Derivation of The Method
Starting with initial values and, we construct a line through the points and, as demonstrated in the picture on the right. In point-slope form, this line has the equation
We find the root of this line – the value of such that – by solving the following equation for :
The solution is
We then use this new value of as and repeat the process using and instead of and . We continue this process, solving for, etc., until we reach a sufficiently high level of precision (a sufficiently small difference between and ).
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“The method of scientific investigation is nothing but the expression of the necessary mode of working of the human mind. It is simply the mode in which all phenomena are reasoned about, rendered precise and exact.”
—Thomas Henry Huxley (1825–95)