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Significant Scots
Matthew Stewart

STEWART, (DR) MATTHEW, an eminent geometrician, and professor of mathematics in the university of Edinburgh, was born at Rothsay, in the island of Bute,—of which his father, the reverend Mr Dugald Stewart, was minister,—in the year 1717. [Memoir by professor Playfair, Trans. R. Soc. Edin. I. 57.] On finishing his course at the grammar school, he was entered at the university of Glasgow in 1734. At college, he became acquainted with Dr Hutcheson and Dr Simson. In the estimation of the latter, he rose, in after life, from the rank of a favourite pupil, to that of an esteemed friend. They were long intimate personal companions, admired the same branches of their common science, and exhibited in their works symptoms of mutual assistance. It is said, indeed, that we are indebted to the friendship and acuteness of Simson, for the suggestion of mathematics as a study suited to the genius of Stewart. At all events, there is every reason to suppose that the love of the latter for the geometry of the ancients, was derived from his intercourse with his instructor. While attending the lectures of Dr Gregory in Edinburgh, in 1741, the attractions of the new analysis were not sufficient to make him neglect his favourite study; and he communicated to his friend his discoveries in geometry, receiving similar communications in return. While Simson was conducting the laborious investigations, which enabled him to revive the porisms of the ancients, Stewart received the progressive benefit of the discoveries, long before they were communicated to the world; and while he probably assisted his friend in his investigations, he was enabled, by investigating the subject in a new direction, to publish, in 1746, his celebrated series of propositions, termed "General Theorems." "They are," says the author’s biographer, "among the most beautiful, as well as most general propositions known in the whole compass of geometry, and are perhaps only equalled by the remarkable locus to the circle in the second book of Apollonius, or by the celebrated theorems of Mr Cotes. The first demonstration of any considerable number of them, is that which was lately communicated to this society [Communicated by Dr Small.] (the Royal Society of Edinburgh); 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."

Meanwhile, Mr Stewart had become a licentiate of the church of Scotland; and through the joint influence of the earl of Bute and the duke of Argyle, had obtained the living of Roseneath. The "General Theorems" made their appearance at a time when they were calculated to have a considerable effect on the prospects of the author. In the summer of 1746, the mathematical chair of Edinburgh became vacant, by the death of Mr Maclaurin. Stewart was not at that period known to the learned world; and Mr Stirling, a gentleman of well known reputation, was requested to become the new professor. This gentleman declined the situation; and, towards the end of the year, when the patrons of the university were looking for another candidate worthy of the important duty, Stewart’s book was published. The author was readily offered the situation, which he accepted. "The duties of this office," says his biographer, "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 which he followed, or of the abilities with which he applied it." This solution appeared in the second volume of the Essays of the philosophical Society of Edinburgh, for the year 1756. To quote again the words of the eminent biographer: "Whoever examines it, 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." In pursuance of his principle of introducing the forms of ancient demonstration, as applicable to those more complicated parts of the science, called the mixed mathematics, for which they had been considered unqualified, he published, in 1761, his "Tracts, Physical and Mathematical, containing an Explanation of several important Points in Physical Astronomy; and a New Method of ascertaining the Sun’s distance from the Earth, by the Theory of Gravitation." "In the first of these," says his biographer, "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 plain geometry, and 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 anywhere to be found." It was the purpose of the three remaining tracts to determine the effect of those forces which disturb the motions of a secondary planet; and, in particular, to determine the distance of the sun, from its effect in disturbing the motions of the moon. Owing to the geometrical method which he adopted, and likewise to the extreme distance of the sun, which makes all the disturbances he produces on the motion of the moon, very near to that point at which increase of distance to infinity would not change their force, he could only proceed on a system of approximation; and in applying the principles of his plan to a practical calculation of the sun’s distance, which he published in 1763, entitled, "Distance of the Sun from the Earth, determined by the Theory of Gravitation, together with several other things relative to the same subject," he was found to have made a very considerable error. He found the distance of the sun to be equal to 29,875 semi-diameters of the earth, or about 118,541,428 English miles. About five years afterwards, there appeared a pamphlet from the pen of Mr Dawson of Sudbury, called "Four Propositions, intended to point out certain Errors in Dr Stewart’s Investigation, which had given a result much greater than the truth." This was followed by a second attack from Mr Lauden, who, like Price in arithmetic, accomplished the difficult task of becComing an enthusiast in mathematics, and, by means of exaggerating errors, and commenting on their atrocity, astonished the world with a specimen of controversial mathematics. The biographer thus states the sources of the mistakes which called forth these animadversions: "As in arithmetic, we neglect those small fractions which, though of inconsiderable amount, would exceedingly embarrass our computations; so, in geometry, it is sometimes necessary to reject those small quantities, which would add little to the accuracy, and much to the difficulty of the investigation. In both cases, however, the same thing may happen; though each quantity thrown out may be inconsiderable in itself, yet the amount of them altogether, and their effect on the last result, may be greater than is apprehended. This was just what had happened in the present case. The problem to be resolved, is, in its nature, so complex, and involves the estimation of so many causes, that, to avoid inextricable difficulties, it is necessary to reject some quantities, as being small in comparison of the rest, and to reason as if they had no existence." Soon after the publication of this essay, Dr Stewart’s health began to decline; and in 1772, he retired to the country, leaving the care of his class to his eminent son, Dugald Stewart, who was elected joint professor with him in 1775. He died on the 23d January, 1785, at the age of sixty-eight. Besides the works above mentioned, he published, "Propositiones Geometricae more veterum Demonstratae ad Geometriam Antiquam Illustrandam et Promovendam Idoneae," 1763.

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