**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.

- We know little of Brouncker's early life.
- 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.
- 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.
- It was a time for Royalists to keep their heads down if they wanted to survive and that is exactly what Brouncker did.
- These studies were mathematics and music, two topics which Brouncker loved.
- 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.
- The years of the Protectorate were the most productive years for Brouncker in terms of mathematics.
- Brouncker was one of those who signed a Declaration acknowledging Monck's rights.
- Brouncker stood in the new parliamentary elections and was elected as Member for Westbury in 1660.
- 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.
- In 1662 the King appointed Brouncker as Chancellor to Queen Anne and Keeper of the Great Seal.
- Brouncker worked hard for the Society ensuring that it was active.
- 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.
- Again nobody could have been more enthusiastic in promoting the aims of the Society than Brouncker so he was a good choice.
- Brouncker now took on a number of roles.
- One of Brouncker's interests was in ships and he built, to his own design, a yacht for King Charles II.
- In 1668 Brouncker was appointed as Controller of the Navy Accounts.
- Things began to go less well for Brouncker from around 1675.
- Brouncker seems to have only infrequent attended the Society in 1677 but he still seems to have wanted to continue as President.
- 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.
- 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.
- 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'.
- Brouncker has gained a somewhat unfortunate reputation.

Born 1620, Castlelyons (N of Cork), Ireland. Died 5 April 1684, Westminster. London, England.

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Algebra, Ancient Indian, Geography, Origin Ireland, Number Theory

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive