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

RULE. Divide the given sum by the cost of one pound at the given rate of exchange, and the quotient will be the face of the bill in pounds.

EXAMPLES FOR PRACTICE.

2. J. Reed, of Cincinnati, proposes to make a remittance to Liverpool of $ 1640, exchange being at 83 per cent. premium ; what will be the face 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 face of the bill he can purchase for that sum ?

Ans. 102£ 5s. 5d.+.

EXCHANGE ON FRANCE.

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.

269. To find the cost of a bill on France.

Divide the face of the bill by the cost of one dollar in francs, and the quotient will be the cost 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 5.173 francs per dollar?

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

270. To find the face of a bill on France, which can be purchased for a given sum,

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

Multiply the given sum by the cost of one dollar in francs, and the product will be the face 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 face 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 face of the bill of exchange they can purchase with the money?

Ans. 3483 francs.

DUODECIMALS.

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, t'a,

1, &c.

Duodecimals decrease from left to right, according to a scale of 12 (Art. 82 ; note). 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, 1 inch or prime, equal to be of a foot, is written 1 in. or 1'. 1 second Ta

1". 1 third

T728 1 fourth

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66

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1".

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1".

TABLE.

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12 fourths
make 1". 12 seconds

make 1'.
12 thirds

1".

12 inches or primes 1ft. NOTE. - The foot expresses 12 linear inches, 144 square inches, or 1728 cubic inches, according to the measure in which the duodecimal is employed.

ADDITION AND SUBTRACTION.

272. Duodecimals are added and subtracted in the same manner as compound numbers.

271. What are duodecimals? How do duodecimals decrease from left to right?

How are the different denominations distinguished from each other? - 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. 54 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" 7'111 take 201ft. 11' 7" 8" 10".

Ans. 195ft. 9' 11" 2" gami.

MULTIPLICATION AND DIVISION. 273. The denomination of the product of any two duodecimals. Ex. 1. What is the product of 9ft. multiplied by 3ft. ?

Ans. 27ft.

OPERATION.

9ft. X 3 = 27ft. 2. What is the product of 7ft. multiplied by 6'? Ans. 3ft. 6'.

OPERATION.

6 of a foot; then 7ft. X * = 42'; 42' = 12

3ft. 6'. 3. What is the product of 5' multiplied by 4'? Ans. 1'8".

OPERATION.

5' = Ps, and 4'=; then == 20"; 20"=1'8". 4. What is the product of 9' multiplied by 11"" ? :

Ans. g 31.

OPERATION.

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9', and 11"' = y; then iz X Its

99". = 12 gli 3.

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99" ;

Thus, feet multiplied by a number denoting feet produce feet; feet, by primes produce primes; primes, by primes produce seconds, &c.; and 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.

274. To multiply duodecimals together. Ex. 1. Multiply 8ft. 6in. by 3ft. 7in. Ans. 30ft. 5'6".

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

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6'.

OPERATION.

We first multiply each of the terms in the multi8ft. 6'

plicand by the 7' in the multiplier; thus, 6' X 7' =

42" 3' and 6". Writing the 6" below, one place 3ft. 7'

to the right, we add the 3' to the product of 8ft. X 7! 4ft. 11' 6"

= 59

4ft. and 11', which we write down. We then 2 5ft. 6' multiply by the 3ft., thus: 61 X 3=18' = ift. and

We write the 6! under primes in the other 3 Oft. 5' 6" partial product, and add the ift. to the product of

the 8ft. x 3, making 25ft, which we write down. The partial products being added, we obtain 30ft. 5' 6".

Note. — The notation of feet, primes, seconds, &c., of the multiplier is retained in the operation to note the different order of units.

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 multiplicand by each term of the multiplier, and give each term of the product the proper index, observing to carry 1 for every 12 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.

Ans. -63ft. 11' 3". 3. Multiply 12ft. 9' by 9ft. 11'.

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, 1ft. 6in. 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 paper will it require to cover the walls ?

Ans. 2520 sq. yd. 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'.

We first divide the 68ft. by 2ft. 8') 6 8ft. 10' 8" (2 5ft. 10' the divisor, and obtain 25ft. 6 6ft. 8

for the quotient. We multiply

the entire divisor by the 25, 2ft. 2 gr

and subtract the product, 66ft. 2ft. 2 8"

8', from the corresponding por

tion of the dividend, and oben 274. The rule for the multiplication of duodecimals ?

yards of

OPERATION.

tain 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 in like manner 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 in like manner as before, and so continue till the division is complete.

EXAMPLES FOR PRACTICE.

2. What must be the length of a board, that is ift. 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 256ft. long. Required the hight of the range to contain 75 cords.

Ans. 8ft. 4in.

INVOLUTION.

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276. Involution is the process of finding any power of a number.

A Power of a number is the product obtained by taking the number, a certain number of times, as a factor. The factor, thus taken, is called the root, or the first power.

The Index or Exponent of a power is a small figure placed at the right of the root, indicating the number of times it is taken as a factor. Thus, 6% indicates the second power of 6; 4, the third power of 4; and (), the fourth power of 4.

The second power of a number is sometimes called its square ; the third power, its cube ; and the fourth power, its bi-quadrate. 277. To raise a number to any required power.

. 3 3, the first power of 3, written 3 or 3'. 3 X 3 9, the second power of 3, written

32. 3 X 3 X 3 27, the third power of 3, 3 X 3 X 3 X 3 81, the fourth power of 3, 3 X 3 X 3 X 3 X 3 243, the fifth power of 3,

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35.

275. The rule ? — 276. What is Involution ? A power? What is the number called that denotes the power? Where is it placed ? - 277. To what is the index in each power equal ?

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