William Brouncker

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Castlelyons (N of Cork), Ireland
5 April 1684
Westminster. London, England

William Brouncker was an Irish mathematician who was a founder and the first president of the Royal Society of London. He worked on continued fractions and calculated logarithms by infinite series.


William Brouncker was the elder son of Sir William Brouncker, a man of high importance who was closely associated with the kings of England and had fought against the Scots in 1639. He served Charles I as one of his privy chamber and acted as vice-chamberlain to his son Charles, Prince of Wales. William Brouncker's mother was Winifred Leigh who came from Newenhan in Warwickshire. In a time in England when King and Parliament would fight the Civil War, the Brouncker family were staunch Royalist supporters.

We know little of Brouncker's early life. We have given 1620 as his date of birth but this is a guess made by historians rather than coming from any specific evidence. His place of birth is also a guess. Whether he was born in Ireland or England is even a matter of debate with no firm evidence to support either hypothesis. About the first we know for certain of Brouncker is that he entered Oxford University when he was sixteen years old and there he studied mathematics, languages and medicine. It is doubtful whether Brouncker learned more than arithmetic at Oxford, for Wallis, giving the status of mathematics at this time, wrote:-
... mathematics, at that time with us, were scarce looked on as academical studies, but rather mechanical - as the business of traders, merchants, seamen, carpenters, surveyors of lands and the like.
This was a difficult time for Brouncker for the political situation in England was in turmoil. When his father fought against the Scots in 1639 Charles I had been ruling for ten years without a Parliament. Charles I, running low on funds with which to continue to fight the Scots, summoned Parliament in 1640 to try to raise money. The English Civil War broke out in 1642, the Scots joined the Parliament forces and Charles I suffered a series of defeats. Brouncker and his father were Royalists and were very definitely on the losing side.

On 12 September 1645 Brouncker's father became Viscount Brouncker of Castle Lyons. He had bought himself into the Irish peerage and according to Samuel Pepys, the diarist (see for example [4]):-
... he gave 1200 pounds to be made an Irish lord, and swore the same day that he had not 12 pence left to pay for his dinner.
The First Viscount Brouncker of Castle Lyons did not live long to enjoy the peerage he bought for he died two months later. At this time William Brouncker, his mathematician son, succeeded him to the title becoming the Second Viscount Brouncker of Castle Lyons. Six months later the King finally lost the Civil War and surrendered. It was a time for Royalists to keep their heads down if they wanted to survive and that is exactly what Brouncker did.

He received the degree of Doctor of Medicine from Oxford University on 23 February 1647 (in fact February at this time was 1646 since the new year began in April but we will give 1647 which is consistent with our present calendar). Charles I was beheaded in January 1649 and in May of that year Brouncker's mother died. The authors of [4] write:-
Brouncker was one of the Royalists who remained quietly in the country pursuing his studies.
These studies were mathematics and music, two topics which Brouncker loved. It was during this time that he worked on a publication which would in fact be his only book. Descartes had written a paper on music in 1618 but it had never been published. Only after Descartes' death in 1650 did a Dutch publisher print it as a 58 page pamphlet with the title Renati Descartes Musicae Compendium . In 1653 Brouncker published his English translation of Descartes' work but he added notes of his own which doubled the size of the work. Mersenne had proposed a scale of 12 equal semitones after Descartes' manuscript had been written and in his notes Brouncker proposed a variation of Mersenne's ideas but he divided the scale into 17 equal semitones. One should not be surprised that all these mathematicians were contributing to musical theory and indeed Brouncker's notes are highly mathematical using algebra and logarithms. One might wonder why Brouncker chose 17 equal semitones and again the reason was a mathematical one for he derived this from taking ratios based on the golden section.

England had became a republic in 1649 when Charles I was beheaded. In 1653 Cromwell had become Protector and he held this role until his death in 1658. The years of the Protectorate were the most productive years for Brouncker in terms of mathematics. He had to keep out of the limelight to avoid paying for his Royalist views so he worked away corresponding with Wallis and solving some difficult mathematical problems which we look at below. Through Wallis, and others with whom he was corresponding, he became involved with a group of scientists who met in Gresham College London. Although we do not know exactly when he first became an active member of this group, we do know that he was taking part in meetings in Gresham College in 1657.

After Cromwell's death in 1658 his son took over but was ineffectual. The situation deteriorated with troops being stationed in Gresham College preventing the scientist continuing their weekly meetings. Monck, who had been appointed as governor in Scotland, marched an army on London and restored order in early 1660. Brouncker was one of those who signed a Declaration acknowledging Monck's rights. Monck called for new elections to Parliament, knowing that the mood of the people would elect Royalists. Brouncker stood in the new parliamentary elections and was elected as Member for Westbury in 1660. The improvement in the situation in London, in particular the troops who had been stationed in Gresham College having left, allowed the scientists to begin meeting again in the College. On Wednesday 28 November 1660 Brouncker was one of a dozen scientists at a meeting in Gresham College which constituted their Society for the Promoting of Physico-Mathematical Experimental Learning which they declared would promote experimental philosophy.

The Convention Parliament, of which Brouncker was a Member, voted to restore the monarchy and Charles II, the son of the executed Charles I, came to the throne in 1660. Brouncker's father had been vice-chamberlain to Charles when he was Prince of Wales and so Brouncker was well known to the new King who soon repaid him for his loyal support. In 1662 the King appointed Brouncker as Chancellor to Queen Catherine (of Braganza) and Keeper of the Great Seal.

The Society at Gresham petitioned King Charles II to recognise it and make a royal grant of incorporation. Brouncker worked hard for the Society ensuring that it was active. In particular he carried out many experiments including some on ballistics, some on the pendulum and a study of the variation of the magnetic needle. The first of these topics he published as Recoil of guns. The Royal Charter which was passed by the Great Seal on 15 July 1662 created the Royal Society of London and the Royal Charter nominated Brouncker as its first President. Given the quality of the founder members of the Society, it might be considered slightly strange that Brouncker, certainly not the most eminent academically, was made President. Well certainly Brouncker was on the best terms with the King and this must have been a major factor, but there were other reasons. Brouncker was a man of independent means, and he was also unmarried, so he had time to devote to the Society which most others would not have had. Again nobody could have been more enthusiastic in promoting the aims of the Society than Brouncker so he was a good choice.

As President, he continued his experimental work. Birch in his History of the Royal Society published in 1667 writes:-
The Lord Viscount Brouncker moved, that the experiments concerning the measure of the first velocity of bodies might be presented, that is what force is required to raise, for instance, one pound weight, one yard high in one second of time. His Lordship was desired to be curator of that experiment.
Brouncker now took on a number of roles. He was president of Gresham College, London, from 1664 to 1667. One of Brouncker's interests was in ships and he built, to his own design, a yacht for King Charles II. This interest in the sea made him an obvious choice to be appointed as a Commissioner to the Navy Board in 1664 and, as with his work for the Royal Society, he took on his duties with great enthusiasm. The diarist Pepys, who was Clerk to the Navy Board, records that he considered Brouncker:-
... a very able person.
In 1668 Brouncker was appointed as Controller of the Navy Accounts. Pepys wrote:-
The truth is [Brouncker] is the best man of them all, and I would be glad, next myself, to serve him ...
Things began to go less well for Brouncker from around 1675. He fell out with Hooke over comments which he had made to Charles II advising against a patent for Hooke's spring-regulated watch. Hooke and others in the Royal Society began to feel that the time had come for a change of President. The yearly elections of 1675 and 1676 were thought by Hooke to be unfair, and the fact that Hooke was not elected to the Council in 1676 made him all the more determined to reform the election process. Brouncker seems to have only infrequent attended the Society in 1677 but he still seems to have wanted to continue as President. Hooke records that on 18 October 1677 Brouncker was in the chair when a proposal was put forward to elect the Council by a vote:-
Lord Brouncker in great passion, raved and went out.
Brouncker was not present at the November meeting which elected Sir Joseph Williamson to succeed him as President.

Brouncker's mathematical achievements includes work on continued fractions and calculating logarithms by infinite series. In 1655 he gave a continued fraction expansion of 4π\Large\frac{4}{\pi}
4π=1+122+322+522+...\Large\frac {4} {\pi} \normalsize = 1 + \Large \frac {1^2}{2+ \Large\frac{3^2}{2 + \Large\frac {5^2}{2+...}}}
This result, written up in around ten pages, was added by Wallis to his treatise Arithmetica Infinitorum and probably first discovered by Brouncker in 1654. Wallis told Huygens of this result and Huygens expressed strong doubts that it was true. However after Brouncker correctly computed the first 10 places in the decimal expansion of π using his continued fraction expansion, Huygens accepted the result.

Probably also in 1654 Brouncker computed the quadrature of the hyperbola although he did not publish this result until 1668. It appeared in a paper published by Brouncker in the Philosophical Transactions of the Royal Society of 1668 but he clearly states that this result is the one referred to by Wallis in 1665. Although not given in this form, what Brouncker proved is equivalent to showing that the integral of 11+x\Large\frac{1}{1+x} between 0 and 1 was
11×2+13×4+15×6+...\Large\frac{1}{1\times 2}\normalsize + \Large\frac{1}{3\times 4}\normalsize + \Large\frac{1}{5\times 6}\normalsize + ...
112+1314+1516+...1 - \Large\frac{1}{2}\normalsize + \Large\frac{1}{3}\normalsize - \Large\frac{1}{4}\normalsize + \Large\frac{1}{5}\normalsize - \Large\frac{1}{6}\normalsize + ...
Brouncker gave a method of solving the diophantine equation
nx2+1=y2nx^{2} + 1 = y^{2}
which evolved during an exchange of letters in 1657-58 discussing a challenge problem posed by Fermat. See our article Pell's equation for more details. It is believed that Euler made an error in naming the equation 'Pell's equation', and that he was intending to acknowledge the outstanding contribution made by Brouncker. It is interesting to think that if Euler had not made this error then Brouncker, instead of being relatively unknown as a mathematician, would be universally known through 'Brouncker's equation'.

In 1659 Brouncker's improvement of Neile's computation of the arc length of the semicubical parabola ay2=x3ay^{2} = x^{3} appeared in Wallis's work De Cycloide et de Corporibus inde Genitis.

Brouncker has gained a somewhat unfortunate reputation. Pepys, who as we have seen was a good friend of Brouncker, thought that he had treated Mrs Turner, one of his lady friends, badly and wrote in his diary:-
I perceive he is a rotten-hearted, false man as any I know ... and, therefore, I must beware of him accordingly, and I hope I shall.
There are other negative comments regarding his character which are believed to be unfair in that the writer has confused Brouncker with his brother Henry Brouncker. Henry was:-
.. ever noted for a hard, covetous, vicious man; but for his worldly craft and skill in gaming few exceeded him.
It seems a cruel twist that two misidentifications, mistaking him for Pell and also for his brother Henry, has led to Brouncker receiving less than his just deserts.

References (show)

  1. J M Dubbey, Biography in Dictionary of Scientific Biography (New York 1970-1990).
    See THIS LINK.
  2. H Hartley (ed.), The Royal Society : Its Origins and Founders (London, 1960).
  3. J Dutka, Wallis's product, Brouncker's continued fraction, and Leibniz's series, Arch. Hist. Exact Sci. 26 (2) (1982), 115-126.
  4. J F Scott and H Hartley, William Viscount Brouncker, Notes and Records. Royal Society of London 15 (1960-61), 147-156.

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Written by J J O'Connor and E F Robertson
Last Update February 2002