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The Viscount Brouncker
William Brouncker, 2nd Viscount Brouncker by Sir Peter Lely.jpg
Portrait of Brouncker (circa 1674) possibly after Sir Peter Lely
Born 1620
Castlelyons, Ireland
Died 5 April 1684(1684-04-05) (aged 64)
Westminster, London, England
Residence England
Alma mater University of Oxford
Known for Brouncker's formula, leadership of Royal Society
Scientific career
Fields Mathematician, civil servant
Institutions Saint Catherine's Hospital
Academic advisors John Wallis
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. He was also a civil servant, serving as a Commissioner of the Royal Navy.


Brouncker was born in Castlelyons, County Cork, the elder son of William Brouncker (1585-1649), 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 his services to the Crown. Although the first Viscount had fought for the Crown 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 Royal Navy in 1664, and his career thereafter can be traced in the Diary of Samuel Pepys; despite their frequent disagreements, Samuel 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 Royal Navy Office started in her private closet: this is unlikely to have improved her relations with Samuel Pepys, whose private apartments were also destroyed in the blaze.

On Brouncker's death in 1684, 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".

Mathematical 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 π/4 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 }}}}}}}}}}}}} {\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}}} {\frac  {1}{1+{\frac  {1^{2}}{2}}}}={\frac  {2}{3}}=1-{\frac  {1}{3}}


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}}.} {\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 }}}}}}}}}}} {\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 }}}}}}}}}}


  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).)

External links

Peerage of Ireland
Preceded by
William Brouncker
Viscount Brouncker
Succeeded by
Henry Brouncker

9 Annotations

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:

And one to the ersthwile Henry Brunkard, 3rd Viscount:

Here are some records from the British House of Commons on the brothers during the civil war.

The following link pints to a short biography that fills out some additional information on the pair.

I am researching a biography of the three Viscount Brunkard's at the moment that I will publish online at 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, 1664-66; Comptroller of the Treasurer's Accounts, 1660-79; 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, 1662-77; president of Gresham College, 1664-7; 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.

Chris Squire UK  •  Link

‘ . . Continued fractions first appeared in the works of the Indian mathematician Aryabhata in the 6th century. He used them to solve linear equations. They re-emerged in Europe in the 15th and 16th centuries and Fibonacci attempted to define them in a general way. The term "continued fraction" first appeared in 1653 in an edition of the book Arithmetica Infinitorum by the Oxford mathematician, John Wallis.

Their properties were also much studied by one of Wallis's English contemporaries, William Brouncker, who along with Wallis, was one of the founders of the Royal Society. At about the same time, the famous Dutch mathematical physicist, Christiaan Huygens made practical use of continued fractions in building scientific instruments. Later, in the eighteenth and early nineteenth centuries, Gauss and Euler explored many of their deep properties . . ‘

‘Chaos in Numberland: The secret life of continued fractions’ by John D. Barrow, a Professor in the Department of Applied Mathematics and Theoretical Physics at the University of Cambridge.

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Chart showing the number of references in each month of the diary’s entries.