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\noindent{\bf Pierre de FERMAT}
\\
b. 17 August 1601 - d. 12 January 1665
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\noindent {\bf Summary}. \ Arguably the greatest French mathematician of
the 17th
century, Fermat was instrumental in giving impetus, with Pascal, to the
theory of probability.
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Fermat, of Basque origin, was born at Beaumont-de-Lomagne,
near Montauban in Gascony, the son of Dominique Fermat, second consul
of the town, and his wife Fran\c{c}oise de Cazelneuve. His family were
leather merchants. He received the elementary part of his schooling, of
which little is known, at home and in a local Franciscan school. In
1631 he received his baccalaureate in law from the University of
Orl\'{e}ans, having studied, it is thought, in Toulouse and Bordeaux.
He worked as a lawyer in the local parliament in Toulouse, becoming
Counsellor ({\it Conseilleur de la Chambre des Requ\^{e}tes du
Parlement}) in 1634. At about this time he married Louise du Long, and
the aristocratic {\it de} was prefixed to his family name. Finding time
outside of his legal profession, which necessitated maintaining a certain
reserve, he became known through his correspondence for his
erudition in the humanities and the classics incorporating a profound
knowledge of Greek and Latin, a mastery of modern European languages and
research in mathematics. He was created King's Counsellor, still in
Toulouse, in 1648.
Despite his intellectual brilliance he was reputedly a quiet, friendly
and kindly man, whose successful public life was paralleled by a happy
family life. His two daughters became nuns. His son, Samuel, a writer,
edited his father's work, published as a two-volume {\it Opera
mathematica} in 1679. Fermat died at the age of 63 (his tombstone says
57), at Castres. There appears to be no definitive biography.
The correspondence between Fermat and Pascal (q.v.), between July and October
1654, which is regarded as laying foundation for the mathematical
theory of probability, came about as follows. The unofficial ``Academy"
which preceded the founding of the Acad\'{e}mie des Sciences de Paris in
1665 was that founded by the Abb\'{e} Marin Mersenne (1588-1648),
indefatigable correspondent on mathematical and scientific matters of
his time. Pascal's father, \'{E}tienne, had moved to Paris in 1631 to
supervise his son's education and became a member of this Academy, which
included Ren\'{e} Descartes (1596-1650), and whose meetings were
initially held at Mersenne's house. \'{E}tienne introduced his son to
the Academy when Blaise Pascal (1623-1662) was fourteen years old.
Pierre de Carcavi, who, like Fermat, was Counsellor in the Parliament of
Toulouse, was in the correspondence circle of Mersenne, and introduced
Fermat into it in 1636, by motivating \'{E}tienne Pascal and Roberval to
write to Fermat. In regard to what we now know as probability theory,
Blaise Pascal's prime concern was the equitable division of stakes, the
``probl\`{e}me des partis", \, or, in English idiom, the ``Problem of
Points". \ This is evident from his letter to the Academy entitled {\it
Celeberrimae Matheseos Academiae Parisiensis}, prior to his
correspondence with Fermat, which alludes to work in progress entitled
{\it Aleae geometria (The Mathematics of Chance)}. \ Sometime later in
that year (1654) in about early July, he wrote to Fermat almost surely
about this problem. That letter has not been found, but the first
surviving letter, from Fermat to Pascal, is about a simple version of
the problem: if a gambler undertakes to throw a six in eight throws, but
stops after the first three throws which have been unsuccessful and does
not continue, what proportion of the total stake should he have? \
Pascal's solution is \ $125/1296 = (5/6)^3 (1/6)$, \ while Fermat's
is \ $1/6$ . \ Fermat's would be correct if a total of $4$, rather
than $8$, throws was originally proposed. With 8 the correct
proportion (probability) would be \ $1 - (5/6)^5$. \ The second
surviving letter, the famous one of 29 July, 1654 from Pascal to Fermat,
discusses a more sophisticated version
of the problem. In the case of two players at each of a number of
trials, each has probability 1/2 of winning the trial. It is agreed
that the first player with \ $n$ \ wins gains the total stake.
The game is interrupted when player \ A \ needs \ $a$ \ trial-wins to
gain the stake, and player \ B \ needs \ $b$ . \ How should the stake
be divided? \ If one considers the maximum number of trials, \ $a + b -
1$ , \ that the game may take to reach conclusion, the problem becomes
one of ``equally likely" outcomes and the calculation of the probability
that \ A \ wins:
$$\Large{\sum_{r=a}^{a+b-1} {{a+b-1}\choose{r}} {\Bigg / } 2^{a+b-1}}$$
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\noindent is made simple, although in practice the game may end earlier.
The solution by Pascal and Fermat (by different methods) even for some
particular cases, was a defining epoch in probability theory.
Fermat's fame in mathematics, however, rests more in the realm of number
theory. Fermat's Last Theorem provided a challenge for centuries to both
professionals and amateurs.
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\begin{thebibliography}{3}
\bibitem{1} David, F.N. (1962). {\it Games, Gods and Gambling}. Griffin,
London. [Contains the letters between
Fermat, Pascal and Carcavi, in English
translation by M. Merrington.]
\bibitem{2} Tannery, P. and Henry, C. (eds.) (1891-1922). {\it Oeuvres de
Fermat.} Gauthier-Villars, Paris, 5 vols.
\bibitem{3}
Todhunter, I. (1865). {\it A History of the Mathematical Theory of
Probability from the Time of Pascal
to that of Laplace}. Reprinted in
1949 and 1961 by Chelsea, New York.
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\hfill{E. Seneta}
\bibitem{4} %%\newline \mbox{} \quad ...
\end{thebibliography}
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