**Abraham de Moivre** was born in Vitry-le-François, which is about halfway between Paris and Nancy, where his father worked as a surgeon. The family was certainly not well off financially, but a steady income meant that they could not be described as poor. De Moivre's parents were Protestants but he first attended the Catholic school of the Christian Brothers in Vitry which was a tolerant school, particularly so given the religious tensions in France at this time. When he was eleven years old his parents sent him to the Protestant Academy at Sedan where he spent four years studying Greek under Du Rondel.

The Edict of Nantes had guaranteed freedom of worship in France since 1598 but, although it made any extension of Protestant worship in France legally impossible, it was much resented by the Roman Catholic clergy and by the local French parliaments. Despite the Edict, the Protestant Academy at Sedan was suppressed in 1682 and de Moivre, forced to move, then studied logic at Saumur until 1684. Although mathematics was not a part of the course that he was studying, de Moivre read mathematics texts in his own time. In particular he read Huygens' treatise on games of chance *De ratiociniis in ludo aleae.* By this time de Moivre's parents had gone to live in Paris so it was natural for him to go there. He continued his studies at the Collège de Harcourt where he took courses in physics and for the first time had formal mathematics training, taking private lessons from Ozanam.

Religious persecution of Protestants became very serious after Louis XIV revoked the Edict of Nantes in 1685, leading to the expulsion of the Huguenots. At this time de Moivre was imprisoned for his religious beliefs in the priory of St Martin. It is unclear how long he was kept there, since Roman Catholic biographers indicate that soon after this he emigrated to England while his Protestant biographers say that he was imprisoned until 27 April 1688 after which he travelled to England. After arriving in London he became a private tutor of mathematics, visiting the pupils whom he taught and also teaching in the coffee houses of London.

By the time he arrived in London de Moivre was a competent mathematician with a good knowledge of many of the standard texts. However after he made a visit to the Earl of Devonshire, carrying with him a letter of introduction, he was shown Newton's *Principia.* He realised instantly that this was a work far deeper than those which he had studied and decided that he would have to read and understand this masterpiece. He purchased a copy, cut up the pages so that he could carry a few with him at all times, and as he travelled from one pupil to the next he read them. Although this was not the ideal environment in which to study the *Principia,* it is a mark of de Moivre's abilities that he was quickly able to master the difficult work. De Moivre had hoped for a chair of mathematics, but foreigners were at a disadvantage in England so although he now was free from religious discrimination, he still suffered discrimination as a Frenchman in England. We describe below some attempts to procure a chair for him.

By 1692 de Moivre had got to know Halley, who was at this time assistant secretary of the Royal Society, and soon after that he met Newton and became friendly with him. His first mathematics paper arose from his study of fluxions in the *Principia* and in March 1695 Halley communicated this first paper *Method of fluxions* to the Royal Society. In 1697 he was elected a fellow of the Royal Society.

In 1710 de Moivre was appointed to the Commission set up by the Royal Society to review the rival claims of Newton and Leibniz to be the discovers of the calculus. His appointment to this Commission was due to his friendship with Newton. The Royal Society knew the answer it wanted! It is also interesting that de Moivre should be given this important position despite finding it impossible to gain a university post.

De Moivre pioneered the development of analytic geometry and the theory of probability. He published *The Doctrine of Chance: A method of calculating the probabilities of events in play* in 1718 although a Latin version had been presented to the Royal Society and published in the *Philosophical Transactions* in 1711. In fact it was Francis Robartes, who later became the Earl of Radnor, who suggested to de Moivre that he present a broader picture of the principles of probability theory than those which had been presented by Montmort in *Essay d'analyse sur les jeux de hazard* (1708). Clearly this work by Montmort and that by Huygens which de Moivre had read while at Saumur, contained the problems which de Moivre attacked in his work and this led Montmort to enter into a dispute with de Moivre concerning originality and priority. Unlike the Newton-Leibniz dispute which de Moivre had judged, the argument with Montmort appears to have been settled amicably. The definition of statistical independence appears in this book together with many problems with dice and other games.

In fact *The Doctrine of Chance* appeared in new expanded editions in 1718, 1738 and 1756. For example in [5] Dupont looks at the "jeu de rencontre" first put forward by Montmort and generalised by de Moivre in Problems XXXIV and XXXV of the 1738 edition. Problem XXXIV reads as follows:-

Any number of letters a, b, c, d, e, f, etc., all of them different, being taken promiscuously as it happens: to find the probability that some of them shall be found in their places according to the rank they obtain in the alphabet; and that others of them shall at the same time be displaced.

Problem XXXV generalises Problem XXXIV by allowing each of the letters *a*, *b*, *c*, ... to be repeated a certain number of times. The "gamblers' ruin" problem appears as Problem LXV in the 1756 edition. Dupont looks at this problem, and Todhunter's solution, in [6]. In fact in *A history of the mathematical theory of probability* (London, 1865), Todhunter says that probability:-

... owes more to[de Moivre]than any other mathematician, with the single exception of Laplace.

The 1756 edition of *The Doctrine of Chance* contained what is probably de Moivre's most significant contribution to this area, namely the approximation to the binomial distribution by the normal distribution in the case of a large number of trials. De Moivre first published this result in a Latin pamphlet dated 13 November 1733 (see [4] for an interesting discussion) aiming to improve on Jacob Bernoulli's law of large numbers. The work contains [1]:-

... the first occurrence of the normal probability integral. He even appears to have perceived, although he did not name, the parameter now called the standard deviation ...

De Moivre also investigated mortality statistics and the foundation of the theory of annuities. An innovative piece of work by Halley had been the production of mortality tables, based on five years of data, for the city of Breslau which he published in 1693. It was one of the earliest works to relate mortality and age in a population and was highly influential in the production of actuarial tables in life insurance. It is almost certain that de Moivre's friendship with Halley led to his interest in annuities and he published *Annuities on lives* in 1724. Later editions appeared in 1743, 1750, 1752 and 1756. His contribution, based mostly on Halley's data, is important because of his [1]:-

... derivation of formulas for annuities based on a postulated law of mortality and constant rates of interest on money. Here one finds the treatment of joint annuities on several lives, the inheritance of annuities, problems about the fair division of the costs of a tontine, and other contracts in which both age and interest on capital are relevant.

In *Miscellanea Analytica* (1730) appears Stirling's formula (wrongly attributed to Stirling) which de Moivre used in 1733 to derive the normal curve as an approximation to the binomial. In the second edition of the book in 1738 de Moivre gives credit to Stirling for an improvement to the formula. De Moivre wrote:-

I desisted in proceeding farther till my worthy and learned friend Mr James Stirling, who had applied after me to that inquiry,[discovered that c = √(2 π)].

De Moivre is also remembered for his formula for

(cos

x+isinx)^{n}

which took trigonometry into analysis, and was important in the early development of the theory of complex numbers. It appears in this form in a paper which de Moivre published in 1722, but a closely related formula had appeared in an earlier paper which de Moivre published in 1707.

Despite de Moivre's scientific eminence his main income was as a private tutor of mathematics and he died in poverty. Desperate to get a chair in Cambridge he begged Johann Bernoulli to persuade Leibniz to write supporting him. He did so in 1710 explaining to Leibniz that de Moivre was living a miserable life of poverty. Indeed Leibniz had met de Moivre when he had been in London in 1673 and tried to obtain a professorship for de Moivre in Germany, but with no success. Even his influential English friends like Newton and Halley could not help him obtain a university post. De Moivre [3]:-

... was the intimate friend of Newton, who used to fetch him each evening, for philosophical discourse at his own house, from the coffee-house(probably Slaughter's), where he spent most of his time.

Indeed de Moivre revised the Latin translation of Newton's *Optics* and dedicated *The Doctrine of Chance* to him. Newton returned the compliment by saying to those who questioned him on the *Principia* [1]:-

Go to Mr De Moivre; he knows these things better than I do.

Clerke writes of his character in [3]:-

He was unmarried, and spent his closing years in peaceful study. Literature, ancient and modern, furnished his recreation; he once said that he would rather have been Molière than Newton; and he knew his works and those of Rabelais almost by heart. He continued all his life a steadfast Christian. After sight and hearing had successively failed, he was still capable of rapturous delight at his election as a foreign associate of the Paris Academy of Sciences on27June1754.

De Moivre, like Cardan, is famed for predicting the day of his own death. He found that he was sleeping 15 minutes longer each night and summing the arithmetic progression, calculated that he would die on the day that he slept for 24 hours. He was right!

**Article by:** *J J O'Connor* and *E F Robertson*

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