On the death of Maclaurin it is stated by Sir Alexander Grant (Story of the Univ. of Edin. II., p. 301) that the patrons desired to appoint Stirling to the Chair but that "his Jacobite principles led him to reject the offered appointment." I am not aware of the authority on which this statement is based, and it would be interesting to have it verified, if correct. In any case Stirling was not appointed and the succession fell to Matthew Stewart.
Gibson History 10 - Matthew Stewart, John Stewart, William Trail
Matthew Stewart was born at Rothesay in 1717. He entered the University of Glasgow in 1734 and in 1741 went to the University of Edinburgh to prepare for entering the Church. He attended the lectures of Maclaurin during Session 1742-3, was licensed for the Ministry and presented to the living of Roseneath in 1745. He became a candidate for the Chair rendered vacant by Maclaurin's death and, as an aid to his candidature, issued his book Some General Theorems of considerable use in the Higher Parts of Mathematics. In 1747 he was elected to the Chair and discharged its duties till 1772 when his health gave way. His son Dugald Stewart undertook to lecture in his stead and in 1775 was appointed joint-professor. Matthew Stewart died on 23rd January 1785.
Besides the General Theorems, Stewart published in 1761 Tracts, Physical and Mathematical and in 1763 Propositiones Geometricae more Veterum Demonstratae.
Among the General Theorems is one which is of considerable importance and which is now known as Stewart's Theorem. It is a curious fact, however, that the credit for the Theorem is due to Simson; for a full discussion of the matter I would refer to Dr Mackay's paper on "Matthew Stewart's Theorem" in the 10th volume of the Proceedings of the Edinburgh Mathematical Society. There are some strange freaks in the nomenclature of theorems; thus the "Simson Line" is not due to Simson nor "Stewart's Theorem" to Stewart. Except as the man whose name is given to a theorem that occurs in elementary geometry, I doubt if Stewart is at all known or has exercised any important influence on the progress of mathematics. His position seems to me to be fairly stated in the following, passage from Chrystal's (unpublished) Inaugural Address, (Grant's Story of Univ. Edin. II. 301). "Though a genius of much lower order than Maclaurin he was nevertheless in his own field an able and original Mathematician. He had been trained by Dr Simson at Glasgow and had imbibed the severe taste of that celebrated expert in the Ancient Geometry. In the cultivation of the Geometrical Analysis of the Ancients Stewart was most successful, and his General Theorems remain much admired monuments of his skill. Like his master Simson he was jealous of the encroachments that Algebra was making on Geometry, and it was his constant aim to reduce to the level of ordinary Geometry problems that were supposed to require the higher calculus. With this view he wrote his Tracts, Physical and Mathematical in which he essayed the application of his pure Geometry to Physical questions. He undoubtedly obtained many important successes in this way; his solution of Kepler's problem being one of the most remarkable. On the whole, however, it was unfortunate for the progress of science in Scotland, that a man of Stewart's limited range should have succeeded the versatile Maclaurin."
Maclaurin's successor in the Chair of Mathematics at the Marischal College, Aberdeen, was John Stewart, whose tenure lasted till 1766, a period of nearly forty years. Stewart had graduated in 1726 so that his promotion came to him early in life. Of his general character and work I can find little information; he was a son of a provost of Aberdeen, and seems to have been known among the students by the nickname "John Triangles." His death is recorded in the Scots Magazine, vol. 28, p. 167, and the circumstances of it are rather painful as his wife and eldest daughter died in the same week and, in the words of the notice, "the three corpses were carried to the grave together."
Stewart published in London in 1745 a translation, with elaborate commentary, of Newton's Quadrature of Curves and Analysis by equations of an infinite number of Terms. Naturally Stewart had his fling at Berkeley who would not, I think, have been greatly discomposed by his arguments; still the commentary is a very careful bit, of work and would be of real advantage to his more thoughtful students. The difference between Stewart's commentary and Maclaurin's Treatise of Fluxions is however very striking; Maclaurin's is the work of an original thinker, thoroughly versed in the writings of his predecessors and extending their results in many directions, while Stewart is content with explaining in minute detail the text of Newton and rarely ventures to go beyond the text. At the same time I should be disposed to conclude that Stewart must have had a good influence on the University teaching, and have contributed materially to the production of the intelligent group of teachers of mathematics who seem to me to have done excellent work in the eighteenth century.
My narrative now brings me to a man who has been styled "the last of the Fathers of Scottish Science." I wonder how many of you can say who he is and who so characterised him. The man is, William Trail, and his sponsor is Sir David Brewster who dedicates his edition of Professor Robison's System of Mechanical Philosophy to Trail, "the last of the Fathers of Scottish Science," and a fellow-student of Robison "whose talents and virtues he admired."
Trail had been a student of Marischal College in 1759-63, had then come to Glasgow where he took the M.A. degree in 1766, and had formed a close intimacy with Simson which continued till Simson's death. In 1766 he was successful in the competition for the Chair of Mathematics at Marischal College, though Playfair and Robert Hamilton were also candidates. He resigned the professorship in 1779 on obtaining preferment in the Irish Church and enjoyed his clerical offices for upwards of fifty years, dying at Bath in 1831 at the age of 85.
It has been said of Trail that he was "a man of great capacity for science, entirely extinguished, together with his taste for its pursuits (as Professor Playfair used to lament) by the sinecure emoluments of the Irish Church."
While in Aberdeen he published Elements of Algebra for the use of Students in Universities. The edition I possess is dated 1789, but it contains an Advertisement, dated Aberdeen, April 1, 1778, stating that it was drawn up for the use of students who were to attend the Lectures of the author, and was not intended to supersede the perusal of a more complete system of Algebra. The book does not bear Trail's name as the author but there is no doubt as to the authorship. Of the book itself little need be said; it is slight but the various operations are clearly described, and the applications of algebra to geometry are on the lines usually followed at that time; in some respects there would be an advantage if our modern school textbooks followed the 18th century books more closely.
Trail's claim to remembrance however depends solely on his Life of Simson. As a biography the book is extraordinarily bad, but it does give a great deal of information about Simson and his geometrical studies that is to be had nowhere else. From an examination of Simson's Adversaria, now in the Glasgow University Library, I can testify how diligently Trail must have studied them; there are some things of interest that he has not transferred to the Life but they are not important, though useful as throwing light on Simson's manner of work. Occasionally letters are given in the Adversaria, not hitherto published, but I doubt if they are worth putting into print.