[23] By square number I mean, a number which has a square root. Thus, 25 is a square number, but 26 is not. [24] The term ‘root’ is frequently used as an abbreviation of square root. The process of forming the numbers to be subtracted may be shortened thus. Let A be the sum of the parts already found, and N a new part: there must then be subtracted 2AN + NN, or (54) 2A + N multiplied by N. The rule, therefore, for forming it is: Double the sum of all the preceding parts, add the new part, and multiply the result by the new part. 162. The process of the last article is as follows: 7,61,76(200 7,61,76(276 4 00 00 70 4 ------- 6 --- 400)3,61,76 47)361 70)3 29 00 329 ------- ----- 400) 32 76 546)3276 140) 32 76 3276 6) ----- ---- 0 0 In the first of these, the numbers are written at length, as we found them; in the second, as in (79), unnecessary ciphers are struck off, and the periods 61, 76, are not brought down, until, by the continuance of the process, they cease to have ciphers under them. The following is another example, to which the reasoning of the last article may be applied. 34,86,78,44,01(50000 34,86,78,44,01(59049 25 00 00 00 00 9000 25 -------------- 40 ---- 100000) 9 86 78 44 01 9 109) 986 9000) 9 81 00 00 00 981 ------------- ------- 100000) 5 78 44 01 11804) 57844 18000) 4 72 16 00 47216 40) ----------- -------- 100000) 1 06 28 01 118089)1062801 18000) 1 06 28 01 1062801 80) ---------- ------- 9) 0 0 163. The rule is as follows: To extract the square root of a number;-- I. Beginning from the right hand, cut off periods of two figures each, until not more than two are left. II. Find the root of the nearest square number next below the number in the first period. This root is the first figure of the required root; subtract its square from the first period, which gives the first remainder. III. Annex the second period to the right of the remainder, which gives the first dividend. IV. Double the first figure of the root; see how often this is contained in the number made by cutting one figure from the right of the first dividend, attending to IX., if necessary; use the quotient as the second figure of the root; annex it to the right of the double of the first figure, and call this the first divisor.