He left school as Dux in 1920 and entered the university of Glasgow with a Patrick Bursary of £32 per annum, having been placed fifth in the Bursary Competition. At the end of his first year at university he shared the top place in the Ordinary Class of Mathematics. He was also top of the Chemistry Class and was offered the Donaldson Bursary on condition that he became a candidate for Honours in Chemistry. This, however, he declined, since his interests lay in mathematics. Three years later in 1924 he graduated with First Class Honours in Mathematics and Natural Philosophy. Further, as was possible at that time, he received in the same year his B.Sc. in Mathematics, Natural Philosophy and Astronomy, having obtained special distinction in the first two of these three subjects.
At that time the premier award in mathematics available to Glasgow students embarking on postgraduate work was the William Bryce Fellowship, which was available every three or four years. This Gillespie received, and he spent the years 1924-27 as a research student at St John's College, Cambridge, working under the supervision of the late Professor E W Hobson; it was through Hobson that Gillespie acquired his lifelong interest in the calculus of variations. He received his Ph.D. in 1932 and published four papers on related subjects in the Proceedings of the Cambridge Philosophical Society and in the Proceedings of the Edinburgh Mathematical Society between the years 1932 and 1935. He was elected a Fellow of the Royal Society of Edinburgh in 1933.
Gillespie was appointed to an Assistantship in the Department of Mathematics at Glasgow University in 1929 by the late Professor T M MacRobert, who was the head of the department at that time. Over the following years Gillespie, together with his junior colleague, the late Dr T S Graham, gradually took over more and more of the responsibility of running the department and arranging the various courses given to students in the Faculties of Arts and Science. Graham's particular concern was with courses on algebra and geometry, while Gillespie's interests were in analysis. Each was concerned to make available as much time as possible for his favoured subject; one of the duties of the head of the department was to strike a fair balance between these claims. Moreover, the training of young Assistants was largely in their hands. This training was carried out, in their earlier years at any rate, with a strictness and attention to detail that is no longer fashionable, although it was very much in the tradition of the department. That it was extremely efficacious in producing excellent and meticulous lecturers cannot be doubted.
A generation of Glasgow students received their lectures on advanced calculus from Gillespie. They and his colleagues will remember his skill and ingenuity in the evaluation of integrals and composing difficult examination questions. The fruits of his teaching are in his books Integration (Edinburgh, 1939) and Partial Differentiation (Edinburgh, 1951) both published in Oliver & Boyd's series of university mathematical texts. The same firm published in 1972 Part I of his book Solving Problems in Advanced Calculus; because of a change of policy by the publishers, Part II was regrettably not published, although it was written at the same time.
At the beginning of the war Gillespie served briefly in the Clyde River Patrol on a part-time basis before joining the Royal Air Force in 1941. For most of the period between then, and the end of the war he was stationed at Prestwick and was concerned with the air traffic control of North Atlantic traffic. After he returned to the University in 1945 he continued to take an interest in the Royal Air Force and served on interviewing boards for the University Air Squadron. Possessed of a wide culture he was also greatly interested in nature and in art. Not many of his friends and colleagues knew that he was a skilful painter since, being a modest man, he kept his talents hidden.
Gillespie was promoted to a Senior Lectureship in 1948. He took his full share in the work of Faculty and Senate, and it was fitting that, when the Universities of Scotland Act of 1966 made it possible, for Lecturers to serve as Assessors on the University Court, he should have been one of the first two such members to be elected.
He retired in 1969 and later he and his wife Maisie, the daughter of the late Professor A A Bowman, left Glasgow and went to live in Edzell. Robert Gillespie gave devoted service to his university and will be greatly missed by his many former colleagues and students.