斯特林James Stirling

斯特林的父亲是Archibald Stirling，他的母亲，即Archibald Stirling的第二任妻子，是安娜·汉密尔顿。斯特林是他们的第三个儿子，他出生在加登的家族庄园，位于苏格兰城镇斯特林以西约20公里处。这个家族是詹姆斯党事业的坚定支持者，这对斯特林的一生有着重大影响。

詹姆斯党的事业就是斯图亚特国王詹姆斯·韦德尔·亚历山大（英国的——苏格兰的斯特林七世：拉丁语中为Jacobus）及其后代的事业，他在1688年革命后被流放。苏格兰于1707年与英格兰和威尔士合并。斯图亚特家族是苏格兰人，但信奉罗马天主教，因此他们只得到有限的支持。然而，他们在法国有一个流亡宫廷，为英国王位提供了一个替代选择，并得到了许多人的大力支持，例如斯特林家族。当斯特林大约17岁时，他的父亲因同情詹姆斯党而被逮捕、监禁并被指控犯有叛国罪。然而，他被宣判无罪。

关于斯特林的童年，或者他在苏格兰的本科岁月，我们一无所知。我们所知道的第一个确切信息是，他于1710年秋天前往牛津，以便在那里注册入学。事实上，斯特林于1711年1月18日作为斯内尔奖学金生在牛津大学贝利奥尔学院注册入学。

斯内尔展览奖学金的条款在[3]中有描述：-

斯内尔展览奖学金由艾尔郡人弗瑞兹·约翰斯内尔（1629？-1679）的遗嘱设立，用于贝利奥尔学院。最初面向苏格兰境内尚未毕业、之后将返回苏格兰担任英格兰教会神职人员的苏格兰学生。提名由格拉斯哥学院进行，候选人条件之一是在格拉斯哥至少学习过一年。

基于这一点，再加上拉姆齐（见[4]）提供的信息——他晚年认识斯特林，并写道斯特林：-

在格拉斯哥大学接受教育

通常会说斯特林确实在格拉斯哥大学学习过（如[1]中所述）。然而这并非绝对确定。我们知道拉姆齐并不总是完全可靠。斯特林的名字并未出现在格拉斯哥的注册学生名单中（并非所有学生的名字都出现在其中，所以这并不十分重要）。特威德尔[3]指出，一位名叫'斯特林'的学生于1710年3月24日在爱丁堡大学注册，未毕业，其签名与这位数学家的签名相似。另一个并非不重要的事实是，斯特林的父亲是爱丁堡的毕业生。若能解决这个以及许多与斯特林生平相关的其他谜题固然很好，但它们可能永远都是谜。

斯特林于1711年10月获得第二份奖学金，即Bishop Warner Exhibition。他本应在注册入学时宣誓，但他对詹姆斯党人的同情使他无法这样做，因此获得豁免。安妮女王于1714年8月去世，德国人乔治一世继承英国王位。1715年爆发了第一次詹姆斯党人叛乱，在1715年11月13日舍里夫缪尔战役打成平局后，叛乱逐渐瓦解。然而，允许斯特林不宣誓的让步被撤销了。当他继续拒绝宣誓时，他失去了奖学金。随后他被指控与参与策划叛乱的詹姆斯党人通信。这段时间他的生活一定很艰难，他甚至因‘诅咒国王乔治’而出现在巡回法庭上，但被宣判无罪。

斯特林现在无法从牛津毕业，但他在那里又待了一段时间。在1717年4月4日伦敦皇家学会的一次会议记录中，当布鲁克·泰勒讲授方程求根和对数时，记录如下：-

牛津大学贝利奥尔学院的斯特林获准出席。

1717年，斯特林发表了他的第一部著作Lineae Tertii Ordinis NeutonianaeⓉ（艾萨克·牛顿的三阶曲线），该书扩展了艾萨克·牛顿的三次平面曲线理论，在艾萨克·牛顿给出的72种曲线之外增加了四种新类型的曲线。该著作在牛津出版，艾萨克·牛顿本人收到了这部献给威尼斯大使尼古拉斯·特龙的著作的副本。

Lineae Tertii Ordinis Neutonianae包含斯特林获得的其他结果。有关于最速降曲线的结果，有关于悬链线的结果（特别是将这一问题与在拱中放置球体的问题联系起来），以及关于正交轨线的结果。正交轨线问题由哥特弗里德·威廉·莱布尼茨提出，除斯特林外，许多数学家也研究过这个问题，包括约翰·伯努利、尼古拉·伯努利一世、尼古拉·伯努利二世和Leonard 莱昂哈德·欧拉。已知斯特林在1716年初解决了这个问题。

1717年，斯特林前往威尼斯。威尼斯大使Tron于1717年6月离开伦敦返回威尼斯，几乎可以肯定斯特林与他同行。斯特林似乎曾被许诺在威尼斯获得一个数学讲席，但由于某种未知原因，这项任命未能实现。斯特林在威尼斯做了什么也不得而知，但他肯定继续了他的数学研究。

斯特林在1719年确实在威尼斯，因为他当时从威尼斯向伦敦皇家学会提交了一篇论文Methodus differentialis Newtoniana illustrata Ⓣ（牛顿的微分方法示例）。该论文由皇家学会收到，并在1719年6月18日的会议上报告。

尼古拉·伯努利一世从1716年到1722年担任帕多瓦大学的讲席。斯特林一定见过尼古拉·伯努利一世并与他相当熟悉，因为1719年，他再次从威尼斯写信给艾萨克·牛顿，提出充当中间人。1721年，斯特林在帕多瓦，我们知道他当时就读于帕多瓦大学。

1722年，斯特林回到了格拉斯哥，也许大约是在他的朋友尼古拉·伯努利一世离开帕多瓦的时候。Tweedie在[5]中讲述了一个故事，说斯特林在意大利期间学到了玻璃工业的秘密，并因担心性命而不得不逃跑，因为玻璃制造商可能曾试图暗杀他，以防止他们的秘密被人知道。从那时到1724年末他做了什么并不清楚，但至少从1722年起，他就有意在伦敦成为一名教师。

1722年8月，科林·麦克劳林在伦敦拜访了艾萨克·牛顿，艾萨克·牛顿给他看了一封斯特林的来信，斯特林在信中写道他打算在伦敦成为一名数学教师。当然，斯特林与艾萨克·牛顿关系友好，这封信几乎肯定是在请求艾萨克·牛顿在这一事业上给予帮助，而艾萨克·牛顿通过告诉科林·麦克劳林关于斯特林的计划来提供这种帮助。

1724年末，斯特林前往伦敦，并在那里待了10年。这十年间，斯特林在数学上非常活跃，与许多数学家通信，并享受着与艾萨克·牛顿的友谊。艾萨克·牛顿提名斯特林为伦敦皇家学会的会士，并于1726年11月3日，斯特林当选。

斯特林在伦敦成为教师的愿望实现了，他被任命到伦敦考文特花园小塔街的威廉·瓦特学院，该学院是[1]：-

……伦敦最成功的学校之一；尽管他不得不借钱购买所需的数学仪器。

该校1727年的招生简章列出了一门由斯特林等人讲授的机械与实验哲学课程。教学大纲包括力学、流体静力学、光学和天文学。

在伦敦期间，斯特林于1730年出版了他最重要的著作Methodus Differentialis Ⓣ（微分方法）。这本书是一部关于无穷级数、求和、插值和求积的论著。$n!$的渐近公式，现在以斯特林公式闻名，斯特林最为人所知的就是这个公式，它作为Methodus Differentialis Ⓣ（微分方法）命题28的例2出现。见THIS LINK。

这本书的主要目的之一是考虑加速级数收敛的方法。斯特林在序言中指出，艾萨克·牛顿曾考虑过这个问题。作为他试图解决的问题的一个例子，斯特林给出了级数$\sum \large\frac{1}{2n(2n-1)}\normalsize$的例子，威廉·布朗克在其关于双曲线下面积的工作中研究过这个级数。斯特林在Methodus DifferentialisⓉ（微分方法）中写道：-

……如果有人要找到这个级数精确到九位的值……他们将需要十亿项；而这个级数比许多其他级数收敛得快得多……

书中给出了他的方法的许多例子，包括哥特弗里德·威廉·莱布尼茨的问题

$\large\frac{\pi}{4}\normalsize = 1 - \large\frac{1}{3}\normalsize + \large\frac{1}{5}\normalsize - \large\frac{1}{7}\normalsize + \large\frac{1}{9}\normalsize - ...$

他还给出了一个处理无穷乘积收敛性的定理。在这部关于加速收敛的著作中，包含了对亚伯拉罕·棣莫弗方法的讨论。

我们上面提到他在Methodus DifferentialisⓉ（微分方法）中研究了插值。例如，他用$T_{1} = 1$定义了级数$T_{n+1} = nT_{n}$。然后他考虑了$T_{3/2}$，介于项$T_{1}$和$T_{2}$之间。用今天的记号，这将是$\Gamma (\large\frac{3}{2}\normalsize )$，而这里的斯特林正在研究伽马函数。他计算了$T_{3/2}$到十位小数。事实上

$T_{3/2} = \Gamma (\large\frac{3}{2}\normalsize ) = \large\frac{1}{2}\normalsize \Gamma (\large\frac{1}{2}\normalsize ) = \large\frac{1}{2}\normalsize √\pi$。

这本书还包含关于Gamma函数和超几何函数的其他结果。

亚伯拉罕·棣莫弗于1730年出版了Miscellanea AnalyticaⓉ（杂分析）。斯特林写信给亚伯拉罕·棣莫弗，指出他在书中阶乘的对数表里犯的一些错误，并告诉亚伯拉罕·棣莫弗关于Methodus DifferentialisⓉ（杂分析）中命题28的例2。亚伯拉罕·棣莫弗能够利用斯特林的想法扩展他早先的结果，并在几个月后出版了Miscellanea Analytica的补编。显然，斯特林和亚伯拉罕·棣莫弗在这段时间经常通信，因为在1730年9月，斯特林在给加布里尔·克拉默的信中叙述了亚伯拉罕·棣莫弗的这一事件和新结果。

斯特林的工作中还有另一个领域我们将要考察，即他关于引力和地球形状的工作。然而，在此之前，我们将看一下斯特林与莱昂哈德·欧拉之间的一次通信，因为这与我们刚刚讨论的关于级数的工作有关。莱昂哈德·欧拉于1736年6月8日从圣彼得堡写信给斯特林。我们引用他信中表达对斯特林工作看法的地方（见[7]或[3]）：-

……我从您那些出色的文章中了解得越多——这些文章我在您的《汇刊》中各处见到过，关于级数的性质，这一研究我确实投入了大量精力——我就越希望结识您，以便能从您本人那里获得更多，同时也能将我自己的思考提交给您评判。但在我写信给您之前，我怀着极大的热切四处寻找您那部关于差分方法的出色著作，不久前我在《莱比锡学报》上看到过它的评论，直到我如愿以偿。如今我已勤勉地通读完毕，我确实惊叹于如此小的一卷书中竟包含如此丰富的大量出色方法，通过这些方法您展示了如何轻松地对缓慢收敛的级数求和，以及如何对非常难以处理的数列进行插值。但特别令我高兴的是第一部分的命题XIV，其中您给出了一种方法，通过这种方法，即使其递进规律尚未确立的级数，也可以仅利用最后几项的关系非常轻松地求和，当然这一方法适用范围极广，用途极大。事实上，这个命题的证明——您似乎有意隐去——给我造成了巨大的困难，直到最后我非常高兴地成功从先前的结果中推导出了它，这就是为什么我至今还未能详细考察所有后续命题的原因

1735年，斯特林回到苏格兰，在那里他被任命为拉纳克郡“苏格兰矿业公司，利德希尔斯”的经理，年薪120英镑。这是一份斯特林做得非常出色的工作，他[8]：-

……作为一位讲求实际的管理者极为成功，由于他采用雇工开矿的方法，矿业公司的状况大为改善。

然而这项工作非常繁重。过了两年他才抽空回复上面我们引用过的莱昂哈德·欧拉的来信。在1738年4月16日从爱丁堡写来的回信中，他解释了自己为何迟迟未复（见[7]或[3]）：——

最近这两年我卷入了大量事务，不得不频繁前往苏格兰，然后再返回伦敦。正是由于这些事务，首先你的信很晚才到我手中，其次，直到今天，几乎都没有时间以应有的专注通读你的信。因为当思考被长时间打断，更不用说被搁置之后，需要耐心才能让头脑重新思考同样的事情。

在同一封信中，斯特林提出愿意提名莱昂哈德·欧拉参选伦敦皇家学会。然而他并没有那样做，可能还是因为矿业公司的工作压力，直到1746年，他才由几位数学家提名，其中不包括斯特林。

看来斯特林从未回复莱昂哈德·欧拉的这第二封信。他在1738年10月26日写信给科林·麦克劳林说，莱昂哈德·欧拉的第二封信：——

……充满许多巧妙的东西，但它很长，我并未完全掌握所有细节。

1745年，斯特林发表了一篇关于矿井通风的论文。他当然没有在矿业公司任职后就放弃数学，在[3]中讨论了斯特林笔记本中未发表的数学工作，这些工作可能写于1730年至1745年之间。

1745年是詹姆斯党人叛乱中最重大的一次发生的年份，科林·麦克劳林在爱丁堡抵御詹姆斯党人的防御中发挥了积极作用。1745年9月17日，年轻的王位觊觎者查尔斯·爱德华率2400人的军队进入爱丁堡。1746年科林·麦克劳林去世，部分原因是前一年战斗的后果，而斯特林被考虑接任他在爱丁堡的讲席。然而，斯特林对詹姆斯党事业的强烈支持意味着这样的任命是不可能的，尤其是在叛乱之后的那一年。

斯特林于1746年当选为柏林皇家科学院的成员。1753年，他辞去了皇家学会的职务，因为他欠学会的钱，再也无力支付年度订阅费。辞职花了他20英镑。

斯特林的一项非数学贡献在[8]中有描述（另见[5]）：-

……他勘测了克莱德河，以期通过一系列船闸使其可通航，从而迈出了使格拉斯哥成为苏格兰商业首都的第一步。市民们并非不知感激，于1752年赠予他一把银茶壶，‘以酬其服务、辛劳与烦劳’。

最后我们必须讨论斯特林的第二项主要数学贡献，即他关于地球形状的工作。1733年12月6日，斯特林向皇家学会宣读了一篇题为Twelve propositions concerning the figure of the Earth的论文。该学会的会议记录写道：-

斯特林被命令致谢，并被要求交流他的命题。

斯特林确实提交了一份扩展版的结果，于1735年以Of the figure of the Earth, and the variation of gravity on the surface发表。在[1]中，该论文被描述为：-

他在其中未经证明地断言地球是一个扁球体，支持艾萨克·牛顿而反对与之竞争的乔凡尼·多美尼科·卡西尼观点。

在接下来的几年里，斯特林被所有人——包括后来自己做出重大贡献的科林·麦克劳林和托马斯·辛普森——认为是该主题上领先的英国专家。正如斯特林未发表的手稿所表明的[3]，他确实比1735年的论文走得远得多，但很可能矿业公司的工作压力使他几乎没有时间打磨这项工作。他在1738年10月26日写给科林·麦克劳林的一封信中解释了为什么尽管有出版的压力他仍未出版：——

去年夏天我收到约翰·梅钦先生的一封信，完全关于地球的形状和新的测量，他似乎认为现在是我发表关于该主题的命题的适当时机，因为每个人都在对此议论纷纷；但我宁愿等到法国人从南方回来，我听说很快就会回来。到目前为止，我还未能将北方的测量与理论调和起来……

事实上，被斯特林称为“南方”的法国厄瓜多尔远征队于1735年出发，但直到1744年才返回。

James Stirling's father was Archibald Stirling and his mother, Archibald Stirling's second wife, was Anna Hamilton. James was their third son and he was born on the family estate at Garden, about 20 km west of the Scottish town of Stirling. The family were strong supporters of the Jacobite cause and this was to have a significant influence on James Stirling's life.

The Jacobite cause was that of the Stuart king, James II (of Britain -- James VII of Scotland: Jacobus in Latin), exiled after the Revolution of 1688, and his descendants. Scotland was united to England and Wales in 1707. The Stuarts were Scottish but Roman Catholics and therefore they had only limited support. They did, however, offer an alternative to the British crown with an exiled court in France which had strong support from many such as the Stirling family. When James Stirling was about 17 his father was arrested, imprisoned and accused of high treason because of his Jacobite sympathies. However he was acquitted of the charges.

Nothing is known of Stirling's childhood or indeed about his undergraduate years in Scotland. The first definite information that we know is that he travelled to Oxford in the autumn of 1710 in order to matriculate there. Indeed Stirling matriculated at Balliol College Oxford on 18 January 1711 as a Snell Exhibitioner.

The terms of the Snell Exhibitions is described in [3]:-

The Snell Exhibitions to Balliol College were established by the will of an Ayrshire man John Snell (1629?-1679). They were originally intended for Scottish students within Scotland who had not graduated and who would subsequently return to Scotland as priests of the Church of England. Nominations were to be made by the College of Glasgow, one of the requirements of candidates being that they should have spent at least one year at Glasgow.

Based on this, together with information from Ramsay (see [4]) who knew Stirling in later life and wrote that he was:-

bred at the University of Glasgow

it is usual to state that indeed Stirling studied at the University of Glasgow (as is done in [1]). However this is not absolutely certain. We know that Ramsay is not always completely reliable. Stirling's name does not appear in the list of students matriculating at Glasgow (not all student's names occur so this is not very significant). Tweddle [3] notes that a student with the name 'James Stirling' matriculated at the University of Edinburgh on 24 March 1710, did not graduate, and has a signature which is similar to that of the mathematician. Another fact, which is not insignificant, is that Stirling's father was a graduate of Edinburgh. It would be nice to solve this and many other puzzles associated with Stirling's life but they may always remain as puzzles.

Stirling was awarded a second scholarship in October 1711, namely the Bishop Warner Exhibition. He should have sworn an oath when matriculating but his Jacobite sympathies would not let him do this and he was excused. Queen Anne died in August 1714 and the German, George I, acceded to the British throne. In 1715 there was the first Jacobite Rebellion, which melted away after the drawn Battle of Sheriffmuir on 13 November 1715. However the concession of allowing Stirling not to swear the oath was withdrawn. He lost his scholarships when he continued to refuse to take the oath. Then he was accused of corresponding with Jacobites who had been involved in planning the rebellion. Life must have been difficult for him at this time and he even appeared at the assizes charged with 'cursing King George' but he was acquitted.

Certainly Stirling could now not graduate from Oxford but he remained there for some time. In the minutes of a meeting of the Royal Society of London on 4 April 1717, when Brook Taylor lectured on extracting roots of equations and on logarithms, it is recorded:-

Mr Stirling of Balliol College Oxford had leave to be present.

In 1717 Stirling published his first work Lineae Tertii Ordinis Neutonianae Ⓣ which extends Newton's theory of plane curves of degree 3, adding four new types of curves to the 72 given by Newton. The work was published in Oxford and Newton himself received a copy of the work which is dedicated to the Venetian ambassador Nicholas Tron.

Lineae Tertii Ordinis Neutonianae contains other results that Stirling had obtained. There are results on the curve of quickest descent, results on the catenary (in particular relating this problems to that of placing spheres in an arch), and results on orthogonal trajectories. The problem of orthogonal trajectories had been raised by Leibniz and many mathematicians worked on the problem in addition to Stirling, including Johann Bernoulli, Nicolaus(I) Bernoulli, Nicolaus(II) Bernoulli, and Leonard Euler. It is known that Stirling solved the problem early in the year 1716.

In 1717 Stirling went to Venice. The Venetian ambassador Tron left London to return to Venice in June 1717 and it is almost certain that Stirling travelled with him. Stirling seems to have been promised a chair of mathematics in Venice but, for some reason that is not known, the appointment fell through. What Stirling did in Venice is also not known but he certainly continued his mathematical research.

Stirling certainly was in Venice in 1719 since he submitted a paper Methodus differentialis Newtoniana illustrata Ⓣ to the Royal Society of London from Venice at that time. The paper was received by the Royal Society and reported to their meeting on 18 June 1719.

Nicolaus(I) Bernoulli occupied the chair at the University of Padua from 1716 until 1722. Stirling must have met Nicolaus(I) Bernoulli and got to know him quite well since, in 1719, he wrote to Newton, again from Venice, offering to act as a go-between. In 1721 Stirling was in Padua and we know that he attended the University of Padua at that time.

In 1722 Stirling returned to Glasgow, perhaps around the time that his friend Nicolaus(I) Bernoulli left Padua. There is a story told by Tweedie in [5] that Stirling learned the secrets of the glass industry while in Italy and had to flee for fear of his life since the glass-makers may have tried to assassinate him to prevent their secrets becoming known. It is not clear what he did between that time and late 1724 but it is clear that, at least from 1722, he had the intention of becoming a teacher in London.

In August 1722 Maclaurin visited Newton in London and Newton showed him a letter from Stirling in which Stirling wrote that he intended to set himself up as a mathematics teacher in London. Certainly Stirling was friendly with Newton and the letter was almost certainly asking for Newton's help in this venture, help which Newton was giving in telling Maclaurin of Stirling's plans.

In late 1724 Stirling travelled to London where he was to remain for 10 years. These were ten years in which Stirling was very active mathematically, corresponding with many mathematicians and enjoying his friendship with Newton. Newton proposed Stirling for a fellowship of the Royal Society of London and, on 3 November 1726, Stirling was elected.

Stirling achieved his aim of becoming a teacher in London when he was appointed to William Watt's Academy in Little Tower Street, Covent Garden, London which was [1]:-

... one of the most successful schools in London; and, although he had to borrow money to pay for the mathematical instruments he needed.

The school's prospectus of 1727 lists a course on mechanical and experimental philosophy given by Stirling and others. The syllabus included mechanics, hydrostatics, optics, and astronomy.

While in London, Stirling published his most important work Methodus Differentialis Ⓣ in 1730. This book is a treatise on infinite series, summation, interpolation and quadrature. The asymptotic formula for $n!$ now known as Stirling's formula for which Stirling is best known, appears as Example 2 to Proposition 28 of the Methodus Differentialis Ⓣ. See THIS LINK.

One of the main aims of the book was to consider methods of speeding up the convergence of series. Stirling notes in the Preface that Newton had considered this problem. As an example of the problem he is trying to solve Stirling gives the example of the series $\sum \large\frac{1}{2n(2n-1)}\normalsize$ which had been studied by Brouncker in his work on the area under a hyperbola. Stirling writes, in Methodus Differentialis Ⓣ, that:-

...if anyone would find an accurate value of this series to nine places ... they would require one thousand million of terms; and this series converges much swifter than many others...

Many examples of his methods are given, including Leibniz's problem of

$\large\frac{\pi}{4}\normalsize = 1 - \large\frac{1}{3}\normalsize + \large\frac{1}{5}\normalsize - \large\frac{1}{7}\normalsize + \large\frac{1}{9}\normalsize - ...$

and he also gives a theorem to treat convergence of an infinite product. Included in this work on accelerating convergence is a discussion of De Moivre's methods.

We mentioned above that he studied interpolation in the Methodus Differentialis Ⓣ. For example he defined the series $T_{n+1} = nT_{n}$ with $T_{1} = 1$. He then considered $T_{3/2}$, between the terms $T_{1}$ and $T_{2}$. In today's notation this would be $\Gamma (\large\frac{3}{2}\normalsize )$ and Stirling here is studying the Gamma function. He calculated $T_{3/2}$ to ten decimal places. In fact

$T_{3/2} = \Gamma (\large\frac{3}{2}\normalsize ) = \large\frac{1}{2}\normalsize \Gamma (\large\frac{1}{2}\normalsize ) = \large\frac{1}{2}\normalsize √\pi$.

The book contains other results on the Gamma function and the Hypergeometric function.

De Moivre published Miscellanea Analytica Ⓣ in 1730. Stirling wrote to De Moivre pointing out some errors that he had made in a table of logarithms of factorials in the book and also telling De Moivre about Example 2 to Proposition 28 of Methodus Differentialis Ⓣ. De Moivre was able to extend his earlier results using Stirling's ideas and published a Supplement to Miscellanea Analytica a few months later. Clearly Stirling and De Moivre regularly corresponded around this time for in September 1730 Stirling relates the episode and new results of De Moivre in a letter to Gabriel Cramer.

There is another area of Stirling's work that we shall examine, namely his work on gravitation and the figure of the Earth. However, before doing so we will look at a correspondence that Stirling had with Euler since this relates to the work we have just discussed on series. Euler wrote to Stirling on 8 June 1736 from St Petersburg. We quote from his letter where he gives his opinion on Stirling's work (see [7] or [3]):-

... the more I have learned from your excellent articles, which I have seen here and there in your Transactions, concerning the nature of series, a study in which I have indeed expended much effort, the more I have wished to become acquainted with you in order that I could receive more from you yourself and also submit my own deliberations to your judgement. But before I wrote to you, I searched all over with great eagerness for your excellent book on the method of differences, a review of which I had seen a short time before in the Acta Lipsiensis, until I achieved my desire. Now that I have read through it diligently, I am truly astonished at the great abundance of excellent methods contained in such a small volume, by means of which you show how to sum slowly converging series with ease and how to interpolate progressions which are very difficult to deal with. But especially pleasing to me was proposition XIV of part 1 in which you give a method by which series, whose law of progression is not even established, may be summed with great ease using only the relation of the last terms, certainly this method extends very widely and is of the greatest use. In fact the proof of this proposition, which you seem to have deliberately withheld, caused me enormous difficulty, until at last I succeeded with very great pleasure in deriving it from the preceding results, which is the reason why I have not yet been able to examine in detail all the subsequent propositions

In 1735 Stirling returned to Scotland where he was appointed manager of the 'Scotch mining company, Leadhills' in Lanarkshire at a salary of £120 per year. This was a job that Stirling did very well, he [8]:-

... proved extremely successful as a practical administrator, the condition of the mining company improving vastly owing to his method of employing labour to work the mines.

However the work was very demanding. It was two years before he got round to replying to Euler's letter from which we quoted above. In the reply, dated 16 April 1738, and written from Edinburgh he explains why he has not replied sooner (see [7] or [3]):-

During these last two years I have been involved in a great many business matters which have required me to go frequently to Scotland, and then return to London. And it was on account of these affairs that first of all your letter came late into my hands and then that, even to this very day, there is scarcely time available for reading through your letter with the attention which it deserves. For after deliberations have been interrupted, not to say neglected, for a long time, patience is required before the mind can be brought to think about the same things once again.

In the same letter Stirling offered to put Euler's name forward for election to the Royal Society of London. He did not do that, however, probably again through pressure of work with the mining company and it was not until 1746 that he was proposed by several mathematicians not including Stirling.

It appears that Stirling never replied to this second letter from Euler. He wrote to Maclaurin on 26 October 1738 saying that Euler's second letter was:-

... full of many ingenious things, but it is long and I am not quite master of all the particulars.

In 1745 Stirling published a paper on the ventilation of mine shafts. He certainly did not give up mathematics when he took up the post in the mining company, and in [3] there is a discussion of unpublished mathematical work in notebooks of Stirling that were probably written between 1730 and 1745.

The year 1745 was the date of the most major of the Jacobite rebellions and Maclaurin played an active role in the defence of Edinburgh against the Jacobites. Charles Edward, the Young Pretender, entered Edinburgh with an army of 2,400 men on 17 September 1745. In 1746 Maclaurin died, partly as a consequence of the battles of the previous year, and Stirling was considered for his chair at Edinburgh. However Stirling's strong support for the Jacobite cause meant that such an appointment was impossible, especially in the year after the rebellion.

Stirling was elected to membership of the Royal Academy of Berlin in 1746. In 1753 he resigned from the Royal Society as he was in debt to the Society and could no longer afford the annual subscriptions. It cost him £20 to resign.

One non-mathematical contribution by Stirling is described in [8] (see also [5]):-

... he surveyed the Clyde with a view to rendering it navigable by a series of locks, thus taking the first step towards making Glasgow the commercial capital of Scotland. The citizens were not ungrateful, and in 1752 presented him with a silver tea-kettle 'for his service, pains, and trouble'.

Finally we must discuss Stirling's second major mathematical contribution, namely his work on the figure of the Earth. On 6 December 1733 Stirling read a paper to the Royal Society entitled Twelve propositions concerning the figure of the Earth. The minutes of the Society state:-

Mr Stirling was ordered thanks, and was desired to communicate his Propositions.

Indeed Stirling did submit an extended version of his results which appeared as Of the figure of the Earth, and the variation of gravity on the surface in 1735. In [1] the paper is described:-

In it he stated, without proof, that the Earth is an oblate spheroid, supporting Newton against the rival Cassinian view.

Certainly Stirling was considered the leading British expert on the subject for the next few years by all including Maclaurin and Simpson who went on to make major contributions themselves. As Stirling's unpublished manuscripts show [3], he did go much further than the 1735 paper but probably the pressure of work at the mining company gave him too little time to polish the work. He explains in a letter to Maclaurin, dated 26 October 1738, why he has not published despite pressure to do so:-

I got a letter this last summer from Mr Machin wholly relating to the figure of the Earth and the new mensuration, he seems to think this a proper time for me to publish my proposition on that subject when everybody is making a noise about it; but I choose rather to stay till the French arrive from the south, which I hear will be very soon. And hitherto I have not been able to reconcile the measurements made in the north to the theory....

In fact the French expedition to Ecuador, referred to by Stirling as 'the south', left in 1735 but did not return until 1744.