We find, as by Art. 266, the value of one pound at the given rate of exchange. The given sum, $4866.663, we divide by the value of a pound, and obtain 1000£. as the required amount of the bill.

RULE. — Divide the given sum by the value of one pound at the given rate of exchange, and the quotient will be the amount in pounds and decimals of a pound.

EXAMPLES FOR PRACTICE.

2. J. Reed, of Cincinnati, proposes to make a remittance to Liverpool of $1640, exchange being at 8 per cent. premium; what will be the amount of the bill he can remit for that sum? Ans. 340£. 1s. 10d. 3. A merchant wishes to remit $500 to England, exchange being at 10 per cent. premium; what will be the amount of the bill he can purchase for that sum ? Ans. 102£. 5s. 5d.+.

EXCHANGE ON FRANCE.

ART. 268. In France accounts are kept in francs and centimes. The centimes are hundredths of a franc. All bills of exchange on France are drawn in francs, and are bought, sold, and quoted, as at a certain number of francs to the dollar.

ART. 269. To find the value in United States currency of a bill on France,

Divide the amount of the bill by the value of one dollar in francs, and the quotient will be the value in dollars.

EXAMPLES FOR PRACTICE.

Ex. 1. What must be paid, in United States currency, for a bill on Paris of 2380 francs, exchange being 5.15 francs per dollar? Ans. $462.13+. 2. How many dollars will purchase a bill on Havre of 30000 francs, exchange being 5.171 francs per dollar?

Ans. $5797.10+. 3. What is the value of a bill on Paris of 62500 francs, exchange being 5.12 francs per dollar? Ans. $12207.03+.

ART. 270. To find the amount of a bill on France, which can be purchased for a given sum of United States currency,

QUESTIONS. Art. 268. How are accounts kept in France? How are all bills of exchange on France drawn? Art. 269. How do you find the value in United States currency of a bill on France? Art. 270. How do you find the amount of a bill on France, which can be purchased for a given sum of United States money?

Multiply the given sum by the value of one dollar in francs and the product will be the amount of the bill in francs.

Ex. 1. Alfred Walker, of New York, pays $2500 for a bill on Paris, exchange being 5.12 francs per dollar. What was the amount of the bill in francs? Ans. 12800. 2. When exchange on France is at 5.13 francs per dollar, a bill of how many francs should $700 purchase? Ans. 3591.

3. Morton and Blanchard, of Boston, wish to remit $675 to Paris, exchange being 5.16 francs per dollar; what will be the amount of the bill of exchange they can purchase with the money? Ans. 3483 francs

ART. 271. DUODECIMALS are a kind of compound numbers in which the unit, or foot, is divided into 12 equal parts, and each of these parts into 12 other equal parts, and so on indefinitely; thus, 1, 114, &c.

144'

Duodecimals decrease from left to right in a twelve-fold ratio ; and the different orders, or denominations, are distinguished from each other by accents, called indices, placed at the right of the numerators. Hence the denominators are not expressed. Thus,

12

1 inch or prime, equal to of a foot, is written 1 in. or 1'. 1 second

1

1 third

1 fourth

144

1".

Hence the following

12 fourths

12 thirds

ADDITION AND SUBTRACTION OF DUODECIMALS. ART. 272. Duodecimals are added and subtracted in the same manner as compound numbers.

QUESTIONS.

Art. 271. What are duodecimals? In what ratio do duodecimals decrease from left to right? How are the different denominations distinguished from each other? Art. 272. How are duodecimals added and subtracted?

EXAMPLES FOR PRACTICE.

1. Add together 12ft. 6' 9", 14ft. 7' 8", 165ft. 11′ 10′′.

Ans. 193ft. 2′ 3′′.

2. Add together 182ft. 11′ 2′′ 4"", 127ft. 7' 8" 11", 291ft. 5′ 11′′ 10"". Ans. 602ft. 0′ 11′′ 1′′. 3. From 204ft. 7′ 9′′ take 114ft. 10' 6''.

Ans. 89ft. 9′ 3′′.

4. From 397ft. 9′ 6′′ 11′′ 7'''' take 201ft. 11′ 7′′ 8′′" 10′′". Ans. 195ft. 9′ 11′′ 2′′′ 9′′".

MULTIPLICATION AND DIVISION OF DUODECIMALS.

ART. 273. To find the denomination of the product of any two numbers in duodecimals, when multiplied together.

Ex. 1. What is the product of 9ft. multiplied by 3ft.?

Ans. 27ft.

2. What is the product of 7ft. multiplied by 6'? Ans. 3ft. 6.

3. What is the product of 5' multiplied by 4'? Ans. 1′ 8′′.

4. What is the product of 9' multiplied by 11""?

Ans. 8'" 3""".

99" ;

18; then 12 × 1728
11
X
99"""128"" 3""".

99 20736

It will be observed in the examples above, that feet multiplied by feet produce feet; feet multiplied by primes produce primes; primes multiplied by primes produce seconds, &c.; and that the several products are of the same denomination as denoted by the sum of the indices of the numbers multiplied together. Hence,

When two numbers are multiplied together, the sum of their indices annexed to their product denotes its denomination.

ART. 274. To multiply duodecimals together.

Ex. 1. Multiply 8ft. 6in. by 3ft. 7in.

QUESTION.

Art. 273.

Ans. 30ft. 5′ 6′′.

Art. 273. How is the denomination of the product denoted when duodecimals are multiplied together?

OPERATION.

8ft.

6'

3ft. 7'

4ft. 11′ 6′′
6'

We first multiply each of the terms. in the multiplicand by the 7' in the multiplier, thus, 7 into 6′ = 67 42′′ 3′ and 6". Placing the 6" under its multiplier, we add the 3′ to the product or 7' into 8ft. 59′ 4ft. and 11', which we write down. We then multiply by the 3ft., thus: 3ft into 6' 18'1ft. and 6. We write the 6 under its multiplier, and add the 1ft. to the product of the 3ft. into 8ft., making 25ft., which we write down. The two products being added together, we obtain 30ft. 5′ 6′′ for the answer.

2 5ft.

3 0ft.

5' 6"

RULE. Write the multiplier under the multiplicand, so that the same denominations shall stand in the same column.

Beginning at the right hand, multiply each term in the multiplicana by each term of the multiplier, and write the first term of each partial product directly under its multiplier, observing to carry a unit for every twelve from each lower denomination to the next higher.

The sum of the several partial products will be the product required. EXAMPLES FOR PRACTICE.

2. Multiply 8ft. 3in. by 7ft. 9in. 3. Multiply 12ft. 9' by 9ft. 11'.

Ans. 63ft. 11′ 3′′.

Ans. 126ft. 5′ 3′′.

4. My garden is 18 rods long and 10 rods wide; a ditch is dug round it 2 feet wide and 3 feet deep; but the ditch not being of a sufficient breadth and depth, I have caused it to be dug 1 foot deeper, and, outside, 1 ft. 6 in. wider. How many solid feet will it be necessary to remove? Ans. 7540.

5. I have a room 12 feet long, 11 feet wide, and 74 feet high. In it are two doors, 6 feet 6 inches high, and 30 inches wide, and the mop-boards are 8 inches high. There are 3 windows, 3 feet 6 inches wide, and 5 feet 6 inches high; how many square yards of paper will it require to cover the walls?

Ans. 252 square yards.

ART. 275. To divide one duodecimal by another. Ex. 1. A certain aisle contains 68ft. 10′ 8′′ of floor. The width of the floor being 2ft. 8', what is its length? Ans. 25ft. 10′.

OPERATION.

We first divide the 68ft. by

2ft. 8′) 6 8ft. 10′ 8′′ (2 5ft. 10′ the divisor, and obtain 25ft.

for the quotient. We multiply the entire divisor by the 25ft., and subtract the prod-. uct, 66ft. 8′, from the corresponding portion of the

QUESTION. — Art. 274. What is the rule for the multiplication of duodeci mals?

dividend, and obtain 2ft. 2′, to which remainder we bring down the 8′′, and dividing, we obtain 10' for the quotient. Multiplying the entire divisor by the 10', we obtain 2ft. 2′ 8′′, which subtract, as before, leaves no remainder. Therefore, 25ft. 10′ is the length of the aisle.

RULE. Find how many times the highest term of the dividend will contain the divisor. By this quotient multiply the entire divisor, and subtract the product from the corresponding terms of the dividend. To the remainder annex the next denomination of the dividend, and divide as before, and so continue till the division is complete.

EXAMPLES FOR PRACTICE.

2. What must be the length of a board, that is 1ft.'9in. wide, to contain 22ft. 2in.? Ans. 12ft. 8in

3. I have engaged E. Holmes to cut me a quantity of wood, It is to be cut 4ft. 6in. in length, and to be "corded" in a range 256 ft. long. Required the height of the range to contain 75 cords. Ans. 8ft. 4in.

ART. 276. INVOLUTION is the method of finding any power of a given quantity.

A power is a quantity produced by taking any given number, a certain number of times, as a factor. The factor, thus taken, is called the root of the power.

The number denoting the power is called the index or exponent of the power, and is a small figure placed at the right of the root. Thus, the second power of 6 is written 62; the third power of 4 is written 43, and the fourth power of is written (3)1.

ART. 277. To raise a number to any required power.

3:

3 x 3

3 × 3 × 3

3, the first power of 3, is written 31 or 3. 9, the second power of 3, is written 32. 27, the third power of 3,

81, the fourth power of 3,

3X 3X 3X3 X 3243, the fifth power of 3,

QUESTIONS. Art. 275.

What is the rule? Art. 276.

66

66 33.

34.

35.

What is Involu

tion? What is a power? What is the number called that denotes the power ? Where is it placed? Art. 277. To what is the index in each power equal?