which depreciated more or less, the value of a colonial £ was less than that of a £ sterling. The depreciation being greater in some colonies than in others, gave rise to the different State currencies. Thus, 8s. In New England, Va., Ky., and Tenn., 6s. or £3=$1. or £=$1. 78. 6d. or £&=$1. 4s. 8d. or £=$1. 5s. or £=$1. The process of changing any sum of U. S. Money to either of these currencies, or the reverse, involves the same principles as the exchanging of sterling money. Hence, to reduce U. S. Money to the currency of a State, Multiply the given sum, expressed in dollars, by the value of $1 expressed in the fraction of a pound; the product will be the value in pounds and decimals of a pound. To reduce a State currency to U. S. Money, Express the given sum in pounds and decimals of a pound, then divide this by the value of $1 expressed in the fraction of a pound; the quotient will be the value in dollars. 1. Reduce $120.50 to N. Eng. currency. 2. $75.25 to N. York currency. 3. $98 to Penn. currency. ANSWERS. £36 38. £30 28. £36 15s. $102.50 $64.625 4. £30 15s. N. E. currency to dollars. $126.00 NOTE. Any sum, in one currency, may be changed to that of nother, by Sim. Proportion (Art. 203); or by short methods. Thus, since 6 shillings New England currency are equal to 8 shilrings New York currency, to change the former to the latter, ADD one-third of the sum; to change the latter to the former, SUBTRACT one-fourth of the sum. REVIEW.-272. In buying or selling bills of exchange, how make the calculations? 273. In what were accounts kept previous to 1786? ART. 274. Any currency may be reduced to U. S. Money, or U. S. Money to any currency, by multiplication or division, as in Art. 271, when the value of a unit of the foreign currency expressed in U. S. Money is known. The unit of French money is the franc, its value being $0.186 Bills of exchange on France are bought and sold at a certain number of francs to the dollar. EXAMPLE.-At $1 for 5.30 francs, what will be the cost of a Bill on Paris for 1166 francs? Ans. $220. In "Ray's Higher Arithmetic" may be found a completo table of all foreign coins, with their value in U. S. Money; also, valuable information respecting exchange with all civilized nations. XXII. DUODECIMALS. ART. 275. Duodecimals are a peculiar order of fractions, which increase and decrease in a twelve-fold ratio. Their name, from the Latin duodecim, signifies twelve. The unit, 1 foot, is divided into 12 equal parts, called inches, or primes, marked thus, ('). Each inch, or prime, is divided into 12 equal parts, called seconds, marked ("). Each second, into 12 equal parts, called thirds, (''). The marks,,"", "", called indices, show the different parts Duodecimals are added and subtracted like Compound Num bers; 12 units of each order making a unit of the next higher. ART. 276. MULTIPLICATION. Duodecimals are used in the measurement of surfaces and solids, as boards, solid walls, &c. 1. Find the superficial contents of a board 7ft. 5in. long, and 4ft. 3in. wide. Length, multiplied by breadth, gives the superficial contents. SOLUTION.-' of a foot; therefore, X4 ft.: X4=29=20 inches, which is 1 ft. 8 in.; write the inches or primes in the order of primes, and carry the 1 ft. Next, 7 ft. X 4 ft.28 ft., to which add the 1 ft. carried, the sum is 29 ft.; which write in the order of feet. OPERATION. Again, 5' and '; therefore, Ans. 31 63 5'x3'=;×31⁄2==15", which is 13"; write the 3 sec. 111 in the order of seconds, and carry the 1 in. Next, 7 ft. X 3 in. 2 7 × 3 = 21=21′, and 1′ carried, make 221 ft. 10. Writing these in their orders, and adding the two products, the entire product is 31 ft. C′ 3′′. The product of any two denominations, is of that denomina tion denoted by the sum of their indices; thus, 3′ 5′ = 15′′, 3'X7=21', 7′′X+=+=+=28". Hence, Seconds multiplied by seconds, give fourths, and so on. REVIEW.-273. Why was the value of a colonial pound less than that of a pound sterling? How reduce U. S. Money to a State currency? 273. How reduce a State currency to U. S. Money? 275. What are duodecimals? Whence their `name? What are Primes? Seconds! Thirds? Fourths? Repeat the table. What are indices? What part of a foot is 1'? 1"? 1"? 1"""'? How aro duodecimals added and subtracted? Rule for Multiplication.-1. Write the multiplier under the multiplicand, placing units of the same order under each other. 2. Multiply, first by the feet, next by the inches, and so on, recollecting that the product will be of that denomination denoted by the sum of their indices. 3. Add the several partial products together, and their sum will be the required product. The primes of the product of two duodecimal factors, are neither linear nor square in., but twelfths of a sq. ft. The primes of the product of three duodecimal factors, are twelfths of a cu. ft. 2. How many square feet in a board 5 ft. 3 in. long, and 1ft. 5 in. wide? Ans. 7 sq. ft. 5' 3". 3. Multiply 5 ft. 7 in. by 1 ft. 10 in. Ans. 10 sq. ft. 2'10". 4. 8ft. 6 in. 9"x 7 ft. 3 in. 5. 8ft. 4 in. 6"x2ft. 7 in. 4". 6. 4ft. 5' 6"x2ft. 3′ 5′′. Ans. Ans. 62 sq. ft. 11′′ 3′′′′. Ans. 21 sq. ft. 10' 5". 10sq.ft. 2′ 2′′ 9'' 6"""". Another method of solution found in “Ray's Higher Arithmetic.” XXIII. INVOLUTION. ART. 277. INVOLUTION is the multiplication of a numbcr by itself one or more times. A POWER is the product obtained by invlution. The ROOT, or first power, is the number multiplied. If the number be taken twice as a factor, the product is the second power; 3X3=9, is the 2d power of 3. If the number be taken 3 times as a factor, the product is the 3d power; 2X2X2=8, is the 3d power of 2. REVIEW.-276. For what are duodecimals used? Of what denomination the product of any two denominations? What is the product of fect by feet? Feet by inches? Inches by inches? Inches by seconds? Seconds by seconds? Rule for multiplication? What do the primes of the product of two duodecimal factors represent? Of three? And, if taken 4 times as a factor, the product is the 4th power; if 5 times, the product is the 5th power, and so on. Hence, the different powers derive their name from the number of times the root is taken as a factor. REM. The given number is called the root, the different powers of the number being derived from it. ART. 278. The second power of a the square; the third power the cube. derived thus: number is called These terms are ILLUSTRATIONS. 1. Take a line, say 3 feet long, its first power is the line itself. 2. If 3 feet be multiplied by itself, the product (Art. 87) will be 3X3 9 square feet. (See diagram of 3 feet square, page 91.) But 3X3 9 is the second power of 3; hence, the 2d power called the square. is 3. If each side of a cube is 3 feet, the cube (Art. 92) contains 3X 3X3=27 cubic ft. (See diagram, p. 94.) But 3 × 3 × 3=27, is the third power of 3; hence, the 3d power is called the cube. ART. 279. The number denoting the power to which the root is to be raised, is the index or exponent of the power. It is placed on the right, a little higher than the root. Thus, 212, the 1st power of 2. 22=2×2=4, the 2d power or square of 2. 24 = To find the second power of 2, use it as a factor twice; thus, 2×2=4. To find the third power of 2, use it three times; thus, 2×2×2=8, and so on. REVIEW.-277. What is involution? What is a power? root, or 1st power? The 2d power? The 8d? The 4th? What is the 277. From what do the different powers derive their name? REM. Why is the given number called the root? 278. What is the second power of a number called? What the 3d? How are these terms derived? 279. What is the index of a power? How find the 2d power of 2? The 3d? The 4th? |