# William Brouncker, 2nd Viscount Brouncker

The Viscount Brouncker
Portrait of Brouncker (circa 1674) possibly after Sir Peter Lely
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
Brouncker's signature as President, signing off the 1667 accounts of the Royal Society, from the minutes book

William Brouncker, 2nd Viscount Brouncker, PRS (1620 – 5 April 1684) was an English mathematician who introduced Brouncker's formula, and was the first President of the Royal Society.

## Life

Brouncker was born in Castlelyons, County Cork, the elder son of William Brouncker, 1st Viscount Brouncker and Winifred, daughter of Sir William Leigh of Newnham. His father was created a Viscount in the Peerage of Ireland in 1645 for services to the Crown. Although the first Viscount had fought in the Anglo-Scots war of 1639, malicious gossip said that he paid the then enormous sum of £1200 for the title and was almost ruined as a result; but in any case he died only a few months afterwards.

William obtained a DM at the University of Oxford in 1647. He was one of the founders and the first President of the Royal Society. In 1662, he became Chancellor to Queen Catherine, then head of the Saint Catherine's Hospital.

He was appointed one of the Commissioners of the Navy in 1664 and his career thereafter can be traced in the Diary of Samuel Pepys; despite their frequent disagreements Pepys on the whole respected Brouncker more than most of his other colleagues, writing in 1668 that "in truth he is the best of them".

### Abigail Williams

Brouncker never married but lived for many years with the actress Abigail Williams (much to Pepys' disgust) and left most of his property to her. She was the daughter of Sir Henry Clere (died 1622), first and last of the Clere Baronets, and the estranged wife of John Williams, otherwise Cromwell, second son of Sir Oliver Cromwell, and first cousin to the renowned Oliver Cromwell. She and John had a son and a daughter. The fire of 1673 which destroyed the Navy Office started in her private closet: this is unlikely to have improved her relations with Samuel Pepys, whose private apartments were destroyed in the blaze.

On Brouncker's death his title passed to his brother Henry, one of the most detested men of the era. William left him nothing in his will : "for reasons I think not fit to mention".

## Works

His mathematical work concerned in particular the calculations of the lengths of the parabola and cycloid, and the quadrature of the hyperbola,[1] which requires approximation of the natural logarithm function by infinite series.[2] 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 generalized continued fractions and, following the work of John Wallis, he provided development in the generalized continued fraction of pi.

### Brouncker's formula

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

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

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

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

and

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

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

Brouncker's formula can also be expressed as[3]

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

## References

1. ^ W. Brouncker (1667) The Squaring of the Hyperbola, Philosophical Transactions of the Royal Society of London, abridged edition 1809, v. i, pp 233–6, link form Biodiversity Heritage Library
2. ^ Julian Coolidge Mathematics of Great Amateurs, chapter 11, pp. 136–46
3. ^ John Wallis, Arithmetica Infinitorum, … (Oxford, England: Leon Lichfield, 1656), page 182. Brouncker expressed, as a continued fraction, the ratio of the area of a circle to the area of the circumscribed square (i.e., 4/π). The continued fraction appears at the top of page 182 (roughly) as: ☐ = 1 1/2 9/2 25/2 49/2 81/2 &c , where the square denotes the ratio that is sought. (Note: On the preceding page, Wallis names Brouncker as: "Dom. Guliel. Vicecon, & Barone Brouncher" (Lord William Viscount and Baron Brouncker).)