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\noindent{\bf Carl Friedrich GAUSS}\\
b. 30 April 1777 - d. 23 February 1855
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\noindent{\bf Summary.} Gauss shaped the treatment of observations into
a practical tool. Various principles which he advocated became an
integral part of statistics and his theory of errors remained a major focus of
probability theory up to the 1930s.
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Gauss was born on 30 April,
1777 in Brunswick, Germany, into a humble family and
attended a squalid school. At the age of ten, he became friendly
with Martin Bartels, later a teacher of Lobachevsky. Bartels, an
assistant schoolmaster in Gauss's school,
studied mathematics together with Gauss and introduced him to
influential friends. From 1792 to 1806 Gauss was financially
supported by the Duke of Brunswick. He was thus able to graduate
from college (1796) and G\"ottingen University (1798). He then
returned to Brunswick and earned his doctorate from Helmstedt
University (1799). Only in 1807 did Gauss become director of the
G\"ottingen astronomical observatory (completed in 1816), and his
further life was invariably connected with that
observatory (and the university of the same city). Gauss was
twice married and had several children, but none became
scientists. He died in G\"ottingen on 23 February, 1855.
Gauss is regarded as one of the greatest mathematicians of all time. He
deeply influenced the development of many branches of mathematics
(e.g., algebra, differential geometry) and initiated the theory of
numbers; he was an illustrious astronomer and geodesist, and
together with Weber he essentially contributed to the study of
terrestrial magnetism. Gauss's importance for developing the
mathematical foundation for the theory of relativity was
``overwhelming" (Einstein, quoted by Dunnington on p.349 without an
exact reference). His command of ancient languages was
exceptional, and, until he discovered the possibility of
constructing a regular 17-gon with a straightedge and compasses, he
had remained undecided whether to pursue mathematics or philology.
He also possessed an admirable style in his mother tongue.
Gauss was vey slow in making known his findings, many of which were
published posthumously. He kept silent about his studies of the
``anti-Euclidean" geometry, as he called it, although he
(successfully) nominated Lobachevsky for Corresponding Membership
of the G\"ottingen Royal Scientific Society. He was showered with honours
from leading academies. In 1849, he became honorary citizen of
Brunswick and G\"ottingen. He had no peers in science and remained
isolated, partly because of his own disposition.
He was
reluctant to refer to other authors and did not befriend younger
scholars (e.g., Jacobi, Dirichlet). Gauss attached great importance
to such problems as the relation of man to God, but thought that
they were insoluble. He believed in enlightened monarchy;
however, in a letter, he wrote about the golden age to be expected
in Hungary after the 1848 revolution.
As an astronomer, Gauss is best known for determining the orbits of
the first minor planets from a scarce number of their observations
and calculating their perturbations, and for contributing to
practical astronomy. He and Bessel
independently originated a new stage in
experimental science by introducing thorough examination of
instruments. Gauss also detected the main systematic errors of
angle measurements in geodesy and outlined means for eliminating
their influence. For about eight years he directly participated in
triangulating Hannover. After 1828, he continued to supervise the
work, which ended in 1844, and he alone performed all the
calculations. His celebrated investigation of curved surfaces and
study of conformal mapping were inspired by geodesy.
Gauss solved several interesting probability-theoretic problems. In
1841, Weber described Gauss's opinion that, in its applications,
probability should be supplemented by ``other knowledge" and that
the theory offers clues for life insurance and for determining the
necessary numbers of jurors and witnesses. Gauss also studied the
laws of infant mortality and for several years directed the widows'
fund at G\"ottingen University.
The treatment of observations occupied Gauss at least from 1794 or
1795. He decided that redundant systems of physically independent
linear equations should be solved according to the principle of
least squares. He applied it in his astronomical and geodetic work
and recommended it to his friends. In 1809 he published its
justification. Issuing from a postulate that the arithmetic mean
of direct measurements of a constant should be assumed as its
value, and making use of the principle of maximum likelihood, he
arrived at the normal distribution of observational errors as their
only possible even and unimodal law. He also supposed a uniform
prior distribution of errors; this, hovever, was already implied by
his postulate.
He substantiated maximum likelihood by the
principle of inverse probability.
Gauss claimed to be the inventor of least squares
although Legendre (q.v.) had introduced it publicly (without
justification) in 1805. For him, priority always meant being first
to discover.
In 1823 (supplemented in 1828) Gauss put foward a new
substantiation of least squares pointing out that an integral
measure of loss (and, more definitely, the principle of minimum
variance) was preferable to maximum likelihood, and abandoning both
his postulate and the uniqueness of the law of error. (The normal
law still holds, more or less, on the strength of the central limit
theorem. In 1888, Bertrand (q.v.) nastily remarked that for small values of
$|x|$ any even law $f(x) = a^2 - b^2 x^2\approx\alpha^2\exp
(-\beta^2 x^2)$.) Also in 1823, Gauss offered, for unimodal
distributions, an inequality of the Bienaym\'e - Chebyshev type,
and another one, for the fourth moment of errors, as well as the
distribution-free formula for the empirical variance, $m^2$, and
for its own variance, $Dm^2$. Owing to an elementary error, his
$Dm^2$ was wrong; first Helmert (1904), then Kolmogorov et al
(1947) corrected it. Gauss set high store by the formula
for $m^2$
which provided an unbiased estimate of $\sigma^2$; however, in
geodetic practice precision is characterized by its biased
estimator, $m$, and Helmert thought that only relative unbiasedness
was important.
Gauss also estimated the precision of the estimators of the
unknowns of his initial linear system, and of linear functions of
these. Partly owing to his apt notation, his method of solving
normal equations by eliminating the unknowns one by one became
standard. He also applied iterative processes (as described
by Dedekind), and he
introduced recursive least squares which mathematicians did not
notice until recently. In 1816, he proved that, for normally
distributed errors $e_i$, the measure of precision $h =
1/\sqrt{2\sigma}$ was best estimated by $(e_1^2 + e_2^2 + \ldots +
e_n^2)$, rather than by $(|e_1|^{k} + |e_2|^{k} + \ldots +
|e_n|^{k})$, for any other integer $k$.
Gauss's contribution to the treatment of observations somewhat
extended by Helmert defined the state of the classical theory of
errors. It seemed perfect, and geodesists hardly paid attention to
statistics, whereas statisticians hardly studied Gauss and thus
missed the opportunity to develop analysis of variance and
regression with less effort. This situation did not begin changing
until well into the 20th Century.
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\begin{thebibliography}{3}
\bibitem{1} C.F. Gauss:
\bibitem{2} 1973-1981 {\it Werke}, Bde 1-12. Hildesheim.
This is a reprint of the previous edition (1870-1930) which
includes
contributions (with separate paging) on the work of Gauss in
geodesy (A. Galle) and in astronomy (M. Brendel), both in Bd. 11,
Tl. 2; and on Gauss as a calculator (Ph. Maennchen, in Bd.10, T1.
2).
\bibitem{3} 1975-1987 {\it Werke}, Ergaenzungsreihe, Bde 1-5.
Hildesheim. Reprints of previously published correspondence of
Gauss.
\bibitem{4} 1964 {\it Abhandlungen zur Methode der kleinsten Quadrate}.
Eds, A. Boersch \& P. Simon. Wuerzburg. Reprint of the edition of
1887. Contains translations from Latin/reprints of all Gauss's
works on the theory of errors (mainly 1809; 1811; 1816; 1823; 1828,
as well as their abstracts compiled by Gauss himself). Russian
transl. of same material with foreword by G.V. Bagratuni; Gauss
(1957). French transl. of essentially the same material, Paris
1855.
\bibitem{5} 1957 - 1958 {\it Izbrannye geodezicheskie sochinenia} (Sel. geod.
works), vols.1-2. Moscow.
\bibitem{6} 1995 {\it Theory of the combination of observations least subject
to error}. Latin and English. Transl. with Afterword by G.W.
Stewart. Philadelphia.
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\bibitem{7} Other authors:
\bibitem{8} Dedekind, R. (1931). Gauss in seiner Vorlesung ueber die Methode
der kleinsten Quadrate (1901). {\it Ges. math. Werke}, Bd.2,
Braunschweig, pp. 293-306.
\bibitem{9} Dunnington, G.W. (1955). {\it C.F. Gauss, Titan of Science}. New
York.
\bibitem{10} Forbes, E.G. (1978). The astronomical work of C.F. Gauss.
{\it Historia Mathematica}, {\bf 5}, 167-181.
\bibitem{11} Hald, A. (1998). {\it A History of Mathematical Statistics from
1750 to 1930}, Wiley, New York. See also the contributions of Helmert (1872),
Plackett (1972), Seal (1967) and Sprott (1978) referenced therein.
\bibitem{12} Reichardt, H., Editor (1957). {\it C.F. Gauss. Gedenkband}. Leipzig.
Includes O. Volk, Astronomie und Geodaesie bei C.F. Gauss, pp.
205-229. The book was reprinted (Berlin, 1960) as {\it C.F. Gauss.
Leben und Werk}.
\bibitem{13} Sartorius von Waltershausen (1965). {\it Gauss zum Gedaechtniss}.
Wiesbaden. First Published in 1856.
\bibitem{14} Sheynin, O.B. (1979; 1993; 1994). Three papers from {\it Archive for
History of Exact Science}. Among the references in the first of these see Loewy (1906),
May (1972), Sofonea (1955) and Subbotin (1956).
\bibitem{15} Sheynin, O.B. (1996). {\it The history of the theory of errors},
this being Deutsche
Hochschulschriften No.1118. Egelsbach a.o.
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\hfill{O.B. Sheynin}
\end{thebibliography}
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