To find what this is when x is any number, for instance, 3000, the best way is to take the first multiplier (2), multiply it by 3000, and take in the next multiplier (0), multiply the result by 3000, and take in the next multiplier (1), and so on to the end, as follows: 2 × 3000 + 0 = 6000; 6000 × 3000 + 1 = 18000001 18000001 × 3000 - 3 = 54000002997 54000002997 × 3000 - 416793 = 162000008574207 Now try the value of the above when _x_ = 30. We have then, for the steps, 60 (2 × 30 + 0), 1801, 54027, and lastly, 1620810-416793, or _x_ = 30 makes the first terms greater than 416793. Now try _x_ = 20 which gives 40, 801, 16017, and lastly, 320340-416793, or _x_ = 20 makes the first terms less than 416793. Between 20 and 30, then, must be a value of _x_ which makes 2_x_⁴ + _x_²-3x equal to 416793. And this is the preliminary step of the process. Having got thus far, write down the coefficients +2, 0, +1,-3, and -416793, each with its proper algebraical sign, except the last, in which let the sign be changed. This is the most convenient way when the last sign is-. But if the last sign be +, it may be more convenient to let it stand, and change all which come before. Thus, in solving _x_³-12_x_ + 1 = 0, we might write -1 0 +12 1 whereas in the instance before us, we write +2 0 +1 -3 416793 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_ = 20_a_ + _b_, _g_ = 20_f_ + _c_, _h_ = 20_g_ + _d_, _i_ = _e_ - 20_h_, _k_ = 20_a_ + _f_, _l_ = 20_k_ + _g_, _m_ = 20_l_ + _h_, _n_ = 20_a_ + _k_, _o_ = 20_n_ + _l_, _p_ = 20_a_ + _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.