be in proportion with the rates assigned between each of them and a third place.

By comparing the par of exchange thus found, with the present course of exchange, a person is enabled to find which way to draw bills, or remit the same to most advantage.

[Questions in this rule are performed by the Rule of Three.]

Arbitration of exchange, is either simple or compound. In simple arbitration, the rates of exchange from one place to two others are given, by which is found the correspondent price between the said two places, called the arbitrated price.

An example or two will make the subject clear.

Ex. 1. If exchange between London and Amsterdam be 34 schil. 9 grotes per £. sterling, and if exchange between London and Genoa be 45 pence per pezza (see Table, p. 236,) what is the par of arbitration between Amsterdam and Genoa:

Here 11. 240 pence: therefore, as

240d.: 34s. 9gr. :: 45d. : 7845gr.

Answer, 78 Flemish grotes, or pence per pezza Genoa.

Ex. 2. If exchange from London to Amsterdam be 33s. 9d. per £. and if exchange from London to Paris be 32d. per crown, what must be the rate of exchange from Amsterdam to Paris?

Ex. 3. If exchange from Paris to London be 32d. per crown, and if exchange from Paris to Amsterdam be 54d. Flemish per crown, what must be the rate of exchange between London and Amsterdam, in order to be on a par with the other two?

Ex. 4. Amsterdam exchanges on London, at 35 schil. 5 gro. per £. sterling; and the exchange between London and Lisbon is 60 pence per milrea, what is the exchange between Amsterdam and

Lisbon?

The course of exchange being given, and the par of arbitration found, we obtain a method of drawing and remitting to advantage.

Ex. 5. If exchange from London to Paris be 32 pence sterling per crown, and to Amsterdam 405 Flemish per £., and if I learn that the course of exchange between Paris and Amsterdam is fallen to 52 pence Flemish per crown;

what may be gained per cent., by drawing on Paris and remitting to Amsterdam?

By Ex. 2, the par of arbitration between Paris and Amsterdam is 54d. Flemish per crown: then

If the course of exchange between Paris and Amsterdam be at 56 Flemish per crown, instead of 52; and if I would gain by the negotiation, I must draw on Amsterdam and remit to Paris: thus

In these cases, credit at one foreign place pays the debt at the other. We might carry the subject of Exchanges to almost any length; but we have said enough to render the theory and practice easy; and from what the pupil has seen he will be able to apply the foregoing principles and rules to the practice of any merchant's counting-house in which he may be situated. We shall, however, give an example in Compound Arbitration.

COMPOUND ARBITRATION.

IN Compound Arbitration, the rate of exchange between three or more places is given, to find how much a remittance passing through them all will amount to at the last place: or to find the arbitrated price, or par of arbitration, between the first and last place.

Examples of this kind may be worked by several successive statings in the Rule of Three, or according to the following Rules.

(1) Distinguish the given rates, or prices, into antecedents and consequents, placing the antecedents in one M

column, and the consequents in another, with the sign of equality between them.

(2) The first antecedent, and the last consequent to which an antecedent is required, must be of the same kind. (3) The second antecedent must be of the same kind with the first consequent, and the third antecedent of the sume kind with the second consequent, &c.

(4) Multiply the antecedents together for a divisor, and the consequents together for a dividend, and the quotient will be the answer required

Ex. If a merchant in London remit 5007. sterling to Spain by way of Holland, at 35 shillings Flemish per pound sterling, thence to France at 58 pence per crown, thence to Venice at 10 crowns for 6 ducats, and thence to Spain at 360 mervadies per ducat; how many piastres of 272 mervadies will the 500l. amount to in Spain ?

and this fraction, reduced to its lowest terms, gives

If the course of direct exchange to Spain were 42 pence sterling, then the 500l. remitted would only amount to 2823 piastres, of course 2875-28234, gives 52, which is the number of piastres gained by the negotiation.

NOTE.

* The fractions are omitted, and on that account the answer by this method will not be quite accurate.

DUODECIMALS.

DUODECIMALS, or Cross Multiplication, is made use of by artificers in measuring their several works, and is performed by means of the following table:

12'' fourths

12" thirds

12" seconds

12' inches

make 1 third.

1 second.

1 inch.

1 foot.

Glaziers, Masons, and others, measure by the square foot.-Painters, Paviors, Plasterers, &c., by the square yard.-Slating, tiling, flooring, &c., by the square of 100 feet.-Brickwork is measured by the rod of 16 feet, the square of which is 2724. See p. 42.

RULE. (1) Arrange the terms of the multiplier under the same denominations of the multiplicand. (2) Multiply each term in the multiplicand, beginning at the lowest,* by the feet in the multiplier, aud write the result of each under its respective term, observing to carry one for every twelve. (3) Multiply, in the same manner, by the inches, and set the result of each term one place removed to the right-hand of those in the multiplicand.† (4) Multiply then by the seconds, setting the result of each term two places removed to the right hand of those in the multiplicand.

Multiply 9 ft. 4 in. 8 sec. by 5 ft. 8 in. 6 sec.

*Hence the origin of the term cross multiplication, the operation being crossways, compared with multiplication in the common way. It is called Duodecimals, because the feet, inches, &c. are divided into twelve parts, whereas in decimals the unit is divided into tenths.

+ Feet

Feet

Feet

multiplied into feet

multiplied into inches

give feet.
give inches.
multiplied into seconds give seconds.
Inches multiplied into inches give seconds.
Inches multiplied into seconds give thirds.
Seconds multiplied into seconds give fourths.

Ex. 1. How much must I pay for a slab of marble 7 ft. 4 in. long, and 2 ft. 1 in. 6 sec. broad, at the rate of 7 s. per square foot?

Ex. 2. What will be the expense of glass for a window that measures, in the clear, 10 ft. 6 in. in height, and 4 ft. 9 in. in width, at 1s. 9d. per foot?

Ex. 3. How much will a room cost in painting, at 9d. per yard; the sides are 18 ft. 10 in. by 10 ft. 3 in., and the two ends are 16 ft. 6 in. by 10 ft. 3 in.?

Ex. 4. What shall I have to pay for statuary marble about my fireplace, at 14s. per foot; the hearth measures 6 ft. 4 in. by 2 ft. 3 in., the three fronts are each 4 ft. 2 in. by 8 in., and the mantle-piece slab is 6 ft. by 9 in.?

Ex. 5. What will the paving of a court-yard come to, at 1s. 2d. per foot, the yard being 74 feet long, and 56 ft. 8 in. wide?

Ex. 6. How much shall I have to pay for slating a house, consisting of two sloping sides, each measuring 24 ft. 5 in. by 15 ft. 9 in. at the rate of 41s. per square of 100 feet?

Ex. 7. What will the tiling of 10 houses come to, the roof of each house consisting of two sides, each 18 feet by 14, and the price of tiling at 28s. per square?

Ex. 8. How many square rods are there in a brick wall 44 ft. 6 in. long, and 7 ft. 4 in. high, and 2 bricks thick?*

Ex. 9. If an oblong garden be 254 ft. 6 in. long, and 184 ft. 8 in. wide, what will a wall cost 10 ft. 6 in. high, and 2 bricks thick, at 15l. 15s. per square rod?

Ex. 10. How much shall I have to pay for the plate-glass of four windows; each window consists of 16 panes, and each pane measures 20 inches by 152 inches at 9s. 6d. per foot?

Ex. 11. How many solid feet of fir are there in a piece of timber 35 ft. 4 in. long, and 133 inches by 14 inches ?+

Ex. 12. How many solid feet of oak are there in a piece 14 feet 3 inches long, and 2 feet 104 by 2 feet 2 inches?

Ex. 13. How many solid feet of fir are there in 46 joists, each 14 feet 3 inches long, 7 inches deep, by 31⁄2 inches broad?

NOTES.

* Bricklayers value their work at the rate of a brick and a half, or three half bricks thick; and if the wall be more or less than this, it must be reduced to that thickness by the following rule:- Multiply the measure found by the number of half bricks, and divide by three:" thus, if the wall be 2 bricks thick, I multiply by 5, and divide the product by 3.

50 feet long, and 9 high, and 2 bricks thick, it

+ Carpenters' rules are divided into eighths, so that in these cases the eighths must be reduced to twelfths, or the whole must be worked by decimals. In this and the following questions, the length, breadth, and thickness, must be multiplied into one another.