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 ship of Darnley with all the family estates and heritable jurisdictions. He married, in 1574, Elizabeth, daughter of Sir William Cavendish [q. v.] (and sister of William, first earl of Devonshire), by whom he had an only daughter, Lady Arabella Stuart [see ]; the fifth earl died in London in 1576, aged 20, and was buried in Henry VII's chapel in Westminster Abbey.

[Letters and Papers, Henry VIII; Cal. State Papers, temp. Eliz. Domestic, Foreign, Spanish, and Venetian Series; Hamilton Papers; Sadler Papers; Reg. P. C. Scotl. vols. i–ii.; Reg. Mag. Sig. Scot. 1543–80; Histories by Knox, Lindsay of Pitscottie, Bishop Leslie, Buchanan, Calderwood, Keith, and Spotiswoode; Diurnal of Occurrents; History of James the Sext, Melville's Memoirs and Richard Bannatyne Memorials (all in the Bannatyne Club); Lord Herries's Memoirs in the Abbotsford Club; Sir William Fraser's Lennox; see also under .]  STEWART, MATTHEW (1717–1785), mathematician, born at Rothesay in Bute in 1717, was the second son of Dugald Stewart (d. 1753), minister of Rothesay, by his wife, Janet Bannatyne (d. 1761). He was educated at the town grammar school, and entered Glasgow University in 1734. There he enjoyed the friendship of Francis Hutcheson (1694–1746) [q. v.] and of Robert Simson [q. v.], the mathematician, to whom he owed his marked predilection for the Greek geometricians. In 1741 he proceeded to Edinburgh University, and studied under Colin Maclaurin [q. v.], but regularly corresponded with Simson on the subject of ancient geometrical methods. Simson was at that time engaged in restoring Euclid's porisms, and Stewart pursued the same subject in a different direction. In 1746 he published ‘General Theorems of considerable use in the higher parts of Mathematics,’ Edinburgh, 8vo. Several of these theorems were in fact porisms, but Stewart avoided the name through fear of seeming to anticipate his friend. Though given without the demonstrations, they placed ‘their discoverer at once among the geometers of the first rank.’

On 6 May 1744 Stewart was licensed to preach by the presbytery of Dunoon, and on 9 May 1745, on the presentation of the Duke of Argyll, he was ordained minister of Roseneath, Dumbartonshire, which charge, however, he resigned on being elected professor of mathematics at Edinburgh University in the beginning of September 1747.

In 1756 he published in the ‘Essays’ of the Philosophical Society of Edinburgh (vol. ii.) a solution of the problem involved in Kepler's second law of planetary motion, remarkable as the first which avoided the use of infinitesimals and employed only elementary geometrical principles. In 1761 Stewart, pursuing his plan of introducing the simplicity of ancient geometrical demonstrations into astronomic investigations, published ‘Tracts, Physical and Mathematical, containing an Explication of several points in Physical Astronomy,’ Edinburgh, 8vo. In these tracts, after laying down the doctrine of centripetal forces in a series of propositions requiring only a knowledge of the elements of plane geometry and of conic sections, he proceeded to determine in the same manner ‘the effect of those forces which disturb the motions of a secondary planet.’ A theorem in which he deduced the motion of the moon's apsides attained an accuracy far surpassing that reached by Newton. The result confirmed that arrived at through algebraical methods by Charles Walmesley [q. v.] in 1749. In 1763 Stewart issued a supplement entitled ‘The Distance of the Sun from the Earth determined by the Theory of Gravity’ (Edinburgh, 8vo), in which he computed the distance at nearly 119 millions of miles. The inaccuracy of this result was due to the difficulty of treating so complex a subject geometrically, Stewart being obliged to neglect so many small quantities in his calculation that the total error seriously affected the result. The nature of his fault was first pointed out in 1769 by John Dawson [q. v.] in a pamphlet entitled ‘Four Propositions’ (Newcastle, 8vo). In 1771 John Landen [q. v.] published an independent refutation of Stewart's conclusions.

On 21 June 1764 Stewart was elected a fellow of the Royal Society. In 1772 the failing state of his health compelled him to retire to his estate at Catrine in Ayrshire, and from 1775 the duties of his mathematical professorship were performed by his son Dugald. He died on 23 Jan. 1785. By his wife Marjory (d. 1771), only daughter of Archibald Stewart, writer to the signet, he was father of Dugald Stewart [q. v.]

Besides the works mentioned, Stewart was the author of ‘Propositiones Geometricæ more veterum demonstratæ,’ Edinburgh, 1763, 8vo; translated in 1801. He also published four propositions extending a theorem in the fourth book of Pappus, in the first volume of the ‘Essays and Observations’ of the Edinburgh Philosophical Society (1754).

[Memoir of Matthew Stewart, by John Playfair [q. v.], in the Transactions of the Royal Society of Edinburgh, i. 57–76; Memoir of Dugald Stewart, by Colonel Matthew Stewart, 1838; Thomson's Hist. of the Royal Society, 