NOTE. In the above rule the amounts named are supposed to be expressed in the currency of the place from which the remittance is made. If in any case an amount is expressed in the currency of the place to which the remittance is made, the terms of the corresponding multiplier must be inverted. The example wrought above may be thus stated: Required to transmit 109150 pence to a second place where one piece of coin is worth 12 at the first place; thence to transmit it to a third where one piece is worth 20 at the second; thence to a fourth place where 4.444 pieces are equal to 1 at the third.

EXAMPLES.

1. A merchant wishes to remit $4888,40 from New York to London, and the exchange is 10 per cent. He finds that he can remit to Paris at 5 francs 15 centimes to the dollar, and to Hamburg at 35 cents per marc banco. Now, the exchange between Paris and London is 25 francs 80 centimes for £1 sterling, and between Hamburg and London 133 marcs banco for £1 sterling. How had he better remit?

1st. To London direct.

The amount to be remitted is $4888,40. The exchange value of £1 is $4,444, and since the exchange is at a premium of 10 per cent, the value of £1 is $4,444+,4444=$4,8884: hence,

$4888,40 X 7.8884 = £1000:

hence, if he remits direct he will obtain a bill for £1000.

2d. Exchange through Paris.

Since 5,15 francs are equal to 1 dollar, the first multiplier will be this amount divided by $1; and since £1 is equal to 25.80 francs, the second multiplier will be £1 divided by this amount. Then by dividing by 5 and multiplying, we find that the amount remitted by the second method would be £975 15s. 81d.

3d. Method through Hamburg.

$4888,40 × 3 × 13.75 35

= 1015.771 = £1015 15s. 5d. Since 1 marc banco is equal to 35 cents, it is 35 hun dredths of a dollar: hence, the first multiplier is 1 marc banco divided by .35, and the second 1 divided by 13.75. The result shows that the best way to remit is through Hamburg, the next best direct, and the most unfavorable through Paris.

2. A merchant in London has sold goods in Amsterdam to the amount of 824 pounds Flemish, which could be remitted to London at the rate of 34s. 4d. Flemish per pound sterling. He orders it to be remitted circuitously at the following rates, viz., to France at the rate of 48d. Flemish per crown; thence to Vienna at 100 crowns for 60 ducats; thence to Hamburg at 100d. Flemish per ducat; thence to Lisbon at 50d. Flemish per crusado of 400 reas; and lastly, from Lisbon to England at 5s. 8d. per milrea: does he gain or lose by the circular exchange?

Hence, the amount gained by circuitous exchange would be £80 6s. 4 d.

DUODECIMALS.

300. DUODECIMALS are denominate fractions in which 1 foot is the unit that is divided.

The unit 1 foot is first supposed to be divided into 12 equal parts, called inches or primes, and marked '.

Each of these parts is supposed to be again divided into 12 equal parts, called seconds, and marked ".

Each second is divided, in like manner, into 12 equal parts, called thirds, and marked "'.

This division of the foot gives

1 inch or prime

1 second is = 11⁄2 of 12

2

1"" third is of 12 of 2 =
= 12 12

2

of a foot.

144 of a foot.

1728 of a foot.

Hence, in duodecimals, the divisions of the foot increase from the lower denominations to the higher, according to the scale of twelves.

301. Duodecimals are added and subtracted like other denominate numbers, 12 of a lesser denomination making one of a greater, as in the following

2. In 250, how many feet and inches?

Ans.

3. In 4367, how many feet?

Ans.

4. In 847", how many feet?

Ans.

QUEST.-300. In Duodecimals, what is the unit that is divided? How is it divided? How are these parts again divided? What are the parts called? 301. How are duodecimals added and subtracted? How many of one denomination make 1 of the next greater?

EXAMPLES IN ADDITION AND SUBTRACTION.

1. What is the sum of 3ft. 6′ 3′′ 2′′'' and 2ft. 1′ 10′′ 11′′/? 2. What is the sum of 8ft. 9′ 7′′ and 6ft. 7′ 3′′′ 4′′/?

3. What is the difference between 9ft. 3′ 5′′′ 6′′'" and 7ft. 34 6/1 7///?

4. What is the difference between 40ft. 6′ 6′′ and 19ft. 7'''? 5. What is the sum of 18ft. 9′ 11′′ 5′′ and 17ft. 6′ 7′′/? 6. What is the difference between 27ft. 7 and 4ft. 9′ 10" 9///?

MULTIPLICATION OF DUODECIMALS.

302. It is known that feet multiplied by feet give square feet in the product. It is now required to show what fractions of the square foot will arise from multiplying feet by the divisions of the foot, and the divisions of the foot by each other.

expresses of of a foot, we

8 12

see that 2 × 8′′ will give 16-twelfths of twelfths of a square foot; that is, one-twelfth and four twelfths of one twelfth, or 4". The 2 feet multiplied by 7' give 14 twelfths of a square foot; that is, 1 square foot and two twelfths, or 2′. The feet multiplied by 6 give 12 square feet.

QUEST.-302. In multiplication how do you set down the multiplier? Where do you begin to multiply? How do you carry from one denomination to another?

of

Again, 9 inches or of a foot multiplied by 8 twelfths

12

of a foot, will give 72 twelfths of twelfths of twelfths of a square foot, which are equal to six twelfths of twelfths, or to 6". Then 9′ × 7′ gives 63 twelfths of twelfths of a square foot, equal to 5′ and 3′: and 9′ × 6 gives 4 square feet and 6'.

303. Hence we see,

1st. That feet multiplied by feet give square feet in the product.

2d. That feet multiplied by inches give twelfths of square feet in the product.

3d. That inches multiplied by inches give twelfths of twelfths of square feet in the product.

4th. That inches multiplied by seconds give twelfths of twelfths of twelfths of square feet in the product.

2. Multiply 9ft. 4in. by eft. 3in. Beginning with the 8 feet, we

OPERATION.

9 4'

8 3/ 74 8'

2

4'

0/

77 0' 0 Ans.

say 8 times 4 are 32', which is equal to 2 feet 8': set down the 8'. Then say 8 times 9 are 72 and 2 to carry are 74 feet: then multiplying by 3' we say, 3 times 4' are 12", equal to 1 inch: set down 0 in the second's place: then to carry make 28′, equal to 2ft. 4'. Therefore the entire product is equal to 77ft.

3 times 9 are 27 and 1

3. How many solid feet in a stick of timber which is 25ft. 6in. long, 2ft. 7in. broad, and 3ft. 3in. thick?

4. Multiply 9ft. 2in. by 9ft. 6in.

5. Multiply 34ft. 10in. by 6ft. 8in.

6. Multiply 70ft. 9in. by 12ft. 3in.

Ans.

Ans.

Ans.

7. How many cords and cord feet in a pile of wood 24 feet long, 4 feet wide, and 3ft. 6in. high?

8. Multiply 6ft. 9′ by 8ft. 6'.

QUEST.-303. Repeat the four principles.

Ans.