De Moivre, A
NAME: Moivre, Abraham de
ACHIEVEMENT(S): 18th century pioneer in the development of analytic trigonometry and the theory of probability.
NICKNAME/ALIAS: None
DATE/PLACE OF BIRTH: 26 May 1667, at Vitry-le-Francois, Champagne, France
DATE/PLACE/AGE AT DEATH: 27 November 1754, in London, England. Aged 87
DISCIPLINE: Mathematician
FIELD: Mathematics
LANGUAGE: French and English
PARENTS: Unknown
SIBLINGS: Unknown
SPOUSE: None
CHILDREN: None
EDUCATION: Catholic Village School; Protestant Academy, Sedan, 1678-82;
Saumur, 1682-4; College de Harcourt, Paris, 1684; tutored by Jacques Ozanam
in Paris, 1684-5.
MAJOR WORK: 'The Doctrine of Chances', 1718
LIFE AND TIMES:
While in Saumur de Moivre studied mathematics, including Christiaan Huyghens'* work on games of chance: 'On Ratiocination in Dice Games' (1657).
In October 1685, Louis XIV* revoked the Edict of Nantes, depriving French Protestants of all religious and civil liberties, and de Moivre, like many other Huguenots, was imprisoned.
On his release in 1686, he fled to England, where he made a modest living tutoring in mathematics, and continued to study recent mathematical texts, the most notable being Newton's* 'Principia Mathematica.' In 1692 he met the astronomer Halley* who was so impressed by him that he read his first paper (on Newton's calculus), to the Royal Society* in 1695. Two years later Halley proposed de Moivre for election to the Society, and he was made a fellow the same year.
During the late 1690’s de Moivre became a close friend of Newton. In 1710 he was appointed to the Grand Commission set up by the Royal Society to try and settle the priority dispute (for calculus) calculus between Leibniz* and Newton.
De Moivre was not able to find a permanent job (he even asked the Swiss mathematician Johann I Bernoulli* to ask Leibniz to intervene on his behalf for a professorship in mathematics at Cambridge) and he continued to eke out a living by tutoring (one of his pupils being Dodson*, who became master of the Royal Mathematical School in London) and acting as a consultant for gambling syndicates and insurance companies for the rest of his life.
In 1718 de Moivre published: 'The Doctrine of Chances', dedicated to Newton (expanded versions were published in 1738 and 1756). Although the modern theory of probability had begun with the correspondence (1654) between Blaise Pascal* and Fermat*, and with treatises by Huyghens and Pierre de Montmort (1708), de Moivre's book greatly advanced the science.
De Moivre's great achievement in 'The Doctrine of Chances' was his statement of the approximation to the binomial probability distribution, which (as the normal or Gaussian distribution) was a significant advance on Bernoulli's law, and he was therefore the first to derive an exact formulation of how "chances" and stable frequencies are related.
He obtained from the binomial expansion of (1+1)[to the power of]n what is now recognized as n![italics], from which in 1733 (appearing in later editions of the 'Doctrine') de Moivre derived the normal frequency curve as an approximation of the binomial law.
This meant that de Moivre was able to sum the terms of the binomial from any point up to the central term: equivalent to the modern normal approximation and the first occurrence of the normal probability integral.
This proved the most important single discovery in the formulation of probability theory and was incorporated into statistical studies for the next two hundred years. De Moivre derived the principles of probability from the mathematical expectation of events, the reverse of present-day practice.
Elsewhere in his 'Doctrine', de Moivre first gave the definition of statistical independence: that the probability of a compound event composed of the intersection of statistically independent events is the product of the probabilities of its components.
In 1730 he published a second important work on probability and other mathematical problems: 'Analytical Miscellany' in which he originated the use of the probability
integral, in which the integrand is the exponential of a negative quadratic.
In this work de Moivre also investigated infinite series and complex numbers, and became one of the first mathematicians to use complex numbers in trigonometry. Today he is mainly remembered for the theorem which bears his name ("De Moivre's Formula", first stated 1722: (cos x + i sin x)n = cos nx + i sin nx) which took trigonometry into analysis, and became an important factor in the early development of complex number theory.
De Moivre also investigated and analysed mortality statistics and laid the mathematical foundations of the theory of annuities, his interest perhaps originating from his friendship with Halley, who in 1693 had written on annuities for the Royal Society.
In 1725 he published 'Annuities on Lives', in which he developed formulae based on a law of mortality and constant rates of interest on money, working out a treatment of joint annuities on several lives, and one for when both age and interest on capital have equal
relevance. This mathematics became a standard part of all commercial applications of annuities in England.
At the end of his life, he earned money by solving problems of chance at coffee houses.
ASSESSMENT: One of the two outstanding giants of the theory of probability in the 18th century.
EXTRA CONNECTIONS: His work was extended by Laplace* and Lagrange*.
May have taught Bayes*
ONELINER:Abraham de Moivre, pioneer of probability math
BIBLIOGRAPHY:
EB, DNB, DSB
Stigler, S.M., The History of Statistics. The Measurement of Uncertainty before 1900, Cambridge, Mass. & London, 1986
Walker, H.M., Walker, "Abraham De Moivre", Scripta Mathematica 2 (1934), 316-333.