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The Viscount Brouncker  

Portrait of Brouncker (circa 1674) possibly after Sir Peter Lely


Born  1620 Castlelyons, Ireland 
Died  5 April 1684(16840405) (aged 64) Westminster, London, England 
Residence  England 
Fields  Mathematician 
Institutions  Saint Catherine's Hospital 
Alma mater  University of Oxford 
Academic advisors  John Wallis 
Known for  Brouncker's formula 
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 AngloScots 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:
 π 4 = 1 1 + 1 2 2 + 3 2 2 + 5 2 2 + 7 2 2 + 9 2 2 + ⋱ {\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
 1 1 + 1 2 2 = 2 3 = 1 − 1 3 {\displaystyle {\frac {1}{1+{\frac {1^{2}}{2}}}}={\frac {2}{3}}=1{\frac {1}{3}}}
and
 1 1 + 1 2 2 + 3 2 2 = 13 15 = 1 − 1 3 + 1 5 . {\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]}
 4 π = 1 + 1 2 2 + 3 2 2 + 5 2 2 + 7 2 2 + 9 2 2 + ⋱ {\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
 ^ 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
 ^ Julian Coolidge Mathematics of Great Amateurs, chapter 11, pp. 136–46
 ^ 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).)
External links
 O'Connor, John J.; Robertson, Edmund F., "William Brouncker, 2nd Viscount Brouncker", MacTutor History of Mathematics archive, University of St Andrews .
 "Brouncker, William". Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
Peerage of Ireland  

Preceded by William Brouncker 
Viscount Brouncker 1645–1684 
Succeeded by Henry Brouncker 
7 Annotations
Nix • Link
Brouncker was the first president of the Royal Society. Though he was trained as a physician, he was best noted for his work in mathematics.
An extensive biography appears at
http://wwwgroups.dcs.stand.ac.uk/~history/Mat...
Cumgranissalis • Link
Brouncher [Brounchar] second Vicount, excellent Mathematitian [per J. Evelyn]
language hat • Link
L&M Companion says:
"As an administrator he was active in all branches of the Board's work, but took a special interest in finance and accounts... His relations with Pepys fluctuated, but were at bottom good. They had common interests in music and science, and despite occasional quarrels over competition for contractors' favours, came to respect each other's ability. 'The truth is,' Pepys wrote (25 Aug. 1668), 'he is the best man of them all.'
Thomas Brunkard • Link
Brouncker and Brounchar are also phonetic spellings of Brunkard. William is frequently confused with his younger brother Henry due to Henry's inheriting of his elder brother's title.
Henry suffers from some bad press in Pepy's so be careful not to confuse them as some other online resources and certain new books about The Royal Society have.
Here is a link to an excellent potrait of William Brunkard, 2nd Viscount:
http://www.npg.org.uk/live/search/portrait.asp?...
And one to the ersthwile Henry Brunkard, 3rd Viscount:
http://www.npg.org.uk/live/search/portrait.asp?...
Here are some records from the British House of Commons on the brothers during the civil war.
http://www.britishhistory.ac.uk/results.asp?qu...
The following link pints to a short biography that fills out some additional information on the pair.
http://www.dcs.warwick.ac.uk/bshm/zingaz/London...
I am researching a biography of the three Viscount Brunkard's at the moment that I will publish online at http://www.brunkard.com in time. Any additional information anyone can add to help in its composition is appreciated.
Bill • Link
William, lord Brouncker, whom bishop Burnet calls a profound mathematician, was chancellor to queen Catherine, keeper of her great seal, and one of the commissioners for executing the office of lord high admiral. Few of his writings are extant. His "Experiments of the recoiling of Guns," and his algebraical paper on the squaring of the hyperbola, are well known. He was the first president of the Royal Society; a body of men, who, since their incorporation, have made a much greater progress in true natural knowledge, than had before been made from the beginning of the world. Ob. 5 April, 1684, Æt. 64.
A Biographical History of England. J. Granger, 1775.
Bill • Link
William [Brouncker], second Lord Viscount Brouncker of Castle Lyons, born about 1620, was the first president of the Royal Society, and a respectable mathematician. Extra Commissioner of the Navy, 166466; Comptroller of the Treasurer's Accounts, 166079; Master of St. Katherine's Hospital in 1681. Died April 5th, 1684.
Wheatley, 1899.
Bill • Link
BROUNCKER or BROUNKER, WILLIAM, second Viscount Brouncker of Castle Lyons in Irish peerage (1620?1684), first president of the Royal Society; M.D. Oxford, 1647; first to introduce continued fractions and to give a series for quadrature of a portion of the equilateral hyperbola; original member of Royal Society, 1662, and first president, 166277; president of Gresham College, 16647; chancellor of Queen Catherine, 1662; commissioner for executing office of lord high admiral, 1664; master of St. Catherine's Hospital, 1681.
Dictionary of National Biography: Index and Epitome. S. Lee, 1906.