Abraham de Moivre

AbrahamdeMoivre (26 May 1667 in Vitry-le-François,Champagne, France – 27 November 1754 in London,England; French pronunciation: [abʁaam də mwavʁ]) wasa French mathematician known for de Moivre’s for-mula, one of those that link complex numbers andtrigonometry, and for his work on the normal distributionand probability theory. He was a friend of Isaac Newton,Edmond Halley, and James Stirling. Among his fellowHuguenot exiles in England, he was a colleague of theeditor and translator Pierre des Maizeaux.De Moivre wrote a book on probability theory, The Doc-trine of Chances, said to have been prized by gamblers.De Moivre first discovered Binet’s formula, the closed-form expression for Fibonacci numbers linking the nthpower of the golden ratio φ to the nth Fibonacci number.

1 Life

Doctrine of chances, 1761

1.1 Early years

Abraham de Moivre was born in Vitry in Champagne onMay 26, 1667. His father, Daniel de Moivre, was a sur-

geon who, though middle class believed in the value ofeducation. Though Abraham de Moivre’s parents wereProtestant, he first attended Christian Brothers’ Catholicschool in Vitry, which was unusually tolerant given reli-gious tensions in France at the time. When he was eleven,his parents sent him to the Protestant Academy at Sedan,where he spent four years studying Greek under Jacquesdu Rondel. The Protestant Academy of Sedan had beenfounded in 1579 at the initiative of Françoise de Bourbon,the widow of Henri-Robert de la Marck.In 1682 the Protestant Academy at Sedan was suppressed,and de Moivre enrolled to study logic at Saumur for twoyears. Although mathematics was not part of his coursework, de Moivre read several works on mathematics onhis own including Elements de Mathematiques by FatherPrestet and a short treatise on games of chance, De Ra-tiociniis in Ludo Aleae, by Christiaan Huygens. In 1684,de Moivre moved to Paris to study physics, and for thefirst time had formal mathematics training with privatelessons from Jacques Ozanam.Religious persecution in France became severe whenKing Louis XIV issued the Edict of Fontainebleau in1685, which revoked the Edict of Nantes, that had givensubstantial rights to French Protestants. It forbade Protes-tant worship and required that all children be baptized byCatholic priests. De Moivre was sent to the Prieure deSaint-Martin, a school that the authorities sent Protestantchildren to for indoctrination into Catholicism.It is unclear when de Moivre left the Prieure de Saint-Martin and moved to England, since the records of thePrieure de Saint-Martin indicate that he left the school in1688, but de Moivre and his brother presented themselvesas Huguenots admitted to the Savoy Church in London onAugust 28, 1687.

1.2 Middle years

By the time he arrived in London, de Moivre was a com-petent mathematician with a good knowledge of many ofthe standard texts. To make a living, de Moivre became aprivate tutor of mathematics, visiting his pupils or teach-ing in the coffee houses of London. De Moivre contin-ued his studies of mathematics after visiting the Earl ofDevonshire and seeing Newton’s recent book, PrincipiaMathematica. Looking through the book, he realized thatit was far deeper than the books that he had studied previ-ously, and he became determined to read and understandit. However, as he was required to take extended walks

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2 2 PROBABILITY

around London to travel between his students, de Moivrehad little time for study, so he tore pages from the bookand carried them around in his pocket to read betweenlessons.According to a possibly apocryphal story, Newton, in thelater years of his life, used to refer people posing mathe-matical questions to him to de Moivre, saying, “He knowsall these things better than I do.”[1]

By 1692, de Moivre became friends with Edmond Halleyand soon after with Isaac Newton himself. In 1695, Hal-ley communicated de Moivre’s first mathematics paper,which arose from his study of fluxions in the PrincipiaMathematica, to the Royal Society. This paper was pub-lished in the Philosophical Transactions that same year.Shortly after publishing this paper, de Moivre also gen-eralized Newton’s noteworthy binomial theorem into themultinomial theorem. The Royal Society became ap-prised of this method in 1697, and it made de Moivrea member two months later.After de Moivre had been accepted, Halley encouragedhim to turn his attention to astronomy. In 1705, deMoivre discovered, intuitively, that “the centripetal forceof any planet is directly related to its distance from thecentre of the forces and reciprocally related to the prod-uct of the diameter of the evolute and the cube of theperpendicular on the tangent.” In other words, if a planet,M, follows an elliptical orbit around a focus F and has apoint P where PM is tangent to the curve and FPM isa right angle so that FP is the perpendicular to the tan-gent, then the centripetal force at point P is proportionalto FM/(R*(FP)3) where R is the radius of the curvatureat M. The mathematician Johann Bernoulli proved thisformula in 1710.Despite these successes, de Moivre was unable to ob-tain an appointment to a chair of mathematics at anyuniversity, which would have released him from his de-pendence on time-consuming tutoring that burdened himmore than it did most other mathematicians of the time.At least a part of the reason was a bias against his Frenchorigins.[2][3][4]

In November 1697 he was elected a Fellow of the RoyalSociety[5] and in 1712 was appointed to a commission setup by the society, alongside MM. Arbuthnot, Hill, Hal-ley, Jones, Machin, Burnet, Robarts, Bonet, Aston, andTaylor to review the claims of Newton and Leibniz as towho discovered calculus. The full details of the contro-versy can be found in the Leibniz and Newton calculuscontroversy article.Throughout his life de Moivre remained poor. It is re-ported that he was a regular customer of Slaughter’s Cof-fee House, St. Martin’s Lane at Cranbourn Street, wherehe earned a little money from playing chess.

1.3 Later years

De Moivre continued studying the fields of probabilityand mathematics until his death in 1754 and several addi-tional papers were published after his death. As he grewolder, he became increasingly lethargic and needed longersleeping hours. He noted that he was sleeping an extra 15minutes each night and correctly calculated the date of hisdeath as the day when the sleep time reached 24 hours,November 27, 1754.[6] He died in London and his bodywas buried at St Martin-in-the-Fields, although his bodywas later moved.

2 Probability

De Moivre pioneered the development of analytic geom-etry and the theory of probability by expanding upon thework of his predecessors, particularly Christiaan Huy-gens and several members of the Bernoulli family. Healso produced the second textbook on probability the-ory, The Doctrine of Chances: a method of calculatingthe probabilities of events in play. (The first book aboutgames of chance, Liber de ludo aleae (OnCasting the Die),was written by Girolamo Cardano in the 1560s, but it wasnot published until 1663.) This book came out in foureditions, 1711 in Latin, and in English in 1718, 1738,and 1756. In the later editions of his book, de Moivreincluded his unpublished result of 1733, which is the firststatement of an approximation to the binomial distribu-tion in terms of what we now call the normal or Gaussianfunction.[7] This was the first method of finding the prob-ability of the occurrence of an error of a given size whenthat error is expressed in terms of the variability of thedistribution as a unit, and the first identification of the cal-culation of probable error. In addition, he applied thesetheories to gambling problems and actuarial tables.An expression commonly found in probability is n! butbefore the days of calculators calculating n! for a largen was time consuming. In 1733 de Moivre proposed theformula for estimating a factorial as n! = cnn+1/2e−n. Heobtained an approximate expression for the constant cbut it was James Stirling who found that c was √(2π) .[8]

Therefore, Stirling’s approximation is as much due to deMoivre as it is to Stirling.De Moivre also published an article called “Annuitiesupon Lives” in which he revealed the normal distribu-tion of the mortality rate over a person’s age. From thishe produced a simple formula for approximating the rev-enue produced by annual payments based on a person’sage. This is similar to the types of formulas used by in-surance companies today. See also de Moivre–Laplacetheorem

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2.1 Priority regarding the Poisson distri-bution

Some results on the Poisson distribution were first in-troduced by de Moivre in De Mensura Sortis seu; deProbabilitate Eventuum in Ludis a Casu Fortuito Penden-tibus in Philosophical Transactions of the Royal Soci-ety, p. 219.[9] As a result, some authors have arguedthat the Poisson distribution should bear the name of deMoivre[10][11]

3 De Moivre’s formula

In 1707 de Moivre derived:

cosx = 12 (cos(nx)+i sin(nx))1/n+ 1

2 (cos(nx)−i sin(nx))1/n

which he was able to prove for all positive integers n.[12]

In 1722 he suggested it in the more well known form ofde Moivre’s Formula:

(cosx+ i sinx)n = cos(nx) + i sin(nx).

In 1749 Euler proved this formula for any real n usingEuler’s formula, which makes the proof quite straight-forward. This formula is important because it relatescomplex numbers and trigonometry. Additionally, thisformula allows the derivation of useful expressions forcos(nx) and sin(nx) in terms of cos(x) and sin(x).

4 Notes[1] Bellhouse, David R. (2011). Abraham De Moivre: Set-

ting the Stage for Classical Probability and Its Applications.London: Taylor & Francis. p. 99. ISBN 978-1-56881-349-3.

[2] Coughlin, Raymond F.; Zitarelli, David E. (1984). Theascent of mathematics. McGraw-Hill. p. 437. ISBN 0-07-013215-1. Unfortunately, because he was not British,De Moivre was never able to obtain a university teachingposition.

[3] Jungnickel, Christa; McCormmach, Russell (1996).Cavendish. Memoirs of the American Philosophical So-ciety 220. American Philosophical Society. p. 52. ISBN9780871692207. Well connected in mathematical circlesand highly regarded for his work, he still could not get agood job. Even his conversion to the Church of Englandin 1705 could not alter the fact that he was an alien.

[4] Tanton, James Stuart (2005). Encyclopedia of Mathemat-ics. Infobase Publishing. p. 122. ISBN 9780816051243.He had hoped to receive a faculty position in mathematicsbut, as a foreigner, was never offered such an appointment.

[5] “Library and Archive Catalogue”. The Royal Society. Re-trieved 3 October 2010.

[6] Cajori, Florian (1991). History of Mathematics (5 ed.).American Mathematical Society. p. 229. ISBN9780821821022.

[7] See:

• Abraham De Moivre (November 12, 1733)“Approximatio ad summam terminorum bi-nomii (a+b)n in seriem expansi” (self-publishedpamphlet), 7 pages.

• English translation: A. De Moivre, The Doctrine ofChances … , 2nd ed. (London, England: H. Wood-fall, 1738), pp. 235-243.

[8] Pearson, Karl. “Historical note on the origin of thenormal curve of errors”. Biometrika 16: 402–404.doi:10.1093/biomet/16.3-4.402.

[9] Johnson, N.L., Kotz, S., Kemp, A.W. (1993) UnivariateDiscrete distributions (2nd edition). Wiley. ISBN 0-471-54897-9, p157

[10] Stigler, Stephen M. “Poisson on the Poisson distribution.”Statistics & Probability Letters 1.1 (1982): 33-35.

[11] Hald, Anders, Abraham de Moivre, and Bruce McClin-tock. “A. de Moivre:'De Mensura Sortis’ or'On theMeasurement of Chance'.” International Statistical Re-view/Revue Internationale de Statistique (1984): 229-262

[12] Smith, David Eugene (1959),A Source Book inMathemat-ics, Volume 3, Courier Dover Publications, p. 444, ISBN9780486646909.

5 References• See de Moivre’s Miscellanea Analytica (London:

1730) p 26–42.

• H. J. R. Murray, 1913. History of Chess. OxfordUniversity Press: 846.

• Schneider, I., 2005, “The doctrine of chances” inGrattan-Guinness, I., ed., Landmark Writings inWestern Mathematics. Elsevier: 105–20

6 Further reading• de Moivre, Abraham at the Wayback Machine

(archived December 19, 2007)

• The Doctrine of Chance at MathPages.

• Biography (PDF), Matthew Maty's Biography ofAbraham De Moivre, Translated, Annotated andAugmented.

• Excerpt from Trigonometric Delights

• de Moivre, On the Law of Normal Probability

4 7 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

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