by Ian Tweddle

© Oxford University Press 2004 All rights reserved

**Stirling, James** (1692-1770), mathematician and mine manager, known within his family as the Venetian, was born in May 1692, the third son of Archibald Stirling (1651-1715), of Garden, Stirlingshire, and his second wife, Anna, eldest daughter of Sir Alexander Hamilton of Haggs and his wife, Dame Mary Murray; there were also five daughters and another son by this marriage and Archibald Stirling had a son by his first marriage. Stirling matriculated at Balliol College, Oxford, as a Snell exhibitioner on 18 January 1711. These exhibitions were intended for undergraduates who had spent at least one year at Glasgow University but there is no documentary evidence of Stirling's having studied at Glasgow and there is reason to believe that he was actually a student at Edinburgh University in 1710. In October 1711 a Bishop Warner exhibition was also secured for him. Stirling had been admitted to Balliol College as a nonjuring student, his family having staunchly Jacobite sympathies, but in the aftermath of the 1715 Jacobite rising this concession was removed and he was deprived of his scholarships, although he remained in Oxford until 1717.

Stirling had formed a friendship with Nicholas Tron, the Venetian ambassador in London (1714-17). At Tron's instigation Stirling then went to the Venetian Republic, probably accompanying Tron, and remained there for several years. Apparently Stirling had the expectation of a professorship of mathematics but this fell through on religious grounds. Little is known about what Stirling did during this period. He did make the acquaintance of Niklaus Bernoulli, professor of mathematics at the University of Padua (1716-22) and there is a record of his being at that university in 1721. He also appears to have been asked to obtain information about the Venetian glass industry for some merchants in England; according to some accounts he had to flee for his life, having discovered some of its closely guarded secrets.

By the second half of 1722 Stirling was back in Scotland but again there is little information on him until he went to London in 1724 or 1725. In 1725 he became a teacher and later a partner in Watts' Academy in Little Tower Street, Covent Garden. He soon made his mark in the scientific community and was elected to the Royal Society on 3 November 1726, having been proposed by Dr John Arbuthnot and recommended by Sir Alexander Cuming. Stirling's affairs generally prospered during the twelve years or so that he spent in London.

Stirling was employed during the summers of 1734-6 by the London-based directors of the Scotch Mines Company at Leadhills in Lanarkshire. In May 1737 he moved permanently from London to Leadhills as the company's chief agent, remaining in this post until the end of his life. By all accounts his management of the mines and his enlightened measures for the welfare of the mining community were highly successful. Unfortunately, mathematical activities could now occupy less and less of his time.

On 30 June 1746 Stirling was elected to membership of the Royal Academy of Berlin, probably at the instigation of Euler, with whom he had conducted a brief mathematical correspondence (1736-8). In the same year, following the death of Colin MacLaurin, Stirling was offered the professorship of mathematics at Edinburgh University but declined it. In 1752 Stirling conducted for the corporation of Glasgow the first of a series of surveys of the River Clyde aimed at improving its navigability and thereby establishing Glasgow as a major commercial centre. For this work he was presented with a silver tea kettle and lamp by the corporation. Stirling resigned from the Royal Society on 17 December 1753 after being pressed for arrears of subscription and refused any concession for his remote domicile. He married rather late in life, some time after 1745, and his wife, Barbara Watson, died in February 1753; they had one daughter, Christian.

Stirling's first published work was his *Lineae tertii ordinis Neutonianae* (1717), which is an account with some extensions of Newton's classification of plane curves determined by third degree polynomial equations in two variables. This little volume was published in Oxford and dedicated to Nicholas Tron. Also included in it are Stirling's solutions to three problems, one of which, on orthogonal trajectories, had been proposed by Leibniz. In 1719 Stirling communicated to the Royal Society from Venice his paper 'Methodus differentialis Newtoniana illustrata' *(PTRS,* 30, 1717-19, 1050-70). It is concerned with interpolation, quadrature, summation, and limits and certainly contains some original work. The theme of this paper was continued and much extended in Stirling's principal contribution to mathematics, the *Methodus differentialis, sive, Tractatus de summatione et interpolatione serierum infinitarum* (1730). Its introductory section contains Stirling's discussion of what are now called 'Stirling numbers' (important in modern combinatorial theory) and in example 2 of proposition 28 there is his version of 'Stirling's formula' for approximating to the logarithm of a large factorial (an early example of an asymptotic series). The results which posterity has chosen to label as Stirling's formula are, however, more directly related to work of De Moivre, whose early investigations led Stirling into this topic and who found a simpler version after learning of Stirling's result. Stirling's book was well received by his contemporaries and was a source of inspiration for later mathematicians. A translation by Francis Holliday entitled *The Differential Method* was published in 1749.

Stirling also wrote a paper entitled 'Of the figure of the earth, and the variation of gravity on the surface' *(PTRS,* 39, 1735-6, 98-105), an important contribution to the theoretical study of the earth's shape and its gravitational forces. In this work he was overtaken by MacLaurin, Thomas Simpson, and A. C. Clairaut. It is, however, clear from his correspondence that he would have contributed substantially more but for his commitments at Leadhills.

Stirling also wrote on non-mathematical subjects. In *A Description of a Machine to Blow Fire by the Fall of Water (PTRS,* 43, 1744-5, 315-17) he described a system employed at Leadhills to produce a blast of air for use in smelters or for ventilation of shafts. Among his manuscripts are many items on weights, measures, and coinage, a short extract from which was published posthumously as 'An account of the money, coins, and weights, used in England, during the reigns of the Saxon princes' *(Transactions of the Society of Antiquaries of Scotland,* 1, 1792, 216-33). Stirling died in Edinburgh on 5 December 1770 and was buried there in Greyfriars churchyard.

IAN TWEDDLE

**Sources **

*James Stirling: a sketch of his life and works along with his scientific correspondence,* ed. C. Tweedie (1922)

W. Fraser, *The Stirlings of Keir* (privately printed, Edinburgh, 1858), 88-102

*Scotland and Scotsmen in the eighteenth century: from the MSS of John Ramsay, esq., of Ochtertyre,* ed. A. Allardyce, 2 (1888), 306-16

I. Tweddle, *James Stirling: 'This about series and such things'* (1988)

W. I. Addison, *The Snell exhibitions: from the University of Glasgow to Balliol College, Oxford* (1901)

*Edinburgh University matriculation album*

A. Grant, *The story of the University of Edinburgh during its first three hundred years,* 2 (1884), 301

W. B. Hendry, 'James Stirling "the Venetian"', *Scotland's Magazine* (Oct 1965), 33-5

A. Stirling, *Gang forward: a Stirling note-book* (1972), 79-83

I. Todhunter, *A history of the mathematical theories of attraction and the figure of the earth,* 2 vols. (1873), vol. 1, pp. 77-82

R. V. Wallis and P. J. Wallis, eds., *Biobibliography of British mathematics and its applications,* 2 (1986), 87

J. Stirling and P. Brown, *A course of mechanical and experimental philosophy* (1727)

I. Tweddle, *James Stirling's methodus differentialis,* Sources in the History of Mathematics and Physical Sciences [forthcoming]

**Archives **

NRA, priv. coll., corresp. and family papers

**Wealth at death **

£40 outstanding owed on bond of £200 to Stirling: NA Scot., commissary of Lanark, CC/6/2

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