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\noindent{\bf Pierre-Simon Marquis de LAPLACE}\\
b. 23 March 1749 - d. 5 March 1827
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\noindent{\bf Summary.} Pierre-Simon Laplace was the most
prominent exponent of 19th
century probability theory. His major probabilistic work, the {\sl
Th\'eorie analytique des probabilit\'es} considerably influenced the
development of mathematical probability and statistics right to the
beginning of the 20th century.
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\noindent{\bf Introduction}


Pierre-Simon Laplace was born in Beaumont-en-Auge
(Normandy). His parents, Pierre and Marie-Anne, n\'{e}e Sochon,
lived in comfortable bourgeouis circumstances. Laplace's scientific
career evolved in a period of political upheaval, but it continued
to flourish in all political systems (1789 French Revolution, 1799
Napoleon's seizure of power, 1815 reestablishment of the monarchy).
Originally destinated to become a priest, Laplace soon discovered
his mathematical talents. Supported by d'Alembert (q.v.), he obtained a
professorship at the {\it\'{E}cole Militaire} in 1771. In 1773 he
was admitted to the {\it Acad{\'e}mie des Sciences de Paris} of
which he became one of the leading members in the 1780's. After the
revolution, Laplace played a decisive role in the commission of
weights and measures aiming at the introduction of the metric
system. Around 1795 he became very influential in the organization
and teaching of the newly established {\it{\'E}cole Polytechnique}
and {\it{\'E}cole Normale}. Laplace served only 6 weeks as
Napoleon's unfortunate Minister of the Interior in 1799, thereafter
he was honorably transferred to the Senate of which he became
Chancellor in 1803. Louis XVIII, too, had the highest esteem for
Laplace. In 1816 he was admitted to the 
{\it Acad{\'e}mie Fran\c{c}aise} and in 1817 he was raised 
to the rank of a Marquis. Laplace
died on 5 March 1827 in Paris, having pursued his scientific
research very actively almost to the end of his life.
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\noindent{\bf Scientific Work in General}

Laplace contributed numerous articles to optics, acoustics, heat
and capillarity. In his purely mathematical papers, he dealt mainly
with difference and differential equations. The focal points of
Laplace's scientific activities were theoretical astronomy and
probability theory, which contained, according to Laplace's
approach, also those parts today considered as ``statistics". From
Laplace's point of view, a differentiation between ``probability
theory" and ``statistics" would not have been appropriate. In 1796
the popular {\it Exposition du syst\`{e}me du monde} appeared and
between 1799 and 1805 the first 4 volumes of the {\it Trait{\'e} de
m{\'e}canique c{\'e}leste}, the fifth volume of which was published
by Laplace in 1825 (({\it {\OE}uvres I-V}). In his work on physics and
astronomy, which made him the leading figure within French natural
sciences, Laplace became the most prominent propagator of the idea
that, in principle, each condition in the world could, according to
the pattern of determining position and velocity of celestial
bodies, be precalculated. Laplace put this strictly deterministic
point of view in concrete form by his proverbial ``Laplacean demon".
The {\it Th{\'e}orie analytique des probabilit\'es} (Ist ed. 1812,
2nd ed. 1814, with an extensive introduction and a chapter on the
probability of testimonies, 3rd ed. 1820 with supplements =
({\OE}uvres VII) was the sum of Laplace's probabilistic work since
1774. The introduction which was added to the {\it Th{\'e}orie
analytique} from the 2nd ed. was also published separately between
1814 and 1825 in 5 editions under the title {\it Essai
philosophique sur les probabilit{\'e}s}.
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\noindent{\bf Philosophy of Probability}

Laplace held the view that man, in contrast to the ``demon", was
capable of achieving only partial knowledge about the causes and
laws which regulate the processes of the cosmos, but he maintained
that probability theory was a means to overcome this defiency. In
accordance with this concept, Laplace put special emphasis on
subjective probabilities depending on the degree of information,
but the frequentistic notion of probability is also used in
Laplace's work in many places. Laplace was convinced of the
universal applicability of probability calculus and he summarized
this opinion, which he shared with all probabilists of the
enlightenment, by the words: ``Probability is basically good sense,
reduced to a calculus."
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\noindent{\bf Analytic Methods of Probability Theory}

Laplace considered his form of probability theory, as described in
the {\it Th{\'e}orie analytique}, important not only because of its
universal applicability but also because of its innovative
analytical methods. Actually, no probabilist before Laplace was
able to offer results which could have been compared with the
analytical content of the ones presented by Laplace. Consequently,
the {\it Th{\'e}orie analytique} was divided into two books, the
first dealing exclusively with the analytical apparatus, in
particular with the application of generating functions to
difference equations and techniques for calculating and
approximating definite integrals. Laplace already used, albeit in a
still rudimentary form, characteristic functions for the
representation of the probabilities of sums of independent random
variables. In its emphasis on the analytical importance of
probabilistic problems, especially in the context of the
``approximation of formula functions of large numbers," Laplace's
work goes beyond the contemporary view which almost exclusively
considered aspects of practical applicability.
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\noindent{\bf Bayesian Methods and Population Statistics}

A considerable part of Laplace's contributions, which would be
considered today as belonging to ``mathematical statistics", was
based on inverse probabilities. By this method, the a posteriori
probability of a certain hypothesis could be calculated from the
results of random experiments, usually under the tacit assumption
of an a priori equiprobability of all possible hypotheses. We are
not sure about whether Laplace began his inquiries with or without
a knowledge of Bayes' (q.v.) fundamental treatise (1764) on this issue. By
the aid of suitable approximations to his resultant formulas - a
problem which Bayes had failed to solve - Laplace showed in several
papers, published between 1774 and 1786, that, on the basis of the
existing data, the probability of a boy's birth is, almost
infallibly, greater than 1/2; that the birth rate for boys in
London is in all likelihood greater than in Paris, and so on. One
can suppose that reports on death rates in French hospitals,
published by the {\it Acad{\'e}mie des Sciences}, were also based
on similar Laplacian calculations. Together with Condorcet (q.v.) and
S\'ejour, Laplace was a member of the commission of the {\it
Acad{\'e}mie des Sciences} which organized, in the 1780's, the
publication of several papers concerning population statistics in
all parts of France, based chiefly on data sampled by La
Michodi{\'e}re. In these statistical investigations the idea of a
micro census, as already used by Graunt (q.v.) was pursued: The ratio
between the number of persons and the number of births per year
within a suitable selection of population must be approximately
equal to the ratio between the total number of persons and the
total number of births per year. By a Bayesian approach, Laplace
calculated the probability of the deviation of the estimated value
for the total number of persons from its actual value, if the
estimation was obtained by equating both ratios. Between 1799 and
1802 a micro census was organized for the whole of France according
to ``Laplace's method" (({\it {\OE}uvres} VII, 398-401). Laplace's
interest in population statistics, however, was apparently less
motivated by social or political concerns, than by the scientific
aim of making evident that the social world can basically be
approached by the same probabilistic methods as the physical.
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\noindent{\bf Central Limit Theorem, Asymptotic Error Theory}

Laplace's main probabilistic result was a fairly general central
limit theorem, which was obtained around 1810. This theorem assures
an approximate normal distribution for practically all sums of
independent random variables in nature and society, if only the
number of the summands is large. This result, although it was
nowhere explicitely formulated, but in each case deduced in the
context of its special applications, was to become a leitmotif of
Laplace's {\it Th{\'e}orie analytique}. On the basis of approximate
normal distributions of linear combinations of errors of
observation, Laplace succeeded in showing that the method of least
squares is, according to various criteria, asymptotically ``most
advantageous" for estimating the parameters of linear models which
occur in the context of astronomical or geodetic observations.
Thus, he presented basic ideas of asymptotic statistics within the
scope of error theory. Error calculus also served Laplace as a
pattern for the determination of natural regularities hidden by
irregular fluctuations, such as weather conditions. An important
example was the ``constant" difference of the air pressures in the
morning and in the afternoon. For an assessment of whether assumed
regularities actually existed, Laplace's central limit theorem
allowed a reasoning similar to the one used in modern tests of
significance, provided that the test statistics were sums of a
large number of independent random variables. On the basis of
central limit theorems, Laplace arrived at a probabilistic
discussion of mean errors of observation, mean gains of gambles or
mean durations of life, and in this context one can find statements
which today would be called weak laws of large numbers.
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\noindent{\bf Probability and Moral Sciences}

In continuation of the work of Condorcet, Laplace reflected on the
field of erroneous human decisions, such as testimonies or
verdicts, within the framework of urn models. In view of the
oversimplified models Laplace expressed certain reservations, but
he emphasized at the same time the advantages of probabilistic
``estimations". In the first supplement of his {\it Th{\'e}orie
analytique}, Laplace calculated the a posteriori probability that
the defendent is actually guilty, if $n$, votes have been cast
against him, under the double presupposition that among $n$ members
of a jury the same probability $x$ of a correct decision in the
case of guilt can be assigned to all of them, and that all values
$x$ are a priority uniformly distributed between 1/2 and 1. On the
basis of these calculations, Laplace gave recommendations for the
composition of, and the majority within, Juries, which he also
published in a pamphlet in 1816 (({\it {\OE}uvres} VII, 529f.) and
repeated in a speech at the {\it Chambre des Pairs} ({\it {\OE}uvres}
XIV, 379-381) in 1821. Laplace's arguments were repeatedly brought
forward in the frequent discussions of the 1820's and early 1830's
about jury systems in France. At the same time, however,
probabilistic reasoning within moral sciences was increasingly
criticized by philosophers and mathematicians. This fell especially
upon Poisson, who amplified Laplace's inquiries on moral questions
by the use of a great deal of statistical data. Following Poisson (q.v.),
there has been little active research in this part of classical
probability theory.
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\noindent{\bf Impact of Laplacian Probability}

To the end of the 19th century, Laplace's {\it Th\'eorie
analytique} remained the most influential book of mathematical
probability theory, which was considered less a part of mathematics
in the narrower sense, but a discipline of ``mathesis mixta".
Reduced as it was by a major field of application of the classical
theory, the moral sciences, and augmented only by problems which
could be mastered within the framework of simple stochastic
techniques, such as the kinetic theory of gases, hardly any
probabilistic concepts were put forward which were new with regard
to Laplace's.

In the field of statistics, Laplace had mainly presented
theoretical concepts in a rather unsystematic way in his {\it
Th\'eorie analytique}. His analytical deductions were written in a
very difficult style, and his mode of reasoning within error theory
became far less popular in comparison with Gauss' (q.v.), which was easier
to understand and to apply. The general relevance for statistics of
Laplacian error theory was appreciated only by the end of the 19th
century. However, it influenced the further development of a
largely analytically oriented probability theory; limit
distributions of sums of independent random variables became a
basis of modern probability theory.

In addition, some basic ideas, chiefly disseminated by Laplace in a
verbal form in his {\it Essai philosophique}, decisively influenced
19th century statistics in producing the expectancy, that all
random fluctuations, in nature and society, could be treated
correspondingly to the pattern of errors of observations. This
concept, together with Laplace's frequent approximations by normal
distributions which, however, he did not investigate as statistical
objects in their own right, paved the way for the later
``Quetelism".
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\noindent{\bf Literature}

\noindent
The ( {\it {\OE}uvres compl\`etes de Laplace}, which appeared between
1878 and 1912 (Paris, Gauthier-Villars) are not really complete.
The most comprehensive scientific biography is C.C. Gillispie,
{\it Pierre-Simon Laplace, 1749-1827, A Life in Exact Science}
Princeton University Press, Princeton, 1997.

\noindent
S.M. Stigler's book  {\it The History of Statistics},
Belknap, Cambridge, MA, 1986 contains a detailed historical
discussion of Laplace's most relevant stochastic contributions.

\noindent
Descriptions of mathematical details can be found in O.B. Sheynin,
P.S. Laplace's Work on Probability,  {\it Archive for History of
Exact Sciences}, {\bf 16} (1976), 137-187, idem, Laplace's Theory of
Error, {\it Archive for History of Exact Sciences}, {\bf 17}, (1977), 1-61, and
concerning Bayesian Methods in A.I. Dale's book  {\it A History of
Inverse Probability}, Springer, New York, 1991. 

\noindent
The intellectual
background of Laplace's probabilistic work together with his
treatment of moral questions is discussed in L. Daston's book
{\it Classical Probability in the Enlightenment}, Princeton University
Press, Princeton, 1988. 

\noindent I. Schneider, (1987). Laplace and Thereafter. In  {\it The
Probabilistic Revolution}, Vol. 1, ed. by L. Kr{\"u}ger  {\it et
al}., MIT-Press, Cambridge, MA,  pp. 191-214, gives
a history of the impact of Laplacian probability theory in the 19th
century.

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\hfill{Hans Fischer}


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