Thomas Simpson

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Thomas Simpson
Born20 August 1710
Died14 May 1761(1761-05-14) (aged 50)

Thomas Simpson FRS (20 August 1710 – 14 May 1761) was a British mathematician and inventor known for the eponymous Simpson's rule to approximate definite integrals. The attribution, as often in mathematics, can be debated: this rule had been found 100 years earlier by Johannes Kepler, and in German it is called Keplersche Fassregel.

Fellow of the Royal Society Elected Fellow of the Royal Society, including Honorary, Foreign and Royal Fellows

Fellowship of the Royal Society is an award granted to individuals that the Royal Society of London judges to have made a 'substantial contribution to the improvement of natural knowledge, including mathematics, engineering science and medical science'.

Simpsons rule numerical integration

In numerical analysis, Simpson's rule is a method for numerical integration, the numerical approximation of definite integrals. Specifically, it is the following approximation for equally spaced subdivisions :

Johannes Kepler German mathematician, astronomer and astrologer

Johannes Kepler was a German astronomer, mathematician, and astrologer. He is a key figure in the 17th-century scientific revolution, best known for his laws of planetary motion, and his books Astronomia nova, Harmonices Mundi, and Epitome Astronomiae Copernicanae. These works also provided one of the foundations for Newton's theory of universal gravitation.



Simpson was born in Sutton Cheney, Leicestershire. The son of a weaver, [1] Simpson taught himself mathematics. At the age of nineteen, he married a fifty-year old widow with two children. [2] As a youth, he became interested in astrology after seeing a solar eclipse. He also dabbled in divination and caused fits in a girl after 'raising a devil' from her. After this incident, he and his wife had to flee to Derby. [3] He moved with his wife and children to London at age twenty-five, where he supported his family by weaving during the day and teaching mathematics at night. [4]

Sutton Cheney village in the United Kingdom

Sutton Cheney is a village in Leicestershire, England, close to the location of the Battle of Bosworth. The population of the civil parish at the 2011 census was 538.

Astrology Interpretation of astronomical events and stellar constellations in relation to earthly conditions

Astrology is the study of the movements and relative positions of celestial objects as a means of divining information about human affairs and terrestrial events. Astrology has been dated to at least the 2nd millennium BCE, and has its roots in calendrical systems used to predict seasonal shifts and to interpret celestial cycles as signs of divine communications. Many cultures have attached importance to astronomical events, and some—such as the Indians, Chinese, and Maya—developed elaborate systems for predicting terrestrial events from celestial observations. Western astrology, one of the oldest astrological systems still in use, can trace its roots to 19th–17th century BCE Mesopotamia, from which it spread to Ancient Greece, Rome, the Arab world and eventually Central and Western Europe. Contemporary Western astrology is often associated with systems of horoscopes that purport to explain aspects of a person's personality and predict significant events in their lives based on the positions of celestial objects; the majority of professional astrologers rely on such systems.

Solar eclipse natural phenomenon wherein the Sun is obscured by the Moon

A solar eclipse occurs when an observer passes through the shadow cast by the Moon which fully or partially blocks ("occults") the Sun. This can only happen when the Sun, Moon and Earth are nearly aligned on a straight line in three dimensions (syzygy) during a new moon when the Moon is close to the ecliptic plane. In a total eclipse, the disk of the Sun is fully obscured by the Moon. In partial and annular eclipses, only part of the Sun is obscured.

From 1743, he taught mathematics at the Royal Military Academy, Woolwich. Simpson was a fellow of the Royal Society. In 1758, Simpson was elected a foreign member of the Royal Swedish Academy of Sciences.

Royal Military Academy, Woolwich military academy in Woolwich, in south-east London

The Royal Military Academy (RMA) at Woolwich, in south-east London, was a British Army military academy for the training of commissioned officers of the Royal Artillery and Royal Engineers. It later also trained officers of the Royal Corps of Signals and other technical corps. RMA Woolwich was commonly known as "The Shop" because its first building was a converted workshop of the Woolwich Arsenal.

Royal Society English learned society for science

The President, Council and Fellows of the Royal Society of London for Improving Natural Knowledge, commonly known as the Royal Society, is a learned society. Founded on 28 November 1660, it was granted a royal charter by King Charles II as "The Royal Society". It is the oldest national scientific institution in the world. The society is the United Kingdom's and Commonwealth of Nations' Academy of Sciences and fulfils a number of roles: promoting science and its benefits, recognising excellence in science, supporting outstanding science, providing scientific advice for policy, fostering international and global co-operation, education and public engagement.

The Royal Swedish Academy of Sciences or Kungliga Vetenskapsakademien is one of the royal academies of Sweden. It is an independent, non-governmental scientific organisation which takes special responsibility for the natural sciences and mathematics, but endeavours to promote the exchange of ideas between various disciplines.

He died in Market Bosworth, and was laid to rest in Sutton Cheney. A plaque inside the church commemorates him.

Early work

Simpson's treatise entitled The Nature and Laws of Chance and The Doctrine of Annuities and Reversions were based on the work of De Moivre and were attempts at making the same material more brief and understandable. Simpson stated this clearly in The Nature and Laws of Chance, referring to De Moivre's Doctroine of Chances: "tho' it neither wants Matter nor Elegance to recommend it, yet the Price must, I am sensible, have put it out of the Power of many to purchase it". In both works, Simpson cited De Moivre's work and did not claim originality beyond the presentation of some more accurate data. While he and De Moivre initially got along, De Moivre eventually felt that his income was threatened by Simpson's work and in his second addition of Annuities upon Lives, wrote in the preface: [5]

"After the pains I have taken to perfect this Second Edition, it may happen, that a certain Person, whom I need not name, out of Compassion to the Public, will publish a Second Edition of his Book on the same Subject, which he will afford at a very moderate Price, not regarding whether he mutilates my Propositions, obscures what is clear, makes a Shew of new Rules, and works by mine; in short, confounds, in his usual way, every thing with a croud of useless Symbols; if this be the Case, I must forgive the indigent Author, and his disappointed Bookseller."


Miscellaneous tracts, 1768 Simpson - Miscellaneous tracts on some curious and very interesting subjects in mechanics, physical-astronomy and speculative mathematics, 1768 - 598998.tif
Miscellaneous tracts, 1768

The method commonly called Simpson's Rule was known and used earlier by Bonaventura Cavalieri (a student of Galileo) in 1639, and later by James Gregory; [6] still, the long popularity of Simpson's textbooks invites this association with his name, in that many readers would have learnt it from them.

Bonaventura Cavalieri Italian mathematician and astronomer (1598–1647)

Bonaventura Francesco Cavalieri was an Italian mathematician and a Jesuate. He is known for his work on the problems of optics and motion, work on indivisibles, the precursors of infinitesimal calculus, and the introduction of logarithms to Italy. Cavalieri's principle in geometry partially anticipated integral calculus. He heavily borrowed upon the works of Indian mathematicians, particularly those in the Indian state of Kerala for his works on calculus.

In the context of disputes surrounding methods advanced by René Descartes, Pierre de Fermat proposed the challenge to find a point D such that the sum of the distances to three given points, A, B and C is least, a challenge popularised in Italy by Marin Mersenne in the early 1640s. Simpson treats the problem in the first part of Doctrine and Application of Fluxions (1750), on pp. 26—28, by the description of circular arcs at which the edges of the triangle ABC subtend an angle of pi/3; in the second part of the book, on pp. 505–506 he extends this geometrical method, in effect, to weighted sums of the distances. Several of Simpson's books contain selections of optimisation problems treated by simple geometrical considerations in similar manner, as (for Simpson) an illuminating counterpart to possible treatment by fluxional (calculus) methods. [7] But Simpson does not treat the problem in the essay on geometrical problems of maxima and minima appended to his textbook on Geometry of 1747, although it does appear in the considerably reworked edition of 1760. Comparative attention might, however, usefully be drawn to a paper in English from eighty years earlier as suggesting that the underlying ideas were already recognised then:

Of further related interest are problems posed in the early 1750s by J. Orchard, in The British Palladium, and by T. Moss, in The Ladies' Diary; or Woman's Almanack (at that period not yet edited by Simpson).

Simpson-Weber triangle problem

This type of generalisation was later popularised by Alfred Weber in 1909. The Simpson-Weber triangle problem consists in locating a point D with respect to three points A, B, and C in such a way that the sum of the transportation costs between D and each of the three other points is minimised. In 1971, Luc-Normand Tellier [8] found the first direct (non iterative) numerical solution of the Fermat and Simpson-Weber triangle problems. Long before Von Thünen's contributions, which go back to 1818, the Fermat point problem can be seen as the very beginning of space economy.

In 1985, Luc-Normand Tellier [9] formulated an all-new problem called the “attraction-repulsion problem”, which constitutes a generalisation of both the Fermat and Simpson-Weber problems. In its simplest version, the attraction-repulsion problem consists in locating a point D with respect to three points A1, A2 and R in such a way that the attractive forces exerted by points A1 and A2, and the repulsive force exerted by point R cancel each other out. In the same book, Tellier solved that problem for the first time in the triangle case, and he reinterpreted the space economy theory, especially, the theory of land rent, in the light of the concepts of attractive and repulsive forces stemming from the attraction-repulsion problem. That problem was later further analysed by mathematicians like Chen, Hansen, Jaumard and Tuy (1992), [10] and Jalal and Krarup (2003). [11] The attraction-repulsion problem is seen by Ottaviano and Thisse (2005) [12] as a prelude to the New Economic Geography that developed in the 1990s, and earned Paul Krugman a Nobel Memorial Prize in Economic Sciences in 2008.


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  1. "Thomas Simpson". Holistic Numerical Methods Institute. Retrieved 8 April 2008.
  2. Stigler, Stephen M. The History of Statistics: The Measurement of Uncertainty before 1900. The Belknap Press of Harvard University Press, 1986.
  3. Simpson, Thomas (1710–1761) Archived 24 August 2004 at the Wayback Machine
  4. Stigler, Stephen M. The History of Statistics: The Measurement of Uncertainty before 1900. The Belknap Press of Harvard University Press, 1986.
  5. Stigler, Stephen M. The History of Statistics: The Measurement of Uncertainty before 1900. The Belknap Press of Harvard University Press, 1986.
  6. Velleman, D. J. (2005). The Generalized Simpson's Rule. The American Mathematical Monthly, 112(4), 342–350.
  7. Rogers, D. G. (2009). Decreasing Creases Archived 4 November 2013 at the Wayback Machine Mathematics Today, October, 167–170
  8. Tellier, Luc-Normand, 1972, "The Weber Problem: Solution and Interpretation”, Geographical Analysis, vol. 4, no. 3, pp. 215–233.
  9. Tellier, Luc-Normand, 1985, Économie spatiale: rationalité économique de l'espace habité, Chicoutimi, Gaëtan Morin éditeur, 280 pages.
  10. Chen, Pey-Chun, Hansen, Pierre, Jaumard, Brigitte, and Hoang Tuy, 1992, "Weber's Problem with Attraction and Repulsion," Journal of Regional Science 32, 467–486.
  11. Jalal, G., & Krarup, J. (2003). "Geometrical solution to the Fermat problem with arbitrary weights". Annals of Operations Research, 123 , 67{104.
  12. Ottaviano, Gianmarco and Jacques-François Thisse, 2005, "New Economic Geography: what about the N?”, Environment and Planning A 37, 1707–1725.