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THE Reverend Dr MATTHEW STEWART, late Professor of Mathematics in the University of Edinburgh, was the son of the Reverend Mr Dugald Stewart, Minister of Rothsay in the Isle of Bute, and was born at that place in the year 1717. After having finished his course at the grammar-school, being intended by his father for the church, he was sent to the University of Glasgow, and was entered there as a student in 1734. His academical studies were prosecuted with diligence and success; and he was so happy as to be particularly distinguished by the friendship of Dr Hutcheson and Dr Simson. With the latter, indeed, he soon became very intimately connected; for though it is said, that his predilection for the mathematics did not instantly appear on his application to the study of that science, yet the particular direction of his talents was probably observed by his master before it was

From the Transactions of the Royal Society of Edinburgh, Vol. I. (1788.)-Ed.

perceived by himself. Accordingly, after being the pupil of Dr Simson, he became his friend; and during all the time that he remained at the University of Glasgow, pursuing the studies of philosophy and theology, he lived in the closest intimacy with that excellent mathematician, and was instructed by him in, what might not improperly be called, the arcana of the ancient geometry. That science was yet involved in some degree of mystery; for though the extent of its discoveries was nearly ascertained, its analysis, or method of investigation, was but imperfectly understood, and seemed inadequate to the discoveries which had been made by it. The learning and genius of Viviani, Fermat, Halley, and of other excellent mathematicians, had already been employed in removing this difficulty; but their efforts had not been attended with complete success. Dr Simson was now engaged in perfecting what they had begun, and in resisting the encroachments, which he conceived the modern analysis to be making upon the ancient. With this view, he had already published a treatise of Conic Sections, and was now preparing a restoration of the Loci Plani of Apollonius, in which that work was to resume its original elegance and simplicity. To these, and other studies of the same kind, he constantly directed the attention of his young friend, while he was delighted, and astonished at the rapidity of his progress.

Mr Stewart's views made it necessary for him to attend the lectures in the University of Edinburgh in 1741; and that his mathematical studies might suffer no interruption, he was introduced by Dr Simson to Mr Maclaurin, who was then teaching, with so much success, both the geometry and the philosophy of Newton. Mr Stewart attended his lectures, and made that proficiency which was to be expected from the abilities of such a pupil, directed by those of so great a master. But the modern analysis, even when thus powerfully recommended, was not able to withdraw his attention from the ancient geometry. He kept up a regular correspondence with Dr Simson, giving him an account of his progress, and of his discoveries in geometry, which were now both numerous and important, and receiving in return many curious communications with respect to the Loci Plani, and the Porisms of Euclid. These last formed the most intricate and paradoxical subject in the history of the ancient mathematics. Every thing concerning them, but the name, had perished. Pappus Alexandrinus has made mention of three books of Porisms written by Euclid, and has given an account of what they contained; but this account has suffered so much from the injuries of time, that the sense of one proposition only is complete. There was no diagram to direct the geometer in his researches, nor any general notion of the subject, or

of the form of the propositions, to serve as a rule for his conjectures. The task, therefore, of restoring these ancient books, which Dr Simson now imposed on himself, exceeded infinitely the ordinary labours of the critic or the antiquary; and it was only by uniting the learning and diligence of these two characters, with the skill of a profound geometer, that he was at last successful in this difficult undertaking. He had begun it as early as the year 1727, but seems to have communicated the whole progress of his discoveries to Mr Stewart alone.

While the second invention of Porisms, to which more genius was perhaps required than to the first discovery of them, employed Dr Simson, Mr Stewart pursued the same subject in a different, and new direction. In doing so, he was led to the discovery of those curious and interesting propositions, which were published, under the title of General Theorems, in 1746. They were given without the demonstrations; but did not fail to place their discoverer at once among the geometers of the first rank. They are, for the most part, Porisms, though Mr Stewart, careful not to anticipate the discoveries of his friend, gave them no other name than that of Theorems. They are among the most beautiful, as well as most general propositions known in the whole compass of geometry, and are perhaps only equalled by the remark


able Locus to the circle in the second book of Apollonius, or by the celebrated theorem of Mr Cotes. The first demonstration of any considerable number of them, is that which was lately communicated to this Society, though I believe there are few mathematicians, into whose hands they have fallen, whose skill they have not often exercised. The unity which prevails among them is a proof, that a single, though extensive view, guided Mr Stewart in the discovery of them all. It seems probable, that, while he aimed at extending geometry beyond the limits it had reached with the ancients, he had begun to consider the Locus ad quatuor rectas, beyond which their analysis had not reached. With this view, he, no doubt, thought of extending the hypotheses of that problem to their utmost generality; that is, to any number of perpendiculars drawn to an equal number of lines, and to any power whatever of these perpendiculars. In doing this, he could not fail to meet with many curious porisms; for a porism is nothing else than that particular case, when the data of a problem are so related to one another, as to render it indefinite, or capable of innumerable solutions. These cases, which rarely occur, except in the construction of very general and complicated problems, must always interest a geometer, because they trace out the di

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