ARBITRATION OF EXCHANGE

368. Arbitration of Exchange is the method by which the currency of one country is changed into that of another, through the medium of one or more intervening currencies, with which the first and last are compared.

369. When there is but one intervening currency, it is called Simple Arbitration; and when there is more than one, it is called Compound Arbitration. The method of performing this is called the Chain Rule.

370. The principle involved in arbitration of exchange is simply this: To pass from one system of values through several others, and find the true proportion between the first and last.

1. Let it be required to remit $6570 to London, by the way of Paris, exchange on Paris being 5 francs 15 centimes for $1, and the exchange from Paris to London 25 francs and 80 centimes for £1: what will be the value of the remittance to London?

ANALYSIS.-$15.15 francs; and 1 franc =

If $1 were remitted to Paris, it would produce there 5.15 francs; and if 1 franc were remitted from Paris to London, it would pro

But $6570 are remitted to Paris; hence, they produce there 6570 × 5.15 francs; and this amount is remitted to London; hence, it produces there,

Rule.--I. Find the value of a single unit of each of the moneys named, in the money of the place next named:

II. Multiply the sum to be remitted by these values in succession, and the product will be the equivalent in the money of the place to which the remittance is to be made.

Examples.

1. A merchant wishes to remit $4888.40 from New York to' London, and the exchange is at a premium of 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 mare 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?

$4888.40 × 3 × 13.75 = £1015.771 = £1015 15s. 5d.

Hence, the best way to remit is through Hamburg, then direct; and the least advantageous, through Paris.

2. A merchant in New York wishes to transmit $1500 tc Vienna, through London and Hamburg: what will be the value when received, if £1 = $4.86, £1 = 14 marcs banco, and 6 marcs banco 8 florins?

3. A merchant at Natchez wishes to pay $10000 in Boston He transmits through New Orleans and New York. From Natchez to New Orleans exchange is % premium, from New Orleans to New York % discount, and from New York to Boston 1% discount: by this exchange, what amount at Natuez will pay the debt?

4. A, of London, draws a bill of £862 10s. on B, of Cadiz, and remits the same to C, of Havre, who, in turn, remits to D, of Amsterdam, and D remits to B, of Cadiz: how much will pay the bill, if 1 Spanish dollar 2 florins 15 stivers, 12 flerins 26 francs, and 24 f. 15 c. = £1?

INVOLUTION.

371. A POWER OF A NUMBER is any product which arises from multiplying the number continually by itself.

THE ROOT, or simple factor, is called the first power: THE SECOND POWER is the product of the root by itself: THE THIRD POWER is the product, when the root is taken 3 times as a factor:

THE FOURTH POWER is the product, when it is taken 4 times: THE FIFTH POWER is the product, when it is taken 5 times. 372. The number denoting how many times the root is taken as a factor, is called the exponent of the power. It is written a little at the right and over the root: thus, if the equal factor or root is 3,

34 = 3 × 3 × 3 × 3 = 81, the fourth power of three.

373. INVOLUTION is the operation of finding the powers of numbers.

NOTE-1. There are three things connected with every power: 1st, The root; 2d, The exponent; and 3d, The power or result of the multiplication.

2. In finding any power, one multiplication gives the 2d power: hence, the number of multiplications is 1 less than the exponent.

Rule.-Multiply the number into itself as many times less as there are units in the exponent, and the last product will be the power.

Examples.

Find the power of the following numbers:

1. The square of 4? 2. The square of 15? 3. The square of 142? 4. The square of 463? 5. The square of 1340? 6. The square of 24.6? 7. The square of .526? 8. The square of 3.125? 9. The square of .0524? 10. The square of ? 11. The square of ? 12. The square of ? 13. The square of 3 ? 14. The square of 25? The square of 75?

15.

247

16. The square of 15?

17. The square of 225?

18. The cube of 6? 19. The cube of 24? 20. The cube of 125? 21. The cube of 136? 22. The 4th power of 12? 23. The 5th power of 9? 24. The value of (4.25)3? 25. The value of (1.8)'? 26. The value of (.45)5? 27. The value of (15)3?. 28. The cube of (5)? 29. The 4th power of ? 30. The value of (21)? 31. The value of (25)1? 32. The value of (243)? 33. The value of (.25)? 34. The value of (142.5)?

EVOLUTION.

374. EVOLUTION is the operation of finding the root of a number; that is, of finding one of its equal factors.

375. The SQUARE ROOT of a number is the factor which, multiplied by itself once, will produce the number.

Thus, 8 is the square root of 64, because 8 x8 = 64.

The sign is called the radical sign. When placed before a number, it denotes that its square root is to be extracted: Thus, 36 6.

=

376. The CUBE ROOT of a number is the factor which, multiplied by itself twice, will produce the number.

Thus, 3 is the cube root of 27, because 3 × 3 × 3 = 27.

We denote the cube root by the sign√, with 3 written over it: thus, 27, denotes the cube root of 27, which is equal to 3. The small figure 3, placed over the radical, is called the index of the root.

The terms Power and Root, are dependent on each other : thus, the power is the product of equal factors; and the root is one of the equal factors.

EXTRACTION OF THE SQUARE ROOT.

377. THE SQUARE ROOT of a number is one of its two equal factors. To extract the square root is to find this factor The first ten numbers and their squares are:

1, 2, 3, 4, 5, 6, 7, 8, 9, 10. 1, 4, 9, 16, 25, 36, 49, 64, 81, 100.

The numbers in the first line are the square roots of those in the second. The numbers 1, 4, 9, 16, 25, 36, &c., having two exact equal factors, are called perfect squares.

A PERFECT SQUARE is a number which has two exact equal factors.

NOTE.-The square root of a number less than 100 will be less than 10; while the square root of a number greater than 100 will be greater than 10: hence, the square root of a number expressed by one or two figures, is a number expressed by one figure.

378. To find the law of the square of a number.

Any number expressed by two or more figures may be re garded as composed of tens and units.

1. What is the square of 36 = 3 tens + 6 units?