In this sketch I can deal only with Gregory's work when at Edinburgh. He seems to have been a teacher of very remarkable powers. A manuscript in Latin of a course on Practical Geometry was left in Edinburgh and was used by his successor in class teaching. An English translation was published by Maclaurin in 1745; the translation contained considerable additions by Maclaurin himself and was long in use. I have read the book and it seems to me to be an excellent piece of work.
In the Simson Collection in Glasgow University Library there is a book with the title Arithmeticae et Algebrae Compendium. In usum juventutis Academiae (Edin. 1736), catalogued under David Gregory's name; but there is no name of the author in the book itself nor any explanation of its origin. It is a well-written textbook and compares not at all unfavourably with Maclaurin's. Some information as to its origin would be welcome. [Further investigation has convinced me that the attribution to Gregory is incorrect (the suggestion, it should be said, was not Simson's) and that Maclaurin is the author or source.]
In 1684 Gregory published a tractate of 50 pages with the title Exercitatio Geometrica de Dimensione Figurarum sive Specimen Methodi Generalis dimetiendi quasvis Figuras. In the Introduction he refers to his uncle's work and states that he has found in such Notebooks of his uncle's as had come into his hands only examples but no systematic treatment of his methods. He had therefore to depend on himself for the discussion of many of the problems dealt with. The tractate shows a thorough mastery of the special processes developed in Gregory's communications to Collins. He applies the method of indivisibles exactly in the manner of Newton and is particularly clear in his explanation of what we should call the element of an integral, ydx, yds, sdx, etc. The problems include practically every type of those that had been handled up to that time. But he is specially good in the treatment of series. He uses freely the binomial theorem for the indices 1/2, -1/2, 1/3, 3/2 and even expands √x(2a - x)3/2/(a - x).
These examples are earlier than the similar cases given by John Craig, in his Methodus figurarum ... quadraturas determinandi (1685). He states but does not prove that the expansion of (x + a)-1 for all values of n may be readily found.
As judged by the Tractate, David Gregory was a worthy successor of his uncle and added to the reputation of the Academic Gregorys.
David Gregory was succeeded in the Chair by his brother, James Gregory (Secundus), born in 1666. He graduated at Edinburgh in 1685, was Professor of Philosophy at St Andrews from 1685 to 1692 and was in that year appointed by the Town Council of Edinburgh to be Professor of Mathematics. Professor Chrystal remarks of him that "he seems to have been an able teacher but did not otherwise add to the reputation of the Gregory family." He held the Chair till 1725 when he retired on a pension that was drawn in part from his successor's salary; he died in 1742.