**James Stirling** was a Scottish mathematician whose most important work _Methodus Differentialis _ in 1730 is a treatise on infinite series, summation, interpolation and quadrature.

- Indeed Stirling matriculated at Balliol College Oxford on 18 January 1711 as a Snell Exhibitioner.
- 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).
- 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.
- However the concession of allowing Stirling not to swear the oath was withdrawn.
- Certainly Stirling could now not graduate from Oxford but he remained there for some time.
- In 1717 Stirling published his first work Lineae Tertii Ordinis Neutonianae Ⓣ(Newton's third order curves) which extends Newton's theory of plane curves of degree 3, adding four new types of curves to the 72 given by Newton.
- Lineae Tertii Ordinis Neutonianae contains other results that Stirling had obtained.
- 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 Ⓣ(Newton's differential method illustrated) to the Royal Society of London from Venice at that time.
- 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.
- 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.
- The school's prospectus of 1727 lists a course on mechanical and experimental philosophy given by Stirling and others.
- While in London, Stirling published his most important work Methodus Differentialis Ⓣ(The differential method) in 1730.
- The asymptotic formula for n!n!n! now known as Stirling's formula for which Stirling is best known, appears as Example 2 to Proposition 28 of the Methodus Differentialis Ⓣ(The differential method).
- Stirling notes in the Preface that Newton had considered this problem.
- In today's notation this would be Γ(π) and Stirling here is studying the Gamma function.
- 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 Ⓣ(Miscellaneous analysis).
- 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.
- 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.
- In 1745 Stirling published a paper on the ventilation of mine shafts.
- 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.
- 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.
- 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.
- 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.
- The French expedition to Ecuador, referred to by Stirling as 'the south', left in 1735 but did not return until 1744.

Born May 1692, Garden (near Stirling), Scotland. Died 5 December 1770, Edinburgh, Scotland.

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Ancient Indian, Astronomy, Origin Scotland

Chapters: 1 2

Parts: 3

Theorems: 4

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