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Thomas Simpson | |
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Born | 20 August 1710 |

Died | 14 May 1761 50) | (aged

**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.

**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'.

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** 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** 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** 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.

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.

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.

**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.

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."

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 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:

- J. Collins A Solution, Given by Mr. John Collins of a Chorographical Probleme, Proposed by Richard Townley Esq. Who Doubtless Hath Solved the Same Otherwise,
*Philosophical Transactions of the Royal Society of London*, 6 (1671), pp. 2093–2096.

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).

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.

*Treatise of Fluxions*(1737)*The Nature and Laws of Chance*(1740)*Essays on Several Curious and Useful Subjects in Speculative and Mix'd Mathematicks*(1740)*The Doctrine of Annuities and Reversions*(1742)*Mathematical Dissertations on a Variety of Physical and Analytical Subjects*(1743)*A Treatise of Algebra*(1745)*Elements of Plane Geometry. To which are added, An Essay on the Maxima and Minima of Geometrical Quantities, And a brief Treatise of regular Solids; Also, the Mensuration of both Superficies and Solids, together with the Construction of a large Variety of Geometrical Problems*(Printed for the Author; Samuel Farrer; and John Turner, London, 1747) [The book is described as being*Designed for the Use of Schools*and the main body of text is Simpson's reworking of the early books of The Elements of Euclid. Simpson is designated*Professor of Geometry in the Royal Academy at Woolwich*.]*Trigonometry, Plane and Spherical*(1748)-
*Doctrine and Application of Fluxions. Containing (besides what is common on the subject) a Number of New Improvements on the Theory. And the Solution of a Variety of New, and very Interesting, Problems in different Branches of the Mathematicks*(two parts bound in one volume; J. Nourse, London, 1750) *Select Exercises in Mathematics*(1752)*Miscellaneous Tracts on Some Curious Subjects in Mechanics, Physical Astronomy and Speculative Mathematics*(1757)

**Adrien-Marie Legendre** was a French mathematician who made numerous contributions to mathematics. Well-known and important concepts such as the Legendre polynomials and Legendre transformation are named after him.

**Abraham de Moivre** was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He was a friend of Isaac Newton, Edmond Halley, and James Stirling. Even though he faced religious persecution he remained a "steadfast Christian" throughout his life. Among his fellow Huguenot exiles in England, he was a colleague of the editor and translator Pierre des Maizeaux.

**Thomas Bayes** was an English statistician, philosopher and Presbyterian minister who is known for formulating a specific case of the theorem that bears his name: Bayes' theorem. Bayes never published what would become his most famous accomplishment; his notes were edited and published after his death by Richard Price.

* The Doctrine of Chances* was the first textbook on probability theory, written by 18th-century French mathematician Abraham de Moivre and first published in 1718. De Moivre wrote in English because he resided in England at the time, having fled France to escape the persecution of Huguenots. The book's title came to be synonymous with

In mathematical analysis, the **maxima and minima** of a function, known collectively as **extrema**, are the largest and smallest value of the function, either within a given range or on the entire domain of a function. Pierre de Fermat was one of the first mathematicians to propose a general technique, adequality, for finding the maxima and minima of functions.

**Alfred Weber** was a German economist, geographer, sociologist and theoretician of culture whose work was influential in the development of modern economic geography.

In geometry, the **Fermat point** of a triangle, also called the **Torricelli point** or **Fermat–Torricelli point**, is a point such that the total distance from the three vertices of the triangle to the point is the minimum possible. It is so named because this problem is first raised by Fermat in a private letter to Evangelista Torricelli, who solved it.

In number theory, a **congruum** is the difference between successive square numbers in an arithmetic progression of three squares. That is, if *x*^{2}, *y*^{2}, and *z*^{2} are three square numbers that are equally spaced apart from each other, then the spacing between them, *z*^{2} − *y*^{2} = *y*^{2} − *x*^{2}, is called a congruum.

**Pierre de Fermat** was a French lawyer at the *Parlement* of Toulouse, France, and a mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he is recognized for his discovery of an original method of finding the greatest and the smallest ordinates of curved lines, which is analogous to that of differential calculus, then unknown, and his research into number theory. He made notable contributions to analytic geometry, probability, and optics. He is best known for his Fermat's principle for light propagation and his Fermat's Last Theorem in number theory, which he described in a note at the margin of a copy of Diophantus' *Arithmetica*.

The **geometric median** of a discrete set of sample points in a Euclidean space is the point minimizing the sum of distances to the sample points. This generalizes the median, which has the property of minimizing the sum of distances for one-dimensional data, and provides a central tendency in higher dimensions. It is also known as the **1-median**, **spatial median**, **Euclidean minisum point**, or **Torricelli point**.

* Ars Conjectandi* is a book on combinatorics and mathematical probability written by Jacob Bernoulli and published in 1713, eight years after his death, by his nephew, Niklaus Bernoulli. The seminal work consolidated, apart from many combinatorial topics, many central ideas in probability theory, such as the very first version of the law of large numbers: indeed, it is widely regarded as the founding work of that subject. It also addressed problems that today are classified in the twelvefold way and added to the subjects; consequently, it has been dubbed an important historical landmark in not only probability but all combinatorics by a plethora of mathematical historians. The importance of this early work had a large impact on both contemporary and later mathematicians; for example, Abraham de Moivre.

A timeline of **probability** and **statistics**

A timeline of **algebra** and **geometry**

This is a timeline of pure and applied mathematics history.

**Ramchundra ** (1821–1880) was a British Indian mathematician. His book, *Treatise on Problems of Maxima and Minima*, was promoted by the prominent mathematician Augustus De Morgan.

**Adequality** is a technique developed by Pierre de Fermat in his treatise *Methodus ad disquirendam maximam et minimam* to calculate maxima and minima of functions, tangents to curves, area, center of mass, least action, and other problems in calculus. According to André Weil, Fermat "introduces the technical term adaequalitas, adaequare, etc., which he says he has borrowed from Diophantus. As Diophantus V.11 shows, it means an approximate equality, and this is indeed how Fermat explains the word in one of his later writings.". Diophantus coined the word παρισότης (*parisotēs*) to refer to an approximate equality. Claude Gaspard Bachet de Méziriac translated Diophantus's Greek word into Latin as *adaequalitas*. Paul Tannery's French translation of Fermat’s Latin treatises on maxima and minima used the words *adéquation* and *adégaler*.

**Enrico Giusti**, is an Italian mathematician mainly known for his contributions to the fields of calculus of variations, regularity theory of partial differential equations, minimal surfaces and history of mathematics. He has been professor of mathematics at the Università di Firenze; he also taught and conducted research at the Australian National University at Canberra, at the Stanford University and at the University of California, Berkeley. After retirement, he devoted himself to the managing of the "Giardino di Archimede", a museum entirely dedicated to mathematics and its applications. Giusti is also the editor-in-chief of the international journal, dedicated to the history of mathematics "Bollettino di storia delle scienze matematiche".

**Luc-Normand Tellier** is a Professor Emeritus in spatial economics of the University of Quebec at Montreal.

In geometry, the **Weber problem**, named after Alfred Weber, is one of the most famous problems in location theory. It requires finding a point in the plane that minimizes the sum of the transportation costs from this point to *n* destination points, where different destination points are associated with different costs per unit distance.

**Giovanni Francesco Fagnano dei Toschi** was an Italian churchman and mathematician, the son of Giulio Carlo de' Toschi di Fagnano, also a mathematician.

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

- Thomas Simpson and his Work on Maxima and Minima at Convergence
. *Encyclopædia Britannica*.**25**(11th ed.). 1911. pp. 135–136.- O'Connor, John J.; Robertson, Edmund F., "Thomas Simpson",
*MacTutor History of Mathematics archive*, University of St Andrews .

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