 22 cents per lb.?

A. 5lb. at 20 cents; 2 at 23; 2 at 26. 18. A jeweller wishes to procure gold of 20 carats fine from gold of 16, 19, 21, and 24 carats fine; what quantity of each must he take? A. 4, 1, 1, 4. 19. We have seen that we can take 3 times, 4 times, 3, 4, or any proportion of each quantity, to form a mixture.

1 1

20. Hence, when the quantity of one simple is given, to find the proportional quantities of any compound whatever, after having found the proportional quantities by the last rule, we have the following

RULE.

21. As the proportional quantity of that piece whose quantity is given is to each proportional quantity, so is the given quantity to the quantities or proportions of the compound required.

22. A grocer wishes to mix one gallon of brandy, worth 15s. per gallon, with rum worth 8s., so that the mixture may be worth 10s per gallon; how much rum must be taken?

23. By the last rule, the differences are 5 to 2; that is, the proportions are 2 of brandy to 5 of rum; hence, he must take 24 gallons of rum for every gallon of brandy. A. 2 gallons.

24. A person wishes to mix 10 bushels of wheat, at 70 cents per bushel, with rye at 48 cents, corn at 36 cents, and barley at 30 cents per bushel, so that a bushel of this mixture may be worth 38 cents:

what quantity of each must be taken? We find by the last rule, that the proportions are 8, 2. 10, and 32.

、

Then, as 8: 2::10: 2 ! bushels of rye.

2

8:10:10: 12 bushels of corn.

8:32:10:40

10:40 bushels of barley.

Answer.

25. How much water must be mixed with 100 gallons of rum, worth 90cts. per gallon, to reduce it to 75cts. per gallon. A. 20gal. 26. A grocer mixes teas at $1.20, $1, and 60 cents, with 20lb. at 40c. per lb.; how much of each sort must he take to make the composition worth 80c. per lb. A. 20 at $1.20, 10 at $1, 10 at 60c.

27. A grocer has currants at 4 cents, 6 cents, 9 cents, and 11 cents per lb.; and he wishes to make a mixture of 240lb., worth 8 cents per [b.; how many currants of each kind must he take? In this example we can find the proportional quantities by linking, as before; then it is plain that their sum will be in the same proportion to any part of their sum, as the whole compound is to any part of the compound, which exactly accords with the principle of Fellowship.

RULE.

28. As the sum of the proportional quantities found by linking, as before: is to each proportional quantity : : so is the whole quantity or compound required: to the required quantity of each.

We will now apply this rule in performing the last question.

wishes

29. A grocer, having sugars at 8c., 12c., and 16c. per lb., to make a composition of 120lb., worth 13c. per lb.; what quantity of each must be taken? A. 30lb. at 8, 30lb. at 12, 60lb. at 16. 30. How much water, at 0 per gal., must be mixed with wine, at 80c. per gal., so as to fill a vessel of 90gal., which may be offered at 50c per gal.? A. 56 gallons of wine, and 33 gallons of water. 31. How much gold, of 15, 17, 18, and 22 carats fine, must be mixed together, to form a composition of 40 ounces of 20 carats fine? A. 5oz. of 15, of 17, of 18, and 25oz. of 22.

ठ

ARITHMETICAL PROGRESSION. CIV. 1. ARITHMETICAL PROGRESSION, OR SERIES, is any rank of numbers more than two, that increase by a constant addition, or decrease by a constant subtraction, of the same number.

2. THE COMMON DIFFERENCE is the number added or subtracted as above.

3. AN ASCENDING SERIES is one formed by a continual addition of the common difference, as 2, 4, 6, 8, 10, &c.

CIV. Q. What is Arithmetical Progression? 1. What the Common Differ ence? 2. An Ascending Series? 3. A Descending Series? 4. Give an ex ample of each 3, 4. What are the terms? 5

4. A DESCENDING ARITHMETICAL SERIES, is one formed by a continual subtraction of the common difference, as 10, 8, 6, 4, 2, &c. 5. THE TERMS are those numbers that form the series, the first and last of which are called the EXTREMES, and the other the MEANS. 6. In Arithmetical Progression there are reckoned five terms, any three of which being given, the remaining two may be found, viz.7. 1. The first term; 2. The last term; 3. The number of terms; 4. The common difference; 5. The sum of all the terms.

8. The first term, the last term, and the number of terms, being given, to find the Common Difference ;

9. A man had 6 sons, whose several ages differed alike the youngest was 3 years old, and the oldest 28; what was the common difference of their ages?

10. The difference between the youngest son and the eldest, evidently shows the increase of the 3 years by all the subsequent additions, till we come to 28 years; and, as the number of these additions are, of course, 1 less than the number of sons (5), it follows, that, if we divide the whole difference (28-3=), 25, by the number of additions (5), we shall have the difference between the ages of each, that is, the common difference. Thus, 28-3-25; then, 25÷5=5 years, the common difference. A. 5 years.

11. Hence, to find the common difference, -Divide the difference of the extremes by the number of terms, less 1, and the quotient will ve the common difference.

12. If the extremes be 3 and 23, and the number of terms 11, what is the common difference?

A. 2.

13. A man is to travel from Boston to a certain place in 6 days, and to go only 5 miles the first day, increasing the distance traveled each day by an equal excess, so that the last day's journey may be 45 miles; what is the daily increase, that is, the common difference? A. 8 miles.

14. If the amount of $1 for 20 years, at simple interest, be $2.20, what is the rate per cent.? In this example, we see the amount of the first year is $1.06 and the last year $2.20, consequently, the extremes are 106 and 220, and the number of terms 20.

A. $.06 6 per cent.

15. A man bought 60 yards of cloth, giving 5 cents for the first yard, 7 for the second, 9 for the third, and so on to the last; what did the last cost? Since, in the last example, we have the common difference given, it will be easy to find the price of the last yard: for, as there are as many additions as there are yards, less 1, that is, 59 additions of 2 cents to be made to the first yard, it follows, that the last yard will cost 2 × 59=118 cents more than the first, and the whole cost of the last, reckoning the cost of the first yard, will be 118 +5=$1.23. A. $1.23.

16. Hence, when the common difference, the first term, and the Q. What is the rule for finding the common difference? 11. For finding the ast term? 16

number of terms, are given, to find the last term.-Multiply the common difference by the number of terms, less 1, and add the first term

to the product.

17. If the first term be 3, the common difference 2, and the number of terms 11, what is the last term? A. 23.

18. A man went from Boston to a certain place in 6 days, traveling the first day 5 miles, the second 8 miles, and each successive day 3 miles farther than the former; how far did he go the last day?

A. 20 miles. 19. What will $1, at 6 per cent., amount to, in 20 years, at simple interest? The common difference is the 6 per cent. ; for the amount of $1, for 1 year, is $1.06, and 1.06+$.06=$1.12, the second year, and so on. A. $2.20.

20. A man bought 10 yards of cloth, in arithmetical progression; for the first yard he gave 6 cents, and for the last yard he gave 24 cents; what was the amount of the whole? In this example, it is plain that half the cost of the first and last yards will be the average price of the whole; thus, 6 cts. +24 cts. =30÷2-15 cts., average price; then, 10 yds. × 15= $1.50, whole cost. A. $1.50.

21. Hence, when the extremes, and the number of terms, are given, to find the sum of all the terms.-Multiply half the sum of the extremes by the number of terms, and the product will be the answer 22. If the extremes be 3 and 273, and the number of terms 40, what is the sum of all the terms?

A. 5520. 23. How many times does a clock strike in 12 hours? A. 78.

A. $5050.

24. A butcher bought 100 oxen, and gave for the first ox $1, for the second $2, for the third $3, and so on to the last; how much did they come to at that rate? 25. What is the sum of the first 1000 numbers, beginning with their natural order, 1, 2, 3, &c. ? A. 500500. 26. If a board, 18 feet long, be 2 feet wide at one end, and come to a point at the other, what are the square contents of the board?

A. 18 feet.

27. If a piece of land, 60 rods in length, be 20 rods wide at one end, and at the other terminate in an angle or point. what number of square rods does it contain?

A. 600.

28. A number of flat stones were laid, 2 yards distant, for the space of 1 mile, from each other, and the first, 2 yards from a certain basket. How far will that man travel who gathers them up singly, and returns with them one by one to the basket? A. 881 miles.

29. A person traveling into the country, went 3 miles the first day, and increased every day's travel 5 miles, till at last he went 58 miles in one day; how many days did he travel?

30. We found, in the example 1, the difference of the extreme divided by the number of terms, less 1, gave the common difference; consequently, if, in this example, we divided (58-3=) 55, the difference of the extremes, by the common difference, 5, the quotient 11, Q. The sum of all the terms? 21.

will be the number of terms, less 1; then, 1+11=12, the number of

terms.

A. 12.

31. Hence, when the extremes and common difference are given, to find the number of terms :-Divide the difference of the extremes by the common difference, and the quotient, increased by 1, will be the

answer.

32. If the extremes be 3 and 45, and the common difference 6, what is the number of terms?

A. 8. 33. A man being asked how many children he had, replied, that the youngest was 4 years old, and the eldest 32, the increase of the family having been 1 in every 4 years; how many had he? A. 8

GEOMETRICAL PROGRESSION.

CV. 1. GEOMETRICAL PROGRESSION, is any rank or series of numbers, which increases by a constant multiplier, or decreases by a constant divisor.

2. Thus, 3, 9, 27, 81, &c., is an increasing geometrical series; and 81, 27, 9, 3, &c., is a decreasing geometrical series.

3. There are five terms in Geometrical Progression, and, like Arithmetical Progression, any three of them being given, the other two may be found, viz :

4. 1. The first term. 2. The last term. 3. The number of terms 4 The sum of all the terms. 5. The ratio.

5. A man purchased a flock of sheep, consisting of 9; and by agreement, was to pay what the last sheep came to, at the rate of $4 for the first sheep, $12 for the second, $36 for the third, and so on, trebling the price to the last; what did the flock cost him?

6. We may perform this example by multiplication; thus, 4×3×3 ×3×3×3×3×3×3 $26,244. A. But this process, you must be sensible, would be, in many cases, a very tedious one; let us see if we cannot abridge it and make it easier.

7. In the above process, we discover that 4 is multiplied by 3 eight times, one time less than the number of terms; consequently, the 8th power of the ratio 3, expressed thus, 38, multiplied by the first term, 4, will produce the last term. But, instead of raising 3 to the 8th power in this manner, we need only raise it to the 4th power, then multiply this 4th power into itself; for, in this way, we do, in fact, use the 3 eight times, raising the 3 to the same power as before; thus, 3481; then 81×81=6561; this, multiplied by 4, the first term, gives $26,244, the same result as before. A. $26,244.

8. Hence, when the first term, ratio, and number of terms, are given, to find the last term.

RULE.

9. Write down some of the leading powers of the ratio, with the

Q. Number of terms? 31.

CV. Q. What is Geometrical Progression? 1. What are the terms? 4 Give exus ví an scending and a descending series. 2.