SECTION II. INVOLUTION AND EVOLUTION. INVOLUTION is the method of raising any number, considered as the root, to any required power. Any number, whether given, or assumed at pleasure, may be called the root, or first power of this number; and its other powers are the products, that result from multiplying the number by itself, and the last product by the same number again; and so on to any number of multiplications. The index, or exponent, is the number denoting the height, or degree of the power, being always greater by one, than the number of multiplications employed in producing the power. It is usually written above the root, as in the following EXAMPLE, where the method of involution is plainly exhibited. Required the fifth power of 8 = the root, or first power. first multiply by 8 then multiply the product 64 82 square, or second power. by 8 &c. 512 = 83 = cube, or third power. 8 4096 84 = biquadrate, or fourth power. 8 32768 65 = Answer. EXAMPLES FOR EXERCISE. What is the second power of 3.05? Ans. 9.3025. Note. When two, or more powers are multiplied together, their product is that power, whose index is the sum of the indices of the factors, or powers multiplied. EVOLUTION is the method of extracting any required root from any given power. Any number may be considered as a power of some other number; and the required root of any given power is that number, which, being multiplied into itself a particular number of times produces the given power; thus, if 81 be the given number, or power, its square, or second root is 9; because 9×9=92-81; and 3 is its biquadrate, or fourth root, because 3x3x3x3=3+=81. Again, if 729 be the given power, and its cube root be required, the answer is 9, for 9 × 9 x9=729; and if the sixth root of that number be required, it is found to be 3, for 3x3x3x3x3x3=729. The required power of any given number, or rout, can always be obtained exactly, by multiplying the number continually into itself; but there are many numbers, from which a proposed root can never be completely extracted;-yet by approximating with decimals, these roots may be found as exact as necessity requires. The roots that are found complete, are denominated rational roots, and those which cannot be found completed, or which only approximate, are called surd, or irrational roots. Roots are usually represented by these characters or exponents; ✔, or which signifies the square root; thus, ✓ 9, or 93-3 1 3/, or cube root; or √64, or 643–4 =4 Likewise 8 signifies the square root of 8 cubed; and, in general, the fractional indices imply that the given numbers are to be raised to such powers as are denoted by their numerators, and that such roots are to be extracted from these powers, as are denoted by their denominators. RULE For extracting the Square Root. Commencing at the unit figure, cut off periods of two figures each, till all the figures of the given number are exhausted.* The first figure of the required root will be the square root of the first period, or of the greatest square root contained in it, if it be not a square itself. Subtract the square of this figure from the first period; to the remainder annex the next period for a dividend; and for part of a divisor, double the part of the root already obtained. Try how often this part of the divisor is contained in the divi *In dividing a decimal, or a number consisting of a whole number with a decimal, into periods, the division must also commence at the unit figure or decimal point, and must be continued both ways, if there be a whole number; and if there be an odd figure at the end of the decimal, a cipher, or if it be a periodical decimal, the figure that would next arise, from its continuation, must be annexed; thus 417.245 will be divided thus, 4'17.24/50: 41.66666, &c. thus, 41', 66'66'66: and .567 thus, 56'70, &c. See the Editor's "Elementary Treatise on Arithmetic, in Theory and Practice," page 219. ED. dend wanting the last figure, and annex the figures thus found to the parts of the root and of the divisor already determined. Thus multiply and subtract as in division; to the remainder bring down the next period; and, adding to the divisor the figure of the root last found, proceed as before.* If any thing remain after continuing the process till all the figures in the given number have been used, proceed in the same manner to find decimals, adding, to find each figure, two ciphers, or if the given number end in an interminate decimal, the two figures that would next arise from its continuation. To extract the root of the fraction, reduce it to its simplest form, if it be not so already, and extract the root of both terms, if they be complete powers: otherwise divide the root of their product by the denominator. The root may also be found by reducing the fraction to a decimal, if it be not such already, and taking the root of the decimal. *The principle on which the preceding rule depends, is, that the square of the sum of two numbers is equal to the squares of the numbers with twice their product. Thus, the square of 34 is equal to the squares of 30 and of 4 with twice the product of 30 and EXAMPLES FOR EXERCISE? Required the square root of 298116. Ans. 546. Required the square root of .000729. Ans. .027. Required the square root of 173. Ans. 4.168333+ TO EXTRACT THE CUBE ROOT. RULE I.-Commencing at the unit figure, cut off periods of three figures each, till all the figures of the given number are exhausted. Then find the greatest cube number contained in the first period, and place the cube root of it in the quotient. Subtract its cube from the first period, and bring down the next three figures; divide the number thus brought down by 300 times the square of the first figure of the root, and it will give the second figure; add 300 times the square of the first figure, 30 times the product of the first and second figures, and the square of the second figure together, for a divisor; then multiply this divisor by the second figure, and subtract the result from the dividend, and then bring down the next period, and so proceed till all the periods are brought down.* 4; that is, to 900+2×30×4+16=1156. Here, in extracting the second root of 1156, we separate it into two parts. 1100 and 56. Thus 1100 contains 900, the square of 30, with the remainder 200; the first part of the root is therefore 30, and the remainder 200+56, or 256. Now according to the principle above mentioned, this remainder must be twice the product of 30, and the part of the root still to be found, together with the square of that part. Now, dividing 256 by 60, the double of 30, we find for quotient 4; then this part being added to 60, the sum is 64, which being multiplied by 4, the product 256, is evidently twice the product of 30 and 4, together with the square of 4. In the same manner the operation may be illustrated in every case. The rule, however, is best demonstrated by Algebra. See my Treatise on the subject, page 231. second edition. ED. * The reason of this rule will appear evident from the following illustration. The cube of 25, for instance, is equivalent to the cube of 20 added to the cube of 5, together with the sum of 300X4X5+30×2×5×5; or, which is the same thing, 25 is equal to 20-+-5, and therefore 25 cubed is equal to 20+5 cubed; but 20-+-5 cubed is equivalent to 8000+300×4×5+30×2×5×5+125, or to 203+(300x4+30×2 X5+5×5)×5=48228544. 20×20×20+3×5×20×20+3×20×25+125 = third power. or, 8000+300x4×5+30×2×25+125. Here the rule is evident. In the same manner, the operation may be illustrated in every case. For a demonstration of this rule in general terms, the reader is referred to the Editor's "Treatise on Algebra, Theoretical and Practical," Ed. To extract the cube root of a fraction, reduce it to a decimal, and ther extract the root; or multiply the numerator by the square of the denominator, find the cube root of the product, and divide by the denominator. The cube root of a mixed number is generally best found by reducing the fractional part to a decimal, if it be not so already, and then extracting the root. It may be also found by reducing the given number to an improper fraction, and then working according to the preceding directions. EXAMPLES. 1. Required the cube root of 48228.544. 127 Divisor 2790 21228 Resolvend. 32 x 300×6=16200 3 x 30x62 =3240 1572.544 Resolvend. Ex. 2. What is the cube root of 62570773? Ans. 397. 2. To Extract the Cube Root by another Method.* 1. By trials find the nearest rational cube to the given number, whether it be greater or less; and call it the assumed cube. 2. Then say, by the Rule of Three, as the sum of the given number and double the assumed cube, is to the sum of the assumed and double the given number, so is the root of the assumed cube, to the root required, nearly. Or, as the first sum is to the difference *This rule is found in Hutton's Mathematics. There has been different rules given for extracting the cube root, among which, this, and another rule given in Pike's |