❝... If \(x^2\) is \(2\) and \(x\) is to be found, it will be said to be \(\sqrt{2}\).
But if you are looking for the approximate root of \(2\), extract the nearest root, which is \(1\), and divide
the remainder by twice the root just discovered. Thus the root of the number \(2\) can be said to be
\(1\frac{1}{2}\), which is less than the true root. Or add \(1\) to the denominator and the root will said to be
\(1\frac{1}{3}\), which is less than the true root. Midway between the two is \(1\frac{5}{8}\), which is very
close to the true root. Alternatively, pairs or zeros may be added to the given square up to infinity, and the
root may be extracted from [a number] extended in this way as from an exact square number. Thus if you are seeking
the root of \(2\), a square, extract, if you you wish, the root of \(\footnotesize 2\, 00\, 00\, 00\, 00\, 00\, 00\, 00\, 00\, 00\, 00\, 00\, 00\, 00\, 00\normalsize\)
[which is] \(\footnotesize 141,421,356,237,309,505\normalsize\). So the root of \(2\) is said to be approximately
\(\frac{141,421,356,237,309,505}{100,000,000,000,000,000}\).❞
—François Viéte (15401603)^{‡}
To put this plainly, find the first approximation as \(1\frac{1}{2}\) above, then calculate its square as \((1\frac{1}{2})^2\)\(=2\frac{1}{4}\).
Find the difference "remainder", as \(22\frac{1}{4}\) \(=\frac{1}{4}\). Divide this value by twice the current approximation and add it to
the approximation: \(1\frac{1}{2}+(\frac{1}{4}/2(1\frac{1}{2}))=1\frac{1}{2}\frac{1}{12}=1\frac{5}{12}\). Repeating these steps again gives:
\((1\frac{5}{12})^2=(2\frac{1}{144})\); \(22\frac{1}{144}=\frac{1}{144}\); \(1\frac{5}{12}+(\frac{1}{144}\div 2(1\frac{5}{12}))\)
\(=1\frac{5}{12}\frac{1}{408}\)\(=1\frac{169}{408}\) \(\approx 1.414215686\). Since the actual value is approximately \(1.414213562\), this is already
accurate to 6 digits.
Viéte also describes similar methods to approximate other various roots in his fascinating book The Analytic Art. I would definitely
suggest picking up a copy of this book if you are interested in mathematics. You can get a copy online from Amazon
here.
^{‡} F. Viéte, The Analytic Art, p.316, The Kent State University Press, 1983, Trans. T. Richard Witmer.
