William Brouncker , born in Castle Lyons (Ireland) in 1620 and deceased with Westminster in 1684 , was a linguist and Mathématicien English.

The Viscount William Brouncker, more known nowadays under the name of Lord Brouncker, obtains a doctorate of Philosophie at the university of Oxford in 1647. He is one of the founders and the first president of the Royal Society, in 1660. In 1662, he becomes chancellor of the Catherine queen, then main of the hospital Holy-Catherine. Its work Mathématiques concerns in particular the correction (measurement of the Length S) of the Parabole and the Cycloïde like on the squaring (measurement of the surface S) of the hyperbole. It is the first, in England , to be interested in the generalized continuous fractions and, following work of Wallis , it gives the development in generalized continuous fraction of 4/π.

Formula of Brouncker

It provides the development of the first Fraction continues generalized of 4/π:

$\{4\; \backslash \; over\; \backslash \; pi\}\; =\; \{1\; +\; \{1^\; \{2\}\; \backslash \; over\; 2\; +\; \{3^\; \{2\}\; \backslash \; over\; 2\; +$

{5^ {2} \ over 2 + {7^ {2} \ over 2 + {9^ {2} \ over 2 + {11^ {2} \ over 2 +…}}}}}}}

Since, many representations in generalized continuous fractions of $k\; \backslash \; pi$, $k/\backslash \; pi$ and $k/(\backslash \; pi^2\; -\; N)$ were obtained (cf).

The values of the reduced of this development are:

*$1\; +\; \{1^\; \{2\}\; \backslash \; over\; 2\}\; =\; \{3\; \backslash \; over\; 2\}$

*$\{1\; +\; \{1^\; \{2\}\; \backslash \; over\; 2\; +\; \{3^\; \{2\}\; \backslash \; over\; 2\}\}\}\; =\; \{15\; \backslash \; over\; 13\}$

*$\{1\; +\; \{1^\; \{2\}\; \backslash \; over\; 2\; +\; \{3^\; \{2\}\; \backslash \; over\; 2\; +\; \{5^\; \{2\}\; \backslash \; over\; 2\}\}\}\}\; =\; \{105\; \backslash \; over\; 76\}$

*etc.

and are exactly the opposite of the sums partial of the formula of Leibniz:

$\{\backslash \; pi\; \backslash \; over\; 4\}\; =\; 1\; -\; \{1\; \backslash \; over\; 3\}\; +\; \{1\; \backslash \; over\; 5\}\; -\; \{1\; \backslash \; over\; 7\}\; +\dots$

*$1\; -\; \{1\; \backslash \; over\; 3\}\; =\; \{2\; \backslash \; over\; 3\}$

*$1\; -\; \{1\; \backslash \; over\; 3\}\; +\; \{1\; \backslash \; over\; 5\}\; =\; \{13\; \backslash \; over\; 15\}$

*$1\; -\; \{1\; \backslash \; over\; 3\}\; +\; \{1\; \backslash \; over\; 5\}\; -\; \{1\; \backslash \; over\; 7\}\; =\; \{76\; \backslash \; over\; 105\}$

*etc.

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