ARBITRATION OF EXCHANGES. This Rule is used by merchants to determine which is the best way of remitting money from one country to another. RULE. The operation is performed by conjoined proportion, or by the Single Rule of Three Direct. EXAMPLES. 1. Suppose a merchant in Richmond has $3530 at Amsterdam, which he can have remitted by way of Lisbon at 840 rees per dollar; thence to Richmond at $1.30cts. per milree. Or by way of Nantz at 5 livers per dollar; thence to Richmond at $1.10cts. per crown of 6 livers. It is required to arbitrate these exchanges, that is, to choose the most advantageous route. $1.00 at Amsterdam=840 rees at Lisbon. $3530x $1.30cts. x 840 rees= 1000 × 1.00 by the way of Lisbon. =3854760.00 =$3854.76cts. Ans. = 1000.00 Again-$1 at Amsterdam =5 livers at Nantz. $3530x 1.10cts. x5.4liv-20968.200 $3494,70. Ans. by Nantz. $3854.76cts.-$3494.70 $360.06cts. in favor of remitting by the way of Lisbon. Performed by the Single Rule of Three. 1st. As $1 : 840 rees :: $3530 .. 2965200 rees. 2d. As 1000 rees: $1.30cts. :: 2965200 rees .. $3854.76cts. Ans. as before, by the way of Lisbon. Again-As $1 : 5.4liv. 5.4liv. $3530 190620lir. 2d. As 6liv.: $1.10 :: 190620 $3494.70cts. Ans. as before by the way of Nantz. . The first method is preferable on account of its brevity. 2. A merchant in London has 500 piasters in Leghorn, for which he can draw directly at 52d. sterling per piaster; but choosing to try a circular route, he sent them to Venice, at 95 piasters for 100 ducats banco; thence to Cadiz, at 350 maravidies per ducat banco ; thence to Lisbon, at 630 rees per piaster of 272 maravidies; thence to Amsterdam, at 48d. Flemish for 400 rees; thence to Paris, at 54d. Flemish per crown; thence to London, at 30d. sterling per What is the arbitrated price between London and Leghorn, per piaster, and what is gained or lost by this circular remittance without reckoning expenses? crown. 500 piasters 500×30×48×630×350×100=15876000000000118 95x400x272 × 95 558144 000 As 500pia.: 118£ 10s. 44d. :: 1pia... 56ąd.+ arbitrated price. Ans. The amount received by the circular route 118£ 10s. 41d.+ 500 piasters at 52d. each = 108 6 8 The merchant gained 10 3. 81 Ans. 3. Amsterdam changes with London at 34s. 3d. Flem. per pound sterling, and with Lisbon at 52d. Flemish for 400 rees; how then ought the exchange to go between London and Lisbon ? Ans. 753d.+per milree. 4. A merchant in Richmond has 225€ sterling in London, which he can draw for at 54d. sterling per dollar; but choosing to try a circular route, he orders it to be sent to Dublin, at 100£ sterling for 109 Irish; thence to Hamburg, at 124 marks banco, for 1£ Irish; thence to Amsterdam, at 33 florins for 40 marks banco; thence to Copenhagen, at 5 rials for 2 rix dollars of Denmark; thence to Bremen, at 3 marks for 1 rix dollar of Denmark; thence to Russia, at 5 marks for two rubles; thence to Bordeaux, at 5 francs per ruble; thence to Cadiz at 18 rials of plate for 10 francs; thence to Lisbon, at 1250 rials of plate, for 100 milrees; thence to Leghorn, at 750 soldi for 88 milrees; thence to Smyrna, at 2 soldi per piaster; thence to Jamaica, at 24d. Jamaica currency, for 1 piaster; and thence to Richmond in Virginia, at 80d. Jamaica currency, for $1 United States currency. What did he gain or lose by the above. circuitous remittance? Ans. The merchant gained $117.42cts.+ 100£ Irish. 100£ sterling 1£ Irish 12 marks at Hamburg. 5 francs in Bordeaux. 1 piaster in Smyrna. INVOLUTION, OR THE RAISING OF POWERS. A Power is the product arising from multiplying any given num ber into itself continually, a certain number of times: Thus2 the root, or first power of 2. 2x2=4, the second power, or square of 2. The number denoting the power is called the index or the exponent of that power. If two or more powers are multiplied together, their product is that power whose index is equal to the sum of the exponents of the factors: thus-16 is the fourth power or biquadrate of 2, and 16× 16=256, the 8th power of 2, or square biquadrate, &c. 749 343 2401 16807 117649 823543 5764801 40353607 8 64 512 4096 32768 262144 209715216777216 134217728 81 729 6561 59049 531441 4782969 43046721 387420489 EXAMPLES. 1. What is the 2d power, or square, of 36? Ans. 1296. Ans. 84.604519. Ans. 84934656. Ans. .000000020511149. 5. What is the 6th power, or square cube of 5.03? Ans. 16196.005304479729. 6. What is the 7th power, or second sursolid, of .029? Ans. 000000000017249876309. 7. What is the 8th power, or square biquadrate, of 9.6? Ans. 72138957.89838336. EVOLUTION, OR THE EXTRACTION OF ROOTS. The root of any number or power being multiplied into itself a certain number of times, will produce that power: thus-2 is the square root of 4, because 2×2=4; 3 is the cube root of 27, because 3×3×3=27; and 4 is the biquadrate root of 256, because 4x4x4x4–256. OF THE SQUARE ROOT. The extraction of the Square Root is the finding of such a num ber, as being multiplied into itself, will produce the given number. RULE. 1. Distinguish the given number into periods of two figures each, beginning at the unit's place or decimal point, and when the decimal does not consist of an even number of places, annex a cipher to it, and equal to the periods of the whole numbers and decimals respectively will be the places of each in the root. 2. Place the greatest square number contained in the first or left hand period under it, and set the root thereof on the right hand of the given number, like a quotient in division. 3. Subtract the said square number from the period above it, and to the remainder annex the next period in the given number to form a resolvend or dividual. 4. Place the double of the root, already found, on the left hand of the resolvend, for a divisor. 5. Seek how often the said divisor is contained in the resolvend, (omitting the unit's figure,) and set the result in the root, and on the right hand of the divisor. 6. Multiply the divisor with the figure annexed to it by the last figure in the root, and subtract the product from the resolvend. 7. Annex the third period in the given number to the remainder for a new resolvend. 8. Double all the figures in the root, now found, for a new divisor, and set it on the left hand of the new resolvend; then find the next figure of the root, as before directed, and continue the operation in the same manner till you have brought down all the periods in the given number. NOTE. The operation may be continued to any degree of exactness by annexing pairs of ciphers to the remainder. See the 4th example, and examine it attentively. PROOF. Square the root and add the remainder (if any) to the product; the result will be equal to the given number, if the operation is right. EXAMPLES. |