威廉·布朗克William Brouncker, 2nd Viscount Brouncker

威廉·布朗克是爵士布朗克的长子，后者是一位重要人物，与英格兰国王关系密切，并在1639年与苏格兰人作战。他曾作为查理一世的枢密内侍之一为其效力，并担任其子威尔士亲王查理的副内侍。布朗克的母亲是来自沃里克郡纽恩汉的温妮弗雷德·利。在英格兰国王与议会即将爆发内战的时代，布朗克家族是坚定的保皇党支持者。

我们对布朗克的早年生活知之甚少。我们把1620年作为他的出生日期，但这是历史学家的猜测，而非来自任何具体证据。他的出生地也是猜测。他究竟出生在爱尔兰还是英格兰，甚至是一个有争议的问题，没有确凿证据支持任何一种假设。关于布朗克我们确切知道的第一件事是，他十六岁时进入牛津大学，在那里学习数学、语言和医学。布朗克在牛津是否学到了算术以外的东西是值得怀疑的，因为约翰·沃利斯在说明当时数学的地位时写道：-

……数学，在当时我们这里，几乎不被视为学院式的研究，而更像是机械性的——是商人、贸易者、海员、木匠、土地测量员之类的事务。

这对布朗克来说是一段艰难的时期，因为英格兰的政治局势动荡不安。当他的父亲在1639年与苏格兰人作战时，查理一世已经统治了十年而没有召开议会。查理一世因继续与苏格兰人作战的资金日渐匮乏，于1640年召集议会试图筹款。1642年英国内战爆发，苏格兰人加入了议会军，查理一世接连遭受失败。布朗克和他的父亲是保皇党人，毫无疑问属于失败的一方。

1645年9月12日，布朗克的父亲成为里昂城堡的布朗克。他为自己购得了爱尔兰贵族爵位，据日记作者皮埃尔·萨米埃尔佩皮斯所述（例如见[4]）：-

……他花了1200英镑成为爱尔兰贵族，并在同一天发誓说他连12便士都不剩来付晚餐的钱。

Castle Lyons的布朗克没能长久享受他买来的爵位，因为两个月后他就去世了。此时，他的数学家儿子布朗克继承了他的爵位，成为第二代Castle Lyons的布朗克。六个月后，国王最终输掉了内战并投降。对于保皇党人来说，如果想生存下去，就该低调行事，而布朗克正是这样做的。

他于1647年2月23日获得牛津大学医学博士学位（事实上，此时的2月按当时历法仍是1646年，因为新年从4月开始，但我们采用与现行历法一致的1647年）。查理一世于1649年1月被斩首，同年5月布朗克的母亲去世。[4]的作者们写道：——

布朗克是留在乡间静心从事研究的保皇党人之一。

这些研究是数学和音乐，布朗克热爱的两个主题。正是在这段时间里，他致力于一部出版物，这实际上将是他唯一的书。勒内·笛卡儿在1618年写了一篇关于音乐的论文，但从未出版。直到1650年勒内·笛卡儿去世后，一家荷兰出版商才将其印成58页的小册子，标题为Renati Descartes Musicae Compendium Ⓣ（勒内·笛卡儿的音乐概要）。1653年，布朗克出版了他对勒内·笛卡儿著作的英译本，但他添加了自己的注释，使该著作的篇幅增加了一倍。马兰·梅森在勒内·笛卡儿的手稿写成后提出了十二平均半音音阶，而在他的注释中，布朗克提出了马兰·梅森思想的一种变体，但他将音阶分为十七平均半音。所有这些数学家都对音乐理论有所贡献，这不应令人惊讶，事实上布朗克的注释使用了代数和对数，极具数学性。人们可能会好奇为什么布朗克选择十七平均半音，原因同样是数学上的，因为他通过基于黄金分割的比率推导出了这一点。

1649年查理一世被斩首后，英格兰成为共和国。1653年克伦威尔成为护国公，并担任此职直到1658年去世。就数学而言，护国公时期是布朗克最多产的岁月。他不得不避开公众视线，以免因保皇党观点而付出代价，因此他埋头与约翰·沃利斯通信，并解决一些我们下面将讨论的困难数学问题。通过约翰·沃利斯以及其他与他通信的人，他参与了一个在伦敦格雷沙姆学院聚会的科学家团体。虽然我们不知道他何时首次成为该团体的活跃成员，但我们确实知道他在1657年参加了格雷沙姆学院的会议。

1658年克伦威尔去世后，他的儿子接任但无能。局势恶化，军队驻扎在格雷沙姆学院，阻止了科学家们继续他们的每周会议。蒙克，曾被任命为苏格兰总督，率军进军伦敦并于1660年初恢复了秩序。布朗克是签署承认蒙克权利的宣言的人之一。蒙克呼吁议会进行新的选举，知道人民的情绪会选出保皇党人。布朗克参加了新的议会选举，并于1660年当选为韦斯特伯里议员。伦敦局势的改善，特别是驻扎在格雷沙姆学院的军队已经离开，使得科学家们能够再次在学院开会。1660年11月28日星期三，布朗克是十几位科学家之一，在格雷沙姆学院的一次会议上，他们成立了促进物理-数学实验学习的学会，他们宣布这将促进实验哲学。

布朗克是议会议员，议会投票决定恢复君主制，被处决的查理一世的儿子查理二世于1660年登基。布朗克的父亲在查理还是威尔士亲王时曾任他的副内侍，因此布朗克为新国王所熟知，国王很快回报了他忠诚的支持。1662年，国王任命布朗克为（布拉干萨的）凯瑟琳王后的掌玺大臣和国玺保管人。

格雷沙姆的学会向国王查理二世请愿，要求承认它并授予皇家特许状。布朗克为学会努力工作，确保它活跃。特别是他进行了许多实验，包括一些关于弹道学的实验，一些关于摆的实验，以及一项关于磁针变化的研究。他将第一个主题发表为Recoil of guns。1662年7月15日由国玺通过的皇家特许状创建了伦敦皇家学会，皇家特许状提名布朗克为其第一任主席。鉴于学会创始成员的素质，布朗克，肯定不是学术上最杰出的，被任命为主席可能被认为有点奇怪。当然，布朗克与国王关系最好，这一定是一个主要因素，但还有其他原因。布朗克是一个有独立收入的人，而且他也未婚，所以他有时间投入到学会中，而大多数其他人则没有。同样，没有人能比布朗克更热情地推动学会的目标，所以他是一个很好的选择。

作为主席，他继续他的实验工作。Birch在其1667年出版的History of the Royal Society中写道：-

布朗克提议，可以展示关于物体第一速度测量的实验，即需要多大的力才能在一秒时间内将例如一磅重物提升一码高。勋爵被要求担任该实验的主持人。

布朗克现在承担了多项角色。他从1664年到1667年担任伦敦格雷欣学院院长。布朗克的兴趣之一在于船舶，他按照自己的设计为国王查理二世建造了一艘游艇。这种对海洋的兴趣使他成为1664年被任命为海军委员会专员的明显人选，并且如同他在皇家学会的工作一样，他以极大的热情承担起职责。日记作家佩皮斯，时任海军委员会书记员，记录下他认为布朗克：-

……一个非常能干的人。

1668年，布朗克被任命为海军账目审计官。佩皮斯写道：-

事实是[布朗克]是他们所有人中最优秀的人，除我自己之外，我愿意为他服务……

大约从1675年起，布朗克的情况开始变得不那么顺利。他因向查理二世进言反对为罗伯特·胡克的弹簧调节手表颁发专利而与罗伯特·胡克发生争执。罗伯特·胡克和皇家学会中的其他人开始觉得更换主席的时候到了。罗伯特·胡克认为1675年和1676年的年度选举不公平，而罗伯特·胡克在1676年未能当选进入理事会这一事实，使他更加决心改革选举程序。布朗克在1677年似乎只是偶尔出席学会会议，但他似乎仍想继续担任主席。罗伯特·胡克记载，1677年10月18日，当有人提出以投票方式选举理事会的建议时，布朗克担任主席：-

布朗克勃然大怒，胡言乱语一番后走了出去。

布朗克没有出席11月的会议，那次会议选举Joseph Williamson爵士接替他担任主席。

布朗克的数学成就包括关于连分数的工作以及用无穷级数计算对数。1655年，他给出了$\Large\frac{4}{\pi}$的连分数展开式。

$\Large\frac {4} {\pi} \normalsize = 1 + \Large \frac {1^2}{2+ \Large\frac{3^2}{2 + \Large\frac {5^2}{2+...}}}$

这一结果写成约十页，被约翰·沃利斯加入他的著作Arithmetica Infinitorum中，可能最早由布朗克在1654年发现。约翰·沃利斯把这个结果告诉了克里斯蒂安·惠更斯，克里斯蒂安·惠更斯表示强烈怀疑其真实性。然而在布朗克用他的连分数展开正确计算出π的十进制展开的前10位之后，克里斯蒂安·惠更斯接受了这个结果。

可能也是在1654年，布朗克计算了双曲线的求积，尽管他直到1668年才发表这一结果。它出现在布朗克于1668年在Philosophical Transactions of the Royal Society上发表的一篇论文中，但他明确说明这一结果就是约翰·沃利斯在1665年所提到的那个。虽然未以这种形式给出，但布朗克所证明的等价于表明$\Large\frac{1}{1+x}$在0与1之间的积分是

$\Large\frac{1}{1\times 2}\normalsize + \Large\frac{1}{3\times 4}\normalsize + \Large\frac{1}{5\times 6}\normalsize + ...$

或

$1 - \Large\frac{1}{2}\normalsize + \Large\frac{1}{3}\normalsize - \Large\frac{1}{4}\normalsize + \Large\frac{1}{5}\normalsize - \Large\frac{1}{6}\normalsize + ...$

布朗克给出了一种解丢番图方程的方法

$nx^{2} + 1 = y^{2}$

它是在1657-58年讨论皮埃尔·德·费马提出的一个挑战问题的通信中发展出来的。更多细节见我们的文章佩尔方程。据信莱昂哈德·欧拉在将该方程命名为“约翰·佩尔的方程”时犯了错误，他本意是要承认布朗克作出的杰出贡献。有趣的是，如果莱昂哈德·欧拉没有犯这个错误，那么布朗克就不会作为数学家相对默默无闻，而会通过“布朗克的方程”而举世闻名。

1659年，布朗克对Neile关于半三次抛物线$ay^{2} = x^{3}$弧长计算的改进出现在约翰·沃利斯的著作De Cycloide et de Corporibus inde Genitis中。

布朗克获得了一个多少有些不幸的名声。佩皮斯，正如我们所见，是布朗克的好朋友，认为他虐待了他的女性朋友之一特纳夫人，并在日记中写道：-

我觉得他是个心地险恶、虚伪的人，和我认识的任何人一样……因此，我必须相应地提防他，我希望我会做到。

还有其他关于他性格的负面评论，这些评论被认为是不公平的，因为作者将布朗克与他的兄弟Henry Brouncker混淆了。Henry是：-

……一向以冷酷、贪婪、邪恶的人而闻名；但在世俗的狡诈和赌博技巧方面，很少有人能超过他。

似乎是一个残酷的转折，两次误认，将他误认为约翰·佩尔以及他的兄弟Henry，导致布朗克没有得到应有的公正评价。

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 $\Large\frac{4}{\pi}$

$\Large\frac {4} {\pi} \normalsize = 1 + \Large \frac {1^2}{2+ \Large\frac{3^2}{2 + \Large\frac {5^2}{2+...}}}$

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 $\Large\frac{1}{1+x}$ between 0 and 1 was

$\Large\frac{1}{1\times 2}\normalsize + \Large\frac{1}{3\times 4}\normalsize + \Large\frac{1}{5\times 6}\normalsize + ...$

or

$1 - \Large\frac{1}{2}\normalsize + \Large\frac{1}{3}\normalsize - \Large\frac{1}{4}\normalsize + \Large\frac{1}{5}\normalsize - \Large\frac{1}{6}\normalsize + ...$

Brouncker gave a method of solving the diophantine equation

$nx^{2} + 1 = y^{2}$

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 $ay^{2} = x^{3}$ 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.