b. 26 May 1667 Vitry in Champagne, France
d. 27 November 1754 London, England
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Moivre received a fine classical education in France. But religious intolerance caused him to flee to England at age 21 where he remained the rest of his life. Henry IV had issued on 13 April 1598 the Edict of Nantes which granted to the Huguenots the right to practice their Protestant faith throughout nearly all of France. But Louis XIV moved toward the removal of these rights and on 17 October 1685 officially revoked the edict. It appears that upon entering England he changed his name from Moivre to de Moivre.
Moivre was the friend of many of the scientific luminaries of the time, in particular, Newton and Halley. He was elected to the Royal Society in 1697, the Berlin Academy in 1735 and finally, in 27 June 1754 to that of Paris, five months before his death.
Selections from his work available here are with few exceptions not translations but rather extracts from texts published in English. One can always refer to the originals as they are widely available. The advantage of those presented here is that the text is more legible and related problems are collected together.
With respect to the theory of probability, Moivre is best known for his
analysis of the duration of play - Gambler's Ruin, but see also related material on the Duration of Play
theory of recurrent series, where is found propositions on the summation of such series
the "normal approximation" to the binomial distribution, (available at University of York here) and
the approximation of n! usually attributed to Stirling.
Moivre's first exposure to probability was through the work of Huygens which he is said to have read at age 15. However, it was apparently with the publication of Montmort's Essai d' analyse sur les jeux de hazard in 1710 that Frances Robartes, a fellow member of the Royal Society, was prompted to pose to Moivre three problems more difficult than those contained in the Essai. Moivre developed new techniques, different from those used by Montmort and Huygens, to solve problems of games. At the behest of Robartes, he submitted his contributions to the Royal Society. This paper, "De Mensura Sortis," was read to the Royal Society in 1711 and published in the Philosophical Transactions the next year in the records of the months January-February-March of 1711.
A small controversy arose with Montmort over the question of priority with Montmort claiming that there was nothing in Moivre's work that he had not previously published in his Essai or had discussed with Nikolaus Bernoulli. From this controversy resulted three papers: the first by Nikolaus Bernoulli and the second by Moivre, both published in the Philosophical Transactions Volume 29, No. 341 regarding what is now called Waldegrave's Problem. The next volume of the Transactions contains a paper of Montmort on the summation of series. With regard to the controversy, the comments of Moivre made in his Doctrine of Chances, 3rd edition may be read and those of Montmort in his Avertissement.
Moivre expanded the "De Mensura Sortis" into the book, The Doctrine of Chances, a work published in 1718 and for which two further editions appeared in 1738 and 1756 respectively.
Fifteen of his papers were published in the Philosophical Transactions between 1695 and 1746 and, in all, Moivre may be credited further with four books (1704, 1718, 1725, 1730), of which the middle two appeared in several editions, and also one pamphlet (1733). All papers published in the Philosophical Transactions are now available through JSTOR. Those papers which bear upon the theory of probability in some manner are highlighted in blue and translations into English of those written in Latin are linked. Where dates form an interval, they refer to the years spanned by the corresponding volume of the Transactions.
"Specimina Quaedam Illustria Doctrinae
Fluxionum Sive Exempla Quibus Methodi Istius Usus et Praestantia in
Solvendis Problematis Geometricis Elucidatur," ex Epistola Peritissimi
Mathematici D. Ab. de Moivre Desumpta, Philosophical Transactions,
Vol. 19, No. 216, pp. 52-57.
"A Method of Raising an
Infinite Multinomial to Any Given Power, or Extracting Any Given Root
of the Same." By Mr. Ab. de Moivre, Philosophical
Transactions, Vol. 19, No. 230, pp. 619 - 625. The first of
a series of five related papers of which the next is that appearing in
"A Method of Extracting the
Root of an Infinite Equation." By A. de Moivre, F.R.S. Philosophical
Transactions, Vol. 20, No. 240, pp. 190-193. The second of
a series of five related papers. The first of which appeared in No. 230
and the third in No. 309.
"Methodus Quadrandi Genera Quaedam Curvarum,
aut ad Curvas Simpliciores Reducendi." per A. De Moivre R. S. S. Philosophical
Transactions, Vol. 23, No. 278, pp. 1113-1127.
Animadversions in D.Georgii Cheynaei
tractatum de fluxionum methodo inversa. London. In the work on the
differential calculus, Fluxionum Methodus inversa, sive
quantitum fluentium leges generaliores of 1702, George Cheyne
claimed some of Moivre's results as his own. Moivre in turn wrote this
pamphlet in response.
"Aequationum Quarundam Potestatis Tertiae, Quintae, Septimae, Nonae, & Superiorum, ad Infinitum Usque Pergendo, in Terminis Finitis, ad Instar Regularum pro Cubicis Quae Vocantur Cardani, Resolutio Analytica," Philosophical Transactions, Vol. 25, No. 309, pp. 2368-2371. The third of a series of five related papers of which the previous appeared in No. 240 and the fourth in No. 373. This paper anticipates the discovery of Moivre's famous trigonometric formula (cos x + i sin x)n = cos nx + i sin nx. which is first stated in 1722.
"De Mensura Sortis, seu, de
Probabilitate Eventuum in Ludis a Casu Fortuito Pendentibus," Philosophical
Transactions Vol. 27, No. 329, pp. 213-264. This has been
translated into English by B. McClintock as "On the measurement of
chance, or, on the probability of events in games depending upon
fortuitous chance" and published in Intern. Statist. Rev. 52,
Generalis Altera Praecedentis Problematis, ope Combinationum et
Serierum infinitarum." per D. Abr. De Moivre. Reg. Soc. Sodalem. Philosophical
Transactions Vol. 29, No. 341, pp. 145-158. This appears
immediately after Nikolaus
Bernoulli's paper Philosophical Transactions Vol. 29,
No. 341, pp. 133-144.
"A Ready Description and Quadrature of a Curve
of the Third Order, Resembling That Commonly Call'd the Foliate."
Communicated by Mr. Abr. De Moivre, F. R. S. Philosophical
Transactions Vol. 29, No. 341, pp. 329-331.
"Proprietates Quaedam Simplices Sectionum
Conicarum ex Natura Focorum deductae; cum Theoremate Generali de
Viribus Centripetis; quorum ope Lex Virium Centripetarum ad Focos
Sectionum Tendentium, Velocitates Corporum in Illis Revolventium, &
Descriptio Orbium Facillime Determinatur." Per Abr. de Moivre. R. S.
Soc. Philosophical Transactions Vol. 30, No. 352, pp.
"De Maximis & Minimis Quae in Motibus
Corporum Coelestium Occurrunt," Philosophical Transactions Vol.
30, No. 360, pp. 952-954. No author is attributed to this paper.
However, within the paper the reader is twice refered to the previous
one appearing in No. 352.
The Doctrine of Chances: or, A Method of Calculating the Probability of Events in Play, 1st Edition.
Fractionibus Algebraicis Radicalitate
Immunibus ad Fractiones Simpliciores Reducendis, Deque Summandis
Terminis Quarundam Serierum Aequali Intervallo a Se Distantibus,"
Auctore Abrahamo de Moivre, S. R. Socio, Philosophical Transactions
Vol 32, No. 373, pp. 162-178. The fourth of a series of
five related papers of which the third appeared in No. 309 and the
fifth in No. 374. This paper shows how to construct a partial fraction
decomposition of reciprocals of polynomials, how to find closed form
sums of certain subseries of infinite series which are given by
reciprocals of polynomials, and the trigonometric solution to the
duration of play problem.
Sectione Anguli," Autore
A. De Moivre, R. S. S. Philosophical Transactions Vol. 32,
No. 374, pp. 228-230. The last of a series of five related papers of
which the previous appeared in No. 374. This paper includes implicitly
the famous trigonometric formula (cos x + i sin x)n
= cos nx + i sin nx.
upon Lives: or, The Valuation of
Annuities upon any Number of Lives; as also, of Reversions. To
added, An Appendix concerning the Expectations of Life, and
Probabilities of Survivorship. Fayram, Motte and Pearson, London.
Miscellanea Analytica de Seriebus et
Quadraturis & Miscellaneis Analyticis Supplementum. A
summary of research conducted between 1721 and 1730. A substantial
portion (pp. 146-229) consisting of seven chapters is entitled
"Responsio ad quasdam Criminationes" and is Moivre's response prompted
by the controversy with Montmort. According to Todhunter nearly all of
the content was incorportated into the Doctrine of Chances. The
supplement contains the derivation of what is now called Stirling's
Formula. A translation of the parts of interest to probability theory is in preparation.
"Approximatio ad Summam
Terminorum Binomii (a + b)n in Seriem expansi." A
pamphet printed on 13 November 1733 for private circulation. A reprint
may be found in a paper by R.C. Archibald "A Rare Pamphlet of De Moivre
and Some of his Discoveries," Isis, 8 (1926), pp.
671-684. Here Moivre obtains what is equivalent to the normal
approximation to the binomial distribution. An English translation of
this pamphlet was eventually incorporated into the 1738 (pp. 235-242)
and the 1756 (pp. 243-250) editions of the Doctrine of Chances
with substantial additions (pp. 250-254 & 334 3rd edition.) Other
small additions by Moivre are easily discerned by comparison of the
reprint with the translation appearing in the 1756 edition.
"De Reductione Radicalium ad simpliciores
terminos, seu de extrahenda radice quacunque data ex Binomio a
+ sqrt(+ b), vel a + sqrt(- b). Epistola.
Gulielmo Jones, Armigero S.P.D. A De Moivre." Philosophical
Transactions, Vol. 40, No. 451, pp. 463-478.
The Doctrine of Chances: or, A Method of
Calculating the Probability of Events in Play, 2nd Edition.
upon Lives: Second edition,
plainer, fuller, and more correct than the former. With several
exhibiting at one View, the Values of Lives, for several Rates of
Interest. Woodfall, London.
"A letter from Mr. Abraham De Moivre, F. R. S.
to William Jones, Esquire, F. R. S. concerning the Easiest Method for
Calculating the Value of Annuties upon Lives, from Tables of
Observations." Philosophical Transactions Vol. 43,
No. 473, pp. 65-78.
Annuities upon Lives, 3rd Edition.
upon Lives, 4th Edition.
The Doctrine of Chances: or, A Method of Calculating the Probability of Events in Play, 3rd Edition, incorporating Annuities upon Lives. Millar, London. Published posthumously. Reprinted by Chelsea, New York 1967. The Appendix to the Chelsea reprint contains several items of interest: The dedication of the 1st edition to Isaac Newton, then President of the Royal Society; an extract appearing both in the Miscellanea Analytica and in Stirling's Methodus Differentialis giving the rule to approximate n!; the life tables of Halley, Kerseboom, de Parcieux and Smart & Simpson; the biography of Moivre by Helen Walker.