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

But if the last sign be +, it may be more convenient to let it stand, and change all which come before. Thus, in solving x³-12x + 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:

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:

We have now before us the row