MOIVRE, ABRAHAM DE

born May 26, 1667, Vitry, Fr. died Nov. 27, 1754, London French mathematician who was a pioneer in the development of analytic trigonometry and in the theory of probability. A French Huguenot, de Moivre was jailed as a Protestant upon the revocation of the Edict of Nantes in 1685. When he was released shortly thereafter, he fled to England. In London he became a close friend of Sir Isaac Newton and the astronomer Edmond Halley. De Moivre was elected to the Royal Society of London in 1697 and later to the Berlin and Paris academies. Despite his distinction as a mathematician, he never succeeded in securing a permanent position but eked out a precarious living by working as a tutor and a consultant on gambling and insurance. De Moivre expanded his paper De mensura sortis (written in 1711), which appeared in Philosophical Transactions, into The Doctrine of Chances (1718). Although the modern theory of probability had begun with the unpublished correspondence (1654) between Blaise Pascal and Pierre de Fermat and the treatise De Ratiociniis in Ludo Aleae (1657; On Ratiocination in Dice Games) by Christiaan Huygens of Holland, de Moivre's book greatly advanced probability study. The definition of statistical independencenamely, that the probability of a compound event composed of the intersection of statistically independent events is the product of the probabilities of its componentswas first stated in de Moivre's Doctrine. Many problems in dice and other games were included, some of which appeared in the Swiss mathematician Jakob (Jacques) Bernoulli's Ars conjectandi (1713; The Conjectural Arts), which was published before de Moivre's Doctrine but after his De mensura. He derived the principles of probability from the mathematical expectation of events, just the reverse of present-day practice. De Moivre's second important work on probability was Miscellanea Analytica (1730; Analytical Miscellany). He was the first to use the probability integral in which the integrand is the exponential of a negative quadratic, He originated Stirling's formula, incorrectly attributed to James Stirling (16921770) of England, which states that for a large number n, n! equals approximately (2pn)1/2e-nnn; that is, n factorial (a product of integers with values descending from n to 1) approximates the square root of 2pn, times the exponential of -n, times n to the nth power. In 1733 he used Stirling's formula to derive the normal frequency curve as an approximation of the binomial law. De Moivre was one of the first mathematicians to use complex numbers in trigonometry. The formula known by his name, (cos x + i sin x)n = cos nx + i sin nx, was instrumental in bringing trigonometry out of the realm of geometry and into that of analysis.