[See Students in 1711 for more information on Charles Gregory]
Neither Charles nor David made any contribution to mathematics that requires mention. But the Snell Exhibition was of essential assistance in the promotion of mathematics when it was conferred on James Stirling who matriculated at Balliol on 18th January 1717.
Mr Tweedie in his book James Stirling: a Sketch of his Life and Works along with his Scientific Correspondence (Oxford: 1922) has dealt so fully and accurately with Stirling's position as a mathematician that I can do nothing more than summarise his statements. I should like however to remark that Mr Tweedie's book is a piece of genuine historical research on a totally different level from that of much more pretentious volumes.
James Stirling was born at Garden in Stirlingshire in 1692 of a family noted for its Royalist - that is, Jacobite - sympathies. It is probable, though there is no documentary evidence for the supposition, that he studied at Glasgow, and his nomination to the Snell Exhibition may have been made by Balliol College to which the right of nomination fell if Glasgow College did not within a certain period send up a scholar or scholars. Stirling's Jacobite sympathies were a source of trouble, but he evidently made a name for himself as a competent mathematician and the story that he was expelled from the University because of his Jacobitism is, in Mr Tweedie's opinion, without foundation. In 1717, on the invitation of Nicolas Tron, the Venetian Ambassador at the English Court to whom he had dedicated his Lineae Tertii Ordinis Neutonianae, he went to Italy in the belief that he would be appointed to a professorship of mathematics in one of the Universities of the Venetian Republic (Padua seems to have been the only "one"). Stirling however was an "Anglican," and, as such, was not acceptable in a Roman Catholic State, so that the proposals for a University Chair fell to the ground. For some time he was financially embarrassed as the Stirling family had suffered for their adhesion to the Stewarts, but the kindly and generous help of Newton brought him relief. It is a pity that so little is known of Stirling's doings in Italy or of the men whom he met; he was certainly in touch with the Bernoullis to the extent that he met Nicholas, as is shown by Stirling's letter to Newton. The connection with Italy is preserved by the epithet "The Venetian" which is applied to him in the Family History of the Stirlings.
On returning from Venice to England he settled in London and was for some years connected with an Academy in Little Tower Street; to this address many letters were sent to him by foreign mathematicians. Mr Tweedie says that "from 1730 onwards Stirling's life in London must have been one of considerable comfort as his 'affairs' became prosperous, while he was a familiar figure at the Royal Society where his opinions carried weight." Still the Academy at Little Tower Street was not at all a satisfactory place for a man of Stirling's abilities, and in 1735 a change came that meant a break with systematic mathematical studies. He was in that year appointed Manager of the Leadhills Mines, and his eminently successful reorganisation and administration of the Mines gave decisive proof that there is no necessary incompatibility between mathematical genius and commercial efficiency. The heavy responsibilities of the managership left little leisure for mathematics, and though his merits as a mathematician were recognised in more ways than one his contributions to mathematics may be said to end with his entry on the serious work at Leadhills. He resigned his membership of the Royal Society in 1754; in 1770 he died and was buried in the Greyfriars' Churchyard.
Stirling made important contributions to mathematics in two different fields (i) in the theory of Higher Plane Curves, and (ii) in the theory of Series. The main features of the Lineae ... Neutonianae are well and briefly summarised by Brill and Max Noether in their "Report on the Development of the Theory of Algebraic Functions" (pp. 128, 129) though they are not quite accurate in their reference to Stirling's examples on his form of Taylor's Theorem. The treatise is, in their judgment, excellent in the treatment of the practical tracing of curves, and it presents the general theory on a basis of fundamental conceptions that have been of essential importance for all later work. He showed that a curve of the nth degree is determined in general by n(n + 3)/2 points, and that parallel lines meet any algebraic curve in the same number of real or imaginary points. The determination of asymptotes, straight or curved, and of the manner in which a curve approaches its asymptotes are excellently treated on the basis of expansions. His proof of Newton's Theorems on cubics is also worthy of notice, but the many notable features that distinguish this first fruit of Stirling's genius can not be detailed here. [For some valuable remarks on Stirling's work on curves I would refer to an article by Wieleitner (Bibliotheca Mathematica, Baud XIV., 55-62.)]
The work by which Stirling is best known is his Methodus Differentialis - not a treatise on the Differential Calculus as that term is now understood but rather on what we call Finite Differences, though that name is inadequate. The book was published in 1730, and contains (i) an Introduction (pp. 1-13); (ii) Part I, Summation of Series (pp. 15-84); (iii) Part II, Interpolation of Series (pp. 85-153). Mr Tweedie has dealt so fully with the special features of Stirling's work in papers in the Proceedings of the Edinburgh Mathematical Society, and in his book on Stirling that it seems unnecessary for me to dwell on them. I would like however to make one remark. If one wishes to get a real insight into the genius of Stirling or of any writer it is absolutely necessary to study the original writings; the too frequent habit of depending on accounts of these writings gives a very imperfect view and it is comparatively uninstructive. It is only by approaching the subject from the standpoint of the writer, and with a knowledge of the limitations prescribed by the state of mathematical science at the time, that the special merits of the writer can be properly estimated. The developments of the Methodus Differentialis have an intimate relation to Gamma Functions and the Hypergeometric Series, but in the study of Stirling we appreciate the genius that enabled him to handle intractable series without the aids that the later developments put at our disposal. It is of course not to be expected that the mathematical student can study at first hand even the majority of the older writers, but I do think that he should make a firsthand acquaintance with some of them; next to Newton I would place Stirling as the man whose work is specially valuable where series are in question. Stirling's work is comparatively small in bulk, and when one had become familiar with the phraseology and manner of statement it would be possible to tackle the bulky volumes that comprise the researches of men like Euler whose work is, I fear, more often quoted than read.