Stirling, James

▪ British mathematician

born 1692, Garden, Stirling, Scotland

died December 5, 1770, Edinburgh

      Scottish mathematician who contributed important advances to the theory of infinite series and infinitesimal calculus.

      No absolutely reliable information about Stirling's undergraduate education in Scotland is known. According to one source, he was educated at the University of Glasgow (Glasgow, University of), while another source indicates that he attended his father's alma mater, the University of Edinburgh (Edinburgh, University of). Beginning in 1711, Stirling matriculated at Balliol College, Oxford (Oxford, University of), England, on various graduate scholarships for which he was initially exempted as a Jacobite (a supporter of the exiled Stuart (Stuart, House of) king, James II) from swearing a loyalty oath to the British crown. Following the Jacobite rebellion of 1715, Stirling's exemption was withdrawn, and his refusal to take the oath resulted in the loss of his scholarships. Although he remained at Oxford until 1717, he could no longer graduate.

      Early in 1717 Stirling published a supplement to Sir Isaac Newton (Newton, Sir Isaac)'s enumeration of 72 forms of the cubic curve (y = ax³ + bx² + cx + d), titled Lineae Tertii Ordines Newtonianae (“Newtonian Third Order Curves”), which he dedicated to the Venetian ambassador to London. Apparently, in June 1717 Stirling accompanied the ambassador on his return to Venice, where Stirling had been promised an academic position. However, the appointment fell through, and it is unclear what he did in Venice other than study mathematics. From Venice he submitted “Methodus Differentialis Newtoniana Illustrata” (1719; “Newton's Differential Method Illustrated”) through Newton to the Royal Society of London. By 1722 Stirling had returned to Glasgow, and late in 1724 or early in 1725 he went to London, where he found employment as a schoolteacher. Through Newton's sponsorship, Stirling was elected a fellow of the Royal Society in 1726.

      It was during this very productive mathematical period in London that Stirling published his most important work, Methodus Differentialis sive Tractatus de Summatione et Interpolatione Serierum Infinitarum (1730; “Differential Method with a Tract on Summation and Interpolation of Infinite Series”), a treatise on infinite series, summation, interpolation, and quadrature. It contains the statement of what is known as Stirling's formula,

n!  (ⁿ/_e)ⁿ√2πn,

although the French mathematician Abraham de Moivre (Moivre, Abraham de) produced corresponding results contemporaneously.

      From 1734 Stirling was temporarily employed by the Scotch Mines Company, Leadhills, Scotland, and in 1737 he took a permanent position with the company as chief agent.

      Stirling's other publications include On the Figure of the Earth, and On the Variation of the Force of Gravity at Its Surface (1735) and A Description of a Machine to Blow Fire by Fall of Water (1745), the latter possibly deriving from glassblowing techniques that he learned in Venice.

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