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\noindent{\bf Abraham DE MOIVRE}\\
b. 26 May 1667 - d. 27 November 1754
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\noindent{\bf Summary.} The first textbook
of a calculus of probabilities to contain a form of local
central limit theorem grew out of the activities of the lonely Huguenot de
Moivre who was forced up to old age to make his living by solving problems of
games of chance and of annuities on lives for his clients whom he used to meet
in a London coffeehouse.
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Due to the fact that he was born as Abraham Moivre and educated in France for
the first 18 years of his life and that he later changed his name and his
nationality in order to become, as a Mr. de Moivre, an English citizen,
biographical interest in him seems to be relatively restricted compared with
his
significance as one of the outstanding mathematicians of his time. Nearly all
contemporary biographical information on de Moivre goes back to his
biography of Matthew Maty, member, later secretary of the Royal Society, and
a close friend of de Moivre's in old age.
Abraham Moivre stemmed from a Protestant family. His father was a Protestant
surgeon from Vitry-le-Fran\c{c}ois in the Champagne. From the age of five to
eleven
he was educated by the Catholic P\'{e}res de la doctrine Chr\`{e}tienne.
Then he
moved to the Protestant Academy at Sedan were he mainly studied Greek. After
the latter was forced to close in 1681 for its profession of faith, Moivre
continued his studies at Saumur between 1682 and 1684
before joining his parents who had meanwhile moved to
Paris. At that time he had studied some books on elementary mathematics and
the first six books of Euclid's elements. He had even tried his hands on the
small tract concerning games of chance of Christiaan Huygens (q.v.),
{\it De ratiociniis
in ludo aleae}, from 1657 without mastering it completely. In Paris he was
taught mathematics by Jacques Ozanam who had made a reputation from a series
of books on practical mathematics and mathematical recreations. Ozanam made
his living as a private teacher of mathematics. He had extended the usual
teachings of the European reckoningmasters and mathematical practitioners by
what was considered as fashionable mathematics in Paris.
Ozanam enjoyed a moderate financial success due
to the many students he attracted. It seems
plausible that young Moivre took him as a model he wanted to follow
when he had to support himself.
After the revocation of the Edict of Nantes in 1685 the protestant faith
was not
tolerated anymore in France. Hundreds of thousands of Huguenots who had
refused to become catholic left France to emigrate in Protestant
countries. Amongst them was Moivre who arrived in England, presumably in
1687. In England he began his
occupation as a tutor in mathematics. Here he added a "de" to his name. The
most plausible reason for this change is that Moivre wanted to take advantage
of the prestige of a (pretended) noble birth in France in dealing with his
clients
many of whom were noblemen. An anecdote from this time which goes back to
(de) Moivre himself tells that he cut out the pages of Newton's
{\it Principia} of
1687 and read them while waiting for his students or walking from one to the
other. True or not, the main function of this anecdote was to demonstrate that
de Moivre was amongst the first true and loyal Newtonians and that as such he
deserved help and protection in order to gain a better position than that of a
humble tutor of mathematics. In 1692 de Moivre met with Edmond Halley and
shortly afterwards with Newton. Halley cared for the publication of de Moivre's
first paper on Newton's doctrine of fluxions in the {\it Phil. Trans.}
for 1695 and
saw to his election to the Royal Society in 1697. Only much later in 1735
de Moivre was elected
fellow of the Berlin Academy of Sciences, and five months before his death the
Paris Academy made him a foreign associate member.
Looking at Newton's influence concerning university positions for mathematics
and natural philosophy in England and Scotland it seemed profitable to de
Moivre to engage in the solution of problems posed by the new infinitesimal
calculus. In 1697 and 1698 he had published the polynomial theorem, a
generalization of Newton's binomial theorem, together with application in the
theory of series. This theorem was the background for a quarrel with the Scotch
physician George Cheyne who had published a book in 1703 on Newton's
method of fluents. De Moivre's critical remarks concerning Cheyne's book
filled another book which was published in 1704. This first book of de Moivre
was no success but stimulated a correspondence with Johann Bernoulli
which lasted until 1714. He had tried to secure the support of
Johann Bernoulli and Leibniz in order to get a professorship on the continent.
De Moivre did not answer Bernoulli's last letter. It seems that de Moivre who
was made a member of the commission in the Royal Society to decide in the
priority dispute between Newton and Leibniz against Leibniz feared to appear
disloyal to the Newtonian cause had he continued this correspondence. At any
rate, the letters of Johann Bernoulli had shown to de Moivre that he lacked the
time and perhaps the mathematical power to compete with a mathematician of
this calibre in the new field of analysis. In addition, when the Lucasian
chair in
Cambridge for mathematics had been given in 1711 on Newton's
recommendation to Nicholas Saunderson de Moivre had realized, that the only
chance for him to survive was to continue his occupation as a tutor and
consultant in mathematical affairs in the world of the coffee houses where he
used to meet his clients; additional income he could draw from the publication
of books and from translations.
So he turned to the calculus of games of chance and probability theory which
was of great interest for many of his students and where he had only a few
competitors. In this respect it was easy for him to become a pioneer in a field
which, apart from two episodes, he could claim for himself. In both cases he
was involved in rather fierce disputes about mutual dependence with other
authors in the field. De Moivre could treat the first of these,
Montmort (q.v.), for
personal and political reasons as an enemy without it
being resented in England
because Montmort was French and France had expelled de Moivre who had
become a naturalized Englishman in 1705 and who had experienced the defeat
of the French armies in the war of Spanish succession with grim satisfaction.
Montmort had published a book on games of chance, the {\it Essay d'Analyse sur
les Jeux de Hazard}, in 1708 and reacted to de Moivre's first publication
in the
field in the second edition of the {\it Essay} which appeared in 1713/14.
The second
opponent was the Englishman Thomas Simpson, who with two books from
1740 and 1742 had plagiarized de Moivre's {\it Doctrine of Chances}
and {\it Annuities on Lives}.
Simpson, a former fortuneteller and weaver from Leistershire with the typical
mentality of a social climber had come to London in 1736. Here he began
immediately to qualify for the market of mathematically interested clients by
turning out a textbook on the theory of fluxions in 1737 which was the
first in a
whole series of mostly very successful mathematical textbooks. De Moivre's
anger and his acrimonious reaction to Simpson who had intruded into his
proper domain is understandable but did not meet with general applause. In a
way fortunately for de Moivre, Simpson was more successful in his efforts to
get a permanent position and so dropped out from the competition for private
clients in London.
Next to his clients it was Montmort who had raised de Moivre's interest in the
theory of games of chance and probability. In the {\it Phil. Trans.}
for 1711 de
Moivre published a longer article on the subject which was followed by his
{\it Doctrine of Chances}. The {\it Doctrine}
appeared in 1718, a second edition from
1738 contained de Moivre's normal approximation to the binomial distribution
which he had found in 1733. The third edition from 1756 contained as a second
part the {\it Annuities on Lives}
which had been published as a monograph for the
first time in 1725.
De Moivre's preoccupation with questions concerning the conduct of a
capitalist society like interest, loan, mortgage, pensions, reversions or
annuities
goes back at least to the 90's of the 17th century from which time a piece of
paper is kept in Berlin containing de Moivre's answers to pertinent
questions of
a client. Halley had reconstructed from the lists of births and deaths in
Breslau
for each of the years 1687-1691 the demographic structure of the population of
Breslau, which he assumed as stationary, in form of a life table.
Halley's life
table was published in the {\it Phil. Trans.} for 1693 together with
applications to
annuities on lives. Besides the formulas for the values of an annuity for
a single
life and for several lives he had calculated a table for the values of
annuities of
a single life for every fifth year of age at an interest rate of 6\%. The
immense
calculation hindered him from doing the same for two and more
lives. De Moivre solved this problem by a simplification. He replaced Halley's
life table by a (piecewise) linear function. Based on such a hypothetical
law of
mortality and fixed rates of interest he could derive formulas for annuities of
single lives and approximations for annuities of joint lives as a function
of the
corresponding annuities on single lives. These results were published together
with the solution of problems of reversionary annuities, annuities on
successive
lives, tontines, and of other contracts which are depending on interest and the
``probability of the duration of life" in his book
{\it Annuities upon lives} which
appeared for the first time in 1725. In the second edition of the
{\it Doctrine of chances} part of the material contained in the
{\it Annuities} together with new
material was incorporated. After three more improved editions of the
{\it Annuities} in 1743, 1750, and 1752 the last version of it was published
in the third edition of the {\it Doctrine}.
The {\it Doctrine} can be considered as the result of a competition between de
Moivre on the one hand and Montmort together with Nicolaus Bernoulli
(q.v.) on the
other. De Moivre's representation of the solution of the then current problems
tended to be more general than that of Montmort. In addition he developed a
series of algebraic and analytic tools for the theory of probability like a
``new algebra" for the solution of the problem of coincidences
which forshadowed
Boolean algebra, the method of generating functions or the theory of recurring
series for the solution of difference equations. Different from Montmort, de
Moivre offered in the {\it Doctrine} an introduction which contains the main
concepts like probability, conditional probability, expectation, dependent and
independent events, the multiplication rule, and the binomial distribution.
De Moivre's greatest mathematical achievement is considered a form of the
central limit theorem which he found in 1733 at the age of 66. There is no
doubt
that de Moivre understood the importance of this special finding. From a
technical point of view de Moivre understood his central limit theorem as a
generalization and a sharpenning of Bernoulli's
{\it Theorema aureum} which was
later named the law of large numbers by Poisson.
Already in his commentary on Huygens (q.v.), Jakob Bernoulli
(q.v.) had introduced the
binomial distribution. With it he had shown that the relative frequency
$h_n$ of an
event with probability $p$ in $n$ independent trials converges in
probability to $p$.
More precisely he had shown that for any given small positive numbers
$\delta$ and $\epsilon$ then
for sufficiently large $n$,
\[ P(|h_{n} - p| \leq \epsilon) > 1 - {\delta}. \]
de Moivre, however, was interested in the precise determination of
these probabilities and, by considering the ratios of binomial
probabilities, he was able to show that for large $n$:
\[ P(|h_{n} - p| \leq s\surd(pq/n)) \approx
\surd(2/{\pi}) \int_{0}^{s} e^{ -x^2/2} dx \]
although he did not use this representation. He calculated the value
of the integral on the right hand side for $s = 1,2,3$.
It is clear that he intuitively understood the importance of what
was later called the standard deviation.
The approximation of the binomial through the normal distribution with its
consequences was the culmination of the {\it Doctrine} from the second
edition on.
This book, especially the last
edition of 1756,was the most complete representation of the new field in the
second
half of the 18th century. That this was felt by the leading mathematicians of
the
next generation is clear in that Lagrange and Laplace (q.v.)
independently planned translations of de Moivre's {\it Doctrine}.
The interest of Lagrange and Laplace in de Moivre's work goes back to de
Moivre's solution of the problem of the duration of play by means of what de
Moivre called recurrent series and what amounts to the solution of a
homogeneous linear difference equation with constant coefficients. In fact, the
most effective analytical tool developed by Laplace for the calculus of
probabilities, the theory of generating functions, is a consequence of
Laplace's occupation with recurrent series.
However, Laplace restricted his praise of de Moivre's theory of probability to
mathematical achievements like the theory of recurrent series, the central
limit
theorem and the hint to the generality which distinguishes the problems and
their solutions chosen by de Moivre from those of his predecessors. Many other
features of the {\it Doctrine of chances}
like the interpretation of the central limit
theorem concerning the relationship of probability and chance remained
unmentioned by Laplace presumably because his generation did not share
anymore de Moivre's views on theology and natural philosophy. De Moivre
seemed to understand very different connotations of the term chance. In the
first
remark to his central limit theorem concerning the tossing of a coin he states,
"that Chance very little disturbs the Events which in their natural Institution
were designed to happen or fail, according to some determinate Law." It seems
clear that chance is used in this sentence as an antithesis to law which
conforms
to what is called statistical law. The existence of laws of this kind is due
to a
"design" which again, according to contemporary convictions, at least amongst
"natural" theologians, relates immediately to divine providence. Chance in
contrast appears as something which obscures this design by "irregularity".
Irregularity had to do with the unpredictability of the outcome in single
trials
and interrelated with that with deviations of a regular pattern according to
which e. g. all six sides of a die should show up in some order in a series
of six throws.
De Moivre did not analyse this irregularity which characterizes chance any
further.
However, his understanding of the concept of
chance can be clarified with the help of remarks on the
central limit theorem which appear only in the third edition of the
{\it Doctrine of Chances}.
He had taken the view that irregularity and unpredictability in a small
number of trials, but not in the long run inherent in the concept of chance,
are
consistent with his repeated reference to divine design and providence. In
order
to understand this we must take into account that de Moivre's and Newton's
creator had not abandoned his creation after its perfection but ruled it
permanently in order to guarantee its existence. Therefore de Moivre's God is
allwise and allpowerful; nothing happens outside his control and involvement.
If one considers the status of probabilities as laws which express God's
design,
two conclusions are forthcoming. First, probabilities are objective
properties of
creation. Secondly, chance, too, as an existing property of the material world
with its irregular and unpredictable aspects, is a manifestation of God's
constant
involvement in the course of his creation and so is objective in the sense
that it
is independent of the human subject and its level of information.
In a way, further progress beyond such a statement is imaginable neither for de
Moivre nor for Newton. Since the supreme goal of scientific enterprise is to
demonstrate the existence of an agent, called God, whose constant activity
permeates the whole cosmos, de Moivre with his interpretation of the interplay
between law or design, which reveals the existence of God, and chance, which
represents his constant activity, has reached this goal. In this sense de
Moivre's
chance has the same status as action at a distance in Newton's physics. Neither
concept is explained by a reduction to more elementary concepts.
This view differed completely from that of Jakob Bernoulli and Laplace. Both
shared a credo in a completely determined world the events of which strictly
follow certain laws which can be described in mathematical terms.
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\begin{thebibliography}{3}
\bibitem{1} Schneider, Ivo (1996). Die R\"{u}ckf\"{u}hrung des Allgemeinen auf den Sonderfall-eine
Neubetrachtung des Grenzwertsatzes f\"{u}r binomiale Verteilungen von Abraham
de Moivre. In: Joseph W. Dauben et alii (Eds.),{\it History of Mathematics:
States of the Art, Flores quadrivii Studies in Honor of Christoph J. Scriba},
Academic Press, San Diego, pp. 263-275.
\bibitem{2} Schneider, Ivo (1994). Abraham de Moivre: pionero de la teor\'{i}a de
probabilidades
entre Jakob Bernoulli y Laplace. In: E. de Bustos, J. C. Garc\'{i}abermejo, E.
P\'{e}rez Sede\~{n}o, A. Rivadulla, J. Urrutia, J. L. Ziofo\'{o} (Eds.),
{\it Perspectivas
actuales de l\'{o}gica y filosof\'{i}a de la ciencia}, Siglo Veintiuno de
Espa\~{n}a Editores, S. A., Madrid, pp. 373-384.
\bibitem{3} Schneider, Ivo (1968). Der Mathematiker Abraham de Moivre (1667-1754),
{\it Archive
for History of Exact Sciences}, {\bf 5}, 177-317.
\bibitem{4} Hald, Anders (1990). {\it A History of Probability and Statistics and their
Applications Before 1750}, John Wiley \& Sons, New York, especially
Chapters 19 to 25.
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\hfill{Ivo Schneider}
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