Авраам де Моивр

Авраам де Моивр
Фрс
Авраам де Моивр Фрс
Рожденный	26 мая 1667 г. ; Vitry-le-François , королевство Франция
Умер	27 ноября 1754 г. (в возрасте 87 лет) ; Лондон , Англия
Альма -матер	Академия Самура ; Harcourt College [ FR ]
Известен для	Формулы Movre ; Закон Хойвра ; От Movre's Martingale ; Разоблачить перемещение ; Принцип включения - эксклюзия ; Генерирующая функция
	Научная карьера
Поля	Математика

Авраам де Моивр Фрс ( Французское произношение: [abʁaam də mwavʁ] ; 26 мая 1667 г. - 27 ноября 1754 года) был французским математиком, известным формулой Де Моивра , формулой, которая связывает сложные числа и тригонометрию , а также своей работой над нормальным распределением и теорией вероятностей .

Он переехал в Англию в молодом возрасте из -за религиозного преследования гугенотов во Франции, которые достигли кульминации в 1685 году с указанием Фонтенбло . ^{[ 1 ]} Он был другом Исаака Ньютона , Эдмонда Хэлли и Джеймса Стерлинга . Среди своих коллег изгнанников гугенот в Англии он был коллегой редактора и переводчика Пьера Де -Майзо .

Де Моивр написал книгу о теории вероятности , доктрине шансов , которая, как говорят, ценилась игроками. Де Моивр впервые обнаружил формулу Бинета , выражение в закрытой форме для чисел Фибоначчи, связывающую силу золотого соотношения φ с числом n th fibonacci. Он также был первым, кто постулирует теорему центрального предела , краеугольный камень теории вероятности.

Жизнь

Ранние годы

Авраам де Моивр родился в vitry-le-françois в Шампанском 26 мая 1667 года. Его отец, Даниэль де Моивр, был хирургом, который верил в ценность образования. Хотя родители Авраама де Моивра были протестантскими, он впервые посещал католическую школу братьев -христианских братьев в Витри, которая была необычайно терпимой, учитывая религиозную напряженность во Франции в то время. Когда ему было одиннадцать, его родители отправили его в протестантскую академию в Седане , где он провел четыре года, изучая греческий под руководством Жак дю Ронделя. Протестантская академия седана была основана в 1579 году на инициативе Франсуаз-де-Бурбон, вдовы Анри-Роберта де ла Марка.

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.

Религиозное преследование во Франции стало суровым, когда король Людовик XIV выпустил указ Фонтенбло в 1685 году, который отменил указ Нант , который дал существенные права на французские протестанты. Это запрещало протестантское поклонение и требовало, чтобы все дети были крещены католическими священниками. Де Моивр был отправлен в Прирею Сен-Мартин-Дес-Хэмпс, школу, которую власти отправили протестантским детям за идеологическую обработку в католицизм.

Неясно, когда де Моивр покинул Прире де Сен-Мартин и переехал в Англию, поскольку записи «Прире де Сен-Мартин» указывают на то, что он покинул школу в 1688 году, но де Моивр и его брат представили себя, когда гугеноты признались в Савойская церковь в Лондоне 28 августа 1687 года.

Средние годы

К тому времени, когда он прибыл в Лондон, де Моивр был компетентным математиком с хорошим знанием многих стандартных текстов. ^{[ 1 ]} Чтобы зарабатывать на жизнь, де Моивр стал частным репетитором по математике , посетив его учеников или преподаванием в кофейнях Лондона. Де Моивр продолжил свои исследования математики после посещения графа Девоншира и просмотра недавней книги Ньютона «Принципа Математика» . Просматривая книгу, он понял, что она была гораздо глубже, чем книги, которые он изучал ранее, и стал полон решимости прочитать и понять это. Однако, поскольку он должен был пройти длительные прогулки по Лондону, чтобы путешествовать между своими учениками, де Моивр имел мало времени для обучения, поэтому он вырвал страницы из книги и нес в своем кармане, чтобы читать между уроками.

Согласно возможному апокрифическому истории, Ньютон, в последующие годы своей жизни, использовал людей, задающих ему математические вопросы к де -Моивру, говоря: «Он знает все эти вещи лучше, чем я». ^{[ 2 ]}

К 1692 году де Моивр подружился с Эдмондом Галли и вскоре после этого с Исааком Ньютоном . В 1695 году Галли сообщил о первой математической статье Де Моивра, которая возникла в результате его изучения потоков в «Принципиальной математике » до Королевского общества . Эта статья была опубликована в философских сделках в том же году. Ньютона Вскоре после публикации этой статьи Де Моивр также обобщал примечательную биномиальную теорему в теорему многономиального . Королевское общество стало проинформированным об этом методе в 1697 году, и 30 ноября 1697 года оно выбрало Де Моивра.

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

^ Jump up to: ^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 $ny+{\tfrac {1-nn}{2\times 3}}ny^{3}+{\tfrac {1-nn}{2\times 3}}{\tfrac {9-nn}{4\times 5}}ny^{5}+{\tfrac {1-nn}{2\times 3}}{\tfrac {9-nn}{4\times 5}}{\tfrac {25-nn}{6\times 7}}ny^{7}+\cdots =a$ $-\ times 5} {5} {5} {\ tfrac {1-nn} {2 \ tfrac {9-nn} {4 \ times 5} {4 \ tfrac {25-nn} {6 \ tis 7} { 7}+\ cdots = a}$ , 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: $y={\tfrac {1}{2}}{\sqrt[{n}]{a+{\sqrt {aa-1}}}}+{\tfrac {1}{2}}{\sqrt[{n}]{a-{\sqrt {aa-1}}}}$ ${\ displayStyle y = {\ tfraac {1} {2}} {\ sqrt [{n} {a+{\ sqrt {aa-1}}}}+{\ tfraac {1} {2} {\ sqrtt [{{\ tfraac {1} {2} {\ sqrtt [{{\ tfraac n}] {a-{\ sqrt {aa-1}}}}}}$ . If y = cos x and a = cos nx , then the result is $\cos x={\tfrac {1}{2}}(\cos(nx)+i\sin(nx))^{1/n}+{\tfrac {1}{2}}(\cos(nx)-i\sin(nx))^{1/n}$ ${\ cossyle \ cos x = {\ tfrac {1} {2} {2} (\ cosin (xcosin (x) sin (x) sin (s)^{1/in}+{\ tfrac {1} {2 } (\} (\ s} (\ s} cos (x) -i \ sin (x)^{1/n}}$
- 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] (PDF). 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 $a+{\sqrt {-b}}$ ${\ displayStyle a+{\ sqrt {-b}}}$ . Solution. Let its root be $x+{\sqrt {-y}}$ ${\ displayStyle x+{\ sqrt {-y}}}$ , then I define ${\sqrt[{n}]{aa+b}}=m$ ${\ displaystyle {\ sqrt [{n}] {aa+b}} = m}$ ; I also define ${\tfrac {n+1}{n}}=p$ ${\ displaystyle {\ tfrac {n+1} {n}} = p}$ [Note: should read: ${\tfrac {n+1}{2}}=p$ ${\ displaystyle {\ tfrac {n+1} {2}} = p}$ ], draw or imagine a circle, whose radius is ${\sqrt {m}}$ ${\ displaystyle {\ sqrt {m}}}$ , and assume in this [circle] some arc A whose cosine is ${\tfrac {a}{{m}^{p}}}$ ${\ displaystyle {\ tfrac {a} {{m}^{p}}}}$ ; let C be the entire circumference. Assume, [measured] at the same radius, the cosines of the arcs ${\tfrac {A}{n}},{\tfrac {C-A}{n}},{\tfrac {C+A}{n}},{\tfrac {2C-A}{n}},{\tfrac {2C+A}{n}},{\tfrac {3C-A}{n}},{\tfrac {3C+A}{n}}$ ${\ displaystyle {\ tfrac {a} {n}}, {\ tfrac {ca} {n}}, {\ tfrac {c+a} {n}}, {\ tfrac {2c-a} {n}} , {\ tfrac {2c+a} {n}}, {\ tfrac {3c-a} {n}}, {\ tfrac {3c+a} {n}}}$ , 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 $x$ ${\ DisplayStyle x}$ , which is related to the quantity $y$ ${\ displayStyle y}$ ; this [i.e., $y$ ${\ displayStyle y}$ ] will always be $m-xx$ ${\ displayStyle m-xx}$ .
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 $\textstyle 2{\frac {21}{125}}$ ${\ DisplayStyle \ TextStyle 2 {\ frac {21} {125}}}$ 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 $\textstyle {\frac {1}{12}}-{\frac {1}{360}}+{\frac {1}{1260}}-{\frac {1}{1680}}$ ${\ displaystyle \ textStyle {\ frac {1} {12}}-{\ frac {1} {360}}+{\ frac {1} {1260}}-{\ frac {1} {1680}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}} }}}}}$ , 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 $\textstyle {\frac {{2.1738}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}$ ${\ displaystyle \ textStyle {\ frac {{2.1738} {(n-1)}^{n-{\ frac {1} {2}}}}} {n^{n}}}}}}}}}}}} {n^{n}}}}}}}}}}} {n^{n}}}}}}}$ Будет больше, чем соотношение, которое средний член биномиала имеет к сумме всех условий, и, перейдя к оставшимся терминам, будет обнаружено, что фактор 2.1676 просто меньше [чем отношение среднего члена к сумме Из всех терминов], и аналогично, что 2,1695 больше, в свою очередь, 2,1682 погружаются немного ниже истинного [значение соотношения]; Учитывая, что, я пришел к выводу, что фактор [равен] 2,168 или ${\textstyle 2{\frac {21}{125}},\ldots }$ ${\ textStyle 2 {\ frac {21} {125}}, \ ldots}$ Примечание: фактором, который искал де Моивр, было: ${\frac {2e}{\sqrt {2\pi }}}=2.16887\ldots$ ${\ displaystyle {\ frac {2e} {\ sqrt {2 \ pi}}} = 2.16887 \ ldots}$ (Lanier & Trotoux, 1998), p. 237
- Байес, Томас (31 декабря 1763 г.). «Письмо покойного преподобного мистера Байеса, от Джона Кантона, Массачусетс и ФРС». Философские транзакции Королевского общества Лондона . 53 : 269–271. doi : 10.1098/rstl.1763.0044 . S2CID 186214800 .
^ (де Моивр, 1730), стр. 170–172.
^
В письме Стерлинга от 19 июня 1729 года де -Моивр, Стерлинг заявил, что написал Александру Камингу «Крипучий циктер Абхинк» (около четырех лет назад [т.е. 1725]) о (среди прочего) приблизительно, используя Исаака Ньютона Метод дифференциалов, коэффициент среднего члена биномиального расширения. Стерлинг признал, что де Моивр решил проблему годами ранее: «…; реагировать на иллюстрацию vir se dubitare a a aliquot ante annos solutum de invenienda uncia media в quavis достойном бинони -solvi sosset в рамках дифференциаций». (...; этот самый прославленный человек [Александр Каминг] ответил, что сомневался в том, может ли эта проблема, решаемая вами несколько лет назад, относительно поведения среднего члена любой силы биномиала, может быть решена дифференциалами.) Стирлинг написал что он тогда начал исследовать проблему, но изначально его прогресс был медленным.
- (Де Моайвер, 1730), с. 170.
- Забелл, SL (2005). Симметрия и ее недовольство: эссе об истории индуктивной вероятности . Нью -Йорк, Нью -Йорк, США: издательство Кембриджского университета. п. 113. ISBN 9780521444705 .
^
См.:
- Стерлинг, Джеймс (1730). Метод дифференциала ... (на латыни). Лондон: Г. Страхан. п. 137. от р. 137: «Если вы хотите подвести итог численных чисел 1, 2, 3, 4, 5 и т. Д. $zl,z-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-{}$ [Примечание: 50, z = log (z)] Добавлен круг логаритмо лучей устройства, то есть 0,39908.99341.79 Дайте сумму поиска и меньшую работу, в которой суммируются многие логарифмы . Если вы хотите, чтобы я был, как много логарифмов натуральных чисел 1, 2, 3, 4, 5 и т. Д., Установите Z-N, чтобы быть последним номером, n составляют ½; $z\log(z)-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-{}$ Добавлено в [половину] логарифм окружности круга, радиус которого - единство [то есть ½ log (2 $π$ )], то есть [добавлен] к этому: 0,39908.99341.79 - даст сумму [которая] искала и чем больше логарифмов [которые] должны быть добавлены, тем меньше работы он [есть].) Примечание: $a=0.434294481903252$ (См. Стр. 135.) = 1/ln (10).
- Английский перевод: Стерлинг, Джеймс (1749). Дифференциальный метод . Перевод Холлидей, Фрэнсис. Лондон, Англия: Э. Кейв. п. 121. [Примечание: принтер неправильно пронумеровал страницы этой книги, так что страница 125 пронумерована как «121», стр. 129.]
^
См.:
- Арчибальд, RC (октябрь 1926 г.). «Редкая брошюра Моивра и некоторые из его открытий». ИГИЛ (на английском и латинском). 8 (4): 671–683. doi : 10.1086/358439 . S2CID 143827655 .
- Английский перевод брошюры появляется в: Моивр, Авраам де (1738). Доктрина шансов ... (2 -е изд.). Лондон, Англия: самостоятельно опубликовано. С. 235–243.

Ссылки

де Моивра См. Аналиновую помощь (Лондон: 1730) стр. 26-42.
HJR Murray , 1913. История шахмат . Издательство Оксфордского университета: стр. 846.
2005, «Учение о шансах» в Grattan-Guinness, I. , ed. Schneider, I. , Elsevier: стр. 105–20

Дальнейшее чтение

«Де Моивр, Авраам» . Архивировано из оригинала 19 декабря 2007 года . Получено 15 июня 2002 года .
«Демоивр Авраама» . Энциклопедия Британская . Тол. 7 (9 -е изд.). 1878. П. 60
Доктрина случайности на математике.
Биография (PDF) , биография Мэтью Мати Авраама де Моивра, переведенная, аннотированная и дополненная .
де Моивр, по закону нормальной вероятности

[mactutor-1] Jump up to: ^a ^b ^c O'Connor, John J.; Robertson, Edmund F., "Abraham de Moivre", MacTutor History of Mathematics Archive, University of St Andrews

[2] 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.

[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

[4] 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.

[5] 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-History-6] Cajori, Florian (1991). History of Mathematics (5 ed.). American Mathematical Society. p. 229. ISBN 9780821821022.

[hsmstex-7] "Biographical details - Did Abraham de Moivre really predict his own death?".

[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 (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.

[16] English translation by Richard J. Pulskamp (2009)

[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 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.

[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. S2CID 186214144.; 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 (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 $\cos B={\tfrac {1}{2}}(\cos(nB)+{\sqrt {-1}}\sin(nB))^{1/n}+{\tfrac {1}{2}}(\cos(nB)+{\sqrt {-1}}\sin(nB))^{-1/n}$

[20] 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.

[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 (in German). 2: 97–102.; see p. 98.

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

[15] Moivre, A. de (1722). "De sectione anguli" [Concerning the section of an angle] (PDF). 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.

[24] English translation by Richard J. Pulskamp (2009) Archived 28 November 2020 at the Wayback Machine

[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 (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.

[27] 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.

[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 (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.)

[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.

[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 ${\textstyle 2{\frac {21}{125}},\ldots }$ " (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 $\textstyle {\frac {{2.1738}{(n-1)}^{n-{\frac {1}{2}}}}{n^{n}}}$ Будет больше, чем соотношение, которое средний член биномиала имеет к сумме всех условий, и, перейдя к оставшимся терминам, будет обнаружено, что фактор 2.1676 просто меньше [чем отношение среднего члена к сумме Из всех терминов], и аналогично, что 2,1695 больше, в свою очередь, 2,1682 погружаются немного ниже истинного [значение соотношения]; Учитывая, что, я пришел к выводу, что фактор [равен] 2,168 или ${\textstyle 2{\frac {21}{125}},\ldots }$ Примечание: фактором, который искал де Моивр, было: ${\frac {2e}{\sqrt {2\pi }}}=2.16887\ldots$ (Lanier & Trotoux, 1998), p. 237
Байес, Томас (31 декабря 1763 г.). «Письмо покойного преподобного мистера Байеса, от Джона Кантона, Массачусетс и ФРС». Философские транзакции Королевского общества Лондона . 53 : 269–271. doi : 10.1098/rstl.1763.0044 . S2CID 186214800 .

[38] Байес, Томас (31 декабря 1763 г.). «Письмо покойного преподобного мистера Байеса, от Джона Кантона, Массачусетс и ФРС». Философские транзакции Королевского общества Лондона . 53 : 269–271. doi : 10.1098/rstl.1763.0044 . S2CID 186214800 .

[23] (де Моивр, 1730), стр. 170–172.

[24] В письме Стерлинга от 19 июня 1729 года де -Моивр, Стерлинг заявил, что написал Александру Камингу «Крипучий циктер Абхинк» (около четырех лет назад [т.е. 1725]) о (среди прочего) приблизительно, используя Исаака Ньютона Метод дифференциалов, коэффициент среднего члена биномиального расширения. Стерлинг признал, что де Моивр решил проблему годами ранее: «…; реагировать на иллюстрацию vir se dubitare a a aliquot ante annos solutum de invenienda uncia media в quavis достойном бинони -solvi sosset в рамках дифференциаций». (...; этот самый прославленный человек [Александр Каминг] ответил, что сомневался в том, может ли эта проблема, решаемая вами несколько лет назад, относительно поведения среднего члена любой силы биномиала, может быть решена дифференциалами.) Стирлинг написал что он тогда начал исследовать проблему, но изначально его прогресс был медленным.
(Де Моайвер, 1730), с. 170.

Забелл, SL (2005). Симметрия и ее недовольство: эссе об истории индуктивной вероятности . Нью -Йорк, Нью -Йорк, США: издательство Кембриджского университета. п. 113. ISBN 9780521444705 .

[41] (Де Моайвер, 1730), с. 170.

[42] Забелл, SL (2005). Симметрия и ее недовольство: эссе об истории индуктивной вероятности . Нью -Йорк, Нью -Йорк, США: издательство Кембриджского университета. п. 113. ISBN 9780521444705 .

[25] См.:
Стерлинг, Джеймс (1730). Метод дифференциала ... (на латыни). Лондон: Г. Страхан. п. 137. от р. 137: «Если вы хотите подвести итог численных чисел 1, 2, 3, 4, 5 и т. Д. $zl,z-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-{}$ [Примечание: 50, z = log (z)] Добавлен круг логаритмо лучей устройства, то есть 0,39908.99341.79 Дайте сумму поиска и меньшую работу, в которой суммируются многие логарифмы . Если вы хотите, чтобы я был, как много логарифмов натуральных чисел 1, 2, 3, 4, 5 и т. Д., Установите Z-N, чтобы быть последним номером, n составляют ½; $z\log(z)-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-{}$ Добавлено в [половину] логарифм окружности круга, радиус которого - единство [то есть ½ log (2 $π$ )], то есть [добавлен] к этому: 0,39908.99341.79 - даст сумму [которая] искала и чем больше логарифмов [которые] должны быть добавлены, тем меньше работы он [есть].) Примечание: $a=0.434294481903252$ (См. Стр. 135.) = 1/ln (10).

Английский перевод: Стерлинг, Джеймс (1749). Дифференциальный метод . Перевод Холлидей, Фрэнсис. Лондон, Англия: Э. Кейв. п. 121. [Примечание: принтер неправильно пронумеровал страницы этой книги, так что страница 125 пронумерована как «121», стр. 129.]

[44] Стерлинг, Джеймс (1730). Метод дифференциала ... (на латыни). Лондон: Г. Страхан. п. 137. от р. 137: «Если вы хотите подвести итог численных чисел 1, 2, 3, 4, 5 и т. Д. $zl,z-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-{}$ [Примечание: 50, z = log (z)] Добавлен круг логаритмо лучей устройства, то есть 0,39908.99341.79 Дайте сумму поиска и меньшую работу, в которой суммируются многие логарифмы . Если вы хотите, чтобы я был, как много логарифмов натуральных чисел 1, 2, 3, 4, 5 и т. Д., Установите Z-N, чтобы быть последним номером, n составляют ½; $z\log(z)-az-{\frac {a}{24z}}+{\frac {7a}{2880z^{3}}}-{}$ Добавлено в [половину] логарифм окружности круга, радиус которого - единство [то есть ½ log (2 $π$ )], то есть [добавлен] к этому: 0,39908.99341.79 - даст сумму [которая] искала и чем больше логарифмов [которые] должны быть добавлены, тем меньше работы он [есть].) Примечание: $a=0.434294481903252$ (См. Стр. 135.) = 1/ln (10).

[45] Английский перевод: Стерлинг, Джеймс (1749). Дифференциальный метод . Перевод Холлидей, Фрэнсис. Лондон, Англия: Э. Кейв. п. 121. [Примечание: принтер неправильно пронумеровал страницы этой книги, так что страница 125 пронумерована как «121», стр. 129.]

[26] См.:
Арчибальд, RC (октябрь 1926 г.). «Редкая брошюра Моивра и некоторые из его открытий». ИГИЛ (на английском и латинском). 8 (4): 671–683. doi : 10.1086/358439 . S2CID 143827655 .

Английский перевод брошюры появляется в: Моивр, Авраам де (1738). Доктрина шансов ... (2 -е изд.). Лондон, Англия: самостоятельно опубликовано. С. 235–243.

[47] Арчибальд, RC (октябрь 1926 г.). «Редкая брошюра Моивра и некоторые из его открытий». ИГИЛ (на английском и латинском). 8 (4): 671–683. doi : 10.1086/358439 . S2CID 143827655 .

[48] Английский перевод брошюры появляется в: Моивр, Авраам де (1738). Доктрина шансов ... (2 -е изд.). Лондон, Англия: самостоятельно опубликовано. С. 235–243.

[ 1 ]

[ 2 ]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]

[11]

[12]

[13]

[14]

[15]

[16]

[17]

[18]

[19]

[20]

[21]

[22]

[23]

[24]

[25]

[26]

v Т и Сэр Исаак Ньютон
Publications	Fluxions (1671) De Motu (1684) Principia (1687) Opticks (1704) Queries (1704) Arithmetica (1707) De Analysi (1711)
Other writings	Quaestiones (1661–1665) "standing on the shoulders of giants" (1675) Notes on the Jewish Temple (c. 1680) "General Scholium" (1713; "hypotheses non fingo" ) Ancient Kingdoms Amended (1728) Corruptions of Scripture (1754)
Contributions	Calculus fluxion Impact depth Inertia Newton disc Newton polygon Newton–Okounkov body Newton's reflector Newtonian telescope Newton scale Newton's metal Spectrum Structural coloration
Newtonianism	Bucket argument Newton's inequalities Newton's law of cooling Newton's law of universal gravitation post-Newtonian expansion parameterized gravitational constant Newton–Cartan theory Schrödinger–Newton equation Newton's laws of motion Kepler's laws Newtonian dynamics Newton's method in optimization Apollonius's problem truncated Newton method Gauss–Newton algorithm Newton's rings Newton's theorem about ovals Newton–Pepys problem Newtonian potential Newtonian fluid Classical mechanics Corpuscular theory of light Leibniz–Newton calculus controversy Newton's notation Rotating spheres Newton's cannonball Newton–Cotes formulas Newton's method generalized Gauss–Newton method Newton fractal Newton's identities Newton polynomial Newton's theorem of revolving orbits Newton–Euler equations Newton number kissing number problem Newton's quotient Parallelogram of force Newton–Puiseux theorem Absolute space and time Luminiferous aether Newtonian series table
Personal life	Woolsthorpe Manor (birthplace) Cranbury Park (home) Early life Later life Apple tree Religious views Occult studies Scientific Revolution Copernican Revolution
Relations	Catherine Barton (niece) John Conduitt (nephew-in-law) Isaac Barrow (professor) William Clarke (mentor) Benjamin Pulleyn (tutor) Roger Cotes (student) William Whiston (student) John Keill (disciple) William Stukeley (friend) William Jones (friend) Abraham de Moivre (friend)
Depictions	Newton by Blake (monotype) Newton by Paolozzi (sculpture) Isaac Newton Gargoyle Astronomers Monument
Namesake	Newton (unit) Newton's cradle Isaac Newton Institute Isaac Newton Medal Isaac Newton Telescope Isaac Newton Group of Telescopes XMM-Newton Sir Isaac Newton Sixth Form Statal Institute of Higher Education Isaac Newton Newton International Fellowship
Categories	Isaac Newton

Базы данных управления авторитетом
International	ISNI VIAF FAST WorldCat
National	Germany United States France BnF data Italy Australia Czech Republic Spain 2 Netherlands 2 Norway Croatia Sweden Israel Belgium
Academics	CiNii Mathematics Genealogy Project zbMATH MathSciNet
People	Trove Deutsche Biographie
Other	IdRef