William Brouncker, 2nd Viscount Brouncker

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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.


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 can be traced in the Diary of Samuel Pepys; despite frequent disagreements Pepys on the whole respected Brouncker more than most of his other colleagues.

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. His title passed to his brother Henry, one of the most detested men of the era.


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 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:

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


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

Brouncker's formula can also be expressed as[1]


  1. 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

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