2. When the area is 160 perches, required the circumference. Ans. 44.839 PROBLEM 6. Given the difference between two numbers, and the dif ference of their squares, to find the numbers. RULE. Divide the difference between the squares by the difference of the numbers, and the quotient will be their sum; then, to half the sum add half the difference, and the sum is the larger; and from half the sum take half the difference, and the remainder is the less. Ex. 1. The difference between two numbers is 20, and the difference of their squares 2000-what are the numbers? Ans. 60 the greater, 40 the less. 2. Said Harry to Charles, My father gave me 12 apples more than he gave brother Jack, and the difference of the squares of our separate parcels was 288-now tell me how many he gave us, ,,and you shall have half of mine. Ans. {Harry's share 18 Further use of the Cube Root. I. To find two mean proportionals between two given numbers. RULE. Divide the greater extreme by the less, and the cube root of the quotient multiplied by the less extreme, gives the less mean; multiply the said cube root by the less mean, and the product will be the greater mean proportional. Ex. 1. What are the two mean proportionals between 7 and 189? Ans. 21 and 63. 2. Required two mean proportionals between 4 and 256. 16 and 64. II. To find the side of a cube that shall be equal in solidity to any given solid, as globe, cylinder, prism, &c. RULE. The cube root of the solid content of any solid body given, is the side of a cube of equal solidity. Ex. 1. If the solid content of a globe be 10648, what is the side of a cube of equal solidity? Ans. 22. 2. Required the side of a cubical vessel that shall contain 80 wine gallons, each 231 cubical inches? Ans. 26.43+ III. The side of a cube being given, to find the side of a cube that shall be double, treble, &c. in quantity to the given cube. RULE. Cube the given side, and multiply it by 2, 3, &c.; the cube root of the product is the side sought. Ex. 1. There is a cubical vessel, whose side is 12 inches; it is required to find the side of another vessel that shall contain three times as much. Ans. 17.306+ 2. Suppose the length of a ship's keel to be 125 feet, the breadth of the midship beam 25 feet, and depth of hold 15 feet-required the dimensions of another ship of the same form, that shall carry three times the burthen. Ans. Length of the keel 180.28+ft, IV. Having the dimensions and capacity of a solid, to find the dimensions of a similar solid of a different capacity. RULE. Like solids are in triplicate proportion to their homologous sides; therefore it will be, as the cube of a dimension is to the cube of any like dimension: so is the given weight to the weight required. EXAMPLES. 1. A brass bullet of 5 inches diameter weighs 20lbs.— required the diameter of a like bullet that weighs 160lbs. Ans. 10 inches. 2. If a ship of 300 tons burthen, be 75 feet long in the keel, I demand the burthen of another ship, whose keel is 100 feet long. Ans. 711 tons. APPLICATION. 3. What is the difference between half a solid foot, and Ans. 3 half feet. a solid half foot? 4. In a cubic foot, how many cubes of 6 inches, and how many of 4, of 3, of 2, and of 1, are contained therein? Ans. 8 of 6in., 27 of 4in., 64 of 3in., 216 of 2in., and 1728 of lin. 5. Suppose a cubical cellar to contain 1728 solid feet: what will one of its cubic sides measure? TABLE OF FOREIGN COINS, With their Federal value, as established by an act of Congress, for estimating We subjoin a few Examples for Practice. 1. Reduce 8764 livres to Federal money. 2. Reduce 10000 francs to dollars. 3. In $1000, how many francs? Ans. $1621.34. Ans. $187.5. 9. In $457.56, how many milreas of Portugal? Ans. 72. Ans. 144. Ans. 369. Ans. 825. Ans. 854. Ans. 4842. NOTE.-The above examples more properly belong to the rule called Exchange. EXCHANGE. EXCHANGE is the giving of the money, weight, or measure of one country, for the like value in money, bills, weight, or measure of another country. Exchange is said to be at par between two countries or places, when the money given in the one is intrinsically equal in value to that received for it in the other. The course of exchange is frequently above or below par. Agio is a term used to signify the difference in some countries, between bank and current money. When remittances are made in cash, exchanges may be readily computed by the intrinsic or legal value of the coins remitted. But as they are generally made, between distant places, by bills of exchange, and they seldom pass according to the intrinsic values of the sums for which they are drawn, the rate of exchange must be taken into the account. EXCHANGE WITH GREAT BRITAIN. In Great Britain they keep their accounts in pounds, shillings, and pence, sterling, and exchange by the £1, which is equal to $4.444, par. NOTE. By an act of Congress, passed November, 1792, the pound sterling was valued at $4.44, in calculating duties, ad valorem, on invoices given in at the Custom-houses. To reduce federal money to sterling, (or English.) RULE 1. Multiply by 9,* and divide by 40.* RULE 2. Multiply by 54, and divide by 12 and by 20. RULE 3. As $1 is to the sum given :: so is 4s. 6d., or 4.5s, or 54d. to the answer in sterling money. RULE 4. To bring federal money to foreign, divide the given sum by the same federal value. EXAMPLES. 1. Reduce $371.75 to sterling. Ans. £83 12s. 10 d. Ans. £170 2s. 2. A of Philadelphia owes B of London $756-what sum sterling will discharge the debt? *£ sterling is equal 240d. 40. $ is equal 54d. 9. M 2 3. Reduce $888.88 to sterling money. Ans. £199 19s. 11d. 2.08qr. 4. Reduce $4536 to English money. To reduce English or sterling money to federal money. RULE 1. Reduce the shillings, pence, and farthings, (if any) to a decimal of a pound, which annex to the pounds. Multiply by 40, and divide the product by 9, and the quotient will be dollars, or dollars, cents, and mills. RULE 2. Reduce the given sum to pence, and divide by 54. As 4s. 6d. is to the sum given :: so is $1 : to the answer in federal money. EXAMPLES. 1. Reduce £83 12s. 104d. English, to federal money. Ans. $371.75. 2. Reduce £170 2s. sterling, to federal money. Ans. $756. 3. Reduce £1020 12s. English, to federal money. Ans. $4536. 4. Reduce £25 sterling, to money of the United States. Ans. $ppp.pp. BILLS OF EXCHANGE. To find the value of a bill of exchange above par RULE. Multiply the sum of the bill by the amount of one dollar, at the given rate per cent. EXAMPLE. What is the amount or value of a bill of exchange, for $750, at 4 per cent. above par? 750 x 1.04-$780 ans. To find the value of a bill of exchange below par. RULE.-Subtract from the given sum one year's interest at the given rate per cent.; and the remainder will be the value required. EXAMPLES. 1. What is the value of a bill of exchange for 780 dollars, at 4 per cent. below par? 780.X.04 $31.20c. then 780-31.20-$748.80. |