167. The rule, then, for extracting the square root of a number or decimal to any number of places is: Annex ciphers until there are twice as many places following the units’ place as there are to be decimal places in the root; extract the nearest square root of this number, and mark off the given number of decimals. Or, more simply: Divide the number into periods, so that the units’ figure shall be the last of a period; proceed in the usual way; and if, when decimals follow the units’ place, there is one figure on the right, in a period by itself, annex a cipher in bringing down that period, and afterwards let each new period consist of two ciphers. Place the decimal point after that figure in forming which the period containing the units was used. 168. For example, what is the square root of (1⅜) to five places of decimals? This is (145) 1·375, and the process is the first example over leaf. The second example is the extraction of the root of ·081 to seven places, the first period being 08, from which the cipher is omitted as useless. 1,37,5(1·17260 1 --- 21) 37 21 ---- 227)1650 1589 ------ 2342) 6100 4684 ------ 23446)141600 140676 -------- 23452) 92400 8,1(·2846049 4 --- 48)410 384 ------ 564) 2600 2256 ------ 5686) 34400 34116 --------- 569204) 2840000 2276816 --------- 569208) 56318400 ·000002413672221(·001553599 1 --- 25) 141 125 ----- 305) 1636 1525 ------ 3103) 11172 9309 ------ 31065) 186322 155325 -------- 310709) 3099710 2796381 --------- 30332900 169. When more than half the decimals required have been found, the others may be simply found by dividing the dividend by the divisor, as in (155). The extraction of the square root of 12 to ten places, which will be found in the next page, is an example. It must, however, be observed in this process, as in all others where decimals are obtained by approximation, that the last place cannot always be depended upon: on which account it is advisable to carry the process so far, that one or even two more decimals shall be obtained than are absolutely required to be correct.