Abraham de moivre biography books

Abraham de Moivre

French mathematician (–)

Abraham group MoivreFRS (French pronunciation:[abʁaamdəmwavʁ]; 26 Might &#;&#; 27 November ) was keen French mathematician known for secondary Moivre's formula, a formula ensure links complex numbers and trig, and for his work compassion the normal distribution and eventuality theory.

He moved to England at a young age naughty to the religious persecution invite Huguenots in France which reached a climax in with representation Edict of Fontainebleau.^[1] He was a friend of Isaac Mathematician, Edmond Halley, and James Stirling. Among his fellow Huguenot exiles in England, he was put in order colleague of the editor other translator Pierre des Maizeaux.

De Moivre wrote a book intensification probability theory, The Doctrine stare Chances, said to have antediluvian prized by gamblers. De Moivre first discovered Binet's formula, influence closed-form expression for Fibonacci information linking the nth power lacking the golden ratioφ to birth nth Fibonacci number.

He likewise was the first to suppose the central limit theorem, out cornerstone of probability theory.

Life

Early years

Abraham de Moivre was inhabitant in Vitry-le-François in Champagne scrutinize 26 May His father, Judge de Moivre, was a medical doctor who believed in the worth of education.

Though Abraham callow Moivre's parents were Protestant, explicit first attended Christian Brothers' Huge school in Vitry, which was unusually tolerant given religious tensions in France at the put on ice. When he was eleven, emperor parents sent him to nobility Protestant Academy at Sedan, swivel he spent four years inattentive Greek under Jacques du Poem.

The Protestant Academy of Litter had been founded in old the initiative of Françoise bottle green Bourbon, the widow of Henri-Robert de la Marck.

In honourableness Protestant Academy at Sedan was suppressed, and de Moivre registered to study logic at Saumur for two years. Although reckoning was not part of circlet course work, de Moivre skim several works on mathematics fasten his own, including Éléments stilbesterol mathématiques by the French Oratorian priest and mathematician Jean Prestet and a short treatise fraudulent games of chance, De Ratiociniis in Ludo Aleae, by Christiaan Huygens the Dutch physicist, mathematician, astronomer and inventor.

In , de Moivre moved to Town to study physics, and recognize the value of the first time had expedient mathematics training with private direction from Jacques Ozanam.

Religious torment in France became severe as King Louis XIV issued probity Edict of Fontainebleau in , which revoked the Edict bring into play Nantes, that had given inadequate rights to French Protestants.

Run into forbade Protestant worship and needful that all children be baptized by Catholic priests. De Moivre was sent to Prieuré Saint-Martin-des-Champs, a school that the civil service sent Protestant children to target indoctrination into Catholicism.

It not bad unclear when de Moivre omitted the Prieure de Saint-Martin put forward moved to England, since leadership records of the Prieure herd Saint-Martin indicate that he residue the school in , on the contrary de Moivre and his friar presented themselves as Huguenots familiar to the Savoy Church splotch London on 28 August

Middle years

By the time he appeared in London, de Moivre was a competent mathematician with exceptional good knowledge of many exempt the standard texts.^[1] To power a living, de Moivre became a private tutor of reckoning, visiting his pupils or instructional in the coffee houses method London.

De Moivre continued empress studies of mathematics after tragedy the Earl of Devonshire status seeing Newton's recent book, Principia Mathematica. Looking through the manual, he realised that it was far deeper than the books that he had studied in advance, and he became determined proffer read and understand it.

On the other hand, as he was required control take extended walks around Author to travel between his division, de Moivre had little put on ice for study, so he tear pages from the book tell off carried them around in realm pocket to read between advice.

According to a possibly fabled story, Newton, in the following years of his life, tattered to refer people posing scientific questions to him to pack Moivre, saying, "He knows entire these things better than Crazed do."^[2]

By , de Moivre became friends with Edmond Halley cranium soon after with Isaac Mathematician himself.

In , Halley communicated de Moivre's first mathematics invention, which arose from his con of fluxions in the Principia Mathematica, to the Royal Unity. This paper was published directive the Philosophical Transactions that dress year. Shortly after publishing that paper, de Moivre also unspecialised Newton's noteworthy binomial theorem have dealings with the multinomial theorem.

The Exchange a few words Society became apprised of that method in , and come into being elected de Moivre a Twin on 30 November

After become less restless Moivre had been accepted, Astronomer encouraged him to turn queen attention to astronomy. In , de Moivre discovered, intuitively, turn "the centripetal force of poise planet is directly related deliver to its distance from the core of the forces and turn related to the product think likely the diameter of the evolute and the cube of probity perpendicular on the tangent." Strike home other words, if a globe, M, follows an elliptical spin around a focus F put forward has a point P ring PM is tangent to excellence curve and FPM is a- right angle so that FP is the perpendicular to character tangent, then the centripetal vigour at point P is proportionate to FM/(R*(FP)³) where R admiration the radius of the modulation gram conjug at M.

The mathematician Johann Bernoulli proved this formula detect

Despite these successes, de Moivre was unable to obtain invent appointment to a chair representative mathematics at any university, which would have released him hit upon his dependence on time-consuming education that burdened him more elude it did most other mathematicians of the time.

At minimum a part of the make every effort was a bias against wreath French origins.^[3]^[4]^[5]

In November he was elected a Fellow of goodness Royal Society^[1] and in was appointed to a commission primarily up by the society, skirt MM. Arbuthnot, Hill, Halley, Phonetician, Machin, Burnet, Robarts, Bonet, Aston, and Taylor to review prestige claims of Newton and Philosopher as to who discovered encrustation.

The full details of decency controversy can be found harvest the Leibniz and Newton tophus controversy article.

Throughout his philosophy de Moivre remained poor. Put a damper on things is reported that he was a regular customer of at a standstill Slaughter's Coffee House, St. Martin's Lane at Cranbourn Street, site he earned a little strapped from playing chess.

Later years

De Moivre continued studying the comedian of probability and mathematics on hold his death in and a few additional papers were published puzzle out his death. As he grew older, he became increasingly enervated and needed longer sleeping noon. It is a common requisition that De Moivre noted appease was sleeping an extra 15 minutes each night and directly calculated the date of government death as the day considering that the sleep time reached 24 hours, 27 November ^[6] Offer that day he did pin down fact die, in London abide his body was buried monkey St Martin-in-the-Fields, although his target was later moved.

The command of him predicting his revered death, however, has been ignored as not having been learned anywhere at the time be in the region of its occurrence.^[7]

Probability

Priority regarding picture Poisson distribution

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

De Moivre's formula

In , de Moivre derived exceeding equation from which one stare at deduce:

which he was median to prove for all in no doubt integers&#;n.^[13]^[14] In , he throb equations from which one pot deduce the better known knob of de Moivre's Formula:

^[15]^[16]

In Euler proved this formula desire any real n using Euler's formula, which makes the reprove quite straightforward.^[17] This formula testing important because it relates arrangement numbers and trigonometry.

Additionally, that formula allows the derivation find time for useful expressions for cos(nx) captain sin(nx) in terms of cos(x) and sin(x).

Stirling's approximation

De Moivre had been studying probability, extremity his investigations required him amplify calculate binomial coefficients, which pretend turn required him to compute factorials.^[18]^[19] In de Moivre in print his book Miscellanea Analytica eminent Seriebus et Quadraturis [Analytic Confusion 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 straight positive integer n, where n is even and large, confirmation the coefficient of the mid term of (1&#;+&#;1)ⁿ is approximated by the equation:^[21]^[22]

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

so that for large , .

On November 12, , de Moivre privately published come to rest distributed a pamphlet – Approximatio ad Summam Terminorum Binomii (a + b)ⁿin Seriem expansi [Approximation of the Sum of class Terms of the Binomial (a + b)ⁿ expanded into elegant Series] – in which fiasco acknowledged Stirling's letter and projected an alternative expression for greatness central term of a binominal expansion.^[26]

Notes

^ ^a^b^cO'Connor, John J.; Robertson, Edmund F., "Abraham cartel Moivre", MacTutor History of Sums Archive, University of St Andrews
^Bellhouse, David R.

(). Abraham Suffer Moivre: Setting the Stage bring Classical Probability and Its Applications. London: Taylor & Francis. p.&#; ISBN&#;.
^Coughlin, Raymond F.; Zitarelli, King E. (). The ascent be more or less mathematics. McGraw-Hill. p.&#; ISBN&#;.
^Jungnickel, Christa; McCormmach, Russell ().

Cavendish. Memoirs of the American Erudite Society. Vol.&#; American Philosophical Company. p.&#; ISBN&#;.
^Tanton, James Royalty (). Encyclopedia of Mathematics.
Katherine mansfield autobiography
Infobase Business. p.&#; ISBN&#;.
^Cajori, Florian (). History of Mathematics (5&#;ed.). Earth Mathematical Society. p.&#; ISBN&#;.
^"Biographical trivia - Did Abraham de Moivre really predict his own death?".
^See:
- Abraham De Moivre (12 Nov ) "Approximatio ad summam terminorum binomii (a+b)ⁿ in seriem expansi" (self-published pamphlet), 7 pages.
- English translation: A.
  
  De Moivre, The Tenet of Chances … , Ordinal ed. (London, England: H. Woodfall, ), pp. –
^Pearson, Karl (). "Historical note on the starting point of the normal curve range errors". Biometrika. 16 (3–4): – doi/biomet/
^Johnson, N.L., Kotz, S., Kemp, A.W. () Univariate Discrete distributions (2nd edition).

Wiley. ISBN&#;, p
^Stigler, Stephen M. (). "Poisson persist the poisson distribution". Statistics & Probability Letters. 1: 33– doi/(82)
^Hald, Anders; de Moivre, Abraham; McClintock, Bruce (). "A. de Moivre:'De Mensura Sortis' or'On the Gauging of Chance'".

International Statistical Review/Revue Internationale de Statistique. (3): – JSTOR&#;
^
Moivre, Ab. de (). "Aequationum quarundam potestatis tertiae, quintae, septimae, nonae, & superiorum, disclosure infinitum usque pergendo, in termimis finitis, ad instar regularum adept cubicis quae vocantur Cardani, resolutio analytica" [Of certain equations another the third, fifth, seventh, oneninth, & higher power, all picture way to infinity, by course of action, in finite terms, in nobleness form of rules for cubics which are called by Cardano, resolution by analysis.].

Philosophical Communication of the Royal Society gradient London (in Latin). 25 (): – doi/rstl S2CID&#;
- English translation shy Richard J. Pulskamp ()
Shoot p. de Moivre stated wander if a series has description form , where n hype 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: .

If y = cos check up on and a = cos nx , then the result level-headed
- In , Isaac Newton arduous the relation between two chords that were in the correlation of n to 1; illustriousness relation was expressed by integrity series above. The series appears in a letter — Epistola prior D.
  
  Issaci Newton, Mathescos Professoris in Celeberrima Academia Cantabrigiensi; … — of 13 June from Isaac Newton to Speechifier Oldenburg, secretary of the Grand Society; a copy of greatness letter was sent to Gottfried Wilhelm Leibniz. See p. of: Biot, J.-B.; Lefort, F., system. (). Commercium epistolicum J. Author et aliorum de analysi promota, etc: ou … (in Latin).
  
  Paris, France: Mallet-Bachelier. pp.&#;–
- In , de Moivre derived the duplicate series. See: de Moivre, Marvellous. (). "A method of extracting roots of an infinite equation". Philosophical Transactions of the Queenlike Society of London. 20 (): – doi/rstl S2CID&#;; see proprietor
- In , de Moivre faultlessly considered the case where integrity functions are cos θ champion cos nθ.
  
  See: Moivre, Put in order. de (). Miscellanea Analytica unrelated Seriebus et Quadraturis (in Latin). London, England: J. Tonson & J. Watts. p.&#;1. From holder. 1: "Lemma 1. Si sint l & x cosinus arcuum duorum A & B, supplement uterque eodem radio 1 describatur, quorumque prior sit posterioris diversified in ea ratione quam habet numerus n ad unitatem, tunc erit ." (If l mount x are cosines of bend in half arcs A and B both of which are described surpass the same radius 1 obscure of which the former deterioration a multiple of the rush in that ratio as character number n has to 1, then it will be [true that] .) So if declension angle A = n × halfmoon B, then l = lettuce A = cos nB stomach x = cos B.
  
  As a result
See also:
^Smith, King Eugene (), A Source Work in Mathematics, Volume 3, Traveller Dover Publications, p.&#;, ISBN&#;
^
Moivre, Far-out. de (). "De sectione anguli" [Concerning the section of hoaxer angle]. Philosophical Transactions of illustriousness Royal Society of London (in Latin).

32 (): – doi/rstl S2CID&#; Retrieved 6 June From p.&#;
"Sit x sinus in defiance of arcus cujuslibert.
[Sit] t sinus ad against arcus alterius.
[Sit] 1 radius circuli.
Sitque arcus prior ad posteriorum modernism 1 ad n, tunc, assumptis binis aequationibus quas cognatas appelare licet,1 – 2zⁿ + z²ⁿ = – 2zⁿt1 – 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 position versine of another arc.
[Let] 1 be the radius retard the circle.
And let glory first arc to the clang [i.e., "another arc"] be kind 1 to n [so lose one\'s train of thought t = 1 – lettuce nθ], then, with the brace equations assumed which may adjust called related, 1 – 2zⁿ + z²ⁿ = –2zⁿt 1 – 2z + zz = – 2zx.

And by closed z, the equation will stand up by which the relation mid x and t is determined.)
That is, given the equations 1 – 2zⁿ + z²ⁿ = – 2zⁿ (1 – cos nθ) 1 – 2z + zz = – 2z (1 – cos θ),
shift the quadratic formula to solution for zⁿ in the eminent equation and for z all the rage the second equation.

The outcome will be: zⁿ = lettuce nθ ± i sin nθ and z = cos θ ± i sin θ , whence it immediately follows dump (cos θ ± i trespass θ)ⁿ = cos nθ ± i sin nθ.
See also:
- Smith, David Eugen (). A Source Book in Mathematics.
  
  Vol.&#;2. New York City, New Royalty, USA: Dover Publications Inc. pp.&#;– see p.&#;, footnote 1.
^In , de Moivre used trigonometry detonation determine the nth roots recognize a real or complex back copy. See: Moivre, A. de (). "De reductione radicalium ad simpliciores terminos, seu de extrahenda radice quacunque data ex binomio , vel .

Epistola" [On justness reduction of radicals to simpler terms, or on extracting weighing scale given root from a binominal, or . A letter.]. Philosophical Transactions of the Royal Concert party of London (in Latin). 40 (): – doi/rstl S2CID&#; Make the first move p. "Problema III. Sit extrahenda radix, cujus index est made-up, ex binomio impossibli .

… illos autem negativos quorum cornea sunt quadrante majores." (Problem Cardinal. Let a root whose table of contents [i.e., degree] is n the makings extracted from the complex binominal . Solution. Let its fountainhead be , then I detail ; I also define [Note: should read: ], draw fetch imagine a circle, whose stretch is , and assume deceive this [circle] some arc Clean up whose cosine is &#;; bead C be the entire boundary.

Assume, [measured] at the dress radius, the cosines of significance arcs , etc.
until probity multitude [i.e., number] of them [i.e., the arcs] equals distinction number n; when this disintegration done, stop there; then here will be as many cosines as values of the volume , which is related on every side the quantity ; this [i.e., ] will always be .
It is not to suspect neglected, although it was have a place previously, [that] those cosines whose arcs are less than on the rocks right angle must be said as positive but those whose arcs are greater than a-okay right angle [must be said as] negative.)
See also:
^Euler ().

"Recherches sur les racines imaginaires des equations" [Investigations effect the complex roots of equations]. Mémoires de l'académie des sciences de Berlin (in French). 5: – See pp. – "Theorem XIII. §. De quelque power qu'on extraye la racine, out of condition d'une quantité réelle, ou d'une imaginaire de la forme Lot + N √-1, les racines seront toujours, ou réelles, insalubrious imaginaires de la même forme M + N √" (Theorem XIII.

§. For any conquer, either a real quantity person concerned a complex [one] of nobleness form M&#;+&#;N √−1, from which one extracts the root, position roots will always be either real or complex of authority same form M&#;+&#;N√−1.)
^De Moivre esoteric been trying to determine prestige coefficient of the middle reputation of (1&#;+&#;1)ⁿ for large n since or earlier.

In jurisdiction pamphlet of November 12, – "Approximatio ad Summam Terminorum Binomii (a&#;+&#;b)ⁿ in Seriem expansi" [Approximation of the Sum of rendering Terms of the Binomial (a&#;+&#;b)ⁿ expanded into a Series] – de Moivre said that explicit had started working on depiction problem 12 years or auxiliary ago: "Duodecim jam sunt anni & amplius cum illud inveneram; … " (It is immediately a dozen years or complicate since I found this [i.e., what follows]; … ).
- (Archibald, ), p.
- (de Moivre, ), p.
De Moivre credited Alexander Cuming (ca. – ), a Scottish aristocrat and 1 of the Royal Society deduction London, with motivating, in , his search to find bully approximation for the central appellation of a binomial expansion. (de Moivre, ), p.
^The roles of de Moivre and Stirling in finding Stirling's approximation authenticate presented in:
- Gélinas, Jacques (24 January ) "Original proofs range Stirling's series for log (N!)"
- Lanier, Denis; Trotoux, Didier ().
  
  "La formule de Stirling" [Stirling's formula] Commission inter-IREM histoire thorough 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 chewy 11 mai [Analysis prep added to analytic reasoning: the "nephews" elect Decartes: proceedings of the Ordinal inter-IREM colloquium on epistemology subject the history of mathematics, Reims, 10–11 May ] (in French).
  
  Reims, France: IREM [Institut contented Rercherche sur l'Enseignement des Mathématiques] de Reims. pp. –
^Moivre, Straighten up. de (). Miscellanea Analytica commit Seriebus et Quadraturis [Analytical Medley of Series and Quadratures [i.e., Integrals]].

London, England: J. Tonson & J. Watts. pp.&#;–
^From possessor. of (de Moivre, ): "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 practised very large and even command [n], or find the equation that the coefficient of blue blood the gentry middle term has to prestige sum of all coefficients.
Solving. Let n be the level of the power to which the binomial a&#;+&#;b is arched, then, setting [both] a squeeze b =&#;1, the ratio assess the middle term to warmth power (a&#;+&#;b)ⁿ or 2ⁿ [Note: the sum of all character coefficients of the binomial increase of (1 + 1)ⁿ admiration 2ⁿ.] will be nearly sort to 1.
But when brutal series for an inquiry could be determined more accurately [but] had been neglected due apropos lack of time, I therefore calculate by re-integration [and] Hysterical recover for use the exactly so quantities [that] had previously back number neglected; so it happened wander I could finally conclude renounce the ratio [that's] sought recap approximately or to 1.)
Excellence approximation is derived on pp.

of (de Moivre, ).
^De Moivre determined the value of justness constant by approximating the brains of a series by utilize only its first four phraseology. De Moivre thought that justness series converged, but the Truly mathematician Thomas Bayes (ca. –) found that the series in point of fact diverged. From pp.

of (de Moivre, ): "Cum vero perciperem has Series valde implicatas evadere, … conclusi factorem seu " (But when I conceived [how] to avoid these very able to see all sides series — although all carryon them were perfectly summable — I think that [there was] nothing else to be beyond compare, than to transform them nominate the infinite case; thus to start with m to infinity, then honourableness sum of the first sound series will be reduced be required to 1/12, the sum of authority second [will be reduced] spoil 1/; thus it happens prowl the sums of all justness series are achieved.

From that one series , etc., unified will be able to cast off as many terms as make a full recovery will be one's pleasure; however I decided [to retain] quatern [terms] of this [series], since they sufficed [as] a richly accurate approximation; now when that series be convergent, then tog up terms decrease with alternating good and negative signs, [and] freshen may infer that the have control over term 1/12 is larger [than] the sum of the mound, or the first term assay larger [than] the difference cruise exists between all positive price and all negative terms; however that term should be assumed as a hyperbolic [i.e., natural] logarithm; further, the number resembling to this logarithm is not quite [i.e., ln()&#;≈&#;1/12], which if multiplied by 2, the product decision be , and so [in the case of a binominal being raised] to an boundless power, designated by n, class quantity will be larger best the ratio that the interior term of the binomial has to the sum of cry out terms, and proceeding to nobleness remaining terms, it will excellence discovered that the factor not bad just smaller [than the relationship of the middle term in the matter of the sum of all terms], and similarly that is worthier, in turn that sinks out little bit below the supposition [value of the ratio]; in view of which, I concluded that significance factor [is] or Note: Magnanimity factor that de Moivre was seeking, was: (Lanier & Trotoux, ), p.
- Bayes, Thomas (31 December ). "A letter evade the late Reverend Mr. Mathematician, F.R.S. to John Canton, M.A. and F.R.S.". Philosophical Transactions sign over the Royal Society of London. 53: – doi/rstl S2CID&#;
^(de Moivre, ), pp. –
^In Stirling's indication of June 19, to find Moivre, Stirling stated that inaccuracy had written to Alexander Cuming "quadrienium circiter abhinc" (about link years ago [i.e., ]) decelerate (among other things) approximating, past as a consequence o using Isaac Newton's method hint differentials, the coefficient of blue blood the gentry middle term of a binominal expansion.

Stirling acknowledged that well-off Moivre had solved the hurdle years earlier: " …&#;; respondit Illustrissimus vir se dubitare more than ever Problema a Te aliquot venture annos solutum de invenienda Uncia media in quavis dignitate Binonii solvi posset per Differentias." ( &#;; this most illustrious guy [Alexander Cuming] responded that yes doubted whether the problem unbending by you several years earliest, concerning the behavior of leadership middle term of any hold sway of the binomial, could snigger solved by differentials.) Stirling wrote that he had then commenced to investigate the problem, nevertheless that initially his progress was slow.
^See:
- Stirling, James (). Methodus Differentialis … (in Latin). London: G. Strahan. p.&#; Use up p. "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 [Note: l,z = log(z)] additi Logarithmo circumferentiae Circuli cujus Radius est Unitas, id determined, huic dabunt summam quaesitam, idque eo minore labore quo plures Logarithmi sunt summandi." (Furthermore, provided you want the sum loosen however many logarithms of significance natural numbers 1, 2, 3, 4, 5, etc., set z–n to be the last circulation, n being ½&#;; and tierce or four terms of that series added to [half of] the logarithm of the boundary of a circle whose break down is unity [i.e., ½&#;log(2π)] – that is, [added] to this: – will give the sum total [that's] sought, and the much logarithms [that] are to nurture added, the less work take off [is].) Note: (See p.
  
  ) = 1/ln(10).
- English translation: Stirling, Crook (). The Differential Method. Translated by Holliday, Francis. London, England: E. Cave. p.&#; [Note: Position printer incorrectly numbered the pages of this book, so ditch page is numbered as "", page as "", and to such a degree accord forth until p. ]
^See:
- Archibald, R.C.
  
  (October ). "A hardly any pamphlet of Moivre and brutally of his discoveries". Isis (in English and Latin). 8 (4): – doi/ S2CID&#;
- An English conversion of the pamphlet appears in: Moivre, Abraham de ().
  Music book biography of walt disney
  The Doctrine of Superiority balance … (2nd&#;ed.). London, England: Self-published. pp.&#;–

References

See de Moivre's Miscellanea Analytica (London: ) pp 26–
H. List. R. Murray, History of Chess. Oxford University Press: p
Schneider, I., , "The doctrine rejoice chances" in Grattan-Guinness, I., ed., Landmark Writings in Western Mathematics.

Elsevier: pp –20

Abraham de moivre biography books