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William Brouncker, 2nd Viscount Brouncker FRS ( – 5 April 1684) was an Anglo-Irish peer and mathematician who served as the president of the Royal Society from 1662 to 1677. Best known for introducing Brouncker's formula, he also worked as a

civil servant The civil service is a collective term for a sector of government composed mainly of career civil service personnel hired rather than elected, whose institutional tenure typically survives transitions of political leadership. A civil service offic ...

, serving as a

commissioner A commissioner (commonly abbreviated as Comm'r) is, in principle, a member of a commission or an individual who has been given a commission (official charge or authority to do something). In practice, the title of commissioner has evolved to incl ...

in the

Royal Navy The Royal Navy (RN) is the naval warfare force of the United Kingdom. It is a component of His Majesty's Naval Service, and its officers hold their commissions from the King of the United Kingdom, King. Although warships were used by Kingdom ...

. Brouncker was a friend and colleague of

Samuel Pepys Samuel Pepys ( ; 23 February 1633 – 26 May 1703) was an English writer and Tories (British political party), Tory politician. He served as an official in the Navy Board and Member of Parliament (England), Member of Parliament, but is most r ...

, and features prominently in the Pepys' diary.

Biography

Brouncker was born in Castlelyons,

County Cork County Cork () is the largest and the southernmost Counties of Ireland, county of Republic of Ireland, Ireland, named after the city of Cork (city), Cork, the state's second-largest city. It is in the Provinces of Ireland, province of Munster ...

, the elder son of William Brouncker (1585–1649), 1st Viscount Brouncker and Winifred, daughter of Sir William Leigh of Newnham. His family came originally from

Melksham Melksham () is a town and civil parish on the Bristol Avon, River Avon in Wiltshire, England, about northeast of Trowbridge and south of Chippenham. The parish population was 18,113 at the 2021 census. History Early history Excavations in ...

Wiltshire Wiltshire (; abbreviated to Wilts) is a ceremonial county in South West England. It borders Gloucestershire to the north, Oxfordshire to the north-east, Berkshire to the east, Hampshire to the south-east, Dorset to the south, and Somerset to ...

. His grandfather Sir Henry Brouncker (died 1607) had been Lord President of Munster 1603–1607, and settled his family in Ireland. His father was created a

viscount A viscount ( , for male) or viscountess (, for female) is a title used in certain European countries for a noble of varying status. The status and any domain held by a viscount is a viscounty. In the case of French viscounts, the title is ...

in the

Peerage of Ireland The peerage of Ireland consists of those Peerage, titles of nobility created by the English monarchs in their capacity as Lordship of Ireland, Lord or Monarchy of Ireland, King of Ireland, or later by monarchs of the United Kingdom of Great B ...

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. He died only a few months afterwards. William obtained a DM at the

University of Oxford The University of Oxford is a collegiate university, collegiate research university in Oxford, England. There is evidence of teaching as early as 1096, making it the oldest university in the English-speaking world and the List of oldest un ...

in 1647. Until 1660 he played no part in public life: being a staunch Royalist, he felt it best to live quietly and devote himself to his mathematical studies. He was one of the founders and the first president of the Royal Society. In 1662, he became

chancellor Chancellor () is a title of various official positions in the governments of many countries. The original chancellors were the of Roman courts of justice—ushers, who sat at the (lattice work screens) of a basilica (court hall), which separa ...

to Queen Catherine, then head of the Saint Catherine's Hospital. He was appointed one of the commissioners of the

in 1664, and his career thereafter can be traced in the '' Diary of Samuel Pepys''; despite their frequent disagreements,

on the whole respected Brouncker more than most of his other colleagues, writing on 25 August 1668 that "the truth is, he is the best man of them all

Although his attendance at the Royal Society had become infrequent, and he had quarrelled with some of his fellow members, he was nonetheless greatly displeased to be deprived of the presidency in 1677. He was commissioner for executing the office of Lord High Admiral of England from 1679.

Abigail Williams

Brouncker never married, but lived for many years with the

actress An actor (masculine/gender-neutral), or actress (feminine), is a person who portrays a character in a production. The actor performs "in the flesh" in the traditional medium of the theatre or in modern media such as film, radio, and television. ...

Abigail Williams (much to Samuel 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 Oliver Cromwell (25 April 15993 September 1658) was an English statesman, politician and soldier, widely regarded as one of the most important figures in British history. He came to prominence during the Wars of the Three Kingdoms, initially ...

. 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 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 almost 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 In mathematics, a parabola is a plane curve which is Reflection symmetry, mirror-symmetrical and is approximately U-shaped. It fits several superficially different Mathematics, mathematical descriptions, which can all be proved to define exactl ...

and

cycloid In geometry, a cycloid is the curve traced by a point on a circle as it Rolling, rolls along a Line (geometry), straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette (curve), roulette, a curve g ...

, and the quadrature of the

hyperbola In mathematics, a hyperbola is a type of smooth function, smooth plane curve, curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected component ( ...

, which requires approximation of the

natural logarithm The natural logarithm of a number is its logarithm to the base of a logarithm, base of the e (mathematical constant), mathematical constant , which is an Irrational number, irrational and Transcendental number, transcendental number approxima ...

function by

infinite series In mathematics, a series is, roughly speaking, an addition of infinitely many terms, one after the other. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathemati ...

. He was the first European to solve what is now known as

Pell's equation Pell's equation, also called the Pell–Fermat equation, is any Diophantine equation of the form x^2 - ny^2 = 1, where ''n'' is a given positive Square number, nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian ...

. He was the first in England to take interest in generalized continued fractions and, following the work of

John Wallis John Wallis (; ; ) was an English clergyman and mathematician, who is given partial credit for the development of infinitesimal calculus. Between 1643 and 1689 Wallis served as chief cryptographer for Parliament and, later, the royal court. ...

, 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: :

\frac \pi 4 = \cfrac

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

\frac = \frac = 1 - \frac

and :

\frac = \frac = 1 - \frac + \frac.

Because of its slow convergence, Brouncker's formula is not useful for practical computations of π. Brouncker's formula can also be expressed asJohn 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).) :

\frac 4 \pi = 1+\cfrac

See Euler's continued fraction formula.

References

External links