Having done this, take the highest figure of the root, properly named, which is 2 tens, or 20. Begin with the first column, multiply by 20, and join it to the number in the next column; multiply that by 20, and join it to the number in the next column; and so on. But when you come to the last column, subtract the product which comes out of the preceding column, or join it to the last column after changing its sign. When this has been done, repeat the process with the numbers which now stand in the columns, omitting the last, that is, the subtracting step; then repeat it again, going only as far as the last column but two, and so on, until the columns present a set of rows of the following appearance:
| a | b | c | d | e |
| f | g | h | i | |
| k | l | m | ||
| n | o | |||
| p | ||||
to the formation of which the following is the key:
- f = 20a + b,
- g = 20f + c,
- h = 20g + d,
- i = e - 20h,
- k = 20a + f,
- l = 20k + g,
- m = 20l + h,
- n = 20a + k,
- o = 20n + l,
- p = 20a + n.
We call this Horner’s Process, from the name of its inventor. The result is as follows:
| 2 | 0 | 1 | -3 | 416793 | (20 |
| 40 | 801 | 16017 | 96453 | ||
| 80 | 2401 | 64037 | |||
| 120 | 4801 | ||||
| 160 | |||||
We have now before us the row
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,[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,