An introduction to the modern theory of equations, by Florian Cajori.

ROOTS OF NUM1ERLCAL EQUATIONS

65

The broken lines indicate the conclusion of the successive transformations. The numbers immediately below a broken line are the coefficients
of the transformed equation. Thus, the second transformed equation is
seen at once to be x3 + 6.6 x2 + 13.52 x -.552 = 0.
Ex. 2. In the equation X3 - 46.6 x2 - 44.6 x - 142.8 = 0 we find that
f(40)= -, f(50) = +. Hence there is a root between 40 and 50. To find
this root, diminish the roots by 40, then find the first figure of the root in
the transformed equation and proceed by Horner's method as already
explained. The work is as follows:
1 - 46.6 - 44.6 - 142.8 1 47.6
40 - 264 - 12344
- 6.308. -.6 - 2486.8
40 1336 11131.4
33.4 1027.4 - 1355.4
40 562.8 1355.4
73.4 1590.2
7 611.8
80.4 2202.0
7 57
87.4 2259
7
94.4.6
95
In the first transformed equation x3 + 73.4 x2 + 1027 4 x - 12486.8 = 0
we only know that the value of x is less than 10; hence the method of
Ex. 1, where we ignored the terms containing x3 and x2, is not applicable.
Since in this transformed equation f(7) - and f(8) = 4-, we know that
7 is the desired digit.
In the second transformed equation we know that x lies between 0 and
1. Hence we find the first digit of x from the equation 2202 x- 1355.4=0.
Since in the third transformation there is no relmainder, we know by
~ 3 that.6 is a root of x3 - 94.4 x2 + 2202 x - 1355.4 = 0 and that 47.6 is a
commensurable root of the given equation.
When the fractional part of the root is being found and the
values of the coefficients 2, 3 'etc., are sufficietly sall, it will
be noticed that the last two terms of each transformdl equation
occurring in Homer's process have opposite signs. This is as it
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