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William Brouncker (2nd Viscount Brouncker)

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Wikipedia

William Brouncker, 2nd Viscount Brouncker, PRS (1620 – 5 April 1684) was an English mathematician.

[edit] Life

Brouncker 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 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.

[edit] Works

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.

[edit] Brouncker's formula

This formula provides a development of π/4 in a generalized continued fraction:

\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

\frac{1}{1+\frac{1^2}{2}} = \frac{2}{3} = 1 - \frac{1}{3}

and

\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.

[edit] External links

Peerage of Ireland
Preceded by William Brouncker Viscount Brouncker 1645–1684 Succeeded by Henry Brouncker
Persondata
Name Brouncker, William Brouncker, 2nd Viscount
Alternative names
Short description
Date of birth 1 January 1620
Place of birth Castlelyons, Ireland
Date of death 5 April 1684
Place of death Westminster, London, England

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References in the diary

A graph of all the references in the diary

1662
Apr: 28
Aug: 13
Sep: 5
1664
Aug: 10
Nov: 18, 19
Dec: 10, 12
1665
Jan: 2, 11, 21
Feb: 11, 15, 18, 21
Mar: 20
Apr: 6, 10, 12
May: 1, 5
Jun: 7
Jul: 6, 13, 25
Aug: 5, 8, 10, 14, 15, 17, 21, 22, 24, 26, 28
Sep: 3, 4, 5, 8, 9, 10, 11, 12, 13, 16, 17, 18, 19, 20, 21, 22, 23, 26
Oct: 2, 12, 23, 28
Nov: 1, 16, 18, 19, 26
Dec: 3, 9, 12, 13, 14, 15, 17, 18, 19, 20, 22, 24, 25, 26, 31
1666
Jan: 2, 4, 5, 10, 12, 13, 18, 22, 24, 28, 29
Feb: 1, 5, 8, 11, 14, 21
Mar: 1, 6, 7, 8, 9, 14, 16, 19, 21, 26
Apr: 5, 7, 11, 12, 21, 22, 23
May: 1, 2, 11, 23
Jun: 4, 6, 7, 27
Jul: 14, 27, 28, 31
Aug: 2, 6, 16, 18, 23, 26
Sep: 7, 11, 17, 22, 25
Oct: 1, 2, 3, 7, 13, 15, 18, 19, 25
Nov: 4, 8, 10, 18, 22, 29
Dec: 15, 16, 24
1667
Jan: 3, 4, 5, 7, 9, 10, 11, 15, 20, 22, 23, 25, 26, 27, 29
Feb: 2, 8, 10, 12, 16, 20, 21, 22, 23, 24, 26, 28
Mar: 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 24, 27
Apr: 3, 10, 13, 23, 24, 25, 29
May: 6, 7, 13, 15, 16, 17, 21
Jun: 11, 14, 15, 18, 24, 27, 28, 30
Jul: 1, 23
Aug: 2, 20, 21, 22, 26, 31
Sep: 8, 10, 18, 23, 25
Oct: 3, 4, 6, 14, 16, 22, 25, 27, 29, 30
Nov: 11, 16, 19, 27, 28
Dec: 23
1668
Jan: 5, 8, 10, 22
Feb: 7, 11, 14, 18, 19, 20, 21
Mar: 3, 4, 5, 9, 12, 13, 14, 18, 24, 26, 31
Apr: 2, 3, 4, 8, 9, 12, 15, 17, 24, 25, 29
May: 2, 8, 13, 15, 27
Jun: 3, 4, 22
Jul: 16
Aug: 4, 9, 20, 25, 27, 28, 30
Sep: 3, 7, 9, 13, 17, 19, 23, 24, 25, 27
Oct: 12, 13, 18, 21, 24, 29
Nov: 3, 17, 26
Dec: 4, 7, 11, 18, 23
1669
Jan: 3, 4, 8, 9, 15