# Taylor, Brook (DNB00)

**TAYLOR**, BROOK (1685–1731), mathematician, born at Edmonton in Middlesex on 18 Aug. 1685, was the eldest son of John Taylor (1665–1729), afterwards of Bifrons in Kent. His grandfather, Nathaniel Taylor (*d*. 1684), recorder of Colchester, represented Bedfordshire in the assembly nominated by Cromwell which met at Westminster on 4 July 1653. Brook Taylor's mother was Olivia, daughter of Sir John Tempest, bart., of Durham. After being educated at home in classics and mathematics, he was admitted to St. John's College, Cambridge, on 3 April 1701 as a fellow-commoner, graduating LL.B. in 1709 and LL.D. in 1714. By this time he had attained great proficiency in mathematics, and commenced a correspondence on the subject with John Keill [q. v.], Savilian professor of astronomy at Oxford. In 1712 he addressed a letter to John Machin containing a solution of the problem involved in Kepler's second law of planetary motion. On 3 April 1712 he was admitted a fellow of the Royal Society, and on 14 Jan. 1714 was elected first secretary, a post which he held till 21 Oct. 1718.

In May 1714 Taylor published a remarkable solution of the problem of the centre of oscillation which he had obtained as early as 1708 (*Phil. Trans*. xxviii. 11), although his claim to priority was unjustly disputed by John Bernoulli. In 1715 he published his ‘Methodus Incrementorum Directa et Inversa’ (London, 4to), which was in reality the first treatise dealing with the calculus of finite differences. It contained the celebrated formula known as ‘Taylor's theorem,’ which was the first general expression for the expansions of functions of a single variable in infinite series, and of which Mercator's expansion of log. (1 + *x*), Sir Isaac Newton's binomial theorem, and his expansions of sin x, cos x, ex, &c., were but particular cases. The importance of the discovery was not fully recognised, however, until it was pointed out by La Grange in 1772. In this work Taylor also applied the calculus for the solution of several problems which had baffled previous investigators. He obtained a formula showing that the rapidity of vibration of a string varies directly as the weight stretching it and inversely as its own length and weight. For the first time he determined the differential equation of the path of a ray of light when traversing a heterogeneous medium. He also discussed the form of the catenary and the determination of the centres of oscillation and percussion. A more useful form of equation for a vibrating string was found by Jean le Rond d'Alembert in 1747, but no further advance was made in the theory of refraction until 1798, when Christian Kramp published his ‘Analyse des Réfractions astronomiques et terrestres.’

In 1715 Taylor published a work entitled ‘Linear Perspective,’ followed in 1719 by a second on the same subject entitled ‘New Principles of Linear Perspective,’ which, in their own field, displayed hardly less originality than the ‘Methodus Incrementorum.’ They contained, among other propositions, the first general enunciation of the principle of vanishing points. The subject had already been partially treated by Guido Ubaldi in his ‘Perspectivæ Libri’ (Pisa, 1600) and by Simon Stevin in his ‘Sciagraphia’ (Leyden, 1608). Taylor's treatises proved somewhat too abstruse for contemporary artists, and, in consequence, Joshua Kirby and Daniel Fournier afterwards attempted to reproduce his principles in a simpler form.

From 1715 his studies took a philosophic and religious bent. He corresponded in that year with the Comte de Montmort on the tenets of Malebranche; and unfinished treatises ‘On the Jewish Sacrifices’ and ‘On the Lawfulness of Eating Blood’ were found among his papers. In 1716 he visited Paris and became acquainted with Bossuet and the Comte de Caylus, and in 1720 he visited Bolingbroke at La Source, near Orleans, and laid the foundation of a lasting friendship.

On 4 April 1729 Taylor's father died, and he succeeded in consequence to the estate of Bifrons. His delicate health, however, now began to give way, and the death of his second wife in 1730 completely prostrated him. He died on 29 Dec. 1731 at Somerset House, and was buried in the churchyard of St. Ann's, Soho. ‘As a mathematician he was the only Englishman, after Newton and Cotes, capable of holding his own with the Bernoullis; but a great part of the effect of his demonstrations was lost through his failure to express his ideas fully and clearly.’ He possessed considerable ability as a musician and artist.

Taylor was twice married: first, in 1721, to Miss Brydges of Wallington in Surrey, who died in 1723 in childbed, leaving no issue. This marriage occasioned an estrangement from his father, which was terminated only by his wife's death. In 1729 he married Sabetta, daughter of John Sawbridge of Olantigh in Kent. She died in childbed, leaving a daughter Elizabeth, who married Sir William Young, bart.

Besides the works mentioned, Taylor was the author of numerous papers in the ‘Philosophical Transactions’ from 1712 onwards. In 1793 an essay written by him towards the close of his life, entitled ‘Contemplatio Philosophica,’ was printed by his grandson, Sir William Young [q. v.], with a sketch of his life prefixed. ‘A Treatise on Logarithms,’ addressed to his friend James Hamilton (afterwards seventh Earl of Abercorn), exists in manuscript. His portrait, engraved in 1714 by Richard Earlom from an original painting in the possession of his daughter, was prefixed to his ‘New Principles of Linear Perspective,’ ed. 1811, and to ‘Contemplatio Philosophica.’

[Life by his grandson, Sir William Young, bart., prefixed to Contemplatio Philosophica with an appendix of original papers and letters; Encyclopædia Britannica, 1888, xxiii. 92; Notes and Queries, II. xii. 519, III. v. 357; Archæologia Cantiana, xiv. 175; Nichols's Literary Anecdotes, 1812, i. 171–3; Biographie Universelle, ed. 1843, xli. 95–9; Monthly Review, 1793, i. 321–37; Gent. Mag. 1793, i. 436; Thomson's History of the Royal Society, 1812, pp. 14, 302, App. p. xxxii; Fétis's Biographie Universelle de Musiciens, viii. 194; Hutton's Phil. and Math. Dict.; Grant's Hist. of Phys. Astronomy, p. 377; Marie's Histoire des Sciences, vii. 231; Athenæum, 1861, ii. 727–8; Edleston's Corresp. of Sir I. Newton, 1850, p. 231; D'Israeli's Works, 1859, iv. 175.]