Extraction of square roots

Oppenheim A. (1955). 2553. Extraction of Square Roots, The Mathematical Gazette, 39 (329) 237. DOI: 10.2307/3608773

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At the end of his article on the extraction of square roots⁽¹⁾ Professor Haldane writes:

The methods here given are probably now of no practical importance. Had they been discovered in the 17th century, as they might have been, they would have saved a good deal of computation.

In point of fact Brook Taylor in 1717 gave "A general Series for expressing the Root of any Quadtratick Equation". It will be found towards the end of his paper

An attempt towards the Improvement of the Method of approximating, in the Extraction of the Roots of Equations in Numbers⁽²⁾.

In modern notation Taylor's solution of the quadratic equation \[xx - akx + akk = 0\] is \[x = k + \frac{k}{c} + \frac{k}{cc'} + \frac{k}{cc'c''} + \cdots\] where $c=a-2$, $c' = c^2 -2$, $c'' = c'^2 -2$, $\cdots$ He gives the example \[1 + \sqrt{2} = \frac{1}{2}-\frac{1}{2\cdot 6}-\frac{1}{2 \cdot 6 \cdot 34}-\frac{1}{2 \cdot 6 \cdot 34 \cdot 1154}-\frac{1}{2 \cdot 6 \cdot 34 \cdot 1154 \cdot 1331714} - \cdots\] and concludes

The Fractions here wrote down giving the Root true to twenty three Places⁽²⁾

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(1) Haldane J.B.S. (1951). The Extraction of Square Roots, The Mathematical Gazette, 35 (312) 89. DOI: 10.2307/3609330 (pdf)
(2) Taylor B. (1717). An Attempt towards the Improvement of the Method of Approximating, in the Extraction of the Roots of Equations in Numbers. By Brook Taylor, Secretary to the Royal Society, Philosophical Transactions of the Royal Society of London, 30 (351-363) 610-622. DOI: 10.1098/rstl.1717.0011