visions of his subject, and are usually distinguished by an elegance and simplicity peculiar to themselves. Such, accordingly, were the propositions which Mr Stewart now communicated to the world. He suppressed his investigations, however, which were geometrical, and which, if given with all the precision required by the forms of the ancient geometry, would probably have occupied several volumes. 4 The history of these geometrical discoveries has led us to neglect the order of time. For Mr Stewart, while engaged in them, had entered into the church, and, through the patronage of the Earl of Bute and the Duke of Argyle, had obtained the living of Roseneath. It was in that retired and romantic situation, that he discovered the greater part of the propositions that have just been mentioned. There, also, he used to receive the visits of his friend Mr Melvil, whose ingenious observations in the Physical and Literary Essays give us cause to regret that he was so early taken from the world of science. * In the summer of 1746, the mathematical chair in the University of Edinburgh became vacant by the death of Mr Maclaurin. The General Theorems had not yet appeared; Mr Stewart was known only to his friends; and the eyes of the public were Observations on Light and Colours, Phys. and Lit. Essays, Vol. II. Art. 4. naturally turned on Mr Stirling, who then resided at Leadhills, and who was well known in the mathematical world. He, however, declined appearing as a candidate for the vacant chair; and several others were named, among whom was Mr Stewart. In the end of this year, the General Theorems were published, and gave to their author a decided superiority above all the other candidates. He was accordingly elected Professor of Mathematics in the University of Edinburgh, in the beginning of September 1747. The duties of this office gave a turn somewhat different to his mathematical pursuits, and led him to think of the most simple and elegant means of explaining those difficult propositions, which were hitherto only accessible to men deeply versed in the modern analysis. In doing this, he was pursuing the object which, of all others, he most ardently wished to attain, viz. the application of geometry to such problems as the algebraic calculus alone had been thought able to resolve. His solution of Kepler's problem was the first specimen of this kind which he gave to the world; and it was impossible to have produced one more to the credit of the method he followed, or of the abilities with which he applied it. When the astronomer, from whom that problem takes its name, discovered the elliptical motion of the planets, and their equable description of areas round the sun, he reduced the problem, of computing the place of a planet for a given time, to that of drawing a line through the focus of an ellipse, that should divide the area of the semi-ellipse in a given ratio. It was soon found, that this problem did not admit of an accurate solution; and that no more was to be expected, than an easy and exact approximation. In this, ever since the days of Kepler, the mathematicians of the first name had been engaged, and the utmost resources of the integral calculus had been employed. But though many excellent solutions had been given, there was none of them at once direct in its method and simple in its principles. Mr Stewart was so happy as to attain both these objects. He founds his solution on a general property of curves, which, though very simple, had perhaps never been observed; and, by a most ingenious application of that property, he shows how the approximation may be continued to any degree of accuracy, in a series of results which converge with prodigious rapidity. Whoever examines this solution will be astonished to find a problem brought down to the level of elementary geometry, which had hitherto seemed to require the finding of fluents and the reversion of series; he will acknowledge the reasonableness of whatever confidence Mr Stewart may be hereafter found to place in those simple methods of investigation, which he could conduct with so much ingenuity and success; and will be convinced, that the solution of a problem, though the most elementary, may be the least obvious, and, though the easiest to be understood, may be the most difficult to be discovered. This solution appeared in the second volume of the Essays of the Philosophical Society of Edinburgh, for the year 1756. In the first volume of the same collection, there are some other propositions of Mr Stewart's, which are an extension of a curious theorem in the fourth book of Pappus. They have a relation to the subject of porisms, and one of them forms the 91st of Dr Simson's Restoration. They are besides very beautiful propositions, and are demonstrated with all the elegance and simplicity of the ancient analysis. It has been already mentioned, that Mr Stewart had formed the plan of introducing into the higher parts of mixed mathematics the strict and simple form of ancient demonstration. The prosecution of this plan produced the Tracts Physical and Mathematical, which were published in 1761. In the first of these, Mr Stewart lays down the doctrine of centripetal forces, in a series of propositions, demonstrated (if we admit the quadrature of curves) with the utmost rigour, and requiring no previous knowledge of the mathematics, except the elements of plane geometry, and of conic sections. The good order of these propositions, added to the clearness and simplicity of the demonstrations, renders this tract the best elementary treatise of physical astronomy that is any where to be found. In the three remaining tracts, our author had it in view to determine, by the same rigorous method, the effect of those forces which disturb the motions of a secondary planet. From this he proposed to deduce, not only a theory of the moon, but a determination of the sun's distance from the earth. The former is well known to be the most difficult subject to which mathematics have been applied. Though begun by Sir Isaac Newton, and explained, as to its principles, with singular success; yet, as to the full detail and particular explanation of each irregularity, it was left by that great philosopher less perfect than any other of his researches. Succeeding mathematicians had been employed about the same subject; the problem of the Three Bodies had been proposed in all its generality, and, in as far as regards the motion of the moon, had been resolved by a direct and accurate approximation. But the intricacy and length of these calculations rendered them intelligible only to those, who were well versed in the higher parts of the mathematics. This was what Dr Stewart proposed to remedy, by giving a theory of the moon that might depend, if possible, on elementary geometry alone, or which should, at least, be the simplest that the nature of things would allow. The Tracts were |