by Niccol˜ Guicciardini

© Oxford University Press 2004 All rights reserved

**Simpson, Thomas** (1710-1761), mathematician, was born on 20 August 1710 at Market Bosworth, Leicestershire, the son of a weaver. It seems that the only formal education he received was from a Richard Smith in a school at Market Bosworth. He moved to Nuneaton, Warwickshire, where he took up the profession of a schoolmaster. He taught himself mathematics, which might seem exceptional but was, however, quite the rule in eighteenth-century Britain: many mathematicians were self-trained. They called themselves 'philomaths' and were often of a provincial background. It is not known exactly which books Simpson read. He was known, because of his reputation as a fortune-teller, as the 'oracle of Nuneaton, Bosworth and the environs'.

About 1730 Simpson married his landlady at Nuneaton, a widow named Swinfield. They had two children, Elizabeth, born in 1736, and Thomas, born in 1738. From 1725 to 1733 he taught mathematics at Nuneaton. He reportedly had to flee to Derby in 1733 after he or his assistant had frightened a girl by dressing up as a devil during an astrology session. In 1736 he was in London, where he continued to work, as a weaver and as a teacher of mathematics. He resided at Spitalfields where he was a tutor at the famous Mathematical Society. Francis Holliday, and perhaps Francis Blake, were among his pupils. With Blake he corresponded on the foundations of the fluxional method in connection with George Berkeley's *Analyst* (1734), a pamphlet consisting of logical criticisms of Newton's and Leibniz's calculi. He was a friend of John Landen, with whom he frequently corresponded. In 1737 he published *A New Treatise of Fluxions.* This was a high-quality textbook devoted to the calculus of fluxions, the Newtonian version of the infinitesimal calculus. The topic was advanced--it was no trivial exercise to write such a book in the 1730s, when the calculus was mastered by only a few mathematicians in Europe.

Simpson also gained a broader reputation as a mathematician by answering some questions in the *Ladies' Diary,* a mathematical periodical where the British philomaths proposed and answered problems. His answers were published from 1736 under pseudonyms such as Marmaduke Hodgson, Hurlothrumbo, Kubernetes, Patrick O'Cavannah, and Anthony Shallow. He also contributed to the *Gentleman's Magazine* in 1736-8, to *Miscellanea Curiosa Mathematica* in 1745-6, and to the *Gentleman's Diary* in 1746. From about 1753 to 1760 he was editor of the *Ladies' Diary,* succeeding Robert Heath, with whom he had a squabble after Heath accused him of plagiarism. The control over this periodical was much valued and there were factions that fought against each other. Heath was a friend of another writer of mathematical textbooks, William Emerson, and their respective conduct as editor and main contributor to the *Ladies' Diary* has been described as contentious.

In 1743 Simpson was appointed assistant to the chief master of mathematics at the newly formed Royal Military Academy at Woolwich, thanks to the recommendation of Martin Folkes, president of the Royal Society, and perhaps also of William Jones (1675-1749). This remained his position until his death. He was elected FRS in 1745 and in 1758 fellow of the Academy of Sciences in Stockholm. His post in Woolwich led him to consider several problems in engineering, concerning in particular fortifications and the building of bridges. About 1760 he had a controversy with John Muller, chief master of mathematics in Woolwich, on the design for a bridge across the Thames at Blackfriars: Muller defended a project with elliptical arches, while Simpson would have preferred semicircular ones.

Simpson was a prolific writer. His most important works are the *Doctrine and Application of Fluxions* (1750), *Mathematical Dissertations* (1743), and *Miscellaneous Tracts* (1757), the first of which is one of the best treatises on fluxions written in the eighteenth century. The second contains a remarkable treatment of the attraction of spheroids, the solids obtained by rotating an ellipse around one of its axes, which improves on results earlier obtained by Colin MacLaurin. The third is mainly concerned with physical astronomy--the precession of the equinoxes, the orbit of the comets, and the motion of the moon. Simpson also contributed several papers to the *Philosophical Transactions of the Royal Society.*

Simpson's style is quite an exception in the panorama of British eighteenth-century mathematics. Unlike many of his countrymen, he leaned towards symbolical and abstract methods based on algebraical manipulations, and he looked with admiration to the analytical results achieved by continental mathematicians. His model was the French savant Alexis-Claude Clairaut. Simpson's most valuable mathematical achievements include his analytical expression of the so-called Newton-Raphson method of approximation of the root of an equation, his study of the methods for solving 'isoperimetrical problems' (methods in what is now called the calculus of variations), and several works on probability and annuities. This last topic occasioned a polemic over priority with Abraham De Moivre. It is an irony that Simpson is nowadays remembered for 'Simpson's rule' for obtaining the area under a curve, a result which he did not claim as his, and which was well known to Isaac Newton. He was also an accomplished writer of introductory textbooks on algebra, geometry, and trigonometry. His slim volume of *Select Exercises for Young Proficients in the Mathematicks* (1752) was widely used; it ran to further editions in 1792 and 1810, and was partly translated into French in 1771.

Simpson died at Market Bosworth, Leicestershire, on 14 May 1761 and was buried at Sutton Cheynell in Leicestershire. After his relatively early death his widow, who died in 1782, was reportedly granted a pension by the crown in recognition of his merit, a rare distinction for a commoner. A tablet was erected on Simpson's grave in 1790 by John Throsby. His papers were given to Henry Watson of the East India Company, who did not publish them. They are now in the library of Columbia University.

NICCOLò GUICCIARDINI

**Sources **

R. V. Wallis and P. J. Wallis, eds., *Biobibliography of British mathematics and its applications,* 2 (1986)

P. J. Wallis, 'Simpson, Thomas', *DSB*

F. M. Clarke, *Thomas Simpson and his times* (1929)

*DNB*

C. Hutton, 'Memoirs of the life and writings of the author', in T. Simpson, *Select exercises for young proficients in the mathematicks,* 2nd edn (1792)

N. Guicciardini, *The development of Newtonian calculus in Britain, 1700-1800* (1989)

I. Todhunter, *A history of the mathematical theories of attraction and the figure of the earth,* 2 vols. (1873)

N. Kollerstrom, 'Thomas Simpson and "Newton's method of approximation": an enduring myth', *British Journal for the History of Science,* 25 (1992), 347-54

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