Page:Passages from the Life of a Philosopher.djvu/148

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132
SOLUTION OF EQUATIONS.

Number of Operation
Cards used.

diminishing the powers by unity at each step) for the value of x; in the given equation.
Continue this until the sign of the resulting number changes from positive to negative. The index of the last power of ten (call it n), which is positive, expresses the number of digits in that part of the root which consists of whole numbers. Call this index n + 1.


4
d. Substitute successively for x in the original equation 0 × 10n, 1 × 10n, 2 × 10n, 3 × 10n,.... 9 × 10n, until a change of sign occurs in the result. The digit previously substituted will be the first figure of the root sought.
5
e.
Transform the original equation into another whose roots are less by the number thus found. The transformed equation will have a real root, the digit, less than 10n.
6
f.
Substitute 1 × 10n-1, 2 × 10n-1, 3 × 10n-1, &c.,

successively for the root of this equation, until a change of sign occurs in the result, as in process 4.

This will give the second figure of the root. This process of alternately finding a new

figure in the root, and then transforming the equation into another (as in process 4 and 5), must be carried on until as many figures as are required, whether whole numbers or decimals, are arrived at.
7
g.
The root thus found must now be used to reduce the original equation to one dimension lower.