INVOLUTION is the raising of Powers from any given number, as a root. 2= 2X2= 2 X 2 X 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 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 third root of 20 is expressed by 3/20; 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 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 3/45 − 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 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 num ber 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) 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 b 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: 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+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) a + 2ab + b2 + 2ac + 2bc + c (a+b+c the root. NOTE. When the root is to be extracted to many places of figures, the work may be considerably shortened, thus: Having proceeded in the extraction after the common method till there be found half the required number of figures in the root, or one figure more; then, for the rest, divide the last remainder by its corresponding divisor, after the manner of the third contraction in Division of Decimals; thus, 2. To find the root of 2 to nine places of figures. RULES FOR THE SQUARE ROOTS OF VULGAR FRACTIONS AND MIXED NUMBERS. FIRST, prepare all vulgar fractions, by reducing them to their least terms, both for this and all other roots. Then, 1. Take the root of the numerator and of the denominator for the respective terms of the root required. And this is the best way if the denominator be a complete power: but if it be not, then, 2. Multiply the numerator and denominator together; take the root of the product: this root being made the numerator to the denominator of the given fraction, or made the denominator to the numerator of it, will form the fractional root required. And this rule will serve whether the root be finite or infinite. 3. Or reduce the vulgar fraction to a decimal, and extract its root. 4. Mixed numbers may be either reduced to improper fractions, and extracted by the first or second rule; or the vulgar fraction may be reduced to a decimal, then joined to the integer, and the root of the whole extracted. By means of the square root also may readily be found the 4th root, or the 8th root, or the 16th root, &c.; that is, the root of any power whose index is some power of the number 2; namely, by extracting so often the square root as is denoted by that power of 2; that is, two extractions for the fourth root, three for the 8th root, and so on. So, to find the 4th root of the number 21035 8, extract the square root two times as follows; 20 |