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%\hfill{Original version, written July 1, 1999, to balance Nicolaus Bernoulli.}
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\noindent{\bf Pierre R\'emond de MONTMORT}\\
b. 27 October 1678 - d. 7 October 1719
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\noindent {\bf Summary}. Montmort's fame rests on his {\it Essay d analyse des
jeux de hazard} , which was virtually contemporaneous with Jacob Bernoulli's
{\it Ars Conjectandi} and Abraham de Moivre's {\it De Mensura Sortis}; and on his
collaboration with Nicolaus Bernoulli.
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The second of three sons of Fran\c{c}ois Reymond, \'Ecuyer,
Sieur de Breviande, and Marguerite R\'{e}mond,
who were of the nobility,
Pierre was born in Paris. He travelled widely in Europe in his youth after giving up the
study of law. By 1699 he had returned to France, and came under the influence
of Father Nicholas de Malebranche, with whom
he studied religion, philosophy, and physics. Over a period of 3 years he and
Fran\c{c}ois Nicole taught themselves the new mathematics. He succeeded his
elder brother as canon of N\^{o}tre-Dame but resigned in 1706 to marry
and settle down at the country estate of Montmort, which he had bought
with the fortune his father had left him in 1699. His marriage was a happy
one, and during this simple and retired life he set to work on the
theory of probability. In 1708, the first edition of his {\it Essay d'analyse
sur les jeux de hazard} appeared
``...where with the courage of Columbus he revealed a new world to
mathematicians..." according to Todhunter. At the time,
Montmort was aware of, and partly motivated by, the work by the
Bernoullis (which had been reviewed in 1705 and 1706) on the book
that was to be published posthumously in 1713 as Jacob Bernoulli's (q.v.)
{\it Ars Conjectandi}, with a preface by Jacob's nephew Nicolaus
Bernoulli (q.v.) (1687-1759). Nicolaus and Montmort had by then
evolved an extensive and fruitful
technical correspondence. Some of it is included [together with a
single letter from Johann Bernoulli (1667-1748)] as the fifth part of the
substantially expanded second edition of Montmort's {\it Essay}, also
published, a few months later, in 1713, and prepared with the aid of Nicolaus
during a 2-month stay at Montmort's estate. It is clear from the correspondence that the
mathematical influence of the Bernoullis (not to mention their
contributions) on the second edition, was substantial. Montmort was
piqued by De Moivre's (q.v.) {\it De Mensura Sortis} (the latin precursor of
the {\it Doctrine of Chances}), which appeared in 1711 and which he
regarded as plagiaristic. It was, in fact, quite scathing in
attacking his own first edition; Montmort retaliated with an {\it
Avertissement} in his second edition. Contrary to popular opinion, the
breach was never properly healed.
The value of Montmort's work is partly in his scholarship. He was
well-versed in the work of chance of his predecessors (Pascal (q.v.),
Fermat (q.v.), Huygens (q.v.)), met Newton on one of a number of visits to England,
corresponded with Leibnitz, but remained on good terms with both sides
during the strife between their followers.
The summation of finite series is an element of Montmort's mathematical
interests which enters into his probability work and distinguishes it
from the earlier purely combinatorial problems arising out of
enumeration of equiprobable sample points. Although the {\it Essay} to a large
extent deals with the analysis of popular gambling games, it focuses on
the mathematical properties and is thus written for mathematicians
rather than gamblers. The Royal Society elected Montmort a Fellow in
1715 and the {\it Acad\'{e}mie Royale des Sciences} made him an
associate member (as he was not a resident of Paris) the following year.
Montmort's best-known contribution to elementary probability is a result
connected with the card games {\it Rencontre, Treize}, and Snap), in
which \ $n$ \ distinct objects are assigned a specific order, while \ $n$
\ matching objects are assigned random order. The probability \ $u_n$ \
of at least one match is now well known to be $\Sigma_{j=1}^{n} (-1)^{j-1}/j!$.
Montmort's general iterative procedure for
calculating \ $u_n$ \ is from \ $nu_n = (n - 1) u_{n-1} + u_{n-2},$ a difference
equation
based on a conditional probability argument (given in a commentary by
Nicolaus Bernoulli) according to the outcome at the first position.
Montmort also worked with Nicolaus on the problem of duration of play in the gambler's
ruin problem, possibly prior to De Moivre, and at the time the most
difficult problem solved in the subject. Finally, in a letter of
September 9, 1713, Nicolaus proposed the following problems to Montmort:
\begin{quote}
{\it Quatri\`{e}me Probl\`{e}me}: \ $A$ \ promises to give an {\it \'{e}cu}
to {\it B}, if with an ordinary die he obtains a six with the first
toss, two {\it \'{e}cus} if he obtains a six with the second toss. . . .
What is the expectation of \ $B$ ?
{\it Cinqui\`{e}me Probl\`{e}me}: \ The same thing if \ $A$ \ promises \
$B$ \ {\it \'{e}cus} in the progression 1, 2, 4, 8, 16, . . . .
\end{quote}
\indent It is clear that the St. Petersburg Paradox, as subsequently treated by
Daniel Bernoulli in 1738 is but an insignificant step away. In his
reply Montmort indicates the solution to Nicolaus' problems and
describes them as being of no difficulty.
Montmort died of smallpox in Paris in 1719.
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\begin{thebibliography}{3}
\bibitem{1} David, F. N. (1962). {\it Games, Gods, and Gambling: The
Origins and History of Probability and Statistical Ideas from the
Earliest Times to the Newtonian Era.} Griffin, London. Chap. 14.
\bibitem{2} Fontenelle, B. (1721). \'{E}loge de M. de Montmort. {\it Histoire de
l'Acad\'{e}mie Royale des Sciences pour l'Ann\'{e}e 1719}, pp. 83-93.
\bibitem{3} Hacking, I. (1974). Montmort, Pierre R\'{e}mond de.
{\it Dictionary of Scientific Biography} (C. C. Gillispie, ed,), Scribner's,
New York. {\bf 9}, 499-500.
\bibitem{4} Hald, A. (1990). {\it A History of Probability and Statistics and Their
Applications Before 1750.} Wiley, New York.
\bibitem{5} Todhunter, I. (1865). {\it A History of the Mathematical
Theory of Probability}. Macmillan, London. Chapter VIII. [Reprinted by
Chelsea, New York, 1949 and 1965.]
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\hfill{E. Seneta}
\end{thebibliography}
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