dividing the index of the power by the index of the root, indicating the division by a fraction. Thus, extract the square root of the 6th power of 2: √/26 = 24 = 212 = 2* = 8. The 6th power of 2, as in the table above, is 64; 1/64 = 8. The Difficult problems in evolution are performed by logarithms, but the square root and the cube root may be extracted directly according to the rules given below. The 4th root is the square root of the square root. 6th root is the cube root of the square root, or the square root of the cube root; the 9th root is the cube root of the cube root; etc. To Extract the Square Root.-Point off the given number into periods of two places each, beginning with units. If there are decimals, point these off likewise, beginning at the decimal point, and supplying as many ciphers as may be needed. Find the greatest number whose square is less than the first left-hand period, and place it as the first figure in the quotient. Subtract its square from the left-hand period, and to the remainder annex the two figures of the second period for a dividend. Double the first figure of the quotient for a partial divisor: find how many times the latter is contained in the dividend exclusive of the right-hand figure, and set the figure representing that number of times as the second figure in the quotient, and annex it to the right of the partial divisor, forming the complete divisor. Multiply this divisor by the second figure in the quotient and subtract the product from the dividend. To the remainder bring down the next period and proceed as before, in each case doubling the figures in the root already found to obtain the trial divisor. Should the product of the second figure in the root by the completed divisor be greater than the dividend, erase the second figure both from the quotient and from the divisor, and substitute the next smaller figure, or one small enough to make the product of the second figure by the divisor less than or equal to the dividend. 3.1415926536 1.77245+ 1 27/214 189 347 2515 2429 3542,8692 35444 160865 354485 1908936 To extract the square root of a fraction, extract the root of numerator To Extract the Cube Root.-Point off the number into periods of 3 figures each, beginning at the right hand, or unit's place. Point off decimals in periods of 3 figures from the decimal point. Find the greatest cube that does not exceed the left-hand period; write its root as the first figure in the required root. Subtract the cube from the left-hand period, and to the remainder bring down the next period for a dividend. Square the first figure of the root; multiply by 300, and divide the product into the dividend for a trial divisor; write the quotient after the first figure of the root as a trial second figure. Complete the divisor by adding to 300 times the square of the first figure, 30 times the product of the first by the second figure, and the square of the second figure. Multiply this divisor by the second figure; subtract the product from the remainder. (Should the product be greater than the remainder, the last figure of the root and the complete divisor are too large; substitute for the last figure the next smaller number, and correct the trial divisor accordingly.) To the remainder bring down the next period, and proceed as before to find the third figure of the root-that is, square the two figures of the root already found; multiply by 300 for a trial divisor, etc. If at any time the trial divisor is greater than the dividend, bring down another period of 3 figures, and place 0 in the root and proceed. The cube root of a number will contain as many figures as there are periods of 3 in the number. Shorter Methods of Extracting the Cube Root.~1. From Wentworth's Algebra: After the first two figures of the root are found the next trial divisor is found by bringing down the sum of the 60 and 4 obtained in completing the preceding divisor, then adding the three lines connected by the brace, and annexing two ciphers. This method shortens the work in long examples, as is seen in the case of the last two trial divisors, saving the labor of squaring 123 and 1234. A further shortening of the work is made by obtaining the last two figures of the root by division, the divisor employed being three times the square of the part of the root already found; thus, after finding the first three figures: 3 x 1232 = 45387|20498963|45.1+ 181548 234416 226935 74813 The error due to the remainder is not sufficient to change the fifth figure of the root. 2. By Prof. H. A. Wood (Stevens Indicator, July, 1890): I. Having separated the number into periods of three figures each, counting from the right, divide by the square of the nearest root of the first period, or first two periods; the nearest root is the trial root. II. To the quotient obtained add twice the trial root, and divide by 3. This gives the root, or first approximation. III. By using the first approximate root as a new trial root, and proceeding as before, a nearer approximation is obtained, which process may be repeated until the root has been extracted, or the approximation carried as far as desired. EXAMPLE.-Required the cube root of 20. The nearest cube to 20 is 38. 32 = 9)20.0 2.2 6 3)8.1 2.7 1st T. R. 2.727.29)20.000 2.743 5.4 3)8.143 2.714, 1st ap. cube root. 2.71427.365796)20.0000000 2.7152534 3)8.1432534 2.7144178 2d ap. cube root. REMARK.-In the example it will be observed that the second term, or first two figures of the root, were obtained by using for trial root the root of the first period. Using, in like manner, these two terms for trial root, we obtained four terms of the root; and these four terms for trial root gave seven figures of the root correct. In that example the last figure should be 7. Should we take these eight figures for trial root we should obtain at least fifteen figures of the root correct. To Extract a Higher Root than the Cube.-The fourth root is the square root of the square root; the sixth root is the cube root of the square root or the square root of the cube root. Other roots are most conveniently found by the use of logarithms. ALLIGATION shows the value of a mixture of different ingredients when the quantity and value of each is known. Let the ingredients be a, b, c, d, etc., and their respective values per unit w, x, y, z, etc. A the sum of the quantities =a+b+c+d, etc. P = mean value or price per unit of A. shows in how many positions any number of things may be arranged in a row; thus, the letters a, b, c may be arranged in six positions, viz. abc, acb, cab, cba, bac, bca. Rule.-Multiply together all the numbers used in counting the things; thus, permutations of 1, 2, and 3 = 1 × 2 × 3 = 6. In how many positions can 9 things in a row be placed? 1 X 2 X 8 X 4 × 5 × 6 × 7 X 8 X 9362880. COMBINATION shows how many arrangements of a few things may be made out of a greater number. Rule: Set down that figure which indicates the greater number, and after it a series of figures diminishing by 1, until as many are set down as the number of the few things to be taken in each combination. Then beginning under the last one set down said number of few things; then going backward set down a series diminishing by 1 until arriving under the first of the upper numbers. Multiply together all the upper numbers to form one product, and all the lower numbers to form another; divide the upper product by the lower one. GEOMETRICAL PROGRESSION. How many combinations of 9 things can be made, taking 3 in each c bination? ARITHMETICAL PROGRESSION, in a series of numbers, is a progressive increase or decrease in each suc sive number by the addition or subtraction of the same amount at each s as 1, 2, 3, 4, 5, etc., or 15, 12, 9, 6, etc. The numbers are called terms, and equal increase or decrease the difference. Examples in arithmetical gression may be solved by the following formulæ : Let a = first term, I last term, d = common difference, n = numbe terms, s = sum of the terms: ents when the quantity spective values per unit -c+d, etc. d= = n 28 s may be arranged in a positions, viz. abc, acb, nting the things; thus, many positions can 9 362880. y be made out of a indicates the greater 71, until as many are in each combination. umber of few things; 1 until arriving under the upper numbers to another; divide the = n(n - 1)' = 2a +(2a 2d d)2+8ds GEOMETRICAL PROGRESSION, in a series of numbers, is a progressive increase or decrease in each cessive number by the same multiplier or divisor at each step, as 1, 2, 16. etc., or 243, 81, 27, 9, etc. The common multiplier is called the ratio. Let a first term, i = last term, r = ratio or constant multiplier, number of terms, m = any term, as 1st, 2d, etc., s = sum of the terms: -1 Estimated Population in Each Year from 1870 to 1909. The above table has been calculated by logarithms as follows: log r = log l log a + (n Pop. 1900.... 76,295,220 log 7.8824988 66 log m= log a (m-1) log r = = log l diff. = .0857703 .00857703 add log for 1890 7.7967285 Compound interest is a form of geometrical progression; the ratio being 1 plus the percentage. *Corrected by addition of 1,260,078, estimated error of the census of 1870, Census Bulletin No. 16, Dec. 12, 1890, |