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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 \&amp; Sons, New York, especially
Chapters 19 to 25.
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\hfill{Ivo Schneider}



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