| 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:
| 2 | 0 | 1 | -3 | 416793(213 |
| 40 | 801 | 16017 | 96453 | |
| 80 | 2401 | 64037 | 274530000 | |
| 120 | 4801 | 69000 | 47339778 | |
| 160 | 4963 | 74127000 | ||
| 162 | 5127 | 75730074 | ||
| 164 | 529300 | 77348376 | ||
| 166 | 534358 | |||
| 1680 | 539434 | |||
| 1686 | 544528 | |||
| 1692 | ||||
| 1698 | ||||
| 1704 | ||||
If we now begin the contraction, it is good to know beforehand on what number of additional root-figures we may reckon. We may be pretty certain of having nearly as many as there are figures in the divisor when we begin to contract—one less, or at least two less. Thus, there being now eight figures in the divisor, we may conclude that the contraction will give us at least six more figures. To begin the contraction, let the dividend stand, cut off one figure from the divisor, two from the column before that, three from the one before that, and so on. Thus, our contraction begins with
| 0002 | 1 | 704 | 5445 | 28 | 7734837 | 6 | 47339778 | |
The first column is rendered quite useless here. Conduct the process as before, using only the figures which are not cut off. But it will be better to go as far as the first figure cut off, carrying from the second figure cut off. We shall then have as follows:
| 1 | 704 | 5445 | 28 | 7734837 | 6 | 47339778(6 |
| 5445 | 5 | 7767570 | 6 | 734354 | ||
| 5465 | 7 | 7800364 | 8 | |||
| 5475 | 9 | |||||
At the next contraction the column 1|704 becomes |001704, and is quite useless. The next step, separately written (which is not, however, necessary in working), is
| 54 | 759 | 780036 | 48 | 734354(0 |