# James Stirling

### Quick Info

Garden (near Stirling), Scotland

Edinburgh, Scotland

**James Stirling**was a Scottish mathematician whose most important work

*Methodus Differentialis*in 1730 is a treatise on infinite series, summation, interpolation and quadrature.

### Biography

**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 Glasgowit 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 propositionsIn 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.

### References (show)

- P J Wallis, Biography in
*Dictionary of Scientific Biography*(New York 1970-1990). See THIS LINK. - Biography in
*Encyclopaedia Britannica.*http://www.britannica.com/biography/James-Stirling - I Tweddle,
*James Stirling: this about series and such things*(Edinburgh, 1988). - J Ramsay,
*Scotland and Scotsmen in the Eighteenth Century*(Edinburgh, 1888). - C Tweedie,
*James Stirling : a Sketch of his Life and Works along with his Scientific Correspondence*(Oxford, 1922). - W B Hendry, James Stirling 'The Venetian',
*Scotland's Magazine*(Oct, 1965), 33-35. - T A Krasotkina, The correspondence of L Euler and J Stirling (Russian),
*Istor.-Mat. Issled.***10**(1957), 117-158. - James Stirling,
*Dictionary of National Biography***LIV**(London, 1898), 379-380. See THIS LINK. - I Tweedle, James Stirling's early work on acceleration of convergence,
*Archive for History of Exact Science***45**(1992), 105-125.

### Additional Resources (show)

Other pages about James Stirling:

Other websites about James Stirling:

### Honours (show)

Honours awarded to James Stirling

### Cross-references (show)

- Student Projects: Indian Mathematics - Redressing the balance: Chapter 11
- Other: 18th January
- Other: 3rd November
- Other: 6th December
- Other: Earliest Known Uses of Some of the Words of Mathematics (H)
- Other: Earliest Known Uses of Some of the Words of Mathematics (M)
- Other: Earliest Known Uses of Some of the Words of Mathematics (S)
- Other: Earliest Uses of Symbols for Constants

Written by J J O'Connor and E F Robertson

Last Update Sept1998

Last Update Sept1998