by Ivo Schneider
© Oxford University Press 2004 All rights reserved
Moivre, Abraham de (1667-1754), mathematician, was born Abraham Moivre on 26 May 1667 at Vitry-le-François, Marne, France, son of the protestant surgeon Daniel Moivre (fl. 1665-1685) and his wife, Anne (fl. 1665-1685). All knowledge of his early life is derived from the biography by Matthew Maty, parts of which, including the years in France, de Moivre dictated to Maty shortly before he died. Apparently he was educated by the Catholic Pères de la Doctrine Chrétienne (1672-7) before he moved in 1678 to the protestant academy at Sedan, where he studied mainly Greek. After the academy closed in 1681 Moivre continued his studies at a protestant academy at Saumur (1682-4). Interested in the new philosophy of Descartes, which was not taught in Saumur, he went to the Collège d'Harcourt in Paris. There, presumably influenced by Descartes, he turned to mathematics. Hitherto he had studied elementary mathematics and, without mastering it completely, Christiaan Huygens's small tract concerning games of chance, De ratiociniis in ludo aleae (1657). In Paris he was taught mathematics by Jacques Ozanam, who made his living as a private teacher of the subject, attracting many students and enjoying a moderate financial success. It seems probable that Moivre later took him as a model when he had to support himself.
After the revocation of the edict of Nantes in 1685, hundreds of thousands of Huguenots who had refused to become Catholic emigrated to protestant countries. Among them was Moivre, who went to England where he and his younger brother Daniel were granted denization in December 1687, and where he began his occupation as a teacher of mathematics. Here both brothers added a 'de' to their names. The most plausible reason for this change is that Abraham for his part wanted the prestige of noble birth in France in dealing with his clients, many of whom were noblemen; Daniel, who became a merchant, presumably felt likewise.
A contemporary anecdote relates that de Moivre cut out the pages of Newton's Principia and read them while waiting for his students or walking from one to the other. True or not, the main function of the story was to place de Moivre among the first true and loyal Newtonians. In 1692 he met with Edmond Halley and shortly afterwards with Newton. Halley oversaw the publication of de Moivre's first paper on Newton's doctrine of fluxions in the Royal Society's Philosophical Transactions (1695); he was elected to the society in November 1697. His election as fellow of the Kurfuerstlich Brandenburgische Sozietaet der Wissenschaften came only in 1735; five months before his death the Académie Royale des Sciences, Paris, made him a foreign associate member.
Newton's influence on mathematics and natural philosophy in the British universities was such that it seemed profitable to de Moivre to attack the problems posed by the new infinitesimal calculus. In 1697 and 1698 he published the polynomial theorem, a generalization of Newton's binomial theorem, together with applications in the theory of series. Criticized by the Scottish physician George Cheyne in Fluxionum methodus inversa (1703), a book on Newton's method of fluents, de Moivre entered into a rather unpleasant fight with Cheyne in the pages of his Animadversiones in G. Cheynaei Tractatum de fluxionem methodo inversa (1704). This at least secured him the attention of Leibniz and Johann Bernoulli, with whom he began a correspondence in 1704, hoping to get support for a professorship on the continent. But after 1712, when he became a member of the commission set up by the Royal Society to support Newton's priority in the dispute between Newton and Leibniz and was drawn into the ensuing quarrels between Newtonians and Leibnizians, which lasted until the 1720s, de Moivre saw no reason to continue his correspondence with Bernoulli. He failed to secure a professorship in Great Britain, however, and was obliged to continue as a tutor and consultant in mathematical affairs and as a translator.
Bernoulli's letters had shown de Moivre how difficult it would be to compete with mathematicians of his calibre in the new field of analysis. So he turned to the calculus of games of chance and probability theory, which was of great interest to many of his students and where he had few competitors. Nevertheless, even outside the domain of probability and chance de Moivre could claim mathematical achievements of lasting value. Some of these were published in his Miscellanea analytica and its supplementum in 1730, such as his expression of cosna as a function of cosa equivalent to (cosa + isina) = cosna + isinna, which he had found between 1707 and 1722. In stochastics he was involved in two rather fierce disputes about priority with the Frenchman Montmort and with Thomas Simpson, who in two books of 1740 and 1742 exploited the content of de Moivre's Doctrine of Chances (1718) and Annuities upon Lives (1725).
In the Philosophical Transactions for 1711 de Moivre published a long article, 'De mensura sortis', which was followed by the Doctrine of Chances. The second, much extended, edition of the Doctrine (1738) contained his normal approximation to the binomial distribution that he had found in 1733. This special case of the central limit theorem he understood as a generalization and a sharpening of Bernoulli's Theorema aureum, which was later named the law of large numbers by Poisson. De Moivre's central limit theorem is considered as his greatest mathematical achievement and shows that he understood intuitively the importance of what was later called the standard deviation. Crucial to this theorem was a form of the so-called Stirling formula for n!, which de Moivre and Stirling had developed in competition which ended in 1730.
De Moivre's representation of the solutions of the then current problems of games of chance tended to be more general than those of Montmort. In addition he developed a series of algebraic and analytic tools for the theory of probability, like a 'new algebra' for the solution of the problem of coincidences which foreshadowed Boolean algebra, the method of generating functions, or the theory of recurrent series for the solution of differential equations. In the Doctrine de Moivre offered an introduction which contains the main concepts such as probability, conditional probability, expectation, dependent and independent events, the multiplication rule, and the binomial distribution. In 1738 and 1756 he interpreted his form of the central limit theorem in terms of natural religion as a proof for the existence and constant engagement of God in his creation. With it were connected the function and role of chance in the world.
De Moivre's preoccupation with matters concerning the conduct of a capitalist society, such as interest, loans, mortgages, pensions, reversions or annuities, dated back at least to the 1690s, from which time a short note survives in Berlin containing his answers to a client's questions. In 1693, using the lists of births and deaths in Breslau for each of the years 1687 to 1691, Edmond Halley had published in the Philosophical Transactions a life table together with applications to annuities on lives, but the amount of calculation involved in extending this to two or more lives turned out to be immense. De Moivre replaced Halley's life table by a (piecewise) linear function, which allowed him to derive formulas for annuities of single lives and approximations for annuities of joint lives as a function of the corresponding annuities on single lives. He published these formulas, together with the solution of problems of reversionary annuities, annuities on successive lives, tontines, and other contracts that depend on interest and the 'probability of the duration of life', in his book Annuities upon Lives. In the second edition of the Doctrine of Chances he incorporated part of the Annuities together with new material. After three more improved editions of the Annuities in 1743, 1750, and 1752, the last version of it was published in the third edition of the Doctrine (1756). The Doctrine, especially, attracted Lagrange and Laplace to de Moivre's work; it derived from de Moivre's solution of the problem of the duration of play by means of what he called recurrent series, which amounted to the solution of a homogeneous linear differential equation with constant coefficients. In fact, the most effective analytical tool that Laplace developed for the calculus of probabilities, the theory of generating functions, is a consequence of his occupation with recurrent series: thus the most important results of de Moivre's Doctrine reappear in Laplace's probability theory represented in a new mathematical form and in a new philosophical context, confirming de Moivre's status as a pioneer in this field.
In his later years, at least, de Moivre was living in the parish of St Anne, Westminster, London. He died in London on 27 November 1754 and was buried on 1 December at St Martin-in-the-Fields, Westminster. He had never married and left his South Sea annuities to be divided among his nieces and nephews.
IVO SCHNEIDER
Sources
M. Maty, 'Mémoire sur la vie et sur les écrits de Mr Abraham de Moivre', Journal Britannique (Sept-Oct 1755); pubd sep. (1760)
I. Schneider, 'Der Mathematiker Abraham de Moivre, 1667-1754', Archive for History of Exact Sciences, 5 (1968-9), 177-317
K. Wollenschläger, 'Der mathematische Briefwechsel zwischen Johann I Bernoulli und Abraham de Moivre', Verhandlungen der Naturforschenden Gesellschaft in Basel, 43 (1933), 151-317
A. Hald, A history of probability and statistics and their applications before 1750 (1990), chaps. 19-25
GM, 1st ser., 24 (1754), 530
parish register (burial), Westminster, St Martin-in-the-Fields, 1 Dec 1754
will, PRO, PROB 11/811/297
Huguenot Society of London [pubns of the Huguenot Society of London]
Archives
Leibniz-Archiv, Hanover, Germany
RS | University of Basel, Bernoulli corresp.
Likenesses
J. Highmore, oils, 1736, RS
Dassier, medal, 1741
Faber, engraving (after Highmore, 1736)
Wealth at death
under £2000; left South Sea annuities of £1600; plus small sundries: will, PRO, PROB 11/811/297
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