Abraham de Moivre

Abraham de Moivre
	Fórmula de De Moivre
Nascimento	26 de maio de 1667; Vitry-le-François
Morte	27 de novembro de 1754 (87 anos); Londres
Nacionalidade	francês
Cidadania	França
Alma mater	Academia de Saumur; Collège d’Harcourt;
Ocupação	matemático, estatístico
Orientador(es)	Jacques Ozanam
Campo(s)	matemática
Obras destacadas	Fórmula de De Moivre
	[edite no Wikidata]

Abraham de Moivre (Vitry-le-François, Champagne, França, 26 de maio de 1667 — Londres, Reino Unido, 27 de novembro de 1754) foi um matemático francês, conhecido pela fórmula de Moivre, uma fórmula que liga números complexos e trigonometria, e por seu trabalho sobre a distribuição normal e a teoria das probabilidades.

Ele se mudou para a Inglaterra ainda jovem devido à perseguição religiosa aos huguenotes na França, que começou em 1685. Ele era amigo de Isaac Newton, Edmond Halley e James Stirling. Entre seus companheiros exilados huguenotes na Inglaterra, ele era colega do editor e tradutor Pierre des Maizeaux.

De Moivre escreveu um livro sobre a teoria da probabilidade, The Doctrine of Chances, que dizem ter sido valorizado pelos jogadores. Ele também foi o primeiro a postular o teorema do limite central, uma pedra angular da teoria da probabilidade.

Vida[editar | editar código-fonte]

Doctrine of chances, 1761

Primeiros anos[editar | editar código-fonte]

Abraham de Moivre nasceu em Vitry-le-François, em Champagne, em 26 de maio de 1667. Seu pai, Daniel de Moivre, era um cirurgião que acreditava no valor da educação. Embora os pais de Abraham de Moivre fossem protestantes, ele primeiro frequentou a escola católica dos Irmãos Cristãos em Vitry, que era excepcionalmente tolerante devido às tensões religiosas na França na época. Quando ele tinha onze anos, seus pais o enviaram para a Academia Protestante de Sedan, onde passou quatro anos estudando grego com Jacques du Rondel. A Academia Protestante de Sedan foi fundada em 1579 por iniciativa de Françoise de Bourbon, a viúva de Henri-Robert de la Marck.

Em 1682, a Academia Protestante de Sedan foi suprimida e de Moivre matriculou-se para estudar lógica em Saumur por dois anos. Embora a matemática não fizesse parte de seu curso, de Moivre leu várias obras sobre matemática por conta própria, incluindo Éléments des mathématiques do sacerdote oratoriano e matemático francês Jean Prestet e um pequeno tratado sobre jogos de azar, De Ratiociniis in Ludo Aleae, de Christiaan Huygens, o físico, matemático, astrônomo e inventor holandês. Em 1684, de Moivre mudou-se para Paris para estudar física e, pela primeira vez, teve treinamento formal em matemática com aulas particulares de Jacques Ozanam.

Em 25 de novembro de 2017, um colloquio foi organizado em Saumur pelo Dr. Conor Maguire, com o patrocínio da Comissão Nacional Francesa da UNESCO, para comemorar o 350º aniversário do nascimento de Abraham de Moivre e o fato de ele ter estudado por dois anos no Academia de Saumur. O colóquio foi intitulado Abraham de Moivre: le Mathématicien, sa vie et son œuvre e cobriu as contribuições importantes de De Moivre para o desenvolvimento de números complexos, consulte a fórmula de De Moivre, e para a teoria da probabilidade, consulte o teorema de De Moivre-Laplace. O colóquio traçou a vida de De Moivre e seu exílio em Londres, onde ele se tornou um amigo altamente respeitado de Isaac Newton. Mesmo assim, ele vivia com recursos modestos que gerava em parte por suas sessões aconselhando jogadores na cafeteria Old Slaughter sobre as probabilidades associadas a seus empreendimentos! Em 27 de novembro de 2016, o professor Christian Genest da Universidade McGill (Montreal) marcou o 262º aniversário da morte de Abraham de Moivre com um colóquio em Limoges intitulado Abraham de Moivre: Génie en exil que discutiu a famosa aproximação de De Moivre da lei binomial que inspirou o teorema do limite central.

A perseguição religiosa na França tornou-se severa quando o rei Luís XIV emitiu o Édito de Fontainebleau em 1685, que revogou o Édito de Nantes, que concedeu direitos substanciais aos protestantes franceses. Proibia o culto protestante e exigia que todas as crianças fossem batizadas por padres católicos. De Moivre foi enviado para o Prieuré Saint-Martin-des-Champs, uma escola para a qual as autoridades mandavam crianças protestantes para serem doutrinadas no catolicismo.

Não está claro quando de Moivre deixou o Prieuré de Saint-Martin e se mudou para a Inglaterra, uma vez que os registros do Prieuré de Saint-Martin indicam que ele deixou a escola em 1688, mas de Moivre e seu irmão se apresentaram como huguenotes admitidos no Igreja Savoy em Londres em 28 de agosto de 1687.

Anos intermediários[editar | editar código-fonte]

Quando chegou a Londres, de Moivre era um matemático competente com um bom conhecimento de muitos dos textos padrão. Para ganhar a vida, de Moivre tornou-se um professor particular de matemática, visitando seus alunos ou ensinando nos cafés de Londres. De Moivre continuou seus estudos de matemática depois de visitar o conde de Devonshire e ver o livro recente de Newton, Principia Mathematica. Olhando através do livro, ele percebeu que era muito mais profundo do que os livros que havia estudado anteriormente e ficou determinado a lê-lo e entendê-lo. No entanto, como era obrigado a fazer longas caminhadas por Londres para viajar entre seus alunos, de Moivre tinha pouco tempo para estudar, então ele rasgou as páginas do livro e as carregou no bolso para ler entre as aulas.

De acordo com uma história possivelmente apócrifa, Newton, nos últimos anos de sua vida, costumava encaminhar as pessoas que lhe faziam perguntas matemáticas para de Moivre, dizendo: "Ele sabe todas essas coisas melhor do que eu".^[1]

Em 1692, de Moivre tornou-se amigo de Edmond Halley e logo depois do próprio Isaac Newton. Em 1695, Halley comunicou o primeiro artigo de matemática de de Moivre, que surgiu de seu estudo de fluxões no Principia Mathematica, para a Royal Society. Este artigo foi publicado na Philosophical Transactions desse mesmo ano. Logo após a publicação deste artigo, de Moivre também generalizada de Newton notável binomial teorema no teorema multinomial. A Royal Society foi informada desse método em 1697 e tornou de Moivre um membro dois meses depois.

Depois que de Moivre foi aceito, Halley o encorajou a voltar sua atenção para a astronomia. Em 1705, de Moivre descobriu, intuitivamente, que “a força centrípeta de qualquer planeta está diretamente relacionada à sua distância do centro das forças e reciprocamente relacionada ao produto do diâmetro do evoluto e do cubo da perpendicular na tangente". Em outras palavras, se um planeta, M, segue uma órbita elíptica em torno de um foco F e tem um ponto P onde PM é tangente à curva e FPM é um ângulo reto, de modo que FP é perpendicular à tangente, então a força centrípeta no ponto P é proporcional a FM / (R * (FP)³) onde R é o raio da curvatura em M. O matemático Johann Bernoulli provou esta fórmula em 1710.

Apesar desses sucessos, de Moivre não conseguiu obter uma nomeação para uma cadeira de matemática em qualquer universidade, o que o teria libertado de sua dependência de tutoriais demorados que o sobrecarregavam mais do que a maioria dos outros matemáticos da época. Pelo menos parte do motivo era um preconceito contra suas origens francesas.^[2]^[3]^[4]

Em novembro de 1697 foi eleito Fellow da Royal Society^[5] e em 1712 foi nomeado para uma comissão criada pela sociedade, ao lado de MM. Arbuthnot, Hill, Halley, Jones, Machin, Burnet, Robarts, Bonet, Aston e Taylor para revisar as afirmações de Newton e Leibniz sobre quem descobriu o cálculo. Os detalhes completos da controvérsia podem ser encontrados no artigo da controvérsia do cálculo de Leibniz e Newton.

Ao longo de sua vida, de Moivre permaneceu pobre. É relatado que ele era um cliente regular do velho Slaughter's Coffee House, St. Martin's Lane na Cranbourn Street, onde ganhava algum dinheiro jogando xadrez.

Anos posteriores[editar | editar código-fonte]

De Moivre continuou estudando os campos de probabilidade e matemática até sua morte em 1754 e vários artigos adicionais foram publicados após sua morte. À medida que envelhecia, tornava-se cada vez mais letárgico e precisava de mais horas de sono. Um comum, embora discutível,^[6] alegação é que ele notou que ele estava dormindo um extra de 15 minutos cada noite e corretamente calculados à data da sua morte como o dia em que o tempo de sono alcançado 24 horas, 27 de novembro de 1754.^[7] Em naquele dia ele de fato morreu, em Londres e seu corpo foi enterrado em St. Martin-in-the-Fields, embora seu corpo tenha sido removido mais tarde

Probabilidade[editar | editar código-fonte]

De Moivre foi o pioneiro no desenvolvimento da geometria analítica e da teoria da probabilidade, expandindo o trabalho de seus predecessores, particularmente Christiaan Huygens e vários membros da família Bernoulli. Ele também produziu o segundo livro sobre teoria da probabilidade, The Doctrine of Chances: a method of calculating the probabilities of events in play. (O primeiro livro sobre jogos de azar, Liber de ludo aleae (On Casting the Die), foi escrito por Girolamo Cardanona década de 1560, mas não foi publicado até 1663.) Este livro saiu em quatro edições, 1711 em latim e em inglês em 1718, 1738 e 1756. Nas edições posteriores de seu livro, de Moivre incluiu seu resultado não publicado de 1733, que é a primeira declaração de uma aproximação da distribuição binomial em termos do que agora chamamos de distribuição normal ou gaussiana.^[8] Este foi o primeiro método de encontrar a probabilidade de ocorrência de um erro de um determinado tamanho quando esse erro é expresso em termos da variabilidade da distribuição como uma unidade, e a primeira identificação do cálculo do erro provável. Além disso, ele aplicou essas teorias a problemas de jogo e tabelas atuariais.

Este foi o primeiro método de encontrar a probabilidade de ocorrência de um erro de um determinado tamanho quando esse erro é expresso em termos da variabilidade da distribuição como uma unidade, e a primeira identificação do cálculo do erro provável. Além disso, ele aplicou essas teorias a problemas de jogo e tabelas atuariais.

Uma expressão comumente encontrada em probabilidade é n! mas antes da época das calculadoras calculando n! para um grande n era demorado. Em 1733, de Moivre propôs a fórmula para estimar um fatorial como n ! = cn ^{(n + 1/2)} e ⁻ⁿ. Ele obteve uma expressão aproximada para a constante c, mas foi James Stirling quem descobriu que c era √2π.^[9]

De Moivre também publicou um artigo chamado "Annuities upon Lives" no qual ele revelou a distribuição normal da taxa de mortalidade de acordo com a idade de uma pessoa. A partir disso, ele produziu uma fórmula simples para aproximar a receita produzida por pagamentos anuais com base na idade de uma pessoa. Isso é semelhante aos tipos de fórmulas usadas pelas seguradoras hoje.

Prioridade em relação à distribuição de Poisson[editar | editar código-fonte]

Alguns resultados da distribuição Poisson foram introduzidos pela primeira vez por de Moivre em De Mensura Sortis seu; de Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus em Philosophical Transactions of the Royal Society, p. 219.^[10] Como resultado, alguns autores argumentaram que a distribuição de Poisson deveria levar o nome de de Moivree.^[11]^[12]

Fórmula de De Moivre[editar | editar código-fonte]

Em 1707, de Moivre derivou uma equação da qual se pode deduzir:

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

que ele foi capaz de provar para todos os inteiros positivos n.^[13]^[14] Em 1722, ele apresentou equações das quais se pode deduzir a forma mais conhecida da Fórmula de Moivre:

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

^[15]^[16]

Em 1749, Euler provou essa fórmula para qualquer n real usando a fórmula de Euler, o que torna a prova bastante direta.^[17] Esta fórmula é importante porque relaciona números complexos e trigonometria. Além disso, esta fórmula permite a derivação de expressões úteis para cos ( nx ) e sin ( nx ) em termos de cos ( x ) e sin ( x ).

Aproximação de Stirling[editar | editar código-fonte]

De Moivre estava estudando probabilidade e suas investigações exigiam que ele calculasse coeficientes binomiais, o que, por sua vez, exigia que calculasse fatoriais.^[18]^[19] Em 1730 de Moivre publicou seu livro Miscellanea Analytica de Seriebus et Quadraturis [Miscelânea Analítica de Séries e Integrais], que incluía tabelas de log (n !).^[20] Para grandes valores de n, de Moivre aproximou os coeficientes dos termos em uma expansão binomial. Especificamente, dado um número inteiro positivo n, onde n é par e grande, então o coeficiente do termo do meio de (1 + 1) ⁿ é aproximado pela equação:^[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}}}

Em 19 de junho de 1729, James Stirling enviou a de Moivre uma carta, que ilustrava como ele calculava o coeficiente do termo do meio de uma expansão binomial (a + b) ⁿ para grandes valores de n.^[23]^[24] Em 1730, Stirling publicou seu livro Methodus Differentialis [O Método Diferencial], no qual incluiu sua série para log ( n !):^[25]

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

,

de modo que para grande $n$ , $n!\approx {\sqrt {2\pi }}({\frac {n}{e}})^{n}$ .

Em 12 de novembro de 1733, de Moivre publicou privadamente e distribuiu um panfleto - Approximatio ad Summam Terminorum Binomii (a + b)ⁿ in Seriem expansi [Aproximação da Soma dos Termos do Binômio (a + b)ⁿ expandido em uma Série] - em que reconheceu a carta de Stirling e propôs uma expressão alternativa para o termo central de uma expansão binomial.^[26]

Referências[editar | editar código-fonte]

↑ 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. [S.l.]: 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. Col: Memoirs of the American Philosophical Society. 220. [S.l.]: 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. [S.l.]: 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.
↑ «Library and Archive Catalogue». The Royal Society. Consultado em 3 de outubro de 2010 ^{[ligação inativa]}
↑ «Biographical details - Did Abraham de Moivre really predict his own death?»
↑ Cajori, Florian (1991). History of Mathematics 5 ed. [S.l.]: American Mathematical Society. p. 229. ISBN 9780821821022
↑
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 (em latim). 25 (309): 2368–2371. doi:10.1098/rstl.1706.0037
- 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 Issac 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 … (em latim). 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 ; 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 (em latim). 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] (em alemão). 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 (em alemão). 2: 97–102 ; see p. 98.
↑ Smith, David Eugene (1959), A Source Book in Mathematics, Volume 3, ISBN 9780486646909, Courier Dover Publications, p. 444
↑
Moivre, A. de (1722). «De sectione anguli» [Concerning the section of an angle]. Philosophical Transactions of the Royal Society of London (em latim). 32 (374): 228–230. doi:10.1098/rstl.1722.0039. Consultado em 6 de junho de 2020. Cópia arquivada em 6 de junho de 2020
- English translation by Richard J. Pulskamp (2009)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 (em latim). 40 (451): 463–478. doi:10.1098/rstl.1737.0081 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 $a+{\sqrt {-b}}$ $a+{\sqrt {-b}}$ . Solution. Let its root be $x+{\sqrt {-y}}$ $x+{\sqrt {-y}}$ , then I define ${\sqrt[{n}]{aa+b}}=m$ ${\sqrt[{n}]{aa+b}}=m$ ; I also define ${\tfrac {n+1}{n}}=p$ ${\tfrac {n+1}{n}}=p$ [Note: should read: ${\tfrac {n+1}{2}}=p$ ${\tfrac {n+1}{2}}=p$ ], draw or imagine a circle, whose radius is ${\sqrt {m}}$ ${\sqrt {m}}$ , and assume in this [circle] some arc A whose cosine is ${\tfrac {a}{{m}^{p}}}$ ${\tfrac {a}{{m}^{p}}}$ ; let C be the entire circumference. Assume, [measured] at the same radius, the cosines of the arcs ${\tfrac {A}{n}},{\tfrac {C-A}{n}},{\tfrac {C+A}{n}},{\tfrac {2C-A}{n}},{\tfrac {2C+A}{n}},{\tfrac {3C-A}{n}},{\tfrac {3C+A}{n}}$ ${\tfrac {A}{n}},{\tfrac {C-A}{n}},{\tfrac {C+A}{n}},{\tfrac {2C-A}{n}},{\tfrac {2C+A}{n}},{\tfrac {3C-A}{n}},{\tfrac {3C+A}{n}}$ , 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 $x$ $x$ , which is related to the quantity $y$ $y$ ; this [i.e., $y$ $y$ ] will always be $m-xx$ $m-xx$ . 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] (em alemão). 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 (em francês). 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 $\textstyle 2{\frac {21}{125}}$ $\textstyle 2{\frac {21}{125}}$ 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 $\textstyle 2{\frac {21}{125}}$ , … " (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 $\textstyle {\frac {1}{12}}-{\frac {1}{360}}+{\frac {1}{1260}}-{\frac {1}{1680}}$ $\textstyle {\frac {1}{12}}-{\frac {1}{360}}+{\frac {1}{1260}}-{\frac {1}{1680}}$ , 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 $\textstyle {\frac {{2.1738}{(n-1)}^{n-{\frac {1}{2}}}}{{n}^{n}}}$ $\textstyle {\frac {{2.1738}{(n-1)}^{n-{\frac {1}{2}}}}{{n}^{n}}}$ 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 $\textstyle 2{\frac {21}{125}}$ $\textstyle 2{\frac {21}{125}}$ , … ) Note: The factor that de Moivre was seeking, was: ${\frac {2e}{\sqrt {2\pi }}}$ ${\frac {2e}{\sqrt {2\pi }}}$ = 2.16887 … (Lanier & Trotoux, 1998), p. 237.
- Bayes, Thomas (31 de dezembro de 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
↑ (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 Issac 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
↑
See:
- Stirling, James (1730). Methodus Differentialis … (em latim). London, England: 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 $zlog(z)-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-$ added to [half of] the logarithm of the circumference of a circle whose radius is unity [i.e., ½log(2π)] – 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; Holliday, Francis, trans. (1749). The Differential Method. 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. (outubro de 1926). «A rare pamphlet of Moivre and some of his discoveries». Isis (em inglês e latim). 8 (4): 671–683. doi:10.1086/358439
- 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

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[1] 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

[2] Coughlin, Raymond F.; Zitarelli, David E. (1984). The ascent of mathematics. [S.l.]: 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

[3] Jungnickel, Christa; McCormmach, Russell (1996). Cavendish. Col: Memoirs of the American Philosophical Society. 220. [S.l.]: 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.

[4] Tanton, James Stuart (2005). Encyclopedia of Mathematics. [S.l.]: 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.

[5] «Library and Archive Catalogue». The Royal Society. Consultado em 3 de outubro de 2010 ^{[ligação inativa]}

[hsmstex-6] «Biographical details - Did Abraham de Moivre really predict his own death?»

[Cajori-History-7] Cajori, Florian (1991). History of Mathematics 5 ed. [S.l.]: American Mathematical Society. p. 229. ISBN 9780821821022

[8] 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.

[9] Abraham De Moivre (12 November 1733) "Approximatio ad summam terminorum binomii (a+b)ⁿ in seriem expansi" (self-published pamphlet), 7 pages.

[10] English translation: A. De Moivre, The Doctrine of Chances … , 2nd ed. (London, England: H. Woodfall, 1738), pp. 235–243.

[9] 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

[JKK157-10] Johnson, N.L., Kotz, S., Kemp, A.W. (1993) Univariate Discrete distributions (2nd edition). Wiley. ISBN 0-471-54897-9, p157

[11] Stigler, Stephen M. (1982). «Poisson on the poisson distribution». Statistics & Probability Letters. 1: 33–35. doi:10.1016/0167-7152(82)90010-4

[12] 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

[13] 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 (em latim). 25 (309): 2368–2371. doi:10.1098/rstl.1706.0037
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 Issac 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 … (em latim). 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 ; 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 (em latim). 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] (em alemão). 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 (em alemão). 2: 97–102 ; see p. 98.

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

[17] 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 Issac 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 … (em latim). Paris, France: Mallet-Bachelier. pp. 102–112

[18] 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 ; see p 192.

[19] 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 (em latim). 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:

[20] Cantor, Moritz (1898). Vorlesungen über Geschichte der Mathematik [Lectures on the History of Mathematics] (em alemão). vol. 3. Leipzig, Germany: B.G. Teubner. p. 624

[21] 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 (em alemão). 2: 97–102 ; see p. 98.

[14] Smith, David Eugene (1959), A Source Book in Mathematics, Volume 3, ISBN 9780486646909, Courier Dover Publications, p. 444

[15] Moivre, A. de (1722). «De sectione anguli» [Concerning the section of an angle]. Philosophical Transactions of the Royal Society of London (em latim). 32 (374): 228–230. doi:10.1098/rstl.1722.0039. Consultado em 6 de junho de 2020. Cópia arquivada em 6 de junho de 2020
English translation by Richard J. Pulskamp (2009)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.

[24] English translation by Richard J. Pulskamp (2009)From p. 229:

[25] 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.

[16] 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 (em latim). 40 (451): 463–478. doi:10.1098/rstl.1737.0081 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] (em alemão). vol. 2. Leipzig, Germany: B.G. Teubner. pp. 76–77

[27] Braunmühl, A. von (1903). Vorlesungen über Geschichte der Trigonometrie [Lectures on the history of trigonometry] (em alemão). vol. 2. Leipzig, Germany: B.G. Teubner. pp. 76–77

[17] Euler (1749). «Recherches sur les racines imaginaires des equations» [Investigations into the complex roots of equations]. Mémoires de l'académie des sciences de Berlin (em francês). 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.)

[18] 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.

[30] (Archibald, 1926), p. 677.

[31] (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.

[19] 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.

[33] Gélinas, Jacques (24 January 2017) "Original proofs of Stirling's series for log (N!)" arxiv.org

[34] 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.

[20] 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

[21] 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).

[22] 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 $\textstyle 2{\frac {21}{125}}$ , … " (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 $\textstyle 2{\frac {21}{125}}$ , … ) Note: The factor that de Moivre was seeking, was: ${\frac {2e}{\sqrt {2\pi }}}$ = 2.16887 … (Lanier & Trotoux, 1998), p. 237.
Bayes, Thomas (31 de dezembro de 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

[38] Bayes, Thomas (31 de dezembro de 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

[23] (de Moivre, 1730), pp. 170–172.

[24] 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 Issac 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

[41] (de Moivre, 1730), p. 170.

[42] 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

[25] See:
Stirling, James (1730). Methodus Differentialis … (em latim). London, England: 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 $zlog(z)-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-$ added to [half of] the logarithm of the circumference of a circle whose radius is unity [i.e., ½log(2π)] – 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; Holliday, Francis, trans. (1749). The Differential Method. 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.]

[44] Stirling, James (1730). Methodus Differentialis … (em latim). London, England: 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 $zlog(z)-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-$ added to [half of] the logarithm of the circumference of a circle whose radius is unity [i.e., ½log(2π)] – 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).

[45] English translation: Stirling, James; Holliday, Francis, trans. (1749). The Differential Method. 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.]

[26] See:
Archibald, R.C. (outubro de 1926). «A rare pamphlet of Moivre and some of his discoveries». Isis (em inglês e latim). 8 (4): 671–683. doi:10.1086/358439

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

[47] Archibald, R.C. (outubro de 1926). «A rare pamphlet of Moivre and some of his discoveries». Isis (em inglês e latim). 8 (4): 671–683. doi:10.1086/358439

[48] 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

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v d e Isaac Newton
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