EXAMPLES. 1. Reduce 1 dwt. to the decimal of a pound troy 201 dwt. 12 0.05 oz. 0004166, &c. lb. answer. 2. Reduce 9d. to the decimal of a pound. 5. Reduce 2-15 lb. to the decimal of a cwt. 6. Reduce 24 yards to the decimal of a mile. 7. Reduce 056 poles to the decimal of an acre. 8. Reduce 1-2 pints of wine to the decimal of a hhd. 9. Reduce 14 minutes to the decimal of a day. 10. Reduce 21 pints to the decimal of a peck. Ans. *03751. Ans. 02734375 lb. Ans. 0010833, &c. £. Ans. 019196 + cwt. Ans. 013636, &c. miles. Ans. 00035 ac. Ans. 00238+ hhd. Ans. 009722, &c. da. Ans. 013125 pec. 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, on the right hand of the dividend next below it; so shall the last quotient be the decimal required. RULE. Prepare the terms by reducing the vulgar fractions to decimals, 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 here to show the method. If of a yard of velvet cost, what willyd. cost? DUODECIMALS, or CROSS MULTIPLICATION, is a rule made use of 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. RULE.-Set down the two dimensions, to be multiplied together, one under the other, so that feet 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 these are of a foot. Then add the two lines together, after the manner of Compound Addit'on, carrying 1 to the feet for 12 inches, when these come to so many. 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= 2×2= 2 × 2 × 2 = 2 × 2 × 2 × 2 2 is the root, or first power of 2. 8 is the 3d power, or cube of 2. And in this manner may be calculated the following Table of the first nine powers of the first nine numbers. 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 same. So 1 is the index or exponent of the 1st power or root, 2 of the 2d power or square, 3 of the 3d 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. When two or more powers are multiplied together, their product will be that power whose index is the sum of the exponents of the factors or powers multi plied. Or the multiplication of the powers, answers to the addition of the indi. ces. Thus, in the following powers of 2. 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 92 = 2 × 2 = 4; and 3 is the cube root or 3d root of 27, because 3 = 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. before the power, Roots are sometimes denoted by writing the character with the index of the root against it. Thus, the third root of 20 is expressed by 20; and the square root or 2d root of it is √/20, the index 2 being always omitted, when the square root is designed. When the power is expressed by several numbers, with the signor 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 is {/45 — 12, or thus, ¡/(45 the numbers in parentheses. 12), inclosing But all roots are now often designed like powers, with fractional indices: thus, the square root of 8 is 8 the cube root of 25 is 253 and the 4th root of TO EXTRACT THE SQUARE ROOT. RULE.* Divide the given number into periods of two figures each, by setting a point over the place of units, another over the place of hundreds, and so on, over every second figure, both to the left hand in integers, and to the right in decimals. Find the greatest square in the first period on the left hand, and set its root on the right hand of the given number, after the manner of a quotient figure in Division. Subtract the square thus found from the said period, and to the remainder annex the two figures of the next following period, for a dividend. Double the root above mentioned for a divisor; and find how often it is contained in the said dividend, exclusive of its right hand figure; and set that quotient figure both in the quotient and divisor. Multiply the whole augmented divisor by this last quotient figure, and subtract the product from the said dividend, bringing down to the next period of the given number, for a new dividend. Repeat the same process over again, viz. find another new divisor, by doubling all the figures now found in the root; from which, and the last dividend, find the next figure of the root as before; and so on through all the periods, to the last. Note. The best way of doubling the root, to form the new divisors, is by adding the last figure always to the last divisor, as appears in the following examples.— Also, after the figures belonging to the given number are all exhausted, the operation may be continued into decimals at pleasure, by adding any number of periods of ciphers, two in each period. The reason for separating the figures of the dividend into periods or portions of two places each, is, that the square of any single figure never consists of more than two places; the square of a number of two figures, of not more than four places, and so on. So that there will be as many figures in the root as the given number contains periods so divided or parted off. And the reason of the several steps in the operation, appears from the algebraic form of the square of any number of terms, whether two, or three, or more. Thus, a+b)* = a® +2ab+ba = aa + 2a + b. b, the square of two terms; where it appears, that a is the first term of the root, and b the second term ; al-o a the first divisor, and the new divisor is 2a + b, or double the first term increased by the second. And hence the manner of extraction is thus: 1st division a) aa + 2ab + b ( a + b the root. 2d divisor 2a + b | 2 ab + ba b2ab+b2 Again, for a root of three parts a, b, c, thus: a + b + c2 = «2 + 20h + 12 + 2ac + 2bc + c2 = a2 + 2a + b . b + 2a + 2b + c . c, the square of three terms; where a is the first term of the root, b the second, and c the third term; also a the first divisor, 2a + b the second, and 2a +26 +e the third, each consisting of the double of the root increased by the next term of the same. And the mode of extraction is thus: 1st divisor a) aa + 2ab + b2 +2ac + 2bc + ca (a+b+c the root. |