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\noindent{\bf Pafnutii Lvovich CHEBYSHEV (or TCH\'{E}BICHEF)}\\
b. 16 May 1821 (o.s.) - d. 26 November 1894 (o.s.)
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\noindent{\bf Summary} Chebyshev is regarded as the founder of the St. Petersburg School
of mathematics, which encompassed path-breaking work in probability theory. The
Chebyshev Inequality carries his name; he intitiated rigorous work on a general
version of the Central Limit Theorem.
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Chebyshev was born in the village of Okatovo,
Borovskii uezd (district of the town of Borovsk), Kaluga \.{g}ubernia.
His education prior to the move of the family to Moscow in 1832 was at the home
of
his parents in Okatovo. His mother taught him reading and writing amongst other
things; but arithmetic and the French language were taught to him by a female
cousin, Avdotiia Kvintillianovna Sukhareva described (Prudnikov, 1976, pp. 18 and
30)
as an educated girl and a governess to the Chebyshev children. It is instructive
to
note now when women's roles are more clearly appreciated, that the history of
mathematics may owe much to this woman; Chebyshev kept a picture of her all his
life.
Nikoforovsky (1992, p. 130) notes that, indeed, Chebyshev completely mastered
French,
and said that he thought out his works in French, and only then composed them in
Russian.
In considering scientific problems, he had the habit of lapsing into French.
This explains much of his subsequent francophilia.
After enrolling at Moscow University in 1837, Chebyshev came under the
mathematical
influence of Nikolai Efimovich Zernov (1804-1862) and Nikolai Dmitrievich
Brashman
(1796-1866). The inspiration of Brashman on Chebyshev devoting himself to
mathematics and Chebyshev's veneration of him, is well-known (he kept a
photograph
which Brashman had given him until his own death). From Zernov Chebyshev
learned
``pure mathematics"; according to Nikoforovsky (1992, p. 126) ``$\cdots$ a solid
knowledge of the foundations of mathematics, enabling him to conduct independent
research". He could hardly have had a better-qualified teacher: Zernov was the
first in Russia to obtain the degree of Doctor of Mathematical Sciences (in
1837).
The year 1841 was notable in that Chebyshev completed his university course, and
began to prepare for his magisterial exams. On 17 June (o.s.) 1841, at a
commemorative ceremony at Moscow University, Brashman read an address entitled
``On the influence of the mathematical sciences on the development of
intellectual
capabilities" which was in part a stimulus for both the probabilistic works:
Buniakovsky's book ``Foundations of the Mathematical Theory of Probabilities" of
1846 (the first probability treatise in the Russian language), and Chebyshev's
magisterial thesis, defended in 1846, but apparently published in 1845: ``An
Essay
on Elementary Analysis of the Theory of Probabilities."
This thesis (for a reprinting see Chebyshev, 1955) gives as its
motivation, dated 17 October (o.s.) 1844:
\begin{quote}
``To show without using transcendental analysis the fundamental theorems of the
calculus of probabilities and their main applications, to serve as a support for
all branches of knowledge, based on observations and evidence - that was the
topic
put to me for realization by his eminence the lord guardian Count Sergei
Grigorievich
Stroganov $\cdots$"
\end{quote}
Sergei Grigorievich Stroganov (1794-1882) was not a supervisor in the normal
technical
sense. An aristocrat and capable general, in the tradition of the times he was
appointed to a sequence of top administrative roles. In particular, in the
period
1835-1847 was guardian [popechitel'] of the Moscow Educational Region.
According to the {\it Entsyklopedicheskii Slovar:} ``Contemporaries describe
these years as a golden era for Moscow University, since he personally became
involved in raising the quality of its professoriate, and had a gift for finding
talented people whom he then supported and protected . . . . . ."
We can see that Chebyshev was fortunate in coming to Moscow University as
student in
1837, and that he was one of those favoured and encouraged by Stroganov,
possibly
through the intervention of Brashman.
In the event the magisterial dissertation was entirely theoretical. Its impact
at the time appears not to have been large. It does give, as promised, an
elementary but rigorous analytical discussion of the then principal aspects of
probability theory, requiring no calculus, only algebra; and directs the reader
to the
use of a supplied table of values for applications. It led to a remarkable
paper
published in 1846 giving an analytical deduction of the Weak Law of Large
Numbers
of Poisson, which, in spite of the eminence of the journal ({\it Crelle's J.
Reine Angew. Mathematik}), passed unnoticed among the French mathematicians to
whom the Law remained an object of controversy. Both the dissertation and paper
display a new feature within probability theory, the estimation of deviation
from
the limit.
The young Chebyshev, unable to find a suitable teaching job in
Moscow after his magisterial dissertation moved to St. Petersburg, where he
started
lecturing on higher algebra and number theory in September, 1847. Buniakovsky
seems to
have taken him under his wing soon after, and they worked together on a new
edition of
Euler's {\it Theory of Numbers}. Buniakovsky lectured on probability theory at
\newline
St. Petersburg University from 1850 to his retirement in 1859; when the course
was taken
over by Chebyshev. He remained a firm supporter of Chebyshev until his own
death,
proposing him for the Academy of Sciences, and acting as an intermediary in St.
Petersburg
for correspondence between Chebyshev and western scientists since Chebyshev was
a
notoriously bad correspondent. Buniakovsky's own francophilia is well-known;
from the
beginning of his mathematical creativity he wrote papers in French, and the
noted
economist and philosopher P.B. Struve in 1918 called him
``a Russian disciple of the French mathematical school". This would
have
accorded well with, and indeed influenced, Chebyshev, strengthening the
background
provided by his cousin Sukhareva, and the general French cultural influences in
the
Russian Empire of the time. Buniakovsky also had a strong interest in the
theory of
mechanisms, a topic on which Chebyshev was to become famous.
The decade from 1850 (when he was appointed extraordinary professor) to 1860 was
one of intensive activity for Chebyshev. In June-November 1852 he was on study
leave
in France, Belgium, Germany and England. From our point of view (statistics and
probability) the most important people he met were Cauchy and Bienaym\'{e} in
Paris.
In a context of interpolation which can be regarded as {\it statistical}
theory, all 3 names became linked through an important paper of Chebyshev of
1859,
apparently originally in French, on {\it linear least squares} and {\it
orthogonal
polynomials}. This followed publication in 1858 of a translation by
Bienaym\'{e} into French of a paper in Russian of 1855 by Chebyshev, with a view
to its use as ammunition for Bienaym\'{e}'s controversy on interpolation with
Cauchy. This translation may have been stimulated by the personal contact
between
Chebyshev and Bienaym\'{e} in 1852, and again in 1856. The French influence in
general, and that of Bienaym\'{e} in particular on Chebyshev's
{\it probabilistic} work in St. Petersburg was facilitated by Chebyshev's friend
Nikolai Vladimirovich Khanykov [1819-1878].
The inequality commonly known as Chebyshev's Inequality, and less commonly as
the
Bienaym\'{e}-Chebyshev Inequality, was published by Chebyshev simultaneouly in
Russian in St. Petersburg, and (in French translation by Khanykov) in
Paris in Liouville's journal ({\it Journal de Math\'{e}matiques Pures et
Appliqu\'{e}es}
{\bf 12}, 177-184), in 1867.
Bienaym\'{e}'s proof was published in 1853 in the {\it C.R.Acad.Sci., Paris}
{\bf
37}, 5-13, within his now best known paper: ``Consid\'{e}rations \`{a} l'appui
de la
d\'{e}couverte de Laplace sur la loi de probabilit\'{e} dans la m\'{e}thode des
moindres carres.". It is reprinted in Liouville's journal immediately preceding
(pp. 158-176) the French version of Chebyshev's paper. There is a small
editorial
comment by Liouville which does not relate to the Inequality. However, the
implication of the juxtaposition of the papers is clear.
Chebyshev spent quite some time after 1867 searching for the essence of
\newline Bienaym\'{e}'s 1853 proof of the Bienaym\'{e}-Chebyshev Inequality,
finally
formulating in 1874 the ``method of Bienaym\'{e}". \ Indeed, this method is
essentially what later came to be called the ``Method of Moments". \ Chebyshev
later used this method to prove the first version of the Central Limit Theorem
for
sums of independent but not identically distributed summands in 1887, his final
and great achievement in probability theory, which was then quickly taken up by
his
students Markov (q.v.) and, in another
\newline direction, Liapunov, thus making the Russian Empire a focus of
probability
theory for the world.
More generally, Chebyshev is credited with founding the great ``Petersburg
School" of
mathematics, with which are associated (in addition to Markov and Liapunov) the
names
of A.N. Korkin, E.I. Zolotarev, K.A. Posse, D.A. Grav\`{e}, G.F. Voronoi, A.V.
Vasiliev, V.A. Steklov, and A.N. Krylov, amongst others; and there were many
\newline ``descendants" of the School in the next generation, such as M.P.
Kravchuk
(Krawtchouk) (q.v.).
It is interesting to note that Chebyshev's eminence was late in being
recognised.
Although his death is noted in a modest anonymous entry in {\it Nature} (August
8,
1895, p.345), he is not mentioned in Vol.XXII (1905) of the {\it Russkii
Biograficheskii Slovar}, even though lesser mathematicians and other Chebyshevs
are.
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\begin{thebibliography}{3}
\bibitem{0}
Chebyshev, P.L. (1955). {\it Izbrannie Trudy}. {[Selected Works.]} AN SSSR,
Moscow.
\bibitem{1} Gnedenko, B.V. and Sheynin, O.B. (1978). The theory of probabilities. [in
Russian].
In: {\it Matematika XIX Veka} \ [Mathematics of the XIX
Century], A.N. Kolmogorov
and A.P. Yushkevich (eds.) Nauka, Moscow. pp. 184-240.
\bibitem{2} Heyde, C.C. and Seneta, E. (1977). {\it I.J. Bienaym\'{e}: Statistical Theory
Anticipated.}
Springer, New York.
\bibitem{3} Nikoforovsky, V.A. (1992). {\it Veroiatnostnii Mir} \ [The World of Probability].
Nauka, Moscow.
\bibitem{4} Prudnikov, V.E. (1976). {\it Pafnuty Lvovich Chebyshev.} \ Nauka, Leningrad.
\bibitem{5} Seneta, E. (1982). Chebyshev (or Tch\'{e}bichef), Pafnuty Lvovich. \
{\it Encyclopedia of Statistical Sciences} (S. Kotz and
N.L.
Johnson, eds). {\bf 1}, 429-431.
\bibitem{6} Youschkevitch, A.P. (1971). Chebyshev, Pafnuty Lvovich. \ {\it Dictionary of}
{\it Scientific Biography} (C.C. Gillispie, ed.), {\bf
3}, 222-232.
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\hfill E. Seneta
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%\hfill { Summary inserted by E.S. 21.3.2000 }
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