During the early years of his residence in Edinburgh he devoted a considerable part of his leisure to the study of botany and geology, and he never lost his interest in these subjects.
Dr Mackay's training was thus unusually broad, and when, as in his case, a liberal education was associated with a genial, upright and sympathetic nature, it is easy to understand why he was so successful as a teacher and gained both the respect and the affection of his pupils. Many of these attained distinction in various fields at home and abroad, and many proofs have been given of the deep affection they cherished for their old master; no furlough was complete that did not include a visit to the wellknown room in Northumberland Street.
From his University days geometry was Dr Mackay's favourite mathematical study; at an early gate he began a careful reading of the Greek geometers in the original tongue, and as the years passed by he acquired a quite unusual familiarity with the whole range of Euclidean geometry. In the course of his researches he was led to give special attention to Pappus, and it is a matter for regret that the appearance of Hultsch's edition induced him to abandon his intention of editing the Collections. It is not possible to enter here on an analysis of his contributions to the history of geometry; most of these are contained in the Proceedings of the Edinburgh Mathematical Society, but attention may be called to his article, Notice sur le Journalisme Mathématique en Angleterre, contributed to the French Association for the Advancement of Science (Congrès de Besançon 1893)  an exceedingly valuable guide to one aspect of English mathematical history.
In 1884 appeared his Elements of Euclid; the book had a very wide circulation, and contained many novel features that were soon more or less successfully adopted by other editors. Dr Mackay was, to say the least, not an enthusiastic supporter of the Perry movement in its bearing on the teaching of geometry; he always contended that the valuable parts of the now methods had been adopted in the better Scotch schools, and he maintained that a standard order of propositions was a practical, though not a theoretical, necessity. His Plane Geometry, which appeared in 1905, was adapted to the newer programmes; though he had not, I think, the same interest in preparing this later textbook, it is nevertheless a sterling piece of work, such as could only have been written by one who was master of the subject.
Other textbooks from his pen are Arithmetical Exercises (1869) and Arithmetic (1899); he also contributed several articles to Chambers's Encyclopaedia and to the Encyclopaedia Britannica.
Dr Mackay was a member of the A.I.G.T. from its origin, and took a warm interest in its work.
He was one of the founders of the Edinburgh Mathematical Society, was its first President, and was for years a leading contributor to its Proceedings. He served for more than one period on the Council of the Royal Society of Edinburgh, and his services were of special value as a member of its Library Committee. For many years he spent the greater part of his summer holiday on the continent, and he was a wellknown figure at the meetings of the French Association for the Advancement of Science. His great linguistic attainments were of much service to the Permanent International Commission for Mathematical Bibliography, of which he was a member. He also took a keen interest in the Franco Scottish Society, and was a member both of the Council and of the Executive.
Dr Mackay's great learning was fittingly recognised by his old University when, in 1887, it conferred on him the honorary degree of LL.D.
In 1904 he retired from active work, but he had no difficulty in finding congenial occupation; he devoted himself to his favourite studies, and was ever ready to place his great knowledge at the disposal of his friends. Until three months ago he enjoyed fairly good health, but early this year a change for the worse took place, and he passed away on the afternoon of Wednesday, 25th March.
Few men in Scotland have had a stronger influence on the development of mathematics in the secondary schools. None has equalled him in the special branch of historical learning which he made peculiarly his own, and his loss is mourned by a wide circle of pupils and friends.
