APPENDIX XI. ON HORNER’S METHOD OF SOLVING EQUATIONS.

2  160  4801  64037    96453

which furnishes our means of guessing at the next, or units’ figure of the root.

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,¹ 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,

As soon as we come to the fractional portion of the root, the process assumes a more² 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,³ 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:

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