values together, may be seen which will be the most advantageous.

Note 1. By this rule the weights, measures, &c. of different countries may be compared. If an allowance for commission, &c. is to be made from place to place, the most certain method will be to find the value of the sum, at each place, by the rule of three, and deduct the commission therefrom as you proceed.

2. The work may sometimes be shortened by subtracting the sum of the logarithms of the antecedents from the sum of the logarithms of the consequents. The remainder will be the logarithm of the answer.

Examples in Simple Arbitration.

(77.) When London exchanges with Paris at £1 sterling for 23 francs 45 cents, and with Amsterdam at 36s. 4d. Flemish per £. sterling; what ought the course of exchange to be between Paris and Amsterdam, that a merchant in London may remit a sum of money to Amsterdam by way of Paris, instead of remitting immedi ately from London thither, without loss; the exchange between Paris and Amsterdam being 3 francs for a certain number of Flemish pence?

23fr. 45c.: 36s. 4d. Flemish :: 3fr.: 55 pence 6 grots. Therefore the course of exchange between Paris and Amsterdam ought to be at 3 francs for 55 pence 6 grots Flemish.

(78.) If Amsterdam exchanges with London at 33s. 7d. per 8. sterling, and with Lisbon at 514d. Flemish for the crusade of 400 reis, how ought the exchange to go between London and Lisbon?

(79.) A merchant of Amsterdam orders his factor at London to remit to his correspondent at Paris at £1 sterling for 23 francs *, and to draw upon Rotterdam for the value at 378. Flemish per . sterling; but, when the order came to hand, the exchange was on Paris at 24 francs per £. sterling. At what rate of exchange ought the factor

France formerly exchanged with England by giving 1 crown for a variable number of pence English: now the French give 23 or 24 francs, and a variable number of cents for 1. sterling. See the QUOTATION to Table IX.

to draw upon Rotterdam to execute his orders without loss to his employer.

(80.) A factor in London is ordered to remit to Venice at 50d. per ducat, and to draw for the value upon Madrid at 42d. per dollar; but, on receipt of the order, bills upon Venice were at 532d. At what rate must he draw upon Spain to compensate this loss?

CLASS II.

(81.) A merchant at London is desirous of transfering a sum of money to Amsterdam in the most advantageous manner, either directly to Amsterdam, or through Paris, at a time when the course of exchange between London and Amsterdam is 34s. 5d. per £. sterling, and between London and Paris 314d. sterling per crown ?— by advice he finds the course of exchange between Paris and Amsterdam to be 52d. Flemish per crown, upon which he remits directly to Amsterdam, and draws for the value upon Paris. What does he gain per cent. by these means; and what would he have lost per cent. had he remitted the money to Amsterdam by way of Paris, and then drawn upon Amsterdam for the value, supposing he had received no advice of the course of exchange between Paris and Amsterdam?

(82.) A Spanish merchant ordered his factor in London to remit the value of 900 ducats to Venice, at 501d. per ducat, and to draw upon him at Madrid for the value at 41d. per piastre. When the order arrived, the exchange at Venice was 51d. per ducat, and at Spain at 424d. per piastre ; whether did the merchant gain or lose by this negociation?

(83.) A merchant in London remitted to Amsterdam 500l. sterling, at the rate of 18d. sterling per guilder; his correspondent at Amsterdam was to remit the same, by order, to Bordeaux, at 3 guilders per crown, rebating

per cent. for his commission; but, when he received this order, the exchange between Amsterdam and Bordeaux was at 3 guilders per crown. The merchant at London, not apprized of this, drew upon Bordeaux at 55d. sterling per crown; whether did he gain or lose, and how much per cent.?

2

(84.) A merchant at Amsterdam was indebted to an other at Paris a bill of 3000 florins current, agio 4 per cent., and exchange at 90 d. per ecu of 60 sols Tournois; but, when this bill became negociable, the exchange was down at 89 d. per crown, and the agio advanced to 5 per cent. Did the Paris merchant gain or lose by this turn of affairs?

Examples in Compound Arbitration.

(85.) Sold goods to a house in Amsterdam to the amount of £824 Flemish, which my correspondent advises me he will remit; but, as the exchange on Amsterdam was so low as 34s. 4d. per £. sterling, I have desired him to remit it to France at 48d. Flemish per crown; thence he orders it to be remitted to Vienna, at 100 crowns for 60 ducats, thence to Hamburgh, at 100d. Flemish per ducat; thence to Lisbon, at 50d. Flemish per crusade of 400 reis; and lastly, from Lisbon to England at 5s. 8d. sterling per mille-reis. Whether shall I gain or lose by the circular exchange?

By the direct Exchange.

34s. 4d. Flemish £1 sterling :: £824 Flem.: £480 sterling. Theu 5601. 6s. 44d.-£480=801. 6s. 44d. advantage by the circular exchange.

(86.) A banker in Paris remits to his factor at Amsterdam, 22641 francs 75 cents, first to London at 24 francs per . sterling; thence to Rome, at 65d. per stampt crown; thence to Venice, at 100 stampt crowns for 142 ducats bank; thence to Leghorn, at 105 ducats bank for 100 piastres; and from Leghorn to Amsterdam, at 87d.

Flemish per piastre. How many guilders bank will be received at Amsterdam, and what will the banker gain, supposing the direct exchange between Paris and Amsterdam to be 51 grots Flemish for 3 francs?

(87.) A merchant in London is desirous to remit £759 sterling to Genoa. He can remit by way of Paris, at 56d. per ecu; thence to Venice, at 100 crowns for 60 ducats bank; thence to Rome, at 140 ducats bank for 100 stampt crowns; and from Rome to Genoa, at 115 stampt crowns for 125 pezzos.-He can likewise remit by way of Amsterdam, at 33s. Flemish per £. sterling: thence to Frankfort, at 2 rix-dollars for 16s. Flemish; thence to Venice, at 12 ducats for 11 rix-dollars; thence to Rome, &c. as above. How many pezzos by each of these methods will the merchant have for his money, aud which method will be the more advantageous?

CLASS II.

(88.) A merchant in London has credit at Leghorn for 7547 piastres, whence he receives advice that a remittance can be made at 52d. per piastre. The merchant upon this orders them to be remitted to Venice, at 95 piastres for 100 ducats bank; thence to Cadiz, at 321 maravedis per ducat; thence to Lisbon, at 631 reis per piastre ; thence to Amsterdam, at 50d. Flemish per crusade; thence to Paris, at 56d. Flemish per ecu; and lastly from Paris to London, at 31 d. per crown. What ought to be the arbitrated price between London and Leghorn; whether will the merchant gain or lose, and how much per cent. by the circular exchange?

(89.) I have ordered my factor at Amsterdam to remit 17577. 158. Flemish (the exchange between London and Amsterdam being 34s. 7d. Flemish per £. sterling) to France at 54d. Flemish per ecu; thence to Venice at 100 crowns for 56 ducats bank; thence to Hamburgh, at 100d. Flemish per ducat; thence to Portugal, at 45d. Flemish per crusade; and thence to London, at 63d. per mille-reis. How much sterling money ought I to receive, allowing my factor per cent. for commission at each place; and whether will be the more advantageous the circular or the direct exchange?

(90.) If 100lb. weight of England make 88lb. at Rouen, 78lb. at Rouen 94lb. at Lyons, 69lb. at Lyons 53lb. at Geneva, 72lb. at Geneva 100lb. at Marseilles, 121lb. at Marseilles 100lb. at Hamburgh, 103lb. at Hamburgh 101lb. at Paris.-What is the difference be tween the weight of a pound at London and Paris?

INVOLUTION.

Definition 1. When any given number is multiplied by itself and that product by the same number, and so on to any assigned number of products, the process is called Involution, or the involving a number to any assigned

power.

2. The given number is called the root, or first power; the first power multiplied by itselfgives the second power, . or square; the second power multiplied by the first, gives the third power, or cube; the third power multiplied by the first, gives the fourth power, or biquadrate, &c.

8. The number denoting the power is called the index, or exponent, of that power. Thus, if a number is to be involved to the fourth power, then 4 is the index of the power.

4. Powers are generally denoted by writing the exponent over the first. Thus the square of 205 is written 2052, the cube 2053; also the fourth power of 705 × 9.15 may be expressed thus, 705×9·154, &c.

Note I. A general rule for the practice of Involution is evidently contained in the 2d definition. A fraction may be involved to any power by a continual multiplication of its terms in a similar manner.

Proposition. To find the power of any number, above the cube, without finding all the intermediate powers.

Rule. Find, by the second definition, two or more such powers of the given number as that the sum of their indices may make the index of the power required. Then multiply these powers continually together, and the last product will be the power required.