Previous to this, like Leslie, Playfair had been twice a candidate for a similar honour, but unsuccessfully. On the first occasion, when only eighteen years old, he had offered himself, with the approbation of his instructors at St. Andrew's, as candidate for the professorship of mathematics in Marischal College, Aberdeen, and had sustained with much credit a competitive examination which lasted eleven days, and embraced nearly the whole range of the exact sciences. Out of six competitors, two only were judged to have surpassed him—the Rev. Dr. Trail, who was appointed to the office, and Dr. Hamilton, who afterwards succeeded to and long filled it with much reputation.
In 1788, Playfair published, in the 'Transactions' of the Royal Society of Edinburgh, a biographical account of Dr. Matthew Stewart, which also contains a singularly clear and interesting account of the labours of Dr. Simson in the restoration of ancient geometry. In this year likewise appeared his paper 'On the Causes which affect the accuracy of Barometrical Measurements,' which is written with all the perspicuity, caution, and sagacity, that constitute the great excellence and the great difficulty of such disquisitions, where scientific principles are employed to give precision to physical observations. In 1790 appeared, in the same 'Transactions,' a paper of still greater interest and delicacy, 'On the Astronomy of the Brahmins,' the publication of which attracted very general attention, both in Europe and in Asia, and gave rise to much discussion and research. This was followed in 1794 by a learned and very beautiful treatise on the 'Origin and Investigation of Porisms,' in which the obscure nature of the very comprehensive and indefinite theorems to which this name was applied by the ancient geometers, is explained with the most lucid simplicity.
In 1797 he composed a sequel to his first paper on the Indian astronomy, in the shape of 'Observations on the Trigonometrical Tables of the Brahmins,' and also a masterly collection of 'Theorems on the Figure of the Earth.' During the course of the last-mentioned year, his friend Dr. James Hutton died; and Playfair, having undertaken to draw up a biographical account of him for the Royal Society, was led to study the doctor's ingenious but crude speculations on the 'Theory of the Earth,' and afterwards to lend them the assistance of his own powerful pen, in his 'Illustrations of the Huttonian Theory.' This work, upon which he bestowed more time and labour than on any of his other productions, did not appear until 1802; and whatever opinion may be formed of the truth or soundness of Dr. Hutton's speculations, it is impossible to doubt that Playfair's illustration of that theory must always be ranked amongst the most brilliant and powerful productions of philosophical genius. Its merits have been universally acknowledged, even by those not convinced by its reasonings, and have extorted, even from the fastidious critics of France, the acknowledgment that "Mr. Playfair writes as well as Buffon, and reasons incomparably better."
In 1805 he quitted the chair of mathematics to succeed Professor Robison in that of natural philosophy. In the contest which ensued upon the appointment of Leslie as his successor in this chair, he took a very active part; and amongst the heaviest blows which Leslie's opponents had to sustain, were those that parted from the hand of Mr. Playfair. In 1807 he was elected a Fellow of the Royal Society, to which learned body he very soon afterwards presented his 'Account of the Lithological Survey of Schehallien;' this was the result of his investigations during the period of Dr. Maskelyne's visit to Schehallien, to measure the attraction of that mountain, on which occasion Playfair shared the shelter of this distinguished astronomer's tent on the side of the mountain, and contracted with him a friendship which lasted during the remainder of their lives.
In 1809 he contributed to the 'Edinburgh Transactions' an excellent paper on 'Solids of the Greatest Attraction,' and in 1812, another, on the 'Progress of Heat in Spherical Bodies.' In 1814 he published, in two volumes octavo, for the use of his class, an elementary work of great value, under the title of 'Outlines of Natural Philosophy.' About the same time he drew up for the 'Encyclopædia Britannica' an introductory 'Dissertation on the Progress of Mathematical and Physical Science,' a treatise distinguished for the soundness of judgment, beauty of writing, and extent of knowledge displayed in it. In 1815, Playfair wrote for the Royal Society of Edinburgh a very interesting memoir of his distinguished predecessor, Dr. Robison. In the course of the same year he undertook, at the age of sixty-eight, a long journey through France and Switzerland into Italy, and did not return for a period of nearly eighteen months, during which time his principal attention was directed to the mineralogical and geological phenomena of the different regions which he visited. On his return from this expedition, he was occupied in drawing up a memoir on the 'Naval Tactics of Clerk of Eldin,' which was published after his death in the 'Philosophical Transactions.'
Playfair had for several years suffered from a recurrence at different times of a painful affection of the bladder, which appeared with increased seventy in the early part of 1819, but was so far got under as to enable him to complete his course of lectures in the spring. It returned, however, in a still more distressing form in the summer, and at last put a period to his life on the 19th of July. Though suffering great pain during the last part of his confinement, he retained not only his intellectual faculties quite unimpaired, but also the serenity and mildness of his spirit, occupying himself until within a few days of his death in correcting the proof-sheets of the 'Dissertation' before noticed.
Besides the previously mentioned works, Playfair was a frequent contributor to the 'Edinburgh Review,' and also wrote the articles 'Æpinus' and 'Physical Astronomy,' in the 'Encyclopædia Britannica.' Francis Jeffrey, of whose elaborate and elegant memoir the above is but a brief summary, speaks of Playfair as being "one of the most learned mathematicians of the age, and among the first, if not the very first, who introduced the beautiful discoveries of the later Continental geometers to the knowledge of his countrymen, and gave their just value and true place in the scheme of European knowledge, to those important improvements by which the whole aspect of abstract science has been renovated since the days of our illustrious Newton;" also, "as possessing in the highest degree all the characteristics both of a fine and powerful understanding, at once penetrating and vigilant, but more distinguished perhaps for the caution and sureness of its march than for the brilliancy or rapidity of its movements; and guided and adorned through all its progress by the most genuine enthusiasm for all that is grand, and the justest taste for all that is beautiful."—Memoir of John Playfair, by Lord Jeffrey.—Encyclopædia Britannica.
