Call the last column the _dividend_, the last but one the _divisor_, and all that come before _antecedents_. See how often the dividend contains the divisor; this gives the guess at the next figure. The guess is a true one,[66] if, on applying Horner’s process, the divisor result, augmented as it is by the antecedent processes, still go as many times in the dividend. For example, in the case before us, 96453 contains 64037 once; let 1 be put on its trial. Horner’s process is found to succeed, and we have for the second process, 2 160 4801 64037 96453 162 4963 69000 27453 164 5127 74127 166 5293 168 As soon as we come to the fractional portion of the root, the process assumes a more[67] methodical form. The equation being of the _fourth_ degree, annex _four_ ciphers to the dividend, _three_ to the divisor, _two_ to the antecedent, and _one_ to the previous antecedent, leaving the first column as it is; then find the new figure by the dividend and divisor, as before,[68] and apply Horner’s process. Annex ciphers to the results, as before, and proceed in the same way. The annexing of the ciphers prevents our having any thing to do with decimal points, and enables us to use the quotient-figures without paying any attention to their _local_ values. The following exhibits the whole process from the beginning, carried as far as it is here intended to go before beginning the contraction, which will give more figures, as in the rule for the square root. The following, then, is the process as far as one decimal place: [66] Various exceptions may arise when an equation has two nearly equal roots. But I do not here introduce algebraical difficulties; and a student might give himself a hundred examples, taken at hazard, without much chance of lighting upon one which gives any difficulty. [67] This form might be also applied to the integer portions; but it is hardly needed in such instances as usually occur. See the article _Involution and Evolution_ in the _Supplement_ to the _Penny Cyclopædia_. [68] After the second step, the trial will rarely fail to give the true figure. 2 0 1 -3 416793(213 40 801 16017 96453 80 2401 64037 ----- 120 4801 ----- 274530000 160 ---- 69000 47339778 --- 4963 74127000 --------- 162 5127 -------- 164 529300 75730074 166 ------ 77348376 1680 534358 ---- 539434 1686 544528 1692 1698 1704 ----