Set down the number whose square root is wanted, say 43. This square root is 6 and a fraction. Set down the integer 6 in the first and third row, and 1 in the second row always. Form the successive rows each from the one before, in the following manner: One row The next row has _b′_, _a′_, _c′_, formed in this order, being thus, _a_ _a′_ = excess of _b′c′_, already formed, over _a_. _b_ _b′_ = quotient of 43 - _a_² divided by _b_. _c_ _c′_ = integer in the quotient of 6 + _a_ divided by _b′_. Thus the second row is formed from the first, as under: 6|1 = excess of 7 × 1 (both just found) over 6. 1|7 = 43 - 6 × 6 divided by 1. --+-- 6|1 = integer of 6 + 6 divided by 7 (just found). The third row is formed from the second, thus: 1 5 = excess of 1 × 6 over 1. 7 6 = 43 - 1 × 1 divided by 7. 1 1 = integer of 6 + 1 divided by 6; and so on. In process of time the second column, 1, 7, 1, occurs again, after which the several columns are repeated in the same order. As a final process, take the set in the lowest line (excluding the first, 6), namely, 1, 1, 3, 1, 5, 1, 3, &c. and use them by the rule given at the beginning of this article, as follows: 1 1 3 1 5 1 3 1 1, &c. 1 1 4 5 29 34 131 165 296 --- --- --- --- ---- ---- ---- ---- ---- 1 2 7 9 52 61 235 296 531 Hence, 6¹⁶⁵/₂₉₆ is very near the square root of 43, not erring by so much as 1 ---------. 296 × 531 If we try it, we shall find (⁶¹⁶⁵/₂₉₆) to be ¹⁹⁴¹/₂₉₆, the square of which is ³⁷⁶⁷⁴⁸¹/₈₇₆₁₆, or 43⁷/₈₇₆₁₆. This rule is of use when it is frequently wanted to use one square root, and therefore desirable to ascertain whether any easy approximation exists by means of a common fraction. For example, √2 is often used. _ √2 = 1 + ... 1|1 1 1|1 1 1|2 2 2 2 2 2 1 2 5 12 29 70 --- --- --- --- --- ---, &c. 2 5 12 29 70 169 Here it appears that 29 1 99 100 - 1 1---- does not err by --------; consequently, ---- or ------- is, 70 70 × 169 70 70 considering the ease of the operation, a fair approximation. In fact, ⁹⁹/₇₀ is 1·4142857 ... the truth being 1·4142135 ... The following is an additional example: __ √19 = 4 + ... 4 | 2 3 3 2 4 4 2 1 | 3 5 2 5 3 1 3 4 | 2 1 3 1 2 8 2 1 3 1 2, &c. 1 1 4 5 14 --- --- --- --- ---, &c. 2 3 11 14 39 APPENDIX IX. ON SOME GENERAL PROPERTIES OF NUMBERS.