SQUARE AND SQUARE ROOT are particular cases of involution and evolution (q.v.), in which the second power and root are alone involved. The process by which the square root of a number is obtained resembles division, differing only by the circumstance that the divisor is changed at each successive step. The rule adopted in arithmetic is deduced from algebra in the following manner: The square of a b is al 2ab which may be written at 4-142a +1)); and to find the square root of the latter we have merely to subtract a portion (at), taking care that it he a square number, and forming a divisor with twice the square root of portion (2a) increased by (h) the remainder of the root (which, in arithmetic, must be found by trial, as in division), and putting (•) the remain der of the root now found, in the quotient, proceed as in division. This mode of obtain ing a divisor from the part of the root already obtained (a). and the part next to be obtained (b), and employing it, must lie repeated till the whole square root is found. In the extraction of the square root in arithmetic it is assumed that the squares of the nine digits are known, and also that the square of a number contains either twice, or one less titan twice, as many digits as the number itself contains, the former being the case when the square number has an even number of digits, the latter when the number of digits is odd. By dividing, then, a number into periods of two each, we can at
once see how many digits its root contains. To illustrate the method of operation adopted in arithmetic and algebra, let the square root of 128,881 be required; remem bering that the square of a+b-Fe is tab + 2(a+ N0+0: 12,83,81(300+50 (or 350) + 9 = 359 a = SOO 12,68,31(300 = =) 3002 = 90000 a + b a+ 0+ c 300 90000 50 = b 38381 2ct + b )38881 in 50 32500 330 (2(5 + 5)6 6381 2(a+ 5) + c = i09 )G.31 X 350X = 6381 6381 •In the common arithmetical mode the zeros are omitted, and we subtract from 12 the square nearest to it, not recognizing the portion of the root, 3, as more than a digit of units till the next period. 88, has been brought down for the second step, when it is evi dent that the 3 is at least 3 tens, and consequently the 6 in the divisor represents 60; similarly, it is only at the commencement of the third step that we find the 5 to repre sent 50, and the 3, 300, A comparison of the above examples will show the agreement and difference between the two modes.