by Lenore Feigenbaum

© Oxford University Press 2004 All rights reserved

**Taylor, Brook** (1685-1731), mathematician, was born on 18 August 1685 in Edmonton, Middlesex, the eldest son of John Taylor (1655-1729), merchant, and his wife, Olivia *(d.* 1716), daughter of Sir Nicholas Tempest, baronet, of Durham. John's puritan father, Nathaniel Taylor *(d.* 1684), was a barrister who had been selected by Cromwell in 1653 to represent the county of Bedford in parliament. In 1694 John Taylor purchased the estate of Bifrons within a large park in the parish of Patrixbourne, near Canterbury. Here he ran his household with an autocratic hand, his austere nature succumbing to one domestic pleasure, music. Among its celebrated practitioners, Lully, Couperon, Babel, and Geminiani were invited to perform at his home. In a painting by Closterman of the eight children of John Taylor about 1698 the young Brook is shown seated with recorder in hand while two of his older sisters prepare to crown him with a laurel wreath.

During his adolescence Taylor became an accomplished musician and artist, talents which would find mathematical expression in later years in his pioneering study of the vibrating string, and in his treatise on linear perspective. A portrait by Goupy depicts the adult Taylor beside his harpsichord, pointing to an open copy of this treatise, with a landscape on the wall behind him, presumably executed by his own hand. After being tutored at home Taylor was admitted as a fellow-commoner to St John's College, Cambridge, on 3 April 1701; he graduated LLB in 1709 and LLD in 1714. He was admitted an advocate in the court of arches in 1714, but no mention of any legal activity on his part has been found.

During his years at Cambridge, Taylor became proficient in mathematics and physics, and he was elected fellow of the Royal Society on 3 April 1712. Two weeks later he was chosen, along with Abraham De Moivre and Francis Aston, to serve on the Royal Society committee charged with adjudicating the priority controversy between Newton and Leibniz over the invention of the calculus. Although the committee's task was completed one week later, allowing Taylor only limited participation, this was his first public act as a partisan of Newton and paved the way for his subsequent activity as a proponent of Newtonian mechanics and the fluxional calculus. John Keill, Savilian professor of astronomy at Oxford and Newton's most outspoken advocate, became Taylor's mentor and friend. In his correspondence with Keill in 1712 and 1713 Taylor discussed many of his important discoveries, which appeared later in his book *Methodus incrementorum.* Two results on the centre of oscillation, composed in 1708, and on the vibrating string, were first published in the *Philosophical Transactions of the Royal Society* (1713).

In response to the Royal Society's interest in experiments that would advance Newtonian physics, Taylor worked on his own and with curators Francis Hauksbee and J. T. Desaguliers to try to determine the laws of capillarity, magnetic force, and thermometry. On 13 January 1715 he was elected secretary of the Royal Society after the death of Richard Waller. His book *Linear Perspective* appeared later that year, written in formal mathematical style with axioms and theorems. Although the abstruse and concise nature of the text made it inaccessible to most artists, the work influenced later writers on the subject and holds a prominent place in the history of perspective. Not only did it contain contributions to the theory of inverse problems and direct construction, but it was the first to call attention to the importance of vanishing points and lines. Taylor published an expanded version, *New Principles of Linear Perspective,* in 1719.

During the year in which his first treatise on perspective appeared Taylor also published his chief mathematical work, *Methodus incrementorum directa et inversa* (1715; 2nd edn, 1717). He felt that his new method of increments, which came to be known as finite differences, would furnish a stronger and more consistent basis for the Newtonian fluxional calculus than Newton himself had given. The first part of the text concerns the fundamental principles of the method and the transformation and solution of finite difference and differential equations. The second part contains applications of both his method and the calculus to problems in mathematics and mechanics. Several of these, including the formulae for the derivatives of the inverse function, the recognition of a singular solution to a differential equation, a comprehensive discussion of the number and type of boundary conditions to be adjoined to finite difference and differential equations, the equation of motion and fundamental period of the vibrating string, and the differential equation for the path of a ray of light in the atmosphere, were first treated by Taylor. Others, like the catenary, isoperimetric problems, and the centres of oscillation and percussion, had been treated by continental mathematicians, especially Huygens, Leibniz, and the brothers Jacob and Johann Bernoulli.

The celebrated series known as the Taylor series occurs in proposition 7, corollary 2 of *Methodus incrementorum.* Taylor proved it using finite differences and the Gregory-Newton interpolation formula and invoked a passage to the limit that modern mathematicians would not consider rigorous. There is no discussion of a remainder term or convergence. Although Taylor was not the first to find the form of the series--he was anticipated by James Gregory, Newton, Leibniz, Johann Bernoulli, and De Moivre--he can be credited with publishing it first, along with a proof based on his theory of finite increments. Moreover he was the first to appreciate its importance and to demonstrate its applicability as an analytical tool: he employed it to generate series solutions to differential equations of all orders, to obtain series representations for integrals, and to find approximations to the roots of ordinary equations. Although the Taylor series about zero came to be associated with Colin MacLaurin, when MacLaurin published his own derivation using the method of undetermined coefficients, he acknowledged his predecessor: 'This theorem was given by Dr Taylor method. increm.' (C. MacLaurin, *A Treatise of Fluxions,* 2, 1742, 611).

Despite praiseworthy comments about Taylor's achievements from Euler, Lagrange, and others, his *Methodus* was not without its detractors. By citing no one but Newton in the text, Taylor incurred the wrath of Leibniz and Johann Bernoulli, both of whom accused him of deliberate obscurity and lack of originality. Bernoulli went further and charged Taylor with plagiarism. Most would agree that Taylor's style is excessively terse and obscure and that he was negligent in failing to acknowledge the work of his continental predecessors, but Taylor's unpublished papers in London (RS, MS 82) and Cambridge (Taylor MSS, St John's College) show the charge of plagiarism to be unfounded. Nevertheless, the controversy between Taylor and Bernoulli escalated, with accusations from each side appearing publicly in the journals and in their private correspondence with others.

Taylor's most frequent correspondent and confidant was the French probabilist Pierre Rémond de Monmort (1678-1719), whom he met on a visit to Paris in 1715. A disciple of Malebranche, Monmort engaged Taylor in an amicable public debate concerning the merits of Newton's gravitational theory over the vortex theory adhered to by many French Cartesians. Realizing later that Taylor could not be swayed, Monmort vowed, 'I shall love you without loving your attractions, and you shall love me without loving our little vortices' (Monmort to Taylor, 5 Nov 1718, St John's College, Taylor MSS). According to Taylor's grandson, in Paris, Taylor 'was eagerly courted by all who had temper to enjoy, or talents to improve, the charms of social intercourse' (Young, 23-4). Among those seeking his society, in addition to the savants of the Académie Royale des Sciences, were the Abbé Conti, the comte de Caylus, Bishop Bossuet, and Lord Bolingbroke, who became his close friend. It was through the Abbé Conti that Leibniz and Bernoulli sent a challenge problem to the English mathematicians, on orthogonal trajectories for families of curves. Newton was in his seventies by then, and it was left to his younger colleague Taylor to salvage the pride of the English. His solution appeared in the *Philosophical Transactions* (30, 1717). Through Monmort, Taylor sent the Leibnizians two more challenges, on the motion of a projectile in a resisting medium and on the integration of rational fractions. Both problems provoked more bitterness, attacks, and recriminations between Taylor and Bernoulli. Having declared himself neutral in the dispute between Newton and Leibniz, Monmort agreed to play the role of intermediary between Taylor and Bernoulli, but to no avail. The feud ended without resolution after Taylor decided to remain silent in the face of further attacks.

Other events in Taylor's life came to occupy his attention during this time. On 21 October 1718 he resigned as secretary of the Royal Society, informing his fellow secretary Edmond Halley that personal matters would keep him away from London. His health deteriorated and he was sent to recuperate in the spa of Aix-la-Chapelle. Indeed the last decade of his life was marked by failing health and severe emotional strain. In 1721 he married Sarah Elizabeth Brydges, of Wallington, Surrey. The marriage caused an estrangement with his father, since she was 'of good family, but of small fortune' (Young, 33) and his father's consent had not been obtained. In 1723 she died in childbirth, along with the child, but the tragic event had a positive consequence, namely reconciliation between father and son. With his father's approval in 1725 Taylor married Elizabeth (Sabetta), daughter of John Sawbridge of Olantigh, Kent. In July 1729 on the death of his father, Brook inherited the family estate of Bifrons, which was to remain in the Taylor family for close to a century. In March of the next year he lost his second wife in childbirth. This time the child, Elizabeth, survived. (In Taylor's will a second daughter, Olive, is mentioned, but it is not known whether she survived to adulthood.) Years later Elizabeth's son, Sir William Young, second baronet, at the request of some members of the Académie Française, composed a short biography of his grandfather and had it printed, along with some of Taylor's correspondence and an unfinished essay entitled *Contemplatio philosophica.* After the death of his second wife, burdened by grief and beset by ill health, Taylor died 'of a decline' (Young, 40) on 30 November 1731 in Somerset House, London. He was buried in London on 2 December 1731, near his first wife, in the churchyard of St Anne's, Soho.

LENORE FEIGENBAUM

**Sources **

W. Young, *Contemplatio philosophica: a posthumous work of the late Brook Taylor ... to which is prefixed a life of the author, by his grandson* (privately printed, London, 1793)

L. Feigenbaum, 'Brook Taylor and the method of increments', *Archive for History of Exact Sciences,* 34 (1985), 1-40

Venn, *Alum. Cant.*

RS, MS 82

St John Cam., Taylor MSS

K. Andersen, *Brook Taylor's work on linear perspective* (1992)

J. L. Heilbron, *Physics at the Royal Society during Newton's presidency* (1983)

*GM,* 1st ser., 1 (1731), 501

will, PRO

W. A. Scott Robertson, 'Patricksbourne church, and Bifrons', *Archaeologia Cantiana,* 14 (1882), 169-84

E. Hasted, 'Patrixborne', *The history and topographical survey of the county of Kent,* 2nd edn, 9 (1800), 277-86

L. Feigenbaum, 'Happy tercentenary, Brook Taylor!', *Mathematical Intelligencer,* 8 (1986), 53-6

J. E. B. Mayor, ed., *Admissions to the College of St John the Evangelist in the University of Cambridge,* pts 1-2: *Jan 1629/30 - July 1715* (1882-93), 156

Burke, *Gen. GB* (1834-8)

**Archives **

RS, letters

St John Cam. | CUL, papers relating to Lucasian professorship

**Likenesses **

J. Closterman, group portrait, oils, 1696? *(The children of John Taylor of Bifrons Park),* NPG

oils, *c.1715,* RS; repro. in Feigenbaum, 'Happy tercentenary, Brook Taylor!'

L. Goupy?, gouache miniature, 1720, NPG *[see illus.]*

R. Earlom, mezzotint (after B. Taylor), BM; repro. in W. Young, *Contemplatio philosophica: a posthumous work of the late Brook Taylor ... to which is prefixed a life of the author, by his grandson* (privately printed, London, 1793)

**Wealth at death **

inherited extensive family estate and neighbouring properties

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