MATTHEW, D.D., an eminent mathematician, was in 1717 born at Rothsay in the Isle of Bute, of which parish his father was minister. Being intended for the church, he passed through the usual course of a grammar-school education, and was in 1734 received as a student into the university of Glasgow. There he had the happiness of having for his preceptors in moral science and mathematics the celebrated professors Hutcheson and Simson, by the latter of whom he was instructed in what may not improperly be called the arcana of the ancient geometry. His views making it necessary for him to remove to Edinburgh, he was introduced by Dr Simson to Mr Maclaurin, that his mathematical studies might suffer no interruption; and he attended the lectures of that great master with such advantage as might be expected from eminent abilities, directed by the judgment of him who made the philosophy and geometry of Newton intelligible to ordinary capacities. From his intimacy with Simson he had however acquired such a predilection for the ancient geometry, as the modern analysis, however powerfully recommended, could not lessen; and he kept up a regular correspondence with his old master, giving him an account of his progress and his discoveries in geometry, and receiving in return many curious communications respecting the Loci Plani, and the porisms of Euclid.

While the second invention of porisms, to which more genius was perhaps required than to the first discovery of them, employed Dr Simson, his pupil 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.

Before this period he had entered the church, and through the patronage of the Duke of Argyle and the Earl of Bute he obtained the living of Roseneath, a retired country parish in the west of Scotland; but in 1747 he was elected to the mathematical chair in the University of Edinburgh, Stewart, which had become vacant the year before by the death of Matthew Mr Maclaurin. The duties of this office gave a turn somewhat different to his 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, namely, 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. On this problem 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; and his 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 ninety-first of Dr Simson's Restoration. They are besides very beautiful propositions, and are demonstrated with all the elegance and simplicity of the ancient analysis.

The prosecution of the plan which he had formed of introducing into the higher parts of mixed mathematics the strict and simple form of ancient demonstration, produced the Tracts Physical and Mathematical, which were published in 1761, and the Essay on the Sun's Distance, which was published in 1763. In this last work it is acknowledged that he employed geometry on a task which geometry cannot perform; but while it is granted that this determination of the sun's distance is by no means free from error, we may venture to affirm that it contains a great deal which will always interest geometers, and will always be admired by them. Few errors in science are redeemed by the display of so much ingenuity, and, what is more singular, of so much sound reasoning. The investigation is everywhere elegant, and will probably be long regarded as a specimen of the most arduous inquiry which has been attempted by mere geometry.

The Sun's Distance was the last work which Dr Stewart published; and though he lived to see several animadversions on it made public, he declined entering into any controversy. His disposition was far from polemical; and he knew the value of that quiet which a literary man should rarely suffer his antagonists to interrupt. He used to say, that the decision of the point in question was now before the public; that if his investigation was right it would never be overturned, and that if it was wrong it ought not to be defended. A few months before he published the essay just mentioned, he gave to the world another work, entitled Propositiones Geometricae more Veterum demonstratae. This title, it is said, was given to it by Dr Simson, who rejoiced in the publication of a work so well calculated to promote the study of the ancient geometry. It consists of a series of geometrical theorems, for the most part new; investigated first by an analysis, and afterwards synthetically demonstrated by the inversion of the same analysis. Dr Stewart's constant use of the geometrical analysis had put him in possession of many valuable propositions which did not enter into the plan of any of the works that have been enumerated. Of these not a few have found a place in the writings of Dr Simson, where they will for ever remain to mark the friendship of these two mathematicians, and to evince the esteem which Simson entertained for the abilities of his pupil. Stewart's health began to decline, and the duties of his office became burdensome to him. In the year 1772 he retired to a small demesne which he possessed in Ayrshire, where he afterwards spent the greater part of his life, and never resumed his labours in the university. But though mathematics had now ceased to be his business, they continued to be his amusement till a very few years before his death, which happened on the 23rd of January 1785, at the age of sixty-eight.

The habits of study, in a man of original genius, are objects of curiosity, and deserve to be remembered. Concerning those of Dr Stewart, his writings have made it necessary to remark, that from his youth he had been accustomed to the most intense and continued application. In consequence of this application, added to the natural vigour of his mind, he retained the memory of his discoveries in a manner that will hardly be believed. He rarely wrote down any of his investigations till it became necessary to do so for the purpose of publication. When he discovered any proposition, he would put down the enunciation with great accuracy, and on the same piece of paper would construct very neatly the figure to which it referred. To these he trusted for recalling to his mind at any future period the demonstration or the analysis, however complicated it might be. Experience had taught him that he might place this confidence in himself without any danger of disappointment; and for this singular power he was probably more indebted to the activity of his invention than the mere tenaciousness of his memory. Though he was extremely studious, he read few books, and verified the observations of M. d'Alembert, that of all men of letters, mathematicians read least of the writings of one another. His own investigations occupied him sufficiently; and indeed the world would have had reason to regret the misapplication of his talents, had he employed in the mere acquisition of knowledge that time which he could dedicate to works of invention.

Stewart, Dugald, the illustrious son of the eminent mathematician whose merits are recorded in the preceding article, was born at Edinburgh on the 22nd of November 1753. As a pupil in the High School of his native city, he was remarkable for the command of language exhibited in his exercises; and at college his turn for mental science began unequivocally to develop itself. In his nineteenth year he attended at Glasgow a course of lectures delivered by Dr Reid, of whose doctrines he was destined to be the most distinguished amender and expounder. In the course of that winter he composed, and read in a literary association, an essay on Dreaming, which afterwards took its place as an interesting section in his principal work.

But his youthful talents and acquirements were speedily subjected to a much more decisive test. Soon after the period of his studies in Glasgow, he assumed, on the decline of his father's health, the sole charge of his classes in the university of Edinburgh, and on the completion of his twenty-first year was appointed to the mathematical professorship, as assistant and successor. His popularity as a teacher in this department was remarkable; but when he was just twenty-five years old, an opportunity was fortunately afforded him of proving his qualifications for communicating knowledge in his favourite branch of philosophy. Dr Ferguson, the professor of moral philosophy, having gone to America on a public mission, Mr Stewart undertook to fill his place during the session of 1778-9; he entered on the duties of the situation within a week after having promised to discharge them; and for six months, besides teaching his two mathematical classes and a new class of astronomy, he delivered a course of lectures on ethics, thinking over every morning the subject of lecture for the day, and addressed his pupils without having written the discourse or made any further preparation. Those who then heard him considered his extemporaneous lectures as possessing an energy and liveliness even exceeding that which distinguished his later addresses. He was soon transferred permanently to the post which he thus had temporarily filled; Dr Ferguson retiring in 1783, and Mr Stewart, then thirty-two years old, being appointed to the chair of moral philosophy in his room. While his fame as a lecturer rapidly spread all over Great Britain, he long abstained from communicating his speculations to the world in any more durable form. His earliest published work, the first volume of his Elements of the Philosophy of the Human Mind, did not appear till 1792. In 1793 he published his Outlines of Moral Philosophy, an unpretending text-book, but which exhibits his powers of generalization in the most favourable point of view. During several subsequent years his only publications were, the life of Dr Smith in 1793, that of Dr Robertson in 1796, and that of Dr Reid in 1802. These lives appeared in the Transactions of the Royal Society of Edinburgh. But, in the mean time, among his pupils he numbered many of those young men whose talents have since illustrated the northern division of the island; and, without naming those who have shone in political or professional life, it is enough to say that his Elements and his Lectures were the spark that first kindled the metaphysical genius of his own successor. As early likewise as 1780, before his first marriage, he had begun to receive into his family a few private pupils of rank; and some years after his second marriage in 1790, he again opened his house for the same purpose, and superintended the education of several who have occupied a prominent place as British statesmen. His ready command over his own mind was shown, not only by the ease with which he discharged the duties involved in these various avocations, but by his adding to his former course of instruction, in 1800, a series of lectures on Political Economy, which however were not continued. Nor is it unimportant to add, that on several occasions when his colleagues were incapacitated from acting, he temporarily gave lectures on natural philosophy, logic, and rhetoric.

In the winter of 1808-9, Mr Stewart, still in the zenith of his fame, but suffering under ill health, and dejected by the recent loss of his younger son, found himself obliged for a time to discharge the duties of his chair by deputy. In the following session his indisposition was more prolonged; and strongly attached to private study, and sensitively alive to his reputation as a public teacher, he resolved to retire altogether from active life. In May 1810, Dr Thomas Brown, his late assistant, was in consequence joined with him as the acting professor.

After his retirement he lived constantly, as he had formerly done occasionally, at Kinneil House, situated on the Firth of Forth, about twenty miles west from Edinburgh. In this retreat he finished his volume of Philosophical Essays, which was published in 1810, and attained very extensive popularity. The second volume of his Elements of the Philosophy of the Human Mind appeared in 1813; and in the end of 1815 that Dissertation on the Progress of Metaphysical and Ethical Philosophy, which, originally written for the Supplement to this Encyclopedia, has now its place as the opening treatise of the present edition. He published nothing further between that time and the year 1829, when he suffered a severe stroke of palsy. But his mind was unshaken and untouched; and his compositions after this time were as vigorous in every respect as those which had preceded. The third volume of the Elements was published in 1827; and in 1828, a few weeks only before his death, his view of the Active and Moral Powers of Man was given to the world, in two volumes octavo. On the 11th of June 1828, he died at Edinburgh, in the seventieth year of his age, and was buried in the Canongate churchyard. A monument to his memory, erected by his friends and admirers, and successfully imitated from one of the most