parts, on the right-hand of the dividend next below it; so shall the last quotient be the decimal required. EXAMPLES. 1. Reduce 17s 91d to the decimal of a pound, 3• £o 0·890625 Ans. 2. Reduce 191 17s 3d to l. Ans. 19.86354166 &c. I. 3. Reduce 15s 6d to the decimal of a d. Ans. 7751, 4. Reduce 7d to the decimal of a shilling. Ans. •625s. 5. Reduce 5 oz 12 dwts 16 gr to lb. 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 į of a yard of velvet cost žl, what will to yd cost ? vd yd 17 — •375 •375 : 4 :: •3125 : .333 &c. or 6 & 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 9f 6 inc. Ans. 43 f. 6. inc. 4. Multiply 12 f 5 inc by 6 f 8 inc. Ans. 82 94 5. Multiply 35 f 41 inc by 12 f 3 inc. Anș. 433 4 6. Multiply 64 f 6 inc by 8 f 9 inc. Ans. 565 8 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. Thus, 2 is the root, or 1st power of 2. 2 x 2 = 4 is the 2d power, or square of 2. 2 x 2 x 2 8 is the 3d power, or cube of 2. 2 x 2 x 2 x 2 = 16 is the 4th power of 2, &c. And in this manner may be calculated the following Table of the first nine powers of the first 9 numbers. 8 64 512 4096 32765262144 2097152 16777216 1342177228 9817290561 590 105314414752909 43046721 387420489 The same. The Index or Exponent of a Power, is the number denoting the height or degree of that power; and it is 1 more than the number of multiplications used in producing the So 1 is the index or exponent of the 1st power or root, 2 of the 2d power or square, 3 of the third power or cube, 4 of the 4th power, and so on. Powers, that are to be raised, are usually denoted by placing the index above the root or first power. So 2 = 4 is the 2d power of 2. 23 = 8 is the 3d power of 2. 24 = 16 is the 4th power of 2. When two or more powers are multiplied together, their product is that power whose index is the sum of the exponents of the factors or powers multiplied. Or the multiplication of the powers, answers to the addition of the indices. Thus, in the following powers of 2, Ist 2d 2 or 21 3d 4th 5th 6th 7th 8th 32 64 123 256 24 25 26 27 28 9th 10th 512 1024 2' 210 Here, 4 X 4 = 16, and 2 + 2 = 4 its index ; OTHER EXAMPLES. 1. What is the 2d power of 45 ? Ans. 2025. Ans: 17.3056. 2. What is the square of 4:16 ? 3. What is the 3d power of 3.5 ? Ans. 42.875. 4. What is the 5th power of .029? Ans. •000000020511149. 5. What is the square of } ? Ans. 4. 6. What is the 3d power of {? Ans. 24. 7. What is the 4th power of f? Ans. EVOLUTION. EVOLUTION EVOLUTION, or the reverse of Involution, is the extracting or finding the roots of any given powers. The root of any number, or power, is such a number, as being multiplied into itself a certain number of times, will produce that power. Thus, 2 is the square root or 2d root of 4, because 22 = 2 x 2 = 4; and 3 is the cube root or 3d root of 27, because 33 = 3 X 3 X 3 = 27. Any power of a given number or root may be found exactly, namely, by multiplying the number continually into itself. But there are many numbers of which a proposed root can never be exactly found. Yet, by means of decimals, we may approximate or approach towards the root, to any degree of exactness. Those roots which only approximate, are called Surd roots; but those which can be found quite exact, are called Rational Roots. Thus, the square root of 3 is a surd root ; but the square root of 4 is a rational root, being equal to 2: also the cube root of 8 is rational, being equal to 2 ; but the cube root of 9 is surd or irrational. Roots are sometimes denoted by writing the character ✓ before the power, with the index of the root against it. Thus, the 3d root of 20 is expressed by 320; and the square root or 2d root of it is </20, the index 2 being always omitted, when only the square root is designed. When the power is expressed by several numbers, with thie sign + or – between them, a line is drawn from the top of the sign over all the parts of it: thus the third root of 45 - 12 is45 12, or thus (45 – 12), inclosing the numbers in parentheses. But all roots are now often designed' like powers, with fractional indices : thus, the square root of 8 is 87, the cube root of 25 is 255, and the 4th root of 45 - 18 is 45 -- 187, or (45-18). TO |