# Moivre, Abraham De

# MOIVRE, ABRAHAM DE

(*b*. Vitry-le-François, France, 26 May 1667; *d*. London, England, 27 November 1754)

*probability*.

De Moivre was one of the many gifted Protestants who emigrated from France to England following the revocation of the Edict of Nantes in 1685. His formal education was French, but his contributions were made within the Royal Society of London. His father, a provincial surgeon of modest means, assured him of a competent but undistinguished classical education. It began at the tolerant Catholic village school and continued at the Protestant Academy at Sedan. After the latter was suppressed for its profession of faith, De Moivre had to study at Saumur. It is said that he read mathematics on the side, almost in secret, and that Christiaan Huygens’ work on the mathematics of games of chance, *De ratiociniis in ludo aleae* (Leiden, 1657), formed part of this clandestine study. He received no thorough instruction in mathematics until he went to Paris in 1684 to read the later books of Euclid and other texts under the supervision of Jacques Ozanam.

His Protestant biographers say that De Moivre, like so many of his coreligionists, was imprisoned during the religious tumult of 1685 and not released until 1688. Other, nearly contemporary sources report him in England by 1686. There he took up his lifelong, unprofitable occupation as a tutor in mathematics. On arrival in London, De Moivre knew many of the classic texts, but a chance encounter with Newton’s *Principia* showed him how much he had to learn. He mastered the book quickly; later he told how he cut out the huge pages and read them while walking from pupil to pupil. Edmond Halley, then assistant secretary of the Royal Society, was sufficiently impressed to take him up after meeting him in 1692; it was he who communicated De Moivre’s first paper, on Newton’s doctrine of fluxions, to the Royal Society in 1695 and saw to his election by 1697. (In 1735 De Moivre was elected fellow of the Berlin Academy of Sciences, but not until 1754 did the Paris Academy follow suit.)

Once Halley had made him known, De Moivre’s talents became esteemed. He was able to dedicate his first book, *The Doctrine of Chances*, to Newton; and the aging Newton would, it is said, turn students away with “Go to Mr. De Moivre; he knows these things better than I do.” He was admired in the verse of Alexander Pope (“Essay on Man” II, 104) and was appointed to the grand commission of 1710, by means of which the Royal Society sought to settle the Leibniz-Newton dispute over the origin of the calculus. Yet throughout his life De Moivre had to eke out a living as tutor, author, and expert on practical applications of probability in gambling and annuities. Despite his powerful friends he found little patronage. He canvassed support in England and even begged Johann I Bernoulli to get Leibniz to intercede on his behalf for a chair of mathematics at Cambridge, but to no avail. He was left complaining of the waste of his time spent walking between the homes of his pupils. At the age of eighty-seven De Moivre succumbed to lethargy. He was sleeping twenty hours a day, and it became a joke that he slept a quarter of an hour more every day and would die when he slept the whole day through.

De Moivre’s masterpiece is *The Doctrine of Chances*. A Latin version appeared as “De mensura sortis” in *Philosophical Transactions of the Royal Society* (1711). Successively expanded versions under the English title were published in 1718, 1738, and 1756. The only systematic treatises on probability printed before 1711 were Huygons’ *De ratiociniis in ludo aleae* and Pierre Rémond de Montmort’s *Essay d’analyse sur les jeux de hazard* (Paris, 1708). Problems which had been posed in these two books prompted De Moivre’s earliest work and, incidentally, caused a feud between Montmort and De Moivre on the subject of originality and priority.

The most memorable of De Moivre’s discoveries emerged only slowly. This is his approximation to the binomial probability distribution, which, as the normal or Gaussian distribution, became the most fruitful single instrument of discovery used in probability
theory and statistics for the next two centuries. In De Moivre’s own time his discovery enormously clarified the concept of probability. At least since the fifteenth century there had been substantial work on games of chance that recognized the existence of stable frequencies in nature. But in the classic work of Huygens and even in that of Montmort, the reader was usually given, in the context of a game or lottery, a set of events of equal probability—a set of what were often called “chances”—and he was asked to derive further probabilities or expectations from this fundamental set. No one had a clear mathematical formulation of how “chances” and stable frequencies are related. Jakob I Bernoulli provided a first answer in part IV of his *Ars conjectandi* (Basel, 1713), where he proved what is now called the weak law of large numbers; De Moivre’s approximation to the binomial distribution was conceived as an attempt to improve on Bernoulli.

In some experiment, let the ratio of favorable to unfavorable “chances” be *p*. In *n* repeated trials of the experiment, let *m* be the number of successes. Consider any interval around *p*, bounded by two limits. Bernoulli proved that the probability that *m*/*n* should lie between these limits increases with increasing *n* and approaches l as *n* grows without bound. But although he could establish the fact of convergence, Bernoulli could not tell at what rate the probability converges. He did obtain some idea of this rate by computing numerical examples for particular values of *n* and *p*, but he was unable to state the principles that underlie his discovery. That was left for De Moivre.

De Moivre’s solution was published as a Latin pamphlet dated 13 November 1733. Introducing his translation of, and comments on, this work at the end of the last edition of *The Doctrine of Chances*, he took “the liberty to say, that this is the hardest Problem that can be proposed on the Subject of Chance” (p. 242). In this problem the probability of getting exactly *m* successes in *n* trials is expressed by the *m* th term in the expansion of (*a* + *b*)^{n}—that is, *a ^{m}b^{n}*

^{−}

*, where*

^{m}*a*is the given ratio of chances and

*b*= 1 −

*a*. Hence the probability of obtaining a proportion of successes lying between the two limits is a problem in “approximating the Sum of the Terms of the Binomial (

*a*+

*b*)

^{n}expanded into a Series” (p. 243).

Working first with the binomial expansion of (1 + 1)^{n}, De Moivre obtained what is now recognized as n! approximated by Stirling’s formula—that is, *cn*^{n+1/2} e^{−n}. He knew the constant *c* only as the limiting sum of an infinite series: “I desisted in proceeding farther till my worthy and learned Friend Mr. James Stirling, who had applied after me to that inquiry,” discovered that (p. 244). Hence what is now called Stirling’s formula is at least as much the work of De Moivre as of Stirling.

With his approximation of *n!* De Moivre was able, for example, to sum the terms of the binomial from any point up to the central term. This summation is equivalent to the modern normal approximation and is, indeed, 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 σ. It was left for Laplace and Gauss to construct the equation of the normal curve in its form

but De Moivre obtained, in a series of examples, expressions that are logically equivalent to this. He understood the rate of the convergence that Bernoulli had discovered and saw that the “error”—that is, the likely difference of the observed frequency from the true ratio of “chances”—decreases in inverse proportion to the square of the number of trials.

De Moivre’s approximation is a theorem in probability theory: given the initial law about the distribution of chances, he could approximate the probability that observed frequencies should lie within any two assigned limits. Unlike some later workers, he did not imagine that his result would solve the converse statistical problem—namely, given the observed frequencies, to approximate the probability that the initial law about the ratio of chances lies within any two limits. But he did think his theorem bore on statistics. After summarizing his theorem, he reasoned:

Conversely, if from numberless Observations we find the Ratio of the Events to converge to a determinate quantity, as to the Ratio of P to Q; then we conclude that this Ratio expresses the determinate Law according to which the Event is to happen. For let that Law be expressed not by the ratio P : Q, but by some other, as R : S; then would the Ratio of the Events converge to this last, and not to the former: which contradicts ourHypothesis[p. 251].

Nowhere in *The Doctrine of Chances* is this converse reasoning put to any serious mathematical use, yet its conceptual value is great. For De Moivre, it seemed to resolve the philosophical paradox of finding regularities within events postulated to be random. As he expressed it in the third edition, “altho’ Chance
produces Irregularities, still the Odds will be infinitely great, that in process of Time, those Irregularities will bear no proportion to the recurreney of that Order which naturally results from ORIGINAL DESIGN” (p. 251).

All the mathematical problems treated by De Moivre before setting out his approximation to the binomial distribution are closely related to earlier work by Huygens and Montmort. They include the first intimation of another approximation to the binomial distribution, now usually named for Poisson. In the normal approximation, the given ratio of chances is constant at *p*; and as *n* increases, so does up. In the Poisson approximation, *np* is constant, so that as *n* grows, *p* tends to zero. It is useful in studying the probabilities of rather infrequent events. Although De Moivre worked out a particular case of the Poisson approximation, he does not appear to have guessed its subsequent uses in probability theory.

Also included in *The Doctrine of Chances* are great advances in problems concerning the duration of play; a clearer formulation of combinatorial problems about chances; the use of difference equations and their solutions using recurring series; and, as illustrated by the work on the normal approximation, the use of generating functions, which, by the time of Laplace, came to play a fundamental role in probability mathematics.

Although no statistics are found in *The Doctrine of Chances*, De Moivre did have a great interest in the analysis of mortality statistics and the foundation of the theory of annuities. Perhaps this originated from his friendship with Halley, who in 1693 had written on annuities for the Royal Society, partly in protest at the inane life annuities still being sold by the British government, in which the age of the annuitant was not considered relevant. Halley had very meager mortality data from which to work; but his article, together with the earlier “political arithmetic” of John Graunt and William Petty, prompted the keeping of more accurate and more relevant records. By 1724, when De Moivre published the first edition of *Annuities on Lives*, he could base his computations on many more facts. Even so, he found it convenient to base most of his computations on Halley’s data, derived from only five years of observation in the city of Breslau; he claimed that other results confirmed the substantial accuracy of those data. In his tables De Moivre found it convenient to suppose that the death rate is uniform after the age of twelve. He did not pretend that the rate is absolutely uniform, as a matter of objective fact, but argued for uniformity partly because of its mathematical simplicity and partly because the mortality records were still so erratically collected that precise curve fitting was unwarranted.

De Moivre’s contribution to annuities lies not in his evaluation of the demographic facts then known but in his 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. This mathematics became a standard part of all subsequent commercial applications in England. Yet the authorship of this work was a matter of controversy. De Moivre’s first edition appeared in 1725; in 1742 Thomas Simpson published *The Doctrine of Annuities and Reversions Deduced From General and Evident Principles*. De Moivre republished in the next year, bitter at what, with some justice, he claimed to be the plagiarization of his work. Since the sale of his books was a real part of his small income, money must have played as great a part as pride in this dispute.

Throughout his life De Moivre published occasional papers on other branches of mathematics. Most of them offered solutions to fairly ephemeral problems in Newton’s calculus; in his youth some of this work led him into yet another imbroglio about authorship, involving some minor figures from Scotland, especially George Cheyne. In these lesser works, however, there is one trigonometric equation the discovery of which is sufficiently undisputed that it is still often called De Moivre’s theorem:

(cos φ + *i* sin φ)^{n} = cos *n*φ + i sin *n*φ

This result was first stated in 1722 but had been anticipated by a related formula in 1707. It entails or suggests a great many valuable identities and thus became one of the most useful steps in the early development of complex number theory.

## BIBLIOGRAPHY

I. Original Works. De Moivre’s two books are *The Doctrine of Chances* (London, 1718; 2nd ed., 1738; 3rd ed., 1756; photo, repr. of 2nd ed., London, 1967; photo. repr. of 3rd ed., together with the biography by Helen M. Walker, New York, 1967); and *A Treatise of Annuities on Lives* (London, 1725), repr. in the 3rd ed. of *The Doctrine of Chances*. Mathematical papers are in *Philosophical Transactions of the Royal Society* between 1695 and 1744 (nos. 216, 230, 240, 265, 278, 309, 329, 341, 345, 352, 360, 373, 374, 451, 473). “De mensura sortis” is no. 329; the trigonometric equation called De Moivre’s formula is in 373 and is anticipated in 309. *Approximatio ad summam**terminorum binomii* (*a* + *b*)^{n}*in seriem expansi* is reprinted by R. C. Archibald, “A Rare Pamphlet of De Moivre and Some of His Discoveries,” in *Isis*, **8** (1926), 671–684. Correspondence with Johann I Bernoulli is published in K. Wollenshläger, “Der mathematische Briefwechsel zwischen Johann I Bernoulli und Abraham de Moivre,” in *Verhandlungen der Naturforschenden Gesellschaft in Basel*, **43** (1933), 151–317. I. Schneider (below) lists all known publications and correspondence of De Moivre.

II. Secondary Literature. Ivo Schneider, “Der Mathematiker Abraham de Moivre,” in *Archive for History of Exact Sciences*, **5** (1968–1969), 177–317, is the definitive study of De Moivre’s life and work. For other biography, see Helen M. Walker, “Abraham de Moivre,” in *Scripta mathematica*, **2** (1934), 316–333, reprinted in 1967 (see above), and Mathew Maty, *Mémoire sur la vie et sur les écrits de Mr. Abraham de Moivre* (The Hague, 1760).

For other surveys of the work on probability, see Isaac Todhunter, *A History of Probability From the Time of Pascal to That of Laplace* (London, 1865; photo. repr. New York, 1949), 135–193; and F. N. David, *Gods, Games and Gambling* (London, 1962), 161–180, 254–267.

Ian Hacking

# Moivre, Abraham de

# Moivre, Abraham de

Abraham de Moivre (1667–1754) was born of French Protestant parents named Moivre. (He was also known as Demoivre; as part of the return address of a letter to Johann Bernoulli he himself wrote his name as deMoivre.) He studied mathematics and physics in Paris under Ozanam, and emigrated to England when he was 21 to escape religious persecution (Walker 1934). Although de Moivre was a mathematical genius of outstanding analytical power and was in contact by correspondence and in person (at the Royal Society) with many of the leading mathematicians of the day, he never succeeded in obtaining a university appointment. Instead, he had to live by tutoring noblemen’s sons and by advising gamblers and speculators who dealt in annuities, which were a popular form of investment in the first half of the eighteenth century (Walford 1871). This mis-fortune for de Moivre is posterity’s gain, for the problems he met in his consulting practice and his successful solution of them provided the material for his two great textbooks. In fact, during his last years de Moivre must have relied heavily on the sales of the later editions of his book on annuity calculations.

De Moivre’s practical text on probability first appeared in 1718 as a translation and revision of his Latin article of 1711. It was dedicated to Isaac Newton, who is accorded the author’s thanks for his writings and conversations. In its final form, published in 1756, this book is notable for its original treatment of the following topics, all of which play a central role in the modern theory of probability:

- The general laws of addition (David & Barton 1962, chapter 2) and multiplication of probabilities (Montucla [1758] 1802, part 5, book 1, chapter 39);
- The binomial distribution law (Cantor 1898, chapter 96);
- Probability-generating functions (Seal 1949a);
- Difference equations involving probabilities and their solution by means of recurring series (Czuber 1900);
- New and general solutions of problems on the duration of play, or “gambler’s ruin” (Todhunter 1865, chapter 9);
- The limiting form of the binomial term

when (a) *n→* ∞ with *np* remaining finite, and (b) *n→* ∞ and *np* → ∞. In case (a) only the term with *x =* 0 was considered (David 1962). In case *(b)* the result

{2л*np* (1-*p*)}^{=1/2} exp[x - *np*^{2}/2 *np*(1-*p*)],

namely, the ordinate of the normal distribution, was obtained explicitly (in different notation). Included in this book were the trigonometrical theorem that goes by de Moivre’s name and his approximation to the logarithm of a factorial which was improved by Stirling’s discovery in the same year that the value of the series contained therein was 27л.

Some of the mathematical derivations of this probability text were published for a wider circle of readers in Latin (1730).

De Moivre’s other textbook laid the foundations of the mathematics of life contingencies (Saar 1923). Although the first edition sold slowly, Thomas Simpson’s plagiaristic text of 1742 spurred de Moivre to a complete revision published in the following year (Young 1908). The success of this edition is indicated by the two further editions, with minor changes, that followed within nine years. In 1756 a final, thoroughly revised edition was printed as the last section of the third edition of *The Doctrine of Chances.* In an appendix, reference is made to a paper published in 1755 by James Dodson, the father of scientific life insurance (Ogborn 1962) and possibly the “friend” who edited the posthumous edition.

The originality of this life contingency textbook is attested by its inclusion of the following:

*(a)* The recursion formula for calculating a life annuity at age *x*, given that at age *x* + 1 (though it is doubtful whether the author envisaged the calculation of the whole set of annuity values by starting at the “oldest age” [Young 1908]);

*(b)* General relations for survivorship and reversionary annuities in terms of single and joint life annuities;

*(c)* Use of the calculus to obtain the value of a continuous annuity-certain;

*(d)* A law of mortality, namely, that of uniform decrements in the number of survivors, which was in substantial agreement with the Breslau table, published by his friend Halley in 1693; and as a result:

*(e)* Easily computable values of single and joint life annuities for limited terms or for life;

*(f)* Expressions for the computation of complex survivorship probabilities;

*(g)* The value of a life annuity with a proportionate payment in the year of death.

All these results originated with de Moivre himself.

The earliest published works of de Moivre in the 1690s were influenced by Newton’s method of fluxions and theory of series (Cantor 1898, chapter 86), and his interest in probability dates only from the first edition of Montmort’s *Essay* (1708). Perhaps his most important contribution, first printed in 1733 as a supplement to the *Miscellanea analytica*, was his improvement of the wide limits obtained by Bernoulli (1713) in his statement of the law of large numbers. For this purpose de Moivre utilized the result mentioned in (6) to obtain the sum of the binomial probabilities from

x = *np*-*l* tox = *np*+*l* with and *l* = *k* (*n*/2)^{1/2},

*k =* 1,2,3, by approximate quadrature (by ordinate summation and the three-eighths rule) of the normal ordinates. While this constitutes the first tabulation of the normal areas at one, two, and three standard deviations from the mean, there is no evidence that de Moivre thought in terms of a continuous probability distribution (Seal 1954; 1957). Nevertheless, this work is clearly the basis for the subsequent demonstration by Laplace (1812, pp. 275-284) that the binomial tends to the normal when *n* is large (Pearson 1924).

It may be added that de Moivre was a pure mathematician little interested in the practical applications of his theory. Although he wrote on life contingencies, only in the final Appendix of the posthumous edition of 1756 is there a brief reference to mortality data later than those of the Breslau table. Actually, the early and middle years of the eighteenth century saw the publication of several collections of mortality statistics that would have been fertile ground for the application of de Moivre’s improved version of Bernoulli’s theorem (Seal 1949 *b*). These data had led to a widespread belief in the divine regularity of demographic ratios, and a few paragraphs in the 1738 and 1756 editions of the *Doctrine* refer to a connection between this belief and Bernoulli’s theorem. Unfortunately, the topic was not pursued by de Moivre or his contemporaries (Westergaard 1932, chapters 7, 10) and cannot be regarded as indicating that de Moivre was interested in theology (Walker 1929) or that he influenced the demographers of the eighteenth and early nineteenth centuries (Pearson 1926).

Hilary L. Seal

*[For the historical context of de Moivre’s work, see the article on the*Bernoulli family. *For discussion of the subsequent development of de Moivre’s ideas, see*Distributions, statistical; Life tables; Probability; *and the biography of*Laplace.]

## WORKS BY DE MOIVRE

*In many reference works, de Moivre is alphabetized under* D;*however, in line with the cataloguing practice of major libraries, we have listed him under* M. *Consistent with this, we have used a lower-case “d” for the particle.*

l711 De mensura sortis seu, de probabilitate eventuum in ludis a casu fortuito pendentibus. Royal Society of London, *Philosophical Transactions* 27:213-264. → Reprinted by Kraus (New York) in 1963.

(1718) 1756 *The Doctrine of Chances: Or, a Method of Calculating the Probabilities of Events in Play.* 3d ed. London: Millar.

(1725) 1752 *Annuities Upon Lives: Or, the Valuation of Annuities Upon Any Number of Lives, as Also, of Reversions.* 4th ed. London: Millar. → The Appendix concerns the expectations of life and the probabilities of survivorship.

1730 *Miscellanea analytica de seriebus et quadraturis. . . .* London: Tonson & Watts.

## SUPPLEMENTARY BIBLIOGRAPHY

Bernoulli, Jakob (1713) 1899 *Wahrscheinlichkeitsrechnung: (Ars conjectandi).* 2 vols. Leipzig: Engel-mann. → First published posthumously in Latin.

Cantor, Moritz 1898 *Vorlesungen über Geschichte der Mathematik.* Volume 3: Von 1668-1758. Leipzig: Teubner.

Czuber, Emanuel (1900) 1906 Calcul des probabilités. Section 1, volume 4, pages 1-46 in *Encyclopedic des sciences mathematiques.* Paris: Gauthier-Villars. → First published in German in *Encyklopäddie der mathématischen Wissenschaften.*

David, F. N. 1962 *Games, Gods and Gambling: The Origins and History of Probability and Statistical Ideas From the Earliest Times to the Newtonian Era.* New York: Hafner.

David, F. N.; and Barton, D. E. 1962 *Combinatorial Chance.* New York: Hafner.

Laplace, Pierre simon de (1812) 1820 *Theorie analytique des probabilités.* 3d ed., rev. Paris: Courcier.

[Montmort, Pierre RÉmond de] (1708) 1713 *Essay d’analyse sur les jeux de hazard.* 2d ed. Paris: Quillau. → First published anonymously.

Montucla, Jean É. (1758) 1802 *Histoire des mathematiques dans laquelle on rend compte de leurs progres depuis leur origine jusqu’d nos jours. . . .* Paris: Agasse.

Ogborn, Maurice Edward 1962 *Equitable Assurances: The Story of Life Assurance in the Experience of the Equitable Life Assurance Society, 1762-1962.* London: Allen & Unwin.

Pearson, Karl 1924 Historical Note on the Origin of the Normal Curve of Errors. *Biometrika* 16:402-404.

Pearson, Karl 1926 Abraham de Moivre. *Nature* 117: 551-552.

Saar, J. DU 1923 De beteekenis van De Moivre’s werk over lijfrenten voor de ontwikkeling van de verzekeringswetenschap. *Verzekerings archief* 4:28-45.

Seal, H. L. 1949 *a* The Historical Development of the Use of Generating Functions in Probability Theory-Vereinigung schweizerischer Versicherungsmathematiker, *Mitteilungen* 49:209-228.

Seal, H. L. 1949 *b* Mortality Data and the Binomial Probability Law. *Skandinavisk aktuarietidskrift* 32: 188-216.

Seal, Hilary L. 1954 A Budget of Paradoxes. *Journal of the Institute of Actuaries Students’ Society* 13: 60-65.

Seal, Hilary L. 1957 A Correction. *Journal of the Institute of Actuaries Students’ Society* 14:210-211.

Simpson, Thomas (1742) 1775 *The Doctrine of Annui-ties and Reversions, Deduced From General and Evident Principles.* . . . 2d ed. London: Printed for J. Nourse.

Todhunter, ISAAC (1865) 1949 A *History of the Mathematical Theory of Probability From the Time of Pascal to That of Laplace.* New York: Chelsea.

Walford, Cornelius 1871 *The Insurance Cyclopaedia.* Volume 1. London: Lay ton. → See especially pages 98-169 on “Annuities.”

Walker, Helen M. 1929 *Studies in the History of Statistical Method: With Special Reference to Certain Educational Problems.* Baltimore: Williams & Wilkins.

Walker, Helen M. 1934 Abraham de Moivre. *Scripta mathematica* 2:316-333.

Westergaard, Harald L. 1932 *Contributions to the History of Statistics.* London: King.

Young, T. E. 1908 Historical Notes Relating to the Discovery of the Formula *a _{x} = vp^{x}* (1 + a

_{x}+1): And to the Introduction of the Calculus in the Solution of Actuarial Problems.

*Journal of the Institute of Actuaries*42:188-205.

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