Abraham de Moivre

b. 26 May 1667 Vitry in Champagne, France
d. 27 November 1754 London, England

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.

The Theory of Probability

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.

Special Problems of the Doctrine of Chances

The Problem of Points or Divison of Stakes. Here is included the full treatment of the problem from the Introduction together with Problem VI concerning 3 players.
Dicing problems in which the number of casts required to have even odds and the expectation of a player given the number of casts.
On the game of Bassette
On the game of Pharaon
On the Duration of Play
On Derangements or the game of Rencontre, which is closely related to the game of Trieze

Publications

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.

1695-1697: "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.
1695-1697: "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 No. 240.
1698: "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.
1702-1703: "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.
1704: 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.
1706-1707: "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.
1710-1712: "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, 237-262, 1984.
1714-1716: "Solutio 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.
1714-1716: "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.
1717-1719: "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. 622-628.
1717-1719: "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.
1718: The Doctrine of Chances: or, A Method of Calculating the Probability of Events in Play, 1st Edition.
1722-1723: "De 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.
1722-1723: "De 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.
1725: Annuities upon Lives: or, The Valuation of Annuities upon any Number of Lives; as also, of Reversions. To which is added, An Appendix concerning the Expectations of Life, and Probabilities of Survivorship. Fayram, Motte and Pearson, London. 1st Edition.
1730: 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. Stirling's Methodus Differentialis was translated into English by Francis Holliday and published as The Differential Method in 1749. The formula is derived on pages 109- 114 (Proposition XXIII). See also the annotated translation by Ian Tweedle published in 2003.
1733: "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.
1737-1738: "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.
1738: The Doctrine of Chances: or, A Method of Calculating the Probability of Events in Play, 2nd Edition.
1743: Annuities upon Lives: Second edition, plainer, fuller, and more correct than the former. With several Tables, exhibiting at one View, the Values of Lives, for several Rates of Interest. Woodfall, London.
1744-1745: "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.
1750: Annuities upon Lives, 3rd Edition. Woodfall, London.
1752: Annuities upon Lives, 4th Edition. Woodfall, London.
1756: 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.