Abraham de Moivre

Abraham de Moivre
FRS
Abraham de Moivre FRS
Born	26 May 1667; Vitry-le-François, Kingdom of France
Died	27 November 1754 (aged 87); London, England
Alma mater	Academy of Saumur; Collège d'Harcourt [fr]
Known for	De Moivre's formula; De Moivre's law; De Moivre's martingale; De Moivre–Laplace theorem; Inclusion–exclusion principle; Generating function
	Scientific career
Fields	Mathematics

Abraham de Moivre FRS (French pronunciation: [abʁaam də mwavʁ]; 26 May 1667 – 27 November 1754) 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 moved to England at a young age due to the religious persecution of Huguenots in France which reached a climax in 1685 with the Edict of Fontainebleau.^[1] He was a friend of Isaac Newton, Edmond Halley, and James Stirling. Among his fellow Huguenot exiles in England, he was a colleague of the editor and translator Pierre des Maizeaux.

De Moivre wrote a book on probability theory, The Doctrine 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 nth power of the golden ratio φ to the nth Fibonacci number. He also was the first to postulate the central limit theorem, a cornerstone of probability theory.

Life

Doctrine of chances, 1756

Early years

Abraham de Moivre was born in Vitry-le-François in Champagne on 26 May 1667. His father, Daniel de Moivre, was a surgeon who believed in the value of education. Though Abraham de Moivre's parents were Protestant, he first attended Christian Brothers' Catholic school in Vitry, which was unusually tolerant given religious 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 Jacques du Rondel. The Protestant Academy of Sedan had been founded 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 two years. Although mathematics was not part of his course work, de Moivre read several works on mathematics on his own, including Éléments des mathématiques by the French Oratorian priest and mathematician Jean Prestet and a short treatise on games of chance, De Ratiociniis in Ludo Aleae, by Christiaan Huygens the Dutch physicist, mathematician, astronomer and inventor. In 1684, de Moivre moved to Paris to study physics, and for the first time had formal mathematics training with private lessons from Jacques Ozanam.

Religious persecution in France became severe when King Louis XIV issued the Edict of Fontainebleau in 1685, which revoked the Edict of Nantes, that had given substantial rights to French Protestants. It forbade Protestant worship and required that all children be baptised by Catholic priests. De Moivre was sent to Prieuré Saint-Martin-des-Champs, a school that the authorities sent Protestant children 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 the Prieure de Saint-Martin indicate that he left the school in 1688, but de Moivre and his brother presented themselves as Huguenots admitted to the Savoy Church in London on 28 August 1687.

Middle years

By the time he arrived in London, de Moivre was a competent mathematician with a good knowledge of many of the standard texts.^[1] To make a living, de Moivre became a private tutor of mathematics, visiting his pupils or teaching in the coffee houses of London. De Moivre continued his studies of mathematics after visiting the Earl of Devonshire and seeing Newton's recent book, Principia Mathematica. Looking through the book, he realised that it was far deeper than the books that he had studied previously, and he became determined to read and understand it. However, as he was required to take extended walks around London to travel between his students, de Moivre had little time for study, so he tore pages from the book and carried them around in his pocket to read between lessons.

According to a possibly apocryphal story, Newton, in the later years of his life, used to refer people posing mathematical questions to him to de Moivre, saying, "He knows all these things better than I do."^[2]

By 1692, de Moivre became friends with Edmond Halley and soon after with Isaac Newton himself. In 1695, Halley communicated de Moivre's first mathematics paper, which arose from his study of fluxions in the Principia Mathematica, to the Royal Society. This paper was published in the Philosophical Transactions that same year. Shortly after publishing this paper, de Moivre also generalised Newton's noteworthy binomial theorem into the multinomial theorem. The Royal Society became apprised of this method in 1697, and it elected de Moivre a Fellow on 30 November 1697.

After de Moivre had been accepted, Halley encouraged him to turn his attention to astronomy. In 1705, de Moivre discovered, intuitively, that "the centripetal force of any planet is directly related to its distance from the centre of the forces and reciprocally related to the product of the diameter of the evolute and the cube of the perpendicular on the tangent." In other words, if a planet, M, follows an elliptical orbit around a focus F and has a point P where PM is tangent to the curve and FPM is a right angle so that FP is the perpendicular to the tangent, then the centripetal force at point P is proportional to FM/(R*(FP)³) where R is the radius of the curvature at M. The mathematician Johann Bernoulli proved this formula in 1710.

Despite these successes, de Moivre was unable to obtain an appointment to a chair of mathematics at any university, which would have released him from his dependence on time-consuming tutoring that burdened him more than it did most other mathematicians of the time. At least a part of the reason was a bias against his French origins.^[3]^[4]^[5]

In November 1697 he was elected a Fellow of the Royal Society^[1] and in 1712 was appointed to a commission set up by the society, alongside MM. Arbuthnot, Hill, Halley, Jones, Machin, Burnet, Robarts, Bonet, Aston, and Taylor to review the claims of Newton and Leibniz as to who discovered calculus. The full details of the controversy can be found in the Leibniz and Newton calculus controversy article.

Throughout his life de Moivre remained poor. It is reported that he was a regular customer of old Slaughter's Coffee House, St. Martin's Lane at Cranbourn Street, where he earned a little money from playing chess.

Later years

De Moivre continued studying the fields of probability and mathematics until his death in 1754 and several additional papers were published after his death. As he grew older, he became increasingly lethargic and needed longer sleeping hours. It is a common claim that De Moivre noted he was sleeping an extra 15 minutes each night and correctly calculated the date of his death as the day when the sleep time reached 24 hours, 27 November 1754.^[6] On that day he did in fact die, in London and his body was buried at St Martin-in-the-Fields, although his body was later moved. The claim of him predicting his own death, however, has been disputed as not having been documented anywhere at the time of its occurrence.^[7]

Probability

De Moivre pioneered the development of analytic geometry and the theory of probability by expanding upon the work of his predecessors, particularly Christiaan Huygens and several members of the Bernoulli family. He also produced the second textbook on probability theory, The Doctrine of Chances: a method of calculating the probabilities of events in play. (The first book about games of chance, Liber de ludo aleae (On Casting the Die), was written by Girolamo Cardano in the 1560s, but it was not published until 1663.) This book came out in four editions, 1711 in Latin, and in English in 1718, 1738, and 1756. In the later editions of his book, de Moivre included his unpublished result of 1733, which is the first statement of an approximation to the binomial distribution in terms of what we now call the normal or Gaussian function.^[8] This was the first method of finding the probability of the occurrence of an error of a given size when that error is expressed in terms of the variability of the distribution as a unit, and the first identification of the calculation of probable error. In addition, he applied these theories to gambling problems and actuarial tables.

An expression commonly found in probability is n! but before the days of calculators calculating n! for a large n was time-consuming. In 1733 de Moivre proposed the formula for estimating a factorial as n! = cn^(n+1/2)e⁻ⁿ. He obtained an approximate expression for the constant c but it was James Stirling who found that c was √2 $π$ .^[9]

De Moivre also published an article called "Annuities upon Lives" in which he revealed the normal distribution of the mortality rate over a person's age. From this he produced a simple formula for approximating the revenue produced by annual payments based on a person's age. This is similar to the types of formulas used by insurance companies today.

Priority regarding the Poisson distribution

Some results on the Poisson distribution were first introduced by de Moivre in De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus in Philosophical Transactions of the Royal Society, p. 219.^[10] As a result, some authors have argued that the Poisson distribution should bear the name of de Moivre.^[11]^[12]

De Moivre's formula

In 1707, de Moivre derived an equation from which one can deduce:

\cos x={\tfrac {1}{2}}(\cos(nx)+i\sin(nx))^{1/n}+{\tfrac {1}{2}}(\cos(nx)-i\sin(nx))^{1/n}

which he was able to prove for all positive integers n.^[13]^[14] In 1722, he presented equations from which one can deduce the better known form of de Moivre's Formula:

(\cos x+i\sin x)^{n}=\cos(nx)+i\sin(nx).\,

^[15]^[16]

In 1749 Euler proved this formula for any real n using Euler's formula, which makes the proof quite straightforward.^[17] This formula is important because it relates complex numbers and trigonometry. Additionally, this formula allows the derivation of useful expressions for cos(nx) and sin(nx) in terms of cos(x) and sin(x).

Stirling's approximation

De Moivre had been studying probability, and his investigations required him to calculate binomial coefficients, which in turn required him to calculate factorials.^[18]^[19] In 1730 de Moivre published his book Miscellanea Analytica de Seriebus et Quadraturis [Analytic Miscellany of Series and Integrals], which included tables of log (n!).^[20] For large values of n, de Moivre approximated the coefficients of the terms in a binomial expansion. Specifically, given a positive integer n, where n is even and large, then the coefficient of the middle term of (1 + 1)ⁿ is approximated by the equation:^[21]^[22]

{n \choose n/2}={\frac {n!}{(({\frac {n}{2}})!)^{2}}}\approx 2^{n}{\frac {2{\frac {21}{125}}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}

On June 19, 1729, James Stirling sent to de Moivre a letter, which illustrated how he calculated the coefficient of the middle term of a binomial expansion (a + b)ⁿ for large values of n.^[23]^[24] In 1730, Stirling published his book Methodus Differentialis [The Differential Method], in which he included his series for log(n!):^[25]

\log _{10}(n+{\frac {1}{2}})!\approx \log _{10}{\sqrt {2\pi }}+n\log _{10}n-{\frac {n}{\ln 10}},

so that for large $n$ , $n!\approx {\sqrt {2\pi }}\left({\frac {n}{e}}\right)^{n}$ .

On November 12, 1733, de Moivre privately published and distributed a pamphlet – Approximatio ad Summam Terminorum Binomii (a + b)ⁿ in Seriem expansi [Approximation of the Sum of the Terms of the Binomial (a + b)ⁿ expanded into a Series] – in which he acknowledged Stirling's letter and proposed an alternative expression for the central term of a binomial expansion.^[26]

Notes

^ ^a ^b ^c O'Connor, John J.; Robertson, Edmund F., "Abraham de Moivre", MacTutor History of Mathematics Archive, University of St Andrews
^ Bellhouse, David R. (2011). Abraham De Moivre: Setting the Stage for Classical Probability and Its Applications. London: Taylor & Francis. p. 99. ISBN 978-1-56881-349-3.
^ Coughlin, Raymond F.; Zitarelli, David E. (1984). The ascent 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 teaching position
^ Jungnickel, Christa; McCormmach, Russell (1996). Cavendish. Memoirs of the American Philosophical Society. Vol. 220. American Philosophical Society. p. 52. ISBN 9780871692207. Well connected in mathematical circles and highly regarded for his work, he still could not get a good job. Even his conversion to the Church of England in 1705 could not alter the fact that he was an alien.
^ Tanton, James Stuart (2005). Encyclopedia of Mathematics. Infobase Publishing. p. 122. ISBN 9780816051243. He had hoped to receive a faculty position in mathematics but, as a foreigner, was never offered such an appointment.
^ Cajori, Florian (1991). History of Mathematics (5 ed.). American Mathematical Society. p. 229. ISBN 9780821821022.
^ "Biographical details - Did Abraham de Moivre really predict his own death?".
^
See:
- Abraham De Moivre (12 November 1733) "Approximatio ad summam terminorum binomii (a+b)ⁿ in seriem expansi" (self-published pamphlet), 7 pages.
- English translation: A. De Moivre, The Doctrine of Chances … , 2nd ed. (London, England: H. Woodfall, 1738), pp. 235–243.
^ Pearson, Karl (1924). "Historical note on the origin of the normal curve of errors". Biometrika. 16 (3–4): 402–404. doi:10.1093/biomet/16.3-4.402.
^ Johnson, N.L., Kotz, S., Kemp, A.W. (1993) Univariate Discrete distributions (2nd edition). Wiley. ISBN 0-471-54897-9, p157
^ Stigler, Stephen M. (1982). "Poisson on the poisson distribution". Statistics & Probability Letters. 1: 33–35. doi:10.1016/0167-7152(82)90010-4.
^ Hald, Anders; de Moivre, Abraham; McClintock, Bruce (1984). "A. de Moivre:'De Mensura Sortis' or'On the Measurement of Chance'". International Statistical Review/Revue Internationale de Statistique. 1984 (3): 229–262. JSTOR 1403045.
^
Moivre, Ab. de (1707). "Aequationum quarundam potestatis tertiae, quintae, septimae, nonae, & superiorum, ad infinitum usque pergendo, in termimis finitis, ad instar regularum pro cubicis quae vocantur Cardani, resolutio analytica" [Of certain equations of the third, fifth, seventh, ninth, & higher power, all the way to infinity, by proceeding, in finite terms, in the form of rules for cubics which are called by Cardano, resolution by analysis.]. Philosophical Transactions of the Royal Society of London (in Latin). 25 (309): 2368–2371. doi:10.1098/rstl.1706.0037. S2CID 186209627.
- English translation by Richard J. Pulskamp (2009)
On p. 2370 de Moivre stated that if a series has the form $ny+{\tfrac {1-nn}{2\times 3}}ny^{3}+{\tfrac {1-nn}{2\times 3}}{\tfrac {9-nn}{4\times 5}}ny^{5}+{\tfrac {1-nn}{2\times 3}}{\tfrac {9-nn}{4\times 5}}{\tfrac {25-nn}{6\times 7}}ny^{7}+\cdots =a$ , where n is any given odd integer (positive or negative) and where y and a can be functions, then upon solving for y, the result is equation (2) on the same page: $y={\tfrac {1}{2}}{\sqrt[{n}]{a+{\sqrt {aa-1}}}}+{\tfrac {1}{2}}{\sqrt[{n}]{a-{\sqrt {aa-1}}}}$ . If y = cos x and a = cos nx , then the result is $\cos x={\tfrac {1}{2}}(\cos(nx)+i\sin(nx))^{1/n}+{\tfrac {1}{2}}(\cos(nx)-i\sin(nx))^{1/n}$
- In 1676, Isaac Newton found the relation between two chords that were in the ratio of n to 1; the relation was expressed by the series above. The series appears in a letter — Epistola prior D. Issaci Newton, Mathescos Professoris in Celeberrima Academia Cantabrigiensi; … — of 13 June 1676 from Isaac Newton to Henry Oldenburg, secretary of the Royal Society; a copy of the letter was sent to Gottfried Wilhelm Leibniz. See p. 106 of: Biot, J.-B.; Lefort, F., eds. (1856). Commercium epistolicum J. Collins et aliorum de analysi promota, etc: ou … (in Latin). Paris, France: Mallet-Bachelier. pp. 102–112.
- In 1698, de Moivre derived the same series. See: de Moivre, A. (1698). "A method of extracting roots of an infinite equation". Philosophical Transactions of the Royal Society of London. 20 (240): 190–193. doi:10.1098/rstl.1698.0034. S2CID 186214144.; see p 192.
- In 1730, de Moivre explicitly considered the case where the functions are cos θ and cos nθ. See: Moivre, A. de (1730). Miscellanea Analytica de Seriebus et Quadraturis (in Latin). London, England: J. Tonson & J. Watts. p. 1. From p. 1: "Lemma 1. Si sint l & x cosinus arcuum duorum A & B, quorum uterque eodem radio 1 describatur, quorumque prior sit posterioris multiplex in ea ratione quam habet numerus n ad unitatem, tunc erit $x={\tfrac {1}{2}}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}+{\tfrac {1}{2}}{\tfrac {1}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}}$ ." (If l and x are cosines of two arcs A and B both of which are described by the same radius 1 and of which the former is a multiple of the latter in that ratio as the number n has to 1, then it will be [true that] $x={\tfrac {1}{2}}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}+{\tfrac {1}{2}}{\tfrac {1}{\sqrt[{n}]{l+{\sqrt {ll-1}}}}}$ .) So if arc A = n × arc B, then l = cos A = cos nB and x = cos B. Hence $\cos B={\tfrac {1}{2}}(\cos(nB)+{\sqrt {-1}}\sin(nB))^{1/n}+{\tfrac {1}{2}}(\cos(nB)+{\sqrt {-1}}\sin(nB))^{-1/n}$
See also:
- Cantor, Moritz (1898). Vorlesungen über Geschichte der Mathematik [Lectures on the History of Mathematics]. Bibliotheca mathematica Teuberiana, Bd. 8-9 (in German). Vol. 3. Leipzig, Germany: B.G. Teubner. p. 624.
- Braunmühl, A. von (1901). "Zur Geschichte der Entstehung des sogenannten Moivreschen Satzes" [On the history of the origin of the so-called Moivre theorem]. Bibliotheca Mathematica. 3rd series (in German). 2: 97–102.; see p. 98.
^ Smith, David Eugene (1959), A Source Book in Mathematics, Volume 3, Courier Dover Publications, p. 444, ISBN 9780486646909
^
Moivre, A. de (1722). "De sectione anguli" [Concerning the section of an angle]. Philosophical Transactions of the Royal Society of London (in Latin). 32 (374): 228–230. doi:10.1098/rstl.1722.0039. S2CID 186210081. Retrieved 6 June 2020.
- English translation by Richard J. Pulskamp (2009) Archived 28 November 2020 at the Wayback Machine
From p. 229:
"Sit x sinus versus arcus cujuslibert.
[Sit] t sinus versus arcus alterius.
[Sit] 1 radius circuli.
Sitque arcus prior ad posteriorum ut 1 ad n, tunc, assumptis binis aequationibus quas cognatas appelare licet, 1 – 2zⁿ + z²ⁿ = – 2zⁿt 1 – 2z + zz = – 2zx. Expunctoque z orietur aequatio qua relatio inter x & t determinatur."
(Let x be the versine of any arc [i.e., x = 1 – cos θ ].
[Let] t be the versine of another arc.
[Let] 1 be the radius of the circle.
And let the first arc to the latter [i.e., "another arc"] be as 1 to n [so that t = 1 – cos nθ], then, with the two equations assumed which may be called related, 1 – 2zⁿ + z²ⁿ = –2zⁿt 1 – 2z + zz = – 2zx. And by eliminating z, the equation will arise by which the relation between x and t is determined.)
That is, given the equations 1 – 2zⁿ + z²ⁿ = – 2zⁿ (1 – cos nθ) 1 – 2z + zz = – 2z (1 – cos θ),
use the quadratic formula to solve for zⁿ in the first equation and for z in the second equation. The result will be: zⁿ = cos nθ ± i sin nθ and z = cos θ ± i sin θ , whence it immediately follows that (cos θ ± i sin θ)ⁿ = cos nθ ± i sin nθ.
See also:
- Smith, David Eugen (1959). A Source Book in Mathematics. Vol. 2. New York City, New York, USA: Dover Publications Inc. pp. 444–446. see p. 445, footnote 1.
^
In 1738, de Moivre used trigonometry to determine the nth roots of a real or complex number. See: Moivre, A. de (1738). "De reductione radicalium ad simpliciores terminos, seu de extrahenda radice quacunque data ex binomio $a+{\sqrt {+b}}$ , vel $a+{\sqrt {-b}}$ . Epistola" [On the reduction of radicals to simpler terms, or on extracting any given root from a binomial, $a+{\sqrt {+b}}$ or $a+{\sqrt {-b}}$ . A letter.]. Philosophical Transactions of the Royal Society of London (in Latin). 40 (451): 463–478. doi:10.1098/rstl.1737.0081. S2CID 186210174. From p. 475: "Problema III. Sit extrahenda radix, cujus index est n, ex binomio impossibli $a+{\sqrt {-b}}$ . … illos autem negativos quorum arcus sunt quadrante majores." (Problem III. Let a root whose index [i.e., degree] is n be extracted from the complex binomial $a+{\sqrt {-b}}$ . Solution. Let its root be $x+{\sqrt {-y}}$ , then I define ${\sqrt[{n}]{aa+b}}=m$ ; I also define ${\tfrac {n+1}{n}}=p$ [Note: should read: ${\tfrac {n+1}{2}}=p$ ], draw or imagine a circle, whose radius is ${\sqrt {m}}$ , and assume in this [circle] some arc A whose cosine is ${\tfrac {a}{{m}^{p}}}$ ; let C be the entire circumference. Assume, [measured] at the same radius, the cosines of the arcs ${\tfrac {A}{n}},{\tfrac {C-A}{n}},{\tfrac {C+A}{n}},{\tfrac {2C-A}{n}},{\tfrac {2C+A}{n}},{\tfrac {3C-A}{n}},{\tfrac {3C+A}{n}}$ , etc.
until the multitude [i.e., number] of them [i.e., the arcs] equals the number n; when this is done, stop there; then there will be as many cosines as values of the quantity $x$ , which is related to the quantity $y$ ; this [i.e., $y$ ] will always be $m-xx$ .
It is not to be neglected, although it was mentioned previously, [that] those cosines whose arcs are less than a right angle must be regarded as positive but those whose arcs are greater than a right angle [must be regarded as] negative.)
See also:
- Braunmühl, A. von (1903). Vorlesungen über Geschichte der Trigonometrie [Lectures on the history of trigonometry] (in German). Vol. 2. Leipzig, Germany: B.G. Teubner. pp. 76–77.
^ Euler (1749). "Recherches sur les racines imaginaires des equations" [Investigations into the complex roots of equations]. Mémoires de l'académie des sciences de Berlin (in French). 5: 222–288. See pp. 260–261: "Theorem XIII. §. 70. De quelque puissance qu'on extraye la racine, ou d'une quantité réelle, ou d'une imaginaire de la forme M + N √-1, les racines seront toujours, ou réelles, ou imaginaires de la même forme M + N √-1." (Theorem XIII. §. 70. For any power, either a real quantity or a complex [one] of the form M + N √−1, from which one extracts the root, the roots will always be either real or complex of the same form M + N√−1.)
^
De Moivre had been trying to determine the coefficient of the middle term of (1 + 1)ⁿ for large n since 1721 or earlier. In his pamphlet of November 12, 1733 – "Approximatio ad Summam Terminorum Binomii (a + b)ⁿ in Seriem expansi" [Approximation of the Sum of the Terms of the Binomial (a + b)ⁿ expanded into a Series] – de Moivre said that he had started working on the problem 12 years or more ago: "Duodecim jam sunt anni & amplius cum illud inveneram; … " (It is now a dozen years or more since I found this [i.e., what follows]; … ).
- (Archibald, 1926), p. 677.
- (de Moivre, 1738), p. 235.
De Moivre credited Alexander Cuming (ca. 1690 – 1775), a Scottish aristocrat and member of the Royal Society of London, with motivating, in 1721, his search to find an approximation for the central term of a binomial expansion. (de Moivre, 1730), p. 99.
^
The roles of de Moivre and Stirling in finding Stirling's approximation are presented in:
- Gélinas, Jacques (24 January 2017) "Original proofs of Stirling's series for log (N!)" arxiv.org
- Lanier, Denis; Trotoux, Didier (1998). "La formule de Stirling" [Stirling's formula] Commission inter-IREM histoire et épistémologie des mathématiques (ed.). Analyse & démarche analytique : les neveux de Descartes : actes du XIème Colloque inter-IREM d'épistémologie et d'histoire des mathématiques, Reims, 10 et 11 mai 1996 [Analysis and analytic reasoning: the "nephews" of Decartes: proceedings of the 11th inter-IREM colloquium on epistemology and the history of mathematics, Reims, 10–11 May 1996] (in French). Reims, France: IREM [Institut de Rercherche sur l'Enseignement des Mathématiques] de Reims. pp. 231–286.
^ Moivre, A. de (1730). Miscellanea Analytica de Seriebus et Quadraturis [Analytical Miscellany of Series and Quadratures [i.e., Integrals]]. London, England: J. Tonson & J. Watts. pp. 103–104.
^ From p. 102 of (de Moivre, 1730): "Problema III. Invenire Coefficientem Termini medii potestatis permagnae & paris, seu invenire rationem quam Coefficiens termini medii habeat ad summam omnium Coefficientium. … ad 1 proxime."
(Problem 3. Find the coefficient of the middle term [of a binomial expansion] for a very large and even power [n], or find the ratio that the coefficient of the middle term has to the sum of all coefficients.
Solution. Let n be the degree of the power to which the binomial a + b is raised, then, setting [both] a and b = 1, the ratio of the middle term to its power (a + b)ⁿ or 2ⁿ [Note: the sum of all the coefficients of the binomial expansion of (1 + 1)ⁿ is 2ⁿ.] will be nearly as ${\frac {2(n-1)^{n-{\frac {1}{2}}}}{{n}^{n}}}$ to 1.
But when some series for an inquiry could be determined more accurately [but] had been neglected due to lack of time, I then calculate by re-integration [and] I recover for use the particular quantities [that] had previously been neglected; so it happened that I could finally conclude that the ratio [that's] sought is approximately ${\frac {2{\frac {21}{125}}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}$ or ${\frac {2{\frac {21}{125}}{(1-{\frac {1}{n}})}^{n}}{\sqrt {n-1}}}$ to 1.)
The approximation ${\frac {{2}{\frac {21}{125}}{(n-1)}^{n-{\frac {1}{2}}}}{{n}^{n}}}$ is derived on pp. 124-128 of (de Moivre, 1730).
^
De Moivre determined the value of the constant $\textstyle 2{\frac {21}{125}}$ by approximating the value of a series by using only its first four terms. De Moivre thought that the series converged, but the English mathematician Thomas Bayes (ca. 1701–1761) found that the series actually diverged. From pp. 127-128 of (de Moivre, 1730): "Cum vero perciperem has Series valde implicatas evadere, … conclusi factorem 2.168 seu ${\textstyle 2{\frac {21}{125}},\ldots }$ " (But when I conceived [how] to avoid these very complicated series — although all of them were perfectly summable — I think that [there was] nothing else to be done, than to transform them to the infinite case; thus set m to infinity, then the sum of the first rational series will be reduced to 1/12, the sum of the second [will be reduced] to 1/360; thus it happens that the sums of all the series are achieved. From this one series $\textstyle {\frac {1}{12}}-{\frac {1}{360}}+{\frac {1}{1260}}-{\frac {1}{1680}}$ , etc., one will be able to discard as many terms as it will be one's pleasure; but I decided [to retain] four [terms] of this [series], because they sufficed [as] a sufficiently accurate approximation; now when this series be convergent, then its terms decrease with alternating positive and negative signs, [and] one may infer that the first term 1/12 is larger [than] the sum of the series, or the first term is larger [than] the difference that exists between all positive terms and all negative terms; but that term should be regarded as a hyperbolic [i.e., natural] logarithm; further, the number corresponding to this logarithm is nearly 1.0869 [i.e., ln(1.0869) ≈ 1/12], which if multiplied by 2, the product will be 2.1738, and so [in the case of a binomial being raised] to an infinite power, designated by n, the quantity $\textstyle {\frac {{2.1738}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}$ will be larger than the ratio that the middle term of the binomial has to the sum of all terms, and proceeding to the remaining terms, it will be discovered that the factor 2.1676 is just smaller [than the ratio of the middle term to the sum of all terms], and similarly that 2.1695 is greater, in turn that 2.1682 sinks a little bit below the true [value of the ratio]; considering which, I concluded that the factor [is] 2.168 or ${\textstyle 2{\frac {21}{125}},\ldots }$ Note: The factor that de Moivre was seeking, was: ${\frac {2e}{\sqrt {2\pi }}}=2.16887\ldots$ (Lanier & Trotoux, 1998), p. 237.
- Bayes, Thomas (31 December 1763). "A letter from the late Reverend Mr. Bayes, F.R.S. to John Canton, M.A. and F.R.S.". Philosophical Transactions of the Royal Society of London. 53: 269–271. doi:10.1098/rstl.1763.0044. S2CID 186214800.
^ (de Moivre, 1730), pp. 170–172.
^
In Stirling's letter of June 19, 1729 to de Moivre, Stirling stated that he had written to Alexander Cuming "quadrienium circiter abhinc" (about four years ago [i.e., 1725]) about (among other things) approximating, by using Isaac Newton's method of differentials, the coefficient of the middle term of a binomial expansion. Stirling acknowledged that de Moivre had solved the problem years earlier: " … ; respondit Illustrissimus vir se dubitare an Problema a Te aliquot ante annos solutum de invenienda Uncia media in quavis dignitate Binonii solvi posset per Differentias." ( ... ; this most illustrious man [Alexander Cuming] responded that he doubted whether the problem solved by you several years earlier, concerning the behavior of the middle term of any power of the binomial, could be solved by differentials.) Stirling wrote that he had then commenced to investigate the problem, but that initially his progress was slow.
- (de Moivre, 1730), p. 170.
- Zabell, S.L. (2005). Symmetry and Its Discontents: Essays on the History of Inductive Probability. New York City, New York, USA: Cambridge University Press. p. 113. ISBN 9780521444705.
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See:
- Stirling, James (1730). Methodus Differentialis … (in Latin). London: G. Strahan. p. 137. From p. 137: "Ceterum si velis summam quotcunque Logarithmorum numerorum naturalam 1, 2, 3, 4, 5, &c. pone z–n esse ultimum numerorum, existente n = ½ ; & tres vel quatuor Termini hujus Seriei $zl,z-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-{}$ [Note: l,z = log(z)] additi Logarithmo circumferentiae Circuli cujus Radius est Unitas, id est, huic 0.39908.99341.79 dabunt summam quaesitam, idque eo minore labore quo plures Logarithmi sunt summandi." (Furthermore, if you want the sum of however many logarithms of the natural numbers 1, 2, 3, 4, 5, etc., set z–n to be the last number, n being ½ ; and three or four terms of this series $z\log(z)-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-{}$ added to [half of] the logarithm of the circumference of a circle whose radius is unity [i.e., ½ log(2 $π$ )] – that is, [added] to this: 0.39908.99341.79 – will give the sum [that's] sought, and the more logarithms [that] are to be added, the less work it [is].) Note: $a=0.434294481903252$ (See p. 135.) = 1/ln(10).
- English translation: Stirling, James (1749). The Differential Method. Translated by Holliday, Francis. London, England: E. Cave. p. 121. [Note: The printer incorrectly numbered the pages of this book, so that page 125 is numbered as "121", page 126 as "122", and so forth until p. 129.]
^
See:
- Archibald, R.C. (October 1926). "A rare pamphlet of Moivre and some of his discoveries". Isis (in English and Latin). 8 (4): 671–683. doi:10.1086/358439. S2CID 143827655.
- An English translation of the pamphlet appears in: Moivre, Abraham de (1738). The Doctrine of Chances … (2nd ed.). London, England: Self-published. pp. 235–243.

References

See de Moivre's Miscellanea Analytica (London: 1730) pp 26–42.
H. J. R. Murray, 1913. History of Chess. Oxford University Press: p 846.
Schneider, I., 2005, "The doctrine of chances" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics. Elsevier: pp 105–20