CASE II. To find the Value of a Decimal in terms of the Inferior Dens minations. MULTIPLY the decimal by the number of parts in the next lower denomination; and cut off as many places for a remainder to the right-hand, as there are places in the given decimal. Multiply that remainder by the parts in the next lower denomination again, cutting off for another remainder as before. Proceed in the same manner through all the parts of the integer; then the several denominations separated on the lefthand, will make up the answer. Note, This operation is the same as Reduction Descending in whole numbers. EXAMPLES. 1. Required to find the value of 775 pounds sterling. CASE III. To reduce Integers or Decimals to Equivalent Decimals of Higher Denominations. DIVIDE by the number of parts in the next higher denomination; continuing the operation to as many higher denominations as may be necessary, the same as in Reduction Ascending of whole numbers. EXAMPLES. 1. Reduce 1 dwt to the decimal of a pound troy. 20 1 dwt 2. Reduce 9d to the decimal of a pound. Ans. .03751. 3. Reduce 7 drams to the decimal of a pound avoird. Ans. 02734375lb. 4. Reduce 26d to the decimal of a 7. Ans. 0010833 &c.l. 5. Reduce 2·15 lb to the decimal of a cwt. Ans. 019196+ cwt. 6. Reduce 24 yards to the decimal of a mile. Ans. 013636 &c. mile. 7. Reduce 056 pole to the decimal of an acre. Ans. 00035 ac. 8. Reduce 12 pint of wine to the decimal of a hhd. Ans. 00238 + hhd. 9. Reduce 14 minutes to the decimal of a day. Ans. 009722 &c. da. 10. Reduce 21 pint to the decimal of a peck. Ans. 013125 pec. 11. Reduce 28" 12" to the decimal of a minute. NOTE, When there are several numbers, to be reduced all to the decimal of the highest : Set the given numbers directly under each other, for dividends, proceeding orderly from the lowest denomination to the highest. Opposite to each dividend, on the left-hand, set such a number for a divisor as will bring it to the next higher name; drawing a perpendicular line between all the divisors and dividends. Begin at the uppermost, and perform all the divisions: only observing to set the quotient of each division, as decimal parts, parts, on the right-hand of the dividend next below it; so shall the last quotient be the decimal required. EXAMPLES. 1. Reduce 17s 94d to the decimal of a pound. Ans. 7751. Ans. 625s. 3. Reduce 15s 6d to the decimal of al. Ans. 46944 &c. lb. RULE OF THREE IN DECIMALS. PREPARE the terms, by reducing the vulgar fractions to decimals, and any compound numbers either to decimals of the higher denominations, or to integers of the lower, also the first and third terms to the same name: Then multiply and divide as in whole numbers. Note, Any of the convenient Examples in the Rule of Three or Rule of Five in Integers, or Vulgar Fractions, may be taken as proper examples to the same rules in Decimals. -The following Example, which is the first in Vulgar Fractions, is wrought out here, to show the method. If of a yard of velvet cost 31, what will yd cost ? DUODECIMALS. DUODECIMALS, or CROSS MULTIPLICATION, is a rule used by workmen and artificers, in computing the contents of their works. Dimensions are usually taken in feet, inches, and quarters; any parts smaller than these being neglected as of no consequence. And the same in multiplying them together, or casting up the contents. The method is as follows. SET down the two dimensions to be multiplied together, one under the other, so that feet may stand under feet, inches under inches, &c. Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each straight under its corresponding term, observing to carry 1 for every 12, from the inches to the feet. In like manner, multiply all the multiplicand by the inches and parts of the multiplier, and set the result of each term one place removed to the right-hand of those in the multiplicand; omitting, however, what is below parts of inches, only carrying to these the proper number of units from the lowest denomination. Or, instead of multiplying by the inches, take such parts of the multiplicand as there are of a foot. Then add the two lines together, after the manner of Compound Addition, carrying 1 to the feet for 12 inches, when these come to so many, 3. Multiply 4 feet 7 inches by 9 f 6 inc. Ans. 43 f. 61⁄2-inc. 4. Multiply 12 f 5 inc by 6 f 8 inc. 5. Multiply 35 f 4 inc by 12f 3 inc. 6. Multiply 64 f 6 inc by 8 f 9 inc. Ans. 82 9 48 Ans. 433 4 Ans. 565 83 INVOLUTION. INVOLUTION. INVOLUTION is the raising of Powers from any given number, as a root. A Power is a quantity produced by multiplying any given number, called the Root, a certain number of times continually by itself. 2 x Thus, 2= 2×2= 2 is the root, or 1st power of 2. 4 is the 2d power, or square of 2. 8 is the 3d power, or cube of 2. x 2 x 2 = 16 is the 4th power of 2, &c. 2 x 2 x 2 = And in this manner may be calculated the following Table of the first nine powers of the first 9 numbers. 19683 39 27 81 243 729 2187 6561 4 16 64 256 1024 4036 16384 65536 262144 5 25 125 6253125 15625 78125 390625 1953125 6 362161296 7776 46656 279936|1679616| 10077696 7 49343 240116807 117649 823543 5764801 40353607 co 64512 4096 32768 262144 2097152 16777216 134217728 981 729056159049531441 4782969 43046721 6721 387420489 The |