William Brouncker, 2nd Viscount Brouncker, PRS (1620 – 5 April 1684) was an English mathematician.
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.
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.
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.
External links
O'Connor, John J.; Robertson, Edmund F., "William Brouncker, 2nd Viscount Brouncker", MacTutor History of Mathematics archive, University of St Andrews.
"Brouncker, William". Dictionary of National Biography. London: Smith, Elder & Co. 1885–1900.
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