# William Brouncker, 2nd Viscount Brouncker: Wikis

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# Encyclopedia

The Viscount Brouncker

The 2nd Viscount Brouncker (1620-1684)
Born 1620
Castlelyons, Ireland
Died 5 April 1684 (aged 64)
Westminster, London, England
Residence England
Fields Mathematician
Institutions Saint Catherine's Hospital
Alma mater University of Oxford
Known for Brouncker's formula

William Brouncker, 2nd Viscount Brouncker, FRS (1620 – 5 April 1684) was an English mathematician.

Brouncker obtained a DM at the University of Oxford in 1647. He was one of the founders and the second President of the Royal Society. In 1662, he became Chancellor to Queen Catherine, then chief of the Saint Catherine's Hospital. His mathematical work concerned in particular the calculations of the lengths of the parabola and cycloid, and the quadrature of the hyperbola, which requires approximation of the natural logarithm function by infinite series. He was the first European to solve what is now known as Pell's equation. He was the first in England to take interest in generalised continued fractions and, following the work of John Wallis, he provided development in the generalised continued fraction of pi.

## Brouncker's formula

This formula provides a development of π/4 in a generalized continued fraction:

$\frac \pi 4 = \cfrac{1}{1+\cfrac{1^2}{2+\cfrac{3^2}{2+\cfrac{5^2}{2+\cfrac{7^2}{2+\cfrac{9^2}{2+\ddots}}}}}}$

The convergents are related to the Leibniz formula for pi: for instance

$\frac{1}{1+\frac{1^2}{2}} = \frac{2}{3} = 1 - \frac{1}{3}$

and

$\frac{1}{1+\frac{1^2}{2+\frac{3^2}{2}}} = \frac{13}{15} = 1 - \frac{1}{3} + \frac{1}{5}.$

Because of its slow convergence Brouncker's formula is not useful for practical computations of π.

Peerage of Ireland
Preceded by
William Brouncker
Viscount Brouncker
1645–1684
Succeeded by
Henry Brouncker