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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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