In 1779, Playfair’s first scientific effort was given to the public, in "An Essay on the Arithmetic of Impossible Quantities," published in the sixty-eighth volume of the Philosophical Transactions. In 1782, an advantageous offer prompted him to give up his living, and become tutor to Mr Ferguson of Raith and his brother Sir Ronald Ferguson. It was at this period that he paid the visit to London in which he met Dr Maskelyne. By that gentleman he was introduced to some literary men, and to institutions of literary or philosophical interest. Some of these roused the calm enthusiasm for philosophical greatness which was one of the principal features of his character. "This," he says, "was the first time that I had seen the Observatory of Greenwich, and I entered with profound reverence into that temple of science, where Flamstead, and Halley, and Bradley, devoted their days and their nights to the contemplation of the Heavens. The shades of these ancient sages seemed still to hover round their former mansions, inspiring their worthy successor with the love of wisdom, and pointing out the road to immortality."

From his thirst after knowledge being untainted by political or local prejudices, Playfair had early turned himself to the important discoveries of the continental algebraists, and was the first man of eminence to introduce them to British notice. He perceived the prejudices entertained on the subject in England, and probably the discovery sharpened his appetite for a subject which he found was almost untouched. Speaking of Dr Maskelyne, he says, "He is much attached to the study of geometry, and I am not sure that he is very deeply versed in the late discoveries of the foreign algebraists. Indeed, this seems to be somewhat the case with all the English mathematicians: they despise their brethren on the continent, and think that everything great in science must be for ever confined to the country that produced Sir Isaac Newton." in the works of the eminent natural philosopher one may search long before he will find anything which shows in explicit terms the exact discipline of mind or system of reasoning, by which he has made it to happen that all he has said, has so much the appearance of being truth; but a petty remark, disconnected with the ordinary pursuits of the philosopher, may often strikingly illustrate the operation of his mind, and the means by which he has disciplined himself to approach as near as possible to truth; and, such a passage occurring in this short diary, we beg to insert it. "An anecdote of some Indians was told, that struck me very much, as holding up but too exact a picture of many of our theories and reasonings from analogy. Some American savages having experienced the effects of gunpowder, and having also accidentally become masters of a small quantity of it, set themselves to examine it, with a design of finding out what was its nature, and how it was to be procured. The oldest and wisest of the tribe, after considering it attentively, pronounced it to be a seed. A piece of ground was accordingly prepared for it, and it was sown in the fullest confidence that a great crop of it was to be produced. We smile at the mistake of these Indians, and we do not consider, that, for the extent of their experience, they reasoned well, and drew as logical a conclusion as many of the philosophers of Europe. Whenever we reason only from analogy and resemblance, and whenever we attempt to measure the nature of things by our conceptions, we are precisely in the situation of these poor Americans." In this Playfair exemplified the propensity to reason from certain qualities perceived to be identical, when it is not known but that other qualities not perceived, may be at variance. The wise American saw colour and form like those of a seed, and from these he drew his conclusion. Had he been a botanist, he would have discovered that the grain consisted of saltpetre and charcoal, instead of kernel; and, whatever else he could have made of it, he would have quickly perceived that it was not a seed. In connexion with this it is to be held in mind, that Playfair was essentially a reasoner, and that he was more celebrated for separating the true from the false in the writings of others, or for establishing and applying truths accidentally stumbled upon by others, than for extensive discoveries of his own.

In 1785, Dr Adam Ferguson exchanged the moral chair in the university for that of mathematics, taught by professor Dugald Stewart, and, being in bad health, chose Playfair as his assistant. He continued, however, to attend his two pupils until 1787, when he took up his residence with his mother, who had for some time lived in Edinburgh. He now commenced a series of papers which appeared in the Transactions of the Royal Society of Edinburgh. The first of these was the life of Dr Matthew Stewart, the late professor of mathematics in the university of Edinburgh; a paper written in his usual flowing, simple, and expressive style. A second was a paper on the causes which affect the accuracy of Barometrical Measurements. A third was Remarks on the Astronomy of the Brahmins. The early eastern astronomy was a subject to which he was very partial, and to which some conceive he has paid more attention than its importance warranted. He fought to a certain extent at disadvantage, from ignorance of the language, and consequently of external evidence as to the authenticity of the remarkable records containing the wisdom of the Brahmins; but he calculated their authenticity from the circumstance, that none but a European acquainted with the refinements of modern science could have made the calculations on which they might have been forged. The death of his brother James, in 1793, interrupted his philosophical pursuits, by forcing on his management some complicated business, along with the education of his brother’s son. In 1795, he published an edition of Euclid’s elements for the use of his class. In this work he adopted the plan of using algebraic signs instead of words, to render the proportions more compact and apparent. The plan has been repeatedly practised since that period, and "Playfair’s Euclid" is a book well known to the boys in most mathematical schools, by whom, however, it is not always so much admired as it is known. In 1797 he suffered a severe attack of rheumatism, during which he sketched an essay on the accidental discoveries which have been made by men of science whilst in pursuit of something else, or when they had no determinate object in view; and wrote the observations on the trigonometrical tables of the Brahmins, and the theorems relating to the figure of the earth, which were afterwards published in the Transactions of the Royal Society of Edinburgh. About the same time, his friend Dr Hutton died, and Playfair, who affectionately intended to have written his memoir, found in the study of his works a vast field in which he afterwards distinguished himself, by the preparation of the "Illustrations of the Huttonian Theory of the Earth." Few observers of nature have possessed the power of describing what they have seen, so as to make their facts and deductions perceivable to ordinary thinkers. Playfair possessed the quality, however, to a rare extent; and it was probably its deficiency in the works of his friend Hutton, which prompted him to prepare the elegant and logical "Analysis of the Volcanic Theory of the Earth," which has been so much admired for its own literary merits, and has been the means of rendering popular an important theory which otherwise might have remained in obscurity. It has been said, that the illustration of a theory of the earth was but a profitless employment for so accurately thinking a philosopher, and that the task aught have been left to more imaginative minds, whose speculations would have afforded equal pleasure to those who delight in forming fabrics of theory on insufficient foundations. It is true, that even the lucid commentary of Playfair does not establish the Huttonian as a general and undeviating theory, in an undoubted and indisputable situation; he seems not to have aimed so high; and from the present state of science, no one can predicate that the elementary formation of the earth, or even of its crust, will ever be shown with chemical exactness. All that can be said is, that in as far as the respective experiments and deductions of the theorists have proceeded, the Huttonian Theory is not directly met by any fact produced on the part of the Neptunians, and the phenomena produced in its favour strongly show—indeed show to absolute certainty in some cases—the present formation of a great part of the crust of the earth to have been the effect of fire, how operating in respect to the whole substance of the globe it is impossible to determine. The defence of a theory of the earth had for some time been unpopular among many philosophers, from the production of such majestic fabrics of theory as those of Whiston and Burnet, which, without a sufficient number of ascertained facts for the analysis of the component parts of any portion of the earth’s surfaces, showed in detail the method of its abstraction from the rest of the universe, and the minutiae of its formation. But Playfair never went beyond rational deduction on the facts which were known to him, limiting the extent of his theories to reasonings on what he knew; and it shows the accuracy of his logic, that, while the experiments of Sir James Hall and others (which were in progress but not complete while he wrote,) have tended to support his explication, especially in justifying his opinion that the reason of calcination in bodies subjected to heat was the necessity of the escape of the gases contained in them, we are aware of none which have contradicted him.

The period between 1797 and 1802 was occupied by Mr Playfair in preparing his Illustrations, and in 1803 his biographical sketch of Hutton was published in the Society Transactions. In 1805 he quitted the mathematical chair, and succeeded professor John Robison in that of natural philosophy; during the same year his mother died at the age of eighty-five, and he retired along with a younger brother, his youngest sister, and two nephews, to Burntisland, that he might devote the summer to uninterrupted preparation for the duties of his new class. In the controversy with the clergymen of Edinburgh, regarding his successor to the chair of mathematics, he took an active part. A letter which he addressed to the provost of Edinburgh, in favour of the election of a scientific man, as opposed to a clergyman, was answered by Dr Inglis, and from the nature of the remarks directed against himself, he considered it necessary to reply. The pamphlet produced under these circumstances, showed that his calm temper might be made dangerous by interference: it is written in considerable asperity of spirit, but without vulgar raillery or much personality, and the serious reproof, mixed with occasional sarcasm which it contains, shows great power to wield the weapons of literary warfare. He next occupied himself in preparing papers on the solids of greatest attraction, and on the progress of heat in spherical bodies, which appeared in the Transactions of the Royal Society of Edinburgh. He also presented to the London Royal Society, of which he was admitted a member in 1807, an Account of the Survey of Schehallien. In 1814, he published for the use of his students his well known Outlines of Natural Philosophy, in two volumes octavo. The first volume of this work treats of Dynamics, Mechanics, Hydrostatics, Hydraulics, Aerostatics, and Pneumatics. The second is devoted to Astronomy. A third volume was intended to have embraced Optics, Electricity, and Magnetism; but the work was never completed. In the following year he presented to the Royal Society of Edinburgh a life of his predecessor, professor Robison. His labours for this institution will be perceived to have been very extensive, and they show him not to have been a mercenary man. He was long its chief support, arranging and publishing the Transactions, and gratuitously acting as secretary. In 1816, he published, in the Supplement to the Encyclopaedia Britannica, a "Dissertation on the Progress of Mathematical and Physical Science since the Revival of Letters in Europe," a work of great erudition and research. This work interrupted a new and much altered edition of his Illustrations of the Huttonian Theory, which he had previously designed, but which unfortunately he was never enabled to complete. "It was intended," says his biographer, "to commence with a description of all the well authenticated facts in geology collected during his extensive reading and personal observation, without any mixture of hypothesis whatever. To this followed the general inferences which may be deduced from the facts, an examination of the various geological systems hitherto offered to the world, and the exclusion of those which involved any contradiction of the principles previously ascertained; while the conclusion would have presented the development of the system adopted by the author, and the application of it to explain the phenomena of geology." Previously to 1815, Mr Playfair had confined his geological observations to Britain and Ireland; nor was he able, from causes public or private, previously to that period, to extend them to the continent. His nephew accompanied him on a tour which he designed to extend as far as he could through Italy, Switzerland, and France. He spent a short time in the philosophical circle of Paris, to which his name could not fail to be an introduction. He then passed to Switzerland, and commenced the most important of his geological notices at Mount Jura, where he found blocks of granite, gneiss, and mica slate, lying loosely on the surface of mountains whose solid substance was entirely calcareous. At Lucerne and Chamouni, he was prevented by adverse weather, from making his intended searches among the interior valleys. Towards winter he was about to return, when he received a letter from the provost of Edinburgh, intimating that the patrons of the university permitted his absence during the ensuing session—a circumstance which enabled him to prolong his tour a whole year. After remaining for a month at Geneva, he entered Italy by the Simplon. In the Academia del Cimento at Florence, his enthusiasm for philosophical history was gratified by an inspection of the instruments made by Galileo, among which was the original telescope, made of two pieces of wood, coarsely hollowed out, and tied together with thread. On the 12th of November he set out for Rome, which he reached on the 18th. There he remained during the winter, occupying himself with researches in the Vatican library, such geological observations as the neighbourhood afforded, and the select English society always to be found in the imperial city, among whom he found many of the friends he had met in England. After the termination of the winter he went to Naples, where a wider field for geological observation lay before him. The observations which he made on this part of his route, not so much connected with the action of the volcano as with the state of the surrounding country, are unbodied in some interesting notes, an abstract of which may be found in the memoir above referred to; but it is to be regretted that the amount of so much accurate observation was not brought to bear on his Analysis of the Theory of the Earth. Mr Playfair returned to Rome, whence, after a second visit to Florence, he proceeded, by such gradations as enabled him accurately to observe the mineralogy of the country, to Geneva. While travelling through Switzerland, he visited, and prepared a short but curious account of the Slide of Alpuach, by which trees are conveyed from the sides of Pilatus into the lake of Lucerne, whence they proceed through the Aur to the Rhine. On his return, he passed through Venice, Lyons, and Paris. In the ensuing summer he retired to Burntisland, where he prepared a memoir on Naval Tactics, in illustration of the discoveries of Clerk of Eldin, which was published after his death. He had intended to publish in detached papers his observations on the remarkable objects of his tour, and to have prepared his Illustrations of the Huttonian Theory of the Earth, but he lived scarcely long enough to commence these labours. For some years he had been afflicted with a strangury, which alarmingly increased in the month of June, 1819, and he died on the ensuing 19th of July. He was buried on the 26th, when the members of the Royal Medical Society, and a numerous body of public and private friends, followed him to the grave.