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 conse quence. 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 multipli cand; 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. 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 = 2x2= 2 X 2 X 2 = 2 x 2 x 2 x 2 = 2 is the root, or 1st power of 2. And in this manner may be calculated the following Table of the first nine powers of the first 9 numbers. 7 49 '343 2401 16807 117649 8235435764801 40353607 8 64 512 4096 32768 262144 2097152 16777216 134217728 981 729 6561 59049 531441 4782969 43046721 387420489 The The Index or Exponent of a Power, is the number de noting 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 first power or root, two 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. 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, OTHER EXAMPLES. 1. What is the 2d power of 45 ? Ans. 2025. Ans. 17.3056. Ans. 42.875.1 Ans. 000000020511149. 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 20; 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 the 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 is 345 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 8, the cube root of 25 is 253, and the 4th root of 45 18 is 45 or (45-18)4. 18), TO TO EXTRACT THE SQUARE ROOT. * 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. * 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 twe or three or more. Thus, 2 (a + b)2=a2 + 2ab+b2 =a2 + (2a + b) b, the square of two terms; where it appears that a is the first term of the root, and the second term; also 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 divisor a) u2 + 2ab + b2 ( a + b the root. Again, for a root of three parts, a, b, c, thus: (a + b + c) 2 a2 +2ab+b2 + 2ac + 2bc + c2± a2 + (2a + b) b + (2a + 26 +c) c, the square of three terms, where a is the first term of the root b, the second, and e the third term; also a the first divisor, 2a + the second, and 2a +26+c 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) az +2ab + b2 + 2ạc + 2bc + c2 (a + b + c the root. |