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William Brouncker, 2nd Viscount Brouncker, PRS (1620 – 5 April 1684) was an Irish born

mathematician A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History On ...

who introduced Brouncker's formula, and was the first president of the

Royal Society The Royal Society, formally The Royal Society of London for Improving Natural Knowledge, is a learned society and the United Kingdom's national academy of sciences. The society fulfils a number of roles: promoting science and its benefits, re ...

. He was also a

civil servant The civil service is a collective term for a sector of government composed mainly of career civil servants hired on professional merit rather than appointed or elected, whose institutional tenure typically survives transitions of political leaders ...

, 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 in ...

of the

Royal Navy The Royal Navy (RN) is the United Kingdom's naval warfare force. Although warships were used by English and Scottish kings from the early medieval period, the first major maritime engagements were fought in the Hundred Years' War against F ...

. He was a friend and colleague of

Samuel Pepys Samuel Pepys (; 23 February 1633 – 26 May 1703) was an English diarist and naval administrator. He served as administrator of the Royal Navy and Member of Parliament and is most famous for the diary he kept for a decade. Pepys had no mariti ...

, and features prominently in the Great Diary.

Biography

Brouncker was born in

Castlelyons Castlelyons () is a small village in the east of County Cork, Ireland. It is also a civil parish in the barony of Barrymore.

County Cork County Cork ( ga, Contae Chorcaí) is the largest and the southernmost county of Ireland, named after the city of Cork, the state's second-largest city. It is in the province of Munster and the Southern Region. Its largest market towns are ...

, the elder son of William Brouncker (1585–1649), 1st

Viscount Brouncker Viscount Brouncker, of Lyons in the Province of Leinster, was a title in the Peerage of Ireland. It was created on 12 September 1645 for the courtier Sir William Brouncker. He was made Baron Brouncker, of Newcastle in the Province of Munster, at ...

and Winifred, daughter of Sir William Leigh of Newnham. His family came originally from

Melksham Melksham () is a town on the River Avon in Wiltshire, England, about northeast of Trowbridge and south of Chippenham. At the 2011 census, the Melksham built-up area had a population of 19,357, making it Wiltshire's fifth-largest settlement af ...

Wiltshire Wiltshire (; abbreviated Wilts) is a historic and ceremonial county in South West England with an area of . It is landlocked and borders the counties of Dorset to the southwest, Somerset to the west, Hampshire to the southeast, Gloucestershire ...

. His grandfather Sir

Henry Brouncker (died 1607) Henry Brouncker (c. 1550 – 3 June 1607), of Erlestoke, Wiltshire and West Ham, Essex, was an English politician whose later career was spent in Ireland. He was born in Wiltshire, a younger son of Henry Brouncker of Melksham and Erlestoke and ...

had been

Lord President of Munster The post of Lord President of Munster was the most important office in the English government of the Irish province of Munster from its introduction in the Elizabethan era for a century, to 1672, a period including the Desmond Rebellions in Munste ...

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. In many countries a viscount, and its historical equivalents, was a non-hereditary, administrative or judicial ...

in the

Peerage of Ireland The Peerage of Ireland consists of those titles of nobility created by the English monarchs in their capacity as Lord or King of Ireland, or later by monarchs of the United Kingdom of Great Britain and Ireland. It is one of the five divisi ...

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 , mottoeng = The Lord is my light , established = , endowment = £6.1 billion (including colleges) (2019) , budget = £2.145 billion (2019–20) , chancellor ...

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 The president of the Royal Society (PRS) is the elected Head of the Royal Society of London who presides over meetings of the society's council. After informal meetings at Gresham College, the Royal Society was officially founded on 28 November ...

. In 1662, he became

chancellor Chancellor ( la, cancellarius) is a title of various official positions in the governments of many nations. The original chancellors were the of Roman courts of justice—ushers, who sat at the or lattice work screens of a basilica or law cou ...

to Queen Catherine, then head of the

Saint Catherine's Hospital In religious belief, a saint is a person who is recognized as having an exceptional degree of holiness, likeness, or closeness to God. However, the use of the term ''saint'' depends on the context and denomination. In Catholic, Eastern Ortho ...

. He was appointed one of the commissioners of the

in 1664, and his career thereafter can be traced in the ''

Diary of Samuel Pepys Samuel Pepys (; 23 February 1633 – 26 May 1703) was an English diarist and naval administrator. He served as administrator of the Royal Navy and Member of Parliament and is most famous for the diary he kept for a decade. Pepys had no marit ...

''; despite their frequent disagreements,

on the whole respected Brouncker more than most of his other colleagues, writing in 1668 that "in truth he is the best of them". 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 Lord is an appellation for a person or deity who has authority, control, or power over others, acting as a master, chief, or ruler. The appellation can also denote certain persons who hold a title of the peerage in the United Kingdom, or are ...

from 1679.

Abigail Williams

Brouncker never married, but lived for many years with the

actress An actor or actress is a person who portrays a Character (arts), character in a performance. The actor performs "in the flesh" in the traditional medium of the theatre or in modern media such as film, radio, and television. The analogous Greek ...

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 Sir Oliver Cromwell ( – 28 August 1655) was an English landowner, lawyer and politician who sat in the House of Commons at various times between 1589 and 1625. He was the uncle of Oliver Cromwell, the Member of Parliament, general, and Lord Pr ...

, and first cousin to the renowned

Oliver Cromwell Oliver Cromwell (25 April 15993 September 1658) was an English politician and military officer who is widely regarded as one of the most important statesmen in English history. He came to prominence during the 1639 to 1651 Wars of the Three Ki ...

. 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

, whose private apartments were also destroyed in the blaze. On Brouncker's death in 1684, his title passed to his brother

Henry Henry may refer to: People *Henry (given name) * Henry (surname) * Henry Lau, Canadian singer and musician who performs under the mononym Henry Royalty * Portuguese royalty ** King-Cardinal Henry, King of Portugal ** Henry, Count of Portugal, ...

, 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 mirror-symmetrical and is approximately U-shaped. It fits several superficially different mathematical descriptions, which can all be proved to define exactly the same curves. One descript ...

and

cycloid In geometry, a cycloid is the curve traced by a point on a circle as it rolls along a straight line without slipping. A cycloid is a specific form of trochoid and is an example of a roulette, a curve generated by a curve rolling on another curve ...

, and the quadrature of the

hyperbola In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth 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, cal ...

, which requires approximation of the

natural logarithm The natural logarithm of a number is its logarithm to the base of the mathematical constant , which is an irrational and transcendental number approximately equal to . The natural logarithm of is generally written as , , or sometimes, if ...

function by

infinite series In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, math ...

. 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 nonsquare integer, and integer solutions are sought for ''x'' and ''y''. In Cartesian coordinate ...

. He was the first in England to take interest in

generalized continued fraction In complex analysis, a branch of mathematics, a generalized continued fraction is a generalization of regular continued fractions in canonical form, in which the partial numerators and partial denominators can assume arbitrary complex values. A ge ...

s and, following the work of

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

, 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 Gottfried Wilhelm (von) Leibniz . ( – 14 November 1716) was a German polymath active as a mathematician, philosopher, scientist and diplomat. He is one of the most prominent figures in both the history of philosophy and the history of math ...

: 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

References

External links

* * {{DEFAULTSORT:Brouncker, William Brouncker, 2nd Viscount 1620 births 1684 deaths 17th-century English mathematicians Founder Fellows of the Royal Society Lords of the Admiralty Presidents of the Royal Society Viscounts in the Peerage of Ireland 17th-century English medical doctors