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\noindent{\bf Joseph BERTRAND}\\
b. 11 March 1822 - d. 3 April 1900
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\noindent{\bf Summary}. Bertrand brought several interesting innovations into the
probability calculus, but it is above all for his role as teacher, as publicist,
and as critic (indeed as polemicist) that he is known to history.
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Joseph Bertrand, who was born and died in Paris, was a prodigy who
fulfilled his childhood promise. He was the son of Alexandre Bertrand, a graduate of the
Ecole Polytechnique who was an expert on sleepwalking, ecstasy and other
extraordinary
states of consciousness. Joseph, orphaned at the age of nine, was raised by
his
uncle, the mathematician J.M.C. Duhamel (1797-1872), who left him free to
study as
he pleased. Joseph Bertrand held all the French records for precocity at the
university; he was allowed to follow lectures at the Ecole Polytechnique at
the
age of 11, and was awarded his doctorate in the mathematical sciences at
sixteen.
He was then admitted as the top candidate at the Ecole Polytechnique, having
at last
reached the age required to compete officially in the entrance examinations.
This
was a unique case, where a pupil had already won his doctorate and was more
qualified and knowledgeable
than his teachers. At the age of 25, he was appointed as an interim
professor of mathematical physics at the Coll\`ege de France, the most
prestigious
French institution of scholarship, to replace J.B. Biot (1771-1862), the last
representative of Laplacian science and of the Soci\'et\'e d'Arcueil. After
Biot's
death in 1862, Bertrand was appointed to the Chair.
Joseph Bertrand became a tutor {\it r\'epetiteur} in analysis at the Ecole Polytechnique
in 1844 (until 1856) and Chair professor in the subject from 1856 to 1895. He thus taught 50
intakes of Polytechnique students, among them some of the most renowned French
scientists of the second half of the 19th century, as well as eventually famous
representatives
of industry, senior administration and the army. Bertrand was an exceptional
teacher,
brilliant, witty, always clear and precise, avoiding obscurities and limiting
his
lectures to what he himself could understand perfectly, or at least give the
impression of understanding perfectly.
On Sturm's death in 1856, he was elected to membership of the Academy of
Sciences. He became its permanent secretary for the mathematical sciences
from 1874 until his
death. In this capacity, he produced two volumes of academic eulogies,
elegant and
witty in the style of the period, which secured his election in 1884 to the
Acad\'emie Fran\c{c}aise.
Bertrand's mathematical work was important and occasionally brilliant, but perhaps not
profound.
It was concerned with branches of mathematics in vogue in the second half
of the 19th century, analysis and geometry of course, but also arithmetic and algebra
for which he had a strong flair (Bertrand's curves, Bertrand series, Bertrand's
postulate). His acknowledged master in all things was Gauss (q.v.), and
his models Abel and Jacobi.
From an early age, Bertrand revealed his notable gift as a polemicist.
His ferocious and deadly irony rapidly assured him of leadership in the numerous
institutions of which he was a member. His natural authority, and an
understanding of the subtleties of arbitration, assisted by carefully selected and
maintained
family alliances assured him of automatic majorities at the Sorbonne, the
Academy of
Sciences, the Ecole Polytechnique and the Coll\`ege de France. Bertrand was
the
brother in law of the benevolent Charles Hermite (1822-1920),
who described him as ``ill-disposed and ill-natured;
with him one could not be totally sure of anything". Bertrand was also
an uncle by marriage of Paul
Appell
(1855-1930)- who was himself the father in law of Emile Borel (q.v)- as well as of
Emile
Picard (1856-1941). For the best part of half a century, Bertrand almost
singly
acted as caretaker of the French mathematical sciences; there is no other
example of
such power, except for that, even greater from all points of view, of the earlier
Sim\'eon-Denis
Poisson (q.v.).
Bertrand early became interested in the calculus of probabilities, which he
taught for 40 years at the Ecole Polytechnique and for several years at the
Coll\`ege
de France. It was he who translated Gauss' papers on the Method of Least
Squares into French; he loved Gauss' algebraic clarity, in contrast to
Laplace's (q.v.) analytic
obscurities. He was interested in combinatorial
analysis, whose profound aesthetics he was one of the first to appreciate in France:
his
students Emile Barbier and D\'esir\'e Andr\'e later contributed to its
development. It
was without any doubt Bertrand who popularized the Method of Expectations in
which the
expectation of the indicator function of an event, rather than its
probability, is
calculated. His large treatise {\it Calcul des probabilit\'es}, the
conclusion of 40 years of carefully
revised teaching, went through two editions (1888 and 1907). Its success
is largely due to the enormous gap in France between the probability books of Poisson
of 1837 and of Cournot(q.v) of 1843, and his own, filled in part only by the
book of H. Laurent in 1873.
Bertrand's book is of great
interest on more than one count, and not only because it served as a reference
text for
all mathematicians interested in the theory of probability at the beginning
of the
20th century. Apart from some regrettably narrow-minded views, as for
example on
Condorcet (q.v.), Canard and Cournot, that is: against the use of mathematics in
economics
and the human sciences, this book is remarkably well written, and could well
serve
to exemplify a certain type of French intellect at the end of the last century.
In it, one can certainly find the famous Bertrand paradox on the different
ways
of drawing a chord at random in a circle; this is a paradox which Bertrand in
his handwritten lecture notes for the Ecole Polytechnique constructed in
stages as a transformation of the famous Buffon needle problem. More importantly, his
Chapter
VI on the gamblers' ruin is at the very foundation of modern research on the
theory of
Brownian motion and the sums of independent random variables. Bachelier (q.v.)
found much
inspiration in it, and through him modern probabilists. Poincar\'e, Hadamard,
Borel
and Paul L\'evy studied Bertrand's treatise, for which
they always showed the greatest respect.
On the other hand, Bertrand remained resolutely hostile to Laplacian or
Bayesian
statistical theory: the chapters on these topics in his book are totally
sceptical.
The ``probability of causes" was, to his mind, too arbitrarily tied to the
prior distribution to be useable in practice under any circumstances. However,
his comments are often acute and lucid.
Bertrand's hostility to Laplace extended to Bienaym\'e (q.v.), who near the end of his life
expressed small regard for some of Bertrand's contributions. Bertrand's book helped
bury the memory of Bienaym\'e's contributions by an inadequate and negative treatment.
Bienaym\'e's probabilistic alter-ego, Chebyshev (q.v.) is not mentioned once. In a letter
to Chebyshev, Catalan writes with uncharacteristic mildness: ``Quel dr\^ ole de livre! [What a
funny book!]".
There were several clashes between Catalan and Bertrand, but few were prepared to take Bertrand on.
However, Bertrand did not always emerge unscathed; the best known instance is ``L'Affaire Carton"
of 1869 when he presented for publication in the {\it Comptes Rendus} of the Academy a paper
by one Jules Carton which purported to give a proof of Euclid's parallels postulate. Liouville and
Bienaym\'e, and then Darboux, Beltrami and Ho\H uel, opposed publication for obvious reasons.
Darboux, a former pupil of Bertrand, finally succeeded in persuading him his position was wrong.
The only areas of application of the calculus of probabilities in which
Bertrand
was apparently interested were artillery accuracy (in which he was the
forerunner of
Gaussian statistics, correlation, axes of inertia), topics in geodesy
(although
his chapters on the Method of Least Squares were not totally convincing), and
particularly problems of insurance, certain aspects of which he attempted to
clarify. In this respect, Bertrand is very representative of his period
and his
``School". In fact, during the second half of the 19th century after a period of decline
in mid-century, graduates
from the
Ecole Polytechnique excelled, in the tradition of Poisson, in the probabilistic study
of the
dispersion of different artillery firings, as can be seen from the lectures
of Henry
(1894). This even became a specialty of the applied military schools
(Metz and
later Fontainebleau), with artillery men creating, within
the milieu of the military, several modern statistical tools
such as chi-square.
It was
graduates of the Ecole Polytechnique who also transformed actuarial methods.
Among
them were H. Laurent,
and E. Dormoy who developed a coefficient for the study of stability and
non-normality in statistical series at the same time as Lexis(q.v.) in Germany.
Most of these
mathematicians had followed the lectures of Bertrand, who had incorporated
the advances of his students into both
his teaching and his treatise.
At a time when French science was no longer central in the world, Joseph
Bertrand played an
arguably indispensable (if somewhat questionable) role, particularly in the
renaissance at the beginning of the 20th century of the French school of probability.
Fortune,too, played
no little part in his ascendancy.
According to the concluding lines of Zerner(1991), a concurrence of circumstances
ensured that an oligarch
became de facto a monarch, even though posterity retains almost nothing of his
mathematical work. One of these
was that Bertrand arrived at adulthood in a period poor in first rate mathematicians.
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\noindent{\bf References}
\noindent
Bru, B. (1996). Le probl\`eme de l'efficacit\'e du tir \`a L'Ecole de
Metz: aspects th\'eoriques et exp\'erimentaux. In B. Belhoste and A.
Picon, eds., {\it L'Ecole d'application de l'artillerie et du g\'enie
de Metz (1802-1870)}, Mus\'ee des plans-reliefs, Paris, pp. 61-70.
\noindent
Cr\'epel, P. (1994). Le calcul des probabilit\'es: de l'arithm\'etique
sociale \`a l'art militaire.; Gispert, H. (1994). De Bertrand \`a Hadamard,
quel enseignement d'analyse pour les polytechniciens? Both in : B. Belhoste,
A. Dahan, A. Picon, Eds., {\it La formation polytechnicienne (1794-1994)},
Dunod, Paris, pp. 197-215 and 181-196.
\noindent
Henry, P.J.P. (1894, 1926). Cours de probabilit\'e du tir, profess\'e
\`a l'Ecole d'application d'artillerie de Fontainebleau en 1894, {\it
M\'emorial de l'Artillerie fran\c{c}aise}, {\bf 5}, 294-447.
\noindent
Jongmans, F.(1996). {\it Eug\`ene Catalan. G\'eom\`etre sans patrie. R\'epublicain
sans r\'epublique.}
Soci\'et\'e Belge des Professeurs de Math\'ematique d'expression fran\c caise,
Mons.
\noindent
Sheynin, O.B. (1994). J.Bertrand's work on probability. {\it Archive for the
History of Exact Sciences,}
{\bf 48,} 155-199.
\noindent
Zerner, M. (1991). "Le r\`egne de Joseph Bertrand (1874-1900)", in
H.Gispert, {\it La France math\'ematique, Cahiers d'histoire et de
philosophie des sciences}, no. 34, diffusion A. Blanchard, Paris,
pp. 298-322.
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\hfill{Bernard Bru and Fran\c cois Jongmans}
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