Abraham de Moivre

How to use this course

Abraham de Moivre is remembered by students for one formula, \((\cos x+i\sin x)^n=\cos nx+i\sin nx\), and by statisticians for something larger: he was the first to see the normal curve emerge from the binomial distribution. He also wrote the standard English textbook of probability for half a century and founded the mathematics of life annuities. He did all of this without a university post, as a French Protestant exile who earned his living as a private tutor.

Each technical lesson begins with a paragraph for the general reader, then gives precise statements in modern notation, derivations, and exercises with folded solutions. De Moivre wrote in Latin and in English prose with little symbolism; where his notation differs from ours, the lesson says so. The order follows his career: algebra and series (1695–1707), probability (1711–1738), trigonometry and analysis (1722–1730), the 1733 limit theorem, and annuities (1725 onward).

Timeline

Champagne, Sedan, Saumur, 1667–1684

Abraham Moivre was born on 26 May 1667 in Vitry-le-François, a small town on the Marne in Champagne. His father Daniel was a surgeon of modest but steady means, and the family was Protestant. The particle “de” appears only later, after he reached England. His first schooling was with the Catholic Christian Brothers in Vitry, a sign of how mixed provincial life still was before the persecutions of the 1680s.

At about eleven he was sent to the Protestant Academy at Sedan, one of the colleges the Huguenots had built under the protection of the Edict of Nantes. He studied Greek there for four years with Jacques du Rondel. In 1682 the crown suppressed the academy, part of a steady campaign against Protestant institutions, and he moved to the academy at Saumur on the Loire to study logic.

A book on dice

Mathematics was not part of his formal course. At Saumur he read on his own the Éléments des mathématiques of Jean Prestet and, more importantly, the short treatise De ratiociniis in ludo aleae (1657) by Christiaan Huygens, the first printed book on probability. Huygens computed fair shares in interrupted games and the odds of dice throws by the idea of expectation. The young reader later turned that slim tract into the largest book on chance of the century.

Paris and the Revocation, 1684–1688

In 1684 Moivre went to Paris. Accounts differ on his formal enrolment; MacTutor places him at the Collège d’Harcourt for physics. What is clear is that he received his first real training in mathematics from Jacques Ozanam, a popular teacher and author of textbooks and books of mathematical recreations. Ozanam made his living by private lessons, the same trade his pupil would practise for the rest of his life.

In October 1685 Louis XIV revoked the Edict of Nantes by the Edict of Fontainebleau. Protestant worship became illegal and Protestant children could be taken for Catholic instruction. Moivre was confined in the priory of Saint-Martin-des-Champs in Paris, which then served as a school for the reconversion of young Protestants. The sources disagree on how long he stayed and on what terms he left: the priory’s records place his departure in 1688, while the register of the French Savoy Church in London records that he and his brother were admitted on 28 August 1687.

Whatever the exact sequence, by his early twenties he was in London, one of perhaps two hundred thousand Huguenots who left France in these years. He never returned. The Paris Academy elected him a foreign associate sixty-seven years later, five months before his death.

A tutor in London, 1688–1710

In London de Moivre lived by teaching mathematics to the sons of the gentry, walking from house to house and meeting pupils in coffee houses. A well-known story says that at the house of the Earl of Devonshire he first saw Isaac Newton’s Principia (1687), bought a copy, and tore it into sheets that he carried in his pocket and read between lessons.

By about 1692 he had come to know Edmond Halley, then assistant secretary of the Royal Society, and through him Newton. In March 1695 Halley presented de Moivre’s first paper, on Newton’s method of fluxions. Two years later came the multinomial theorem (Lesson 04), and on 30 November 1697 he was elected a Fellow of the Royal Society. He was thirty.

Newton’s circle

De Moivre became one of the small group around Newton. He corrected the Latin translation of the Opticks, published in 1706. In 1704 he wrote a sharp Latin critique of George Cheyne’s treatise on the inverse method of fluxions, defending Newton’s methods against a careless popularizer. Newton’s regard for him is attested by a story, possibly embellished, that he would send questioners about the Principia to de Moivre as the man who knew it better than he did.

Friendship with the powerful did not produce a post. He became a naturalized English subject in 1705, but he hoped in vain for a university chair. In 1710 Johann Bernoulli, at his request, wrote to Gottfried Wilhelm Leibniz asking for help toward a professorship. Nothing came of it. Foreign birth counted against him at Cambridge and Oxford, and he remained a tutor for life.

The multinomial theorem

For the general reader. Newton had shown how to expand \((a+b)^n\). De Moivre showed how to expand a sum of any number of terms raised to a power, and even an infinite series raised to a power. The coefficients count arrangements, and the result is the first general tool of combinatorial algebra.

Proof

Expanding the product of \(n\) factors, each term arises by choosing one \(x_j\) from each factor. The monomial \(x_1^{k_1}\cdots x_m^{k_m}\) arises once for each word of length \(n\) containing \(k_1\) letters \(x_1\), …, \(k_m\) letters \(x_m\). There are \(n!\) orderings of \(n\) labelled letters, and permuting equal letters among themselves gives the same word, so the number of words is \(n!/(k_1!\cdots k_m!)\).

Powers of an infinite series

De Moivre’s 1697 paper was titled A method of raising an infinite multinomial to any given power, or extracting any given root of the same. In modern terms, if \(f(z)=a_0+a_1z+a_2z^2+\cdots\) then the coefficient of \(z^N\) in \(f(z)^n\) is

So the coefficient is a sum over partitions of \(N\) into at most \(n\) parts. This is the bookkeeping behind every later computation of the chance of throwing a given total with several dice.

Powers and angles: de Moivre’s formula

For the general reader. Multiplying an angle by \(n\) and raising a quantity to the \(n\)-th power turn out to be the same operation in disguise. De Moivre used this to solve certain equations of high degree with trigonometric tables. The modern statement, with \(i=\sqrt{-1}\), packs the idea into one line.

The 1707 equations

In 1707 de Moivre considered the equations of odd degree that express \(\cos n\phi\) as a polynomial in \(\cos\phi\). In modern notation these are \(T_n(x)=y\), where \(T_n\) is the Chebyshev polynomial defined by \(T_n(\cos\phi)=\cos n\phi\); for instance \(T_3(x)=4x^3-3x\). He gave the root in the form

a generalization of Cardano’s formula for the cubic. When \(|y|\le1\) the quantities under the \(n\)-th roots are complex, and the root is real only because of the identity below. In De sectione anguli (1722) he made the link with angle multiplication explicit, though always in terms of cosines and versed sines, never with \(i\) written as we do.

Proof. For \(n\ge0\) use induction and the addition formulas: \((\cos x+i\sin x)(\cos nx+i\sin nx)=\cos(n+1)x+i\sin(n+1)x\). For negative \(n\) note that \((\cos x+i\sin x)^{-1}=\cos x-i\sin x=\cos(-x)+i\sin(-x)\). Leonhard Euler stated the formula in this form and extended it to real exponents, and with \(e^{ix}=\cos x+i\sin x\) it reduces to \((e^{ix})^n=e^{inx}\).

Roots and factorization

The \(n\) solutions of \(z^n=\cos\theta+i\sin\theta\) are \(z_k=\cos\frac{\theta+2\pi k}{n}+i\sin\frac{\theta+2\pi k}{n}\), \(k=0,\dots,n-1\). Roger Cotes had found the factorization of \(x^{2n}\pm1\) into real quadratic factors. De Moivre, in Miscellanea analytica (1730), gave the general version:

Proof. The left side equals \((x^n-e^{in\theta})(x^n-e^{-in\theta})\). The roots of \(x^n=e^{in\theta}\) are \(e^{i(\theta+2\pi k/n)}\), those of \(x^n=e^{-in\theta}\) their conjugates. Pairing each root with its conjugate gives \((x-e^{i\alpha})(x-e^{-i\alpha})=x^2-2x\cos\alpha+1\).

The Doctrine of Chances

For the general reader. Gamblers wanted to know what a bet was worth. De Moivre turned their questions into a systematic theory, with rules for combining probabilities, a clear definition of independent events, and general methods in place of clever tricks. His book was the probability textbook of the English-speaking world for most of the eighteenth century.

The book

De Moivre’s Latin memoir De mensura sortis (“On the measurement of chance”) appeared in the Philosophical Transactions for 1711. The English book The Doctrine of Chances: or, a Method of Calculating the Probabilities of Events in Play followed in 1718, dedicated to Newton, with enlarged editions in 1738 and 1756. Its introduction states the multiplication rule for independent events in words: the probability that several independent events all happen is the product of their probabilities. It treats the problem of points, the dice and card games of the day, the matching problem (jeu de rencontre), and the duration of play.

The gambler’s ruin

Two players A and B play repeated independent rounds. A wins a round with probability \(p\), B with \(q=1-p\), and the loser pays the winner one counter. A starts with \(a\) counters, B with \(b\).

De Moivre’s argument. Give A’s counters the values \(q/p,\ (q/p)^2,\dots,(q/p)^a\) and B’s the values \((q/p)^{a+1},\dots,(q/p)^{a+b}\), stacked so that the counter that changes hands is always the one at the boundary. A round then becomes a fair game in value: A wins the counter worth \(r^{x+1}\) with probability \(p\) or loses the one worth \(r^{x}\) with probability \(q\), where \(r=q/p\) and \(x\) is A’s current number of counters, and \(p\,r^{x+1}=q\,r^{x}\). Since every round is fair, A’s expected final wealth equals his initial wealth. Equating \(P_A\sum_{j=1}^{a+b}r^j\) with \(\sum_{j=1}^{a}r^j\) and summing the geometric series gives the formula. In modern terms he had found the martingale \(r^{X_t}\).

How many trials for an even chance?

An event has probability \(p\) per trial. The chance it occurs at least once in \(n\) trials is \(1-(1-p)^n\). Setting this equal to \(\tfrac12\) gives \(n=\log2/(-\log(1-p))\). In De mensura sortis de Moivre observed that for small \(p\) this is close to \(n\approx(\log 2)/p\approx0.7/p\). The same computation, read the other way, gives the Poisson approximation \((1-p)^n\approx e^{-np}\); for this reason some historians argue that the Poisson distribution first appears in his work.

Recurring series and generating functions

For the general reader. Many probability problems lead to sequences where each term is built from the few before it. De Moivre called these recurring series. He showed that the whole infinite sequence can be packed into a single fraction, and that unpacking the fraction gives a closed formula for every term. This is the method of generating functions.

De Moivre called the coefficients of the recurrence the scale of relation. Suppose a sequence satisfies

Proof. Multiply \(U(z)\) by \(Q(z)=1-c_1z-\cdots-c_kz^k\). For \(n\ge k\) the coefficient of \(z^n\) in \(Q(z)U(z)\) is \(u_n-c_1u_{n-1}-\cdots-c_ku_{n-k}=0\), so \(QU\) is a polynomial of degree less than \(k\). Conversely, \(P/Q\) has a power series whose coefficients satisfy the same relation for the same reason.

If \(Q(z)=\prod_j(1-\alpha_jz)\) with distinct \(\alpha_j\), partial fractions give \(U(z)=\sum_j\frac{A_j}{1-\alpha_jz}\), hence \(u_n=\sum_jA_j\alpha_j^n\). De Moivre used this to solve the duration-of-play problem, in which the probability that a game ends at a given round obeys a linear recurrence.

Factorials and the central term

For the general reader. To find the chance of exactly 5000 heads in 10,000 tosses, one must handle numbers like \(10000!\), far beyond hand calculation. De Moivre found an accurate formula for such factorials. His friend James Stirling supplied the missing constant, \(\sqrt{2\pi}\), and the formula has carried Stirling’s name ever since.

The history

In Miscellanea analytica (1730) de Moivre studied the ratio of the middle term of \((1+1)^n\) to the sum \(2^n\). In the Supplementum he gave a series for \(\log n!\) of the form below, with the constant expressed as a series \(1-\frac1{12}+\frac1{360}-\frac1{1260}+\cdots\) which he evaluated numerically. James Stirling, who corresponded with him in 1729, showed in his Methodus differentialis (1730) that the constant is \(\log\sqrt{2\pi}\). De Moivre acknowledged this openly. Thomas Bayes later showed that de Moivre’s series for the constant does not converge, although its first terms give an excellent approximation.

Derivation

Let \(d_n=\log n!-(n+\tfrac12)\log n+n\). Then

So \(d_n\) decreases, \(0<d_n-d_{n+1}<\frac{t^2}{3(1-t^2)}=\frac{1}{12}\big(\frac1n-\frac1{n+1}\big)\), and \(d_n\) converges to a limit \(C\) with \(C<d_n<C+\frac1{12n}\). To identify \(C\), take Wallis’s product in the form \(\lim_n\frac{2^{4n}(n!)^4}{((2n)!)^2(2n+1)}=\frac{\pi}{2}\). Substituting \(n!\approx e^{C}n^{n+1/2}e^{-n}\) gives \(e^{2C}/4=\pi/2\), so \(e^{C}=\sqrt{2\pi}\).

The normal approximation, 1733

For the general reader. Toss a coin many times and draw a bar chart of how often each number of heads comes up. The bars trace a bell-shaped curve. De Moivre found the equation of that curve in 1733 and used it to show how tightly the proportion of heads clusters around one half. It was the first appearance of the normal distribution and the first case of the central limit theorem.

In November 1733 de Moivre circulated privately a short Latin pamphlet, Approximatio ad summam terminorum binomii \((a+b)^n\) in seriem expansi. He wanted to improve Jacob Bernoulli’s law of large numbers (1713), which proved convergence of frequencies but gave hopeless bounds on the number of trials needed. An English version appeared in the 1738 and 1756 editions of The Doctrine of Chances.

The local step

Let \(n\) be even and \(M=\binom{n}{n/2}\). For the term at distance \(l\) from the middle,

using \(\log\frac{1-x}{1+y}\approx-x-y\) for small \(x,y\). Combined with Lesson 08, the term is approximately \(\frac{2}{\sqrt{2\pi n}}e^{-2l^2/n}\). This is the curve drawn at the top of the page.

Summing

Summing over \(|l|\le L\) and replacing the sum by an integral, de Moivre obtained the probability that the number of heads lies within \(L\) of \(n/2\):

He evaluated it by expanding the exponential in series, for \(L=\tfrac12\sqrt n\), \(\sqrt n\) and \(\tfrac32\sqrt n\). In modern terms these are 1, 2 and 3 standard deviations, and the answers are about 0.68, 0.95 and 0.997. He called \(\tfrac12\sqrt n\) the modulus, the unit by which deviations should be measured. He also indicated the extension to unequal chances \(p\ne q\).

De Moivre proved the symmetric case \(p=\tfrac12\) and sketched the general one. Pierre-Simon Laplace gave the general proof in his Théorie analytique des probabilités (1812), hence the name de Moivre–Laplace theorem. The curve was later attached to Carl Friedrich Gauss’s theory of errors and became the “Gaussian”, but de Moivre found it first.

De Moivre drew a philosophical conclusion: the regularity of frequencies in the long run reflects a stable underlying order, which he attributed to design. The square-root law he discovered, that fluctuations grow like \(\sqrt n\) while totals grow like \(n\), is the quantitative content.

Annuities upon lives

For the general reader. Governments and towns in the seventeenth and eighteenth centuries raised money by selling life annuities: pay a lump sum now and receive a yearly income until death. Pricing them requires combining interest with the chances of survival. Halley had shown how to do it with a mortality table, but the arithmetic was enormous. De Moivre found a simple model of mortality that made the calculation short.

Halley’s method

In 1693 Halley published a life table built from the burial and birth records of Breslau, with a method for valuing annuities. If \(l_x\) is the number of survivors at age \(x\), \(i\) the rate of interest and \(v=1/(1+i)\), then an annuity of 1 per year, paid at the end of each year while a life now aged \(x\) survives, is worth

De Moivre’s hypothesis

In Annuities upon Lives (1725) de Moivre observed that over most of adult life Halley’s numbers of survivors decrease roughly in arithmetic progression. He assumed that \(l_x\) is proportional to \(86-x\) from about age 12 onward, with 86 the limiting age. The remaining span \(n=86-x\) he called the complement of life.

Derivation. We need \(\sum_{t=1}^{n}v^t-\frac1n\sum_{t=1}^{n}t\,v^t\). With \((Ia)_{\overline{n}|}=\sum tv^t=\frac{(1+i)a_{\overline{n}|}-nv^n}{i}\) (sum by parts), the result is \(a_{\overline{n}|}-\frac{(1+i)a_{\overline{n}|}}{ni}+\frac{v^n}{i}=\frac{1-v^n}{i}+\frac{v^n}{i}-\frac{(1+i)a_{\overline{n}|}}{ni}\), which is the formula. One table of annuities certain then values every life annuity.

The same book treats annuities on two or more lives, reversions and tontines. De Moivre approximated the curve of survivors by straight segments where a single line fitted poorly, a piecewise-linear method that kept the arithmetic short. His later editions (1743, 1750, 1752) and the 1756 printing with The Doctrine of Chances made the work a standard reference for the new insurance societies.

Priority, patrons and poverty, 1710–1740

In 1712 the Royal Society appointed de Moivre to the committee that examined the dispute between Newton and Leibniz over the invention of the calculus. The committee, drawn from Newton’s supporters, found for Newton. De Moivre’s presence there reflected his closeness to Newton rather than impartiality, and it sits awkwardly with the fact that Leibniz and Johann Bernoulli had tried to help him two years earlier.

His own priority quarrels concerned probability. Pierre Rémond de Montmort had published his Essay d’analyse sur les jeux de hazard in 1708, with a second edition in 1713 that included correspondence with Nicolaus Bernoulli. Montmort felt that De mensura sortis took insufficient account of his work, and de Moivre felt the reverse. The exchange was sharp in print but ended on civil terms. Later, de Moivre accused Thomas Simpson of taking his annuity methods in cheaper books, and complained of it in the preface to his 1743 edition.

The relationship with James Stirling was the opposite. Stirling wrote to him in June 1729 about the central binomial term, and de Moivre printed both his own result and Stirling’s improvement, crediting his worthy and learned friend. He was elected to the Berlin Academy in 1735.

None of this paid. He supported himself mainly by tutoring, and he was known to sit at Slaughter’s Coffee House in St Martin’s Lane, where he played chess for small stakes and answered questions. He never married. The second edition of The Doctrine of Chances (1738) and the revisions of Annuities were his main sources of reputation, if not of income.

Last years and legacy, 1740–1754

De Moivre worked on new editions into his eighties. On 27 June 1754 the Paris Academy of Sciences elected him a foreign associate, the recognition his native country had withheld. He died in London on 27 November 1754, aged 87, and was buried at St Martin-in-the-Fields. A famous story says that he noticed he was sleeping a quarter of an hour longer each night, summed the progression, and predicted that he would die on the day he slept for 24 hours. The story circulated after his death and has no contemporary documentation. The third edition of The Doctrine of Chances appeared in 1756.

Legacy

• Probability and statistics: the normal approximation, the square-root law and the idea of measuring deviations in a natural unit, which Laplace generalized and Gauss tied to the theory of errors. The central limit theorem in its modern forms descends from the 1733 pamphlet.
• Method: generating functions and recurring series became the standard analytic tools of combinatorics and of the probability of Laplace.
• Analysis: the de Moivre–Stirling formula for \(n!\), and the angle-multiplication formula through which every student first meets complex numbers in polar form.
• Actuarial science: “de Moivre’s law” of uniform mortality remains the first example in actuarial textbooks, and his annuity methods served the life insurance offices founded in Britain in the eighteenth century.
• Influence: Thomas Bayes read him closely; Laplace built on him; Euler gave his trigonometric results their modern form.

Connections

People linked to Abraham de Moivre across this collection. A name without a link has no page here yet. See Abraham de Moivre in the network of all 337 lives →

Teachers and advisors

• Jacques Ozanam 1684-1685

Influenced by

• Christiaan Huygens
• Edmond Halley
• Jacob Bernoulli
• Roger Cotes

Influenced

• Leonhard Euler
• Pierre-Simon Laplace
• Thomas Bayes

Colleagues and collaborators

• Brook Taylor
• Gottfried Wilhelm Leibniz
• Johann Bernoulli 1710
• James Stirling 1729-1738

Rivals

• Pierre Rémond de Montmort 1711-1719
• Thomas Simpson

Friends

• Edmond Halley 1692-1742
• Isaac Newton 1692-1727

Also mentioned in this life

• Gerolamo Cardano
• Pafnuty Chebyshev
• Fibonacci
• Carl Friedrich Gauss
• Siméon Denis Poisson
• John Wallis

Fields

• probability theory 23 lives
• analysis 20 lives
• trigonometry 9 lives

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Abraham de Moivre, MacTutor History of Mathematics.
2. Abraham de Moivre, Wikipedia.
3. R. J. Pulskamp, Abraham de Moivre: works and translations.
4. I. Schneider, Abraham de Moivre, Encyclopedia of Mathematics.
5. D. R. Bellhouse, Abraham de Moivre: Setting the Stage for Classical Probability and Its Applications, CRC Press, 2011.
6. H. M. Walker, Abraham de Moivre, in the 1967 reprint of The Doctrine of Chances, Chelsea.

Mathematics

1. A. de Moivre, A method of raising an infinite multinomial to any given power, Phil. Trans. 19 (1697).
2. A. de Moivre, De mensura sortis, Phil. Trans. 27 (1711); English translation by B. McClintock, International Statistical Review 52 (1984).
3. A. de Moivre, The Doctrine of Chances, London, 1718; 2nd ed. 1738; 3rd ed. 1756.
4. A. de Moivre, Annuities upon Lives, London, 1725.
5. A. de Moivre, Miscellanea analytica de seriebus et quadraturis, London, 1730.
6. A. Hald, A History of Probability and Statistics and Their Applications before 1750, Wiley, 1990.
7. S. M. Stigler, The History of Statistics, Harvard University Press, 1986, chapter 2.
8. P.-S. Laplace, Théorie analytique des probabilités, Paris, 1812.