the length of the days, length of the wall, and its height and breadth.

As the answer to the question is to be in days, we make the days the third term.

Were all the circumstances of the question alike, except the number of men and the number of days, the question would consist in finding in how many days 24 men would perform the same labor that 6 men had done in 16 days; that is, if 6 men had built a certain wall in 16 days, how many days would it take 24 men to perform the same labor? This would furnish the following proportion.

6x16
24

24 men: 6 men :: 16 days : = 4 days.

Or, if this were the question, If a certain number of men, by laboring 9 hours a day, perform a piece of work in 16 days, how many days would it take the same men to do the labor by working 8 hours a day? the following would be the propor

tion.

9x16
8 hours: 9 hours :: 16 days: = 18 days.

8

Or, if this were the question, If a certain number of men build a wall 20 feet long in 16 days, how long would it take the same men to build a wall 200 feet long? — the following would be the statement.

20 feet 200 feet: 16 days
:

16×200
20

=160 days.

Or, if only the days and height of the wall were considered, this would be the statement.

8X16
6

21 days.

6 feet: 8 feet :: 16 days : Lastly, were we to consider only the days and the thickness of the wall, it would furnish the following statement.

4 feet 6 feet :: 16 days:
:

6×16
4

= 24 days.

We see, by this mode of resolving the question, that 16 days. must have to the true answer the ratio compounded of the ratios

That 24 men have to 6 men;

That 8 hours have to 9 hours;

That 20 feet have to 200 feet;

That 6 feet have to 8 feet; and
That 4 feet have to 6 feet.

Stating the above in Compound Proportion, we have

20 feet

200 feet: 16 days: 90 days = Answer.

6 feet : 8 feet

4 feet : 6 feet

The continued product of all the second terms by the third term, and this divided by the continued product of the first terms, will produce the answer.

3. If 5 compositors in 16 days, 11 hours long, can compose 25 sheets of 24 pages in each sheet, and 44 lines in a page, and 40 letters in a line, in how many days 10 hours long may 9 compositors compose a volume, to be printed on the same letter, consisting of 36 sheets, 16 pages to a sheet, 50 lines to a page, and 45 letters in a line? Ans. 12 days.

12 days, Ans.

RULE. Make that number which is of the same kind as the answer required the third term; and of the remaining numbers, take any two that are of the same kind, and consider whether an answer depending upon these alone would be greater or less than the third term, and place them as directed in Simple Proportion. Then take any other two, and consider whether an answer depending only upon them would be greater or less than the third term, and arrange them accordingly; and so on,

until all are used. Multiply the continued product of the second terms by the third, and divide by the continued product of the first, and you produce the answer.

NOTE. -All the following questions are to be performed not only by the Rule, but by analysis. The pupil should also apply the cancelling rule.

4. If $100 gain $6 in one year, how much would $500 gain in four months ? Ans. $10. 5. If $100 gain $6 in one year, what must be the sum to gain $10 in 4 months? Ans. $500. 6. How long will it take $500 to gain $10, if $100 gain $6 in one year? Ans. 4 months. 7. If $500 gain $10 in 4 months, what is the rate per

cent.? Ans. 6 per cent. 8. If 8 men spend $32 in 13 weeks, what will 24 men spend in 52 weeks? Ans. $384.

9. If 12 men can build a wall 30 feet long, 6 feet high, and 3 feet thick, in 15 days, when the days are 12 hours long, in what time will 60 men build a wall 300 feet long, 8 feet high, and 6 feet thick, when they work only 8 hours a day?

Ans. 120 days. 10. If 16 horses consume 84 bushels of grain in 24 days, how many bushels will suffice 32 horses 48 days?

Ans. 336 bushels. 11. If the carriage of 5cwt. 3qr. 150 miles cost $24.58, what must be paid for the carriage of 7cwt. 2qr. 25lb. 64 miles, at the same rate? Ans. $14.08,6. 12. If 74oz. of bread be bought for 4ąd. when corn is 4s. 2d. per bushel, what weight of it may be bought for 1s. 2d. when the price per bushel is 5s. 6d. ? Ans. 161 oz.

13. If 496 men, in 5 days of 11 hours each, dig a trench of 7 degrees of hardness 465 feet long, 33 wide, 21 deep, in how many days of 9 hours long will 24 men dig a trench of 4 degrees of hardness 3374 feet long, 5g wide, and 31 deep? Ans. 132 days.

SECTION LIV.

CHAIN RULE.

THE CHAIN RULE consists in joining many proportions together, and by the relation which the several antecedents have

to their consequents the proportion between the first antecedent and the last consequent is discovered.

This rule may often be abridged by cancelling equal quantities on both sides, and abbreviating commensurables.

NOTE. The first numbers in each part of the question are called antecedents, and the following consequents.

1. If 20lb. at Boston make 23lb. at Antwerp, and 155lb. at Antwerp make 180lb. at Leghorn, how many pounds at Boston are equal to 144lb. at Leghorn?

RULE. Write the numbers alternately, that is, the antecedents at the left hand, and the consequents at the right; and, if the last number stands at the left hand, multiply the numbers of the left-hand column continually together for a dividend, and those at the right for a divisor; but, if the last number stands at the right hand, multiply the numbers of the righthand column continually together for a dividend, and those at the left for a divisor; and the quotient will be the answer.

NOTE. The demonstration for this rule is the same as for Compound Proportion.

2. If 12lb. at Boston make 10lb. at Amsterdam, and 10lb. at Amsterdam make 12lb. at Paris, how many pounds at Boston are equal to 80lb. at Paris? Ans. 80lb.

3. If 25lb. at Boston are equal to 221b. at Nuremburg, and 88lb. at Nuremburg are equal to 92lb. at Hamburg, and 461b. at Hamburg are equal to 491b. at Lyons, how many pounds at Boston are equal to 98lb. at Lyons? Ans. 100lb.

NOTE. The pupil may cancel all the following questions in a similar

manner.

4. If 24 shillings in Massachusetts are equal to 32 shillings in New York, and if 48 shillings in New York are equal to 45 shillings in Pennsylvania, and if 15 shillings in Pennsylvania are equal to 10 shillings in Canada, how many shillings in Canada are equal to 100 shillings in Massachusetts ?

Ans. 83 shillings.

5. If 17 men can do as much work as 25 women, and 5 women do as much as 7 boys, how many men would it take to do the work of 75 boys? Ans. 363 men.

6. If 10 barrels of apples will pay for 5 cords of wood, and 20 cords of wood for 4 tons of hay, how many barrels of apples will it take to purchase 50 tons of hay?

Ans. 500bbl.

7. If 100 acres in Bradford be worth 120 in Haverhill, and 50 in Haverhill be worth 65 in Methuen, how many acres in Bradford are equal to 150 in Methuen ?

Ans. 96 acres.

8. If 10lb. of cheese are equal in value to 7lb. of butter, and 11lb. of butter to 2 bushels of corn, and 1 bushels of corn to 8 bushels of rye, and 4 bushels of rye to one cord of wood, how many pounds of cheese are equal in value to 10 cords of wood? Ans. 4324lb.

SECTION LV.

PARTNERSHIP, OR COMPANY BUSINESS.

PARTNERSHIP is the association of two or more persons in business, with an agreement to share the profits and losses in proportion to the amount of the capital stock contributed by each.

EXAMPLES.

1. Three men, A, B, and C, enter into partnership for two years, with a capital of $1080. A puts in $240, B-$ 360, and