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\noindent{\bf Adrien-Marie LEGENDRE}\footnote{\\Recent research has shown that the portrait
employed in the hard copy edition of Statisticians of the Centuries and elsewhere is actually that of the politician Louis Legendre (1752--1797).} \\
b. 18 September 1752 -- d. 9 January 1833
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\noindent
{\bf Summary.} In 1805, Legendre published the first description of
the method of least squares as an algebraic fitting procedure. It was
subsequently justified on statistical grounds by Gauss and Laplace.
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Adrien-Marie Legendre was born in Paris (France). He was the son
of
well-to-do parents and could afford to devote himself full-time
to
scientific research until the rigours of the French Revolution
dissipated
the family fortunes and made it necessary for him to work for a
living as a
minor official in a variety of administrative posts.
The French Revolution began in May 1789, King Louis XVI was executed
in
January 1793 and Queen Marie-Antoinette in October 1793. the
subsequent
Reign of Terror hastened the deaths of many scientists including
those of
Antoine Lavoisier and the Marquis de Condorcet (q.v.). It is
fortunate for
the development of science in France that Fourier, Laplace (q.v.), and
Legendre amongst others were able to escape with their lives.
Following the completion of the Revolution in France, the French
government,
latterly under the control of Napol\'{e}on Bonaparte, fought a series
of
wars which drew in most of the countries of Europe. Thus it was in
the year
of the battles of Austerlitz and Jena, that Legendre, then 52 years
old,
published his monograph outlining the fundamental results on the
method of
least squares.
Legendre was educated at the {\it Coll\`{e}ge Mazarin}, graduating in
1770.
Between 1775 and 1780 he taught mathematics at the {\it \'{E}cole
Militaire}.
The geodetic work published by Legendre during this period was so
important
that it merited four chapters in Todhunter's {\it History of
Mathematical
Theories of Attraction and the Figure of the Earth (1873)}. Thus,
when in
1787, the {\it Acad\'{e}mie des Sciences} was asked to nominate three
members of a commission to connect the maps of Britain and France by
determining the relationship between the meridians of Greenwich and
Paris,
it chose Legendre, M\'{e}chain, and J.D.Cassini. In 1791 the {\it %
Acad\'{e}mie} again nominated the same individuals as its members of a
commission to determine the length of the {\it m\`{e}tre} which was
to form
the basis of a new decimal system of mensuration. However, Legendre
resigned
from the second commission in March 1792.
In 1799 Laplace became Minister of the Interior and Legendre
succeeded him
as the examiner in mathematics of students assigned to the artillery
corps.
In 1815 Legendre voluntarily resigned this position on half pay but
was
subsequently stripped of his pension when he refused to vote for the
official candidate in an election to a seat in the {\it Institut de
France}.
Legendre died in Paris in 1833.
In the second half of the eighteenth century astronomers, geodesists,
and
other practical scientists were obliged to perfect their own methods
for
solving systems of equations in which there were more observations
than
unknowns. From antiquity until 1750 the only possible solution to this
problem was to arrange that there should be as many equations as
there were
unknowns. In 1750 Tobias Mayer suggested that the unknowns
should be
determined by setting certain sums of equations equal to zero; in 1760
Boscovich (q.v.) proposed that the unknowns should be determined by
minimising the sum of the absolute errors subject to an adding-up
constraint; and in 1786 Laplace suggested that the unknowns should be
determined by minimising the largest absolute error. Legendre's
method of
least squares clearly represents a further contribution in this
practical
tradition as his argument is entirely algebraic and has no statistical
content.
Legendre's discussion of the method of least squares is to be found
in the
first four pages (pp.72-75) of a nine-page appendix attached to his
1805
work on the determination of the orbits of comets. The remaining five
pages
of this appendix are concerned with an application of the method of
least
squares to the determination of the ellipticity of the Earth, and
hence the
length of the {\it m\`{e}tre}.
By contrast with the Boscovich and minimax procedures which, in
practice,
seem to have been restricted to the case of two unknowns, the method
proposed by Legendre could be applied to any number of linear
equations in
any lesser number of unknowns. Instead of choosing values for the
unknowns
to minimise the largest absolute error or to minimise the sum of the
absolute errors, he chose these values to minimise the sum of the
squared
errors. Legendre wrote down the first order conditions for the
minimisation
of this function. He then noted that each of these conditions may be
obtained by multiplying all the terms in each of the equations by the
coefficient of one of the unknowns and summing the result. He was
thus able
to establish that there were the same number of equations as there
were
unknowns, and hence that the first order conditions could be solved
for the
unknowns by a standard procedure which he did not describe.
Legendre asserted that the solution found by this procedure
corresponds to a
minimum of the sum of squares function . But he did not prove this
result
except in the special case when there is an exact fit and the least
squares
values of the unknowns are associated with a set of zero errors.
Further, if
the least squares solution produces any large errors then Legendre
suggested
that the corresponding observations should be discarded and new least
squares values computed by deleting the corresponding terms from the
calculations. This statement presumably represents an extension of
current
practice to the method of least squares. The problem of deciding when
an
error is large enough to warrant deletion was not discussed. Legendre
gave
very little justification for choosing the sum of the squared errors
as his
optimality criterion other than its computational simplicity. However
he did
note that it yields the arithmetic mean when there are a set of direct
observations on a single unknown quantity. Subsequently, in the
nonstatistical section of his {\it Theoria Motus Corporum Coelestium
(1809)}%
, Gauss (q.v.) essentially reproduced Legendre's deliberations but
offering
the sum of the fourth powers, the sum of the sixth powers, etc as
possible
alternatives to the sum of the squared errors. Of these criteria, the
simplest is indeed the sum of the squared errors, which Gauss says he
had
used in practical calculations since 1795, much to the annoyance of
Legendre
who published a venomous response in 1820.
The method of least squares would seem to have been hanging in the
air at
this time as there are several claimants for the privilege of having
first
invented it including Gauss, Cotes, Simpson, and Huber. Most
of these
claims can be rejected as having been made with the benefit of
hindsight and
in ignorance of the difficulty that eighteenth century mathematicians
would
have experienced in conceiving of minimising a sum of {\it squared}
errors
rather than a sum of absolute errors. However, the claim for priority
made
by Gauss at the time would seem to have been substantiated in the
opinion of
Laplace, Plackett (1972), Stigler (1977, 1981, 1986), and other
authorities,
but Legendre retains the priority of publication, for what that is
worth.
Further, Gauss's practical fitting procedure was designed for use with
nonlinear problems and its precise nature is still open to question,
again
see Stigler (1981) for details.
As noted above, Legendre's derivation of the method of least squares
was
entirely algebraic; statistical justifications for this fitting
procedure
were subsequently provided by Gauss, Laplace, Cauchy (q.v.), and
Thiele
(q.v.) amongst others.
Legendre's system of orthogonal polynomials have also found
applications in
statistics.
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\noindent {\bf References}
\noindent Legendre, A.M. (1805). {\it Nouvelles M\'{e}thodes
pour
la D\'{e}termination des Orbites des Com\`{e}tes}, Firmin Didot, Paris; second edition Courcier, Paris, 1806.
Pages
72-75 of the appendix reprinted in Stigler (1986, p.56). English
translation
of these pages by H.A. Ruger and H.M. Walker in D.E. Smith, {\it A
Source Book
of Mathematics}, McGraw-Hill Book Company, New York, 1929, pp.576-579.
\noindent Legendre, A.M. (1820). Note par M.*** Second supplement to
the
third edition of Legendre (1805), pp.79-80 in a separate pagination.
English
translation by Stigler (1977).
\noindent Plackett, R.L. (1972). The discovery of the method of
least squares, {\it Biometrika}, {\bf 59}, 239-251.
\noindent Stigler, S.M. (1977). An attack on Gauss published
by
Legendre in 1820, {\it Historia Mathematica}, {\bf 4}, 31-35.
\noindent Stigler, S.M. (1981). Gauss and the invention of
least
squares, {\it Annals of Statistics}, {\bf 9}, 465-474.
\noindent Stigler, S.M. (1986). {\it The History of
Statistics: the
Measurement of Uncertainty before 1900}, Harvard University Press,
Cambridge, Massachusetts.
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\hfill{R.W. Farebrother}
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