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 × 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 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 81, the cube root of 25 is 25, and the 4th root of 45-18 is (45 - 18). 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. 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)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 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: 1st divisor a) a2 + 2ab+b2 (a+b the root. Again, for a root of three parts, a, b, c, thus: (a+b+c) s =a2+2ab+b2+2ac+2bc+c2 = a3 -(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 e the third term; also a the first divisor, 2a+b the second, and 2a +26+ the third, each consisting of the double of the root increased by the next term of the same. And the mode of extraction agrees with the rule. See farther, Case 2, of Evolution in the Algebra. For an approximation observe that vas+na. all cases where n is small in respect of a. VOL. I. 12 4a3 + 3n nearly in 4a2+ n 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 it 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. EXAMPLES. 1. To find the square root of 29506624. 29506624 (5432 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 Deci. mals; thus, 2. To find the root of 2 to nine places of figures. 2 (1.41421356 the root. 3. What is the square root of 2025? Ans. 45. Ans. 4.16. Ans. •027. Ans. 1.732050. Ans. 2.236068. 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; which 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. 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 4th 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: |