His Dissertation on the Progress of Mathematical and Physical Science since the Revival of Letters in Europe, contributed to the 4th and later editions of the Encyclopaedia Britannica, is, as respects the mathematical sections at least, a remarkably able and accurate narrative for the time at which it was written. Even the disastrous controversy over the rise of the Calculus is handled with a freedom from prejudice that is a sure guarantee of the genuine scientific spirit; it is, I think (with the possible exception of Maclaurin's Fluxions), the first direct statement in English of the essential elements in the case that is free from a decidedly national bias. He rendered great service to the Royal Society in its early days, being General Secretary for many years, and his Obituary Notices of Matthew Stewart, Hutton and Robison are still frequently cited.
His Elements of Euclid was long in use in, Scottish Schools; in it he uses the Parallel Axiom now known by his name, though he expressly states that it had been "assumed by others, particularly by Ludlam in his very useful little tract entitled Rudiments of Mathematics" (p. 439 of the 7th Ed.). His article on Porisms in the 3rd vol. of the Transactions is one of the frequently quoted expositions of that much debated subject. In his efforts to broaden the outlook of mathematicians and to arouse an interest in the historical development of science he had an able colleague in Leslie who, though less balanced and more prejudiced in his judgments, deserves to be gratefully remembered for his interest in the philosophical treatment of the elements of mathematics. It is perhaps worth noting that Leslie's Elements of Geometry was translated into French and German, and had for some years a considerable circulation on the Continent.
There can be no doubt, I think, that in the closing years of the 18th and the early years of the 19th century, Edinburgh was pre-eminent in Scotland for its active and enlightened interest in science, and the Royal Society was the centre from which that interest was maintained. As an indication of the sympathetic and broadminded spirit of Playfair, the Secretary, it may not be out of place to note that it was through his good offices that the first contributions of Wallace, Leslie's successor in the Chair of Mathematics, and of James Ivory, afterwards known for his work on the attraction of ellipsoids, appeared in the 4th volume of the Transactions of the Society. Both men owed much to Playfair's kindly interest, and they justified it by their subsequent contributions to mathematical learning.