and third period. Proceed in this manner till all the periods are brought down. If the last remainder is 0, the number is a perfect square, and the root found, its exact square root. If the last remainder is less than twice the root, plus 1, the number is not a perfect square. In the latter case, annex periods of two zeros each to the remainder, and continue the calculation. The approximation to the true value of the root may be made thus with any required degree of accuracy. The result obtained contains one decimal figure for every period of zeros annexed to the remainder. If the proposed number is a decimal fraction, render the number of figures even, and separate them into periods of two figures each, beginning with the place of tenths. Then find the square root as in the case of a whole number. This root contains one decimal figure for every period of decimal figures in the number. If the first period consists of zeros, the first figure of the root is a zero; if the first and second periods are zeros, the first and second figures of the root are zeros, and so on. If the proposed number consists of an integral part and a decimal part, render the number of decimal figures even; separate the number into periods of two figures, reckoning the periods from the decimal point to the left in the integral part, and to the right in the decimal part; extract the square root as if the number were integral, and point off, from the result, one decimal figure for each period of decimal figures in the number, and of zeros annexed to the remainder. To extract the square root of a vulgar fraction:. If the terms of the fraction are square numbers, or capable of reduction to square numbers, extract the square root of the numerator for the numerator of the root, and the square root of the denominator for its denominator. If the denominator only of the fraction is a square number, find the approximate square root of the numerator, the exact square root of the denominator, and divide the former by the latter. If neither the numerator nor denominator is a square number, reduce this case to the last, by multiplying the terms of the fraction by its denominator; or reduce the vulgar to a decimal which the different figures of a number that Lamber were known, it would be power into the parts due to the ys and thus to find these figures, and Let a, b, c, d, represent the values of the consecutive figures of a number, a being the highest figure. Then Whence the cube of a number is equal to the cube of the first figure, plus three times the square of the first figure into the second figure, plus three times the first figure into the square of the second figure, plus the cube of the second figure; plus three times the square of the number expressed by the first and second figures into the third figure, plus three times the number expressed by the first and second figures into the square of the third figure, plus the cube of the third figure; plus, &c. Taking into consideration the 10, 100, &c. by which the absolute values of the figures of the root are multiplied, in order to give them their relative values, it is found that the figure expressing the simple units of the root gives three or two significant figures, or one only, to the power (Art. 38.); that the figure expressing the tens (which may be represented generally by 10n, or n× 10, the cube of which is n3 × 1000) gives zeros in the places of the three last figures of the power, and significant figures in the sixth, fifth, and fourth places, or in the fifth and fourth places, or in the fourth place only of the power; that the figure expressing the hundreds (n × 100, the cube of which is n3 × 1000000) gives zeros in the places of the six last figures of the power, and significant figures in the ninth, eighth, and seventh places, or in the eighth and seventh places, or in the seventh place only. . ... Whence, if a number be separated into periods of three figures each, the significant figures arising from the cube of the simple units of the cube root of the number are found in the lowest period or period on the right; the significant figures arising from the cube of the tens in the next period; . . . and the fraction, and find the square root of the latter by the proper rule. 129. Examples in the formation of the square, and the extraction of the square root of numbers :— Find the squares of the numbers which follow, viz.,— 2. 441. 3. 9409. 4. 34596. 5. 335241. 6. 649636. 5. 9. 09. 10. 002. Extract the square root of each of the following numbers, 1. 169. 130. If the manner in which the different figures of a number enter into the cube of that number were known, it would be possible to decompose the power into the parts due to the several figures of the root, and thus to find these figures, and consequently the root itself. .... Let a, b, c, d, represent the values of the consecutive figures of a number, a being the highest figure. Then (a+b)s=a3 +3a2b+3ab2 + b3 (a+b+c)s=a3 +3a2b+3ab2 + b3 (a+b+c+d+ .. )3=a3+3a2b+3a b2 + b3 +3(a+b)°c + 3 (a + b)c2 + c3 +3(a+b)2c+3(a + b) c2 + c3 + Whence the cube of a number is equal to the cube of the first figure, plus three times the square of the first figure into the second figure, plus three times the first figure into the square of the second figure, plus the cube of the second figure; plus three times the square of the number expressed by the first and second figures into the third figure, plus three times the number expressed by the first and second figures into the square of the third figure, plus the cube of the third figure; plus, &c. Taking into consideration the 10, 100, &c. by which the absolute values of the figures of the root are multiplied, in order to give them their relative values, it is found that the figure expressing the simple units of the root gives three or two significant figures, or one only, to the power (Art. 38.); that the figure expressing the tens (which may be represented generally by 10n, or n × 10, the cube of which is n3 × 1000) gives zeros in the places of the three last figures of the power, and significant figures in the sixth, fifth, and fourth places, or in the fifth and fourth places, or in the fourth place only of the power; that the figure expressing the hundreds (n x 100, the cube of which is n3 × 1000000) gives zeros in the places of the six last figures of the power, and significant figures in the ninth, eighth, and seventh places, or in the eighth and seventh places, or in the seventh place only. Whence, if a number be separated into periods of three figures each, the significant figures arising from the cube of the simple units of the cube root of the number are found in the lowest period or period on the right; the significant figures arising from the cube of the tens in the next period; . . . and the |