ATy A'Ax-
Note that this result corresponds to the following choice of the L matrix
L— A'
After solving his set of equations for the three unknowns (py. d and/, Legendre obtained for
the flattening the value1/148. which he recognizes as being too large. He therefore recom
puted his least-squares adjustment, but now with constrained to the at that time adopted
value for the Earth's flattening. As a result he obtained a value for d. which was now close to
the value obtained earlier by Laplace and on which the actual definition of the meter was
based. The reason for Legendre having to constrain lies in the poor resolution of his data,
lite total arc length of his data covered only about 10 degrees.
Although Legendre did not give a clear motivation for his 'least-squares' criterion, he did
realize its potential. His method used all the observations, had an objective criterion and
most importantly, residted - as opposed to Boscovich' method - in a solvable linear system of
equations, l ite method met with almost immediate success. Within ten year after Legendre's
publication, the method of least-squares became a standard tool in astronomy and geodesy
in various European countries, and within twenty years, also the probabilistic foundations of'
the method were largely completed, the main contributors being Laplace and Gauss.
1 Based on a presentation given at the occasion of the official opening of the Geodetic-Astronomical
Observatory Westerbork, 24 Sept. 1999.
2 Mathematical geodesy covers the development of theory and its implementation as is needed in order to
process, analyse, integrate and validate the various geodetic data. It concerns itself with the calculus of
observations (adjustment and estimation theory), with the validation of mathematical models (testing and
reliability theory) and with the analysis of spatial and temporal phenomena (interpolation and prediction
theory). Founders of the Dutch School of Mathematical Geodesy, internationally also known as the 'Delft
School', are the professors J.M. Tienstra (1895-1951) and W. Baarda.
3 Stigler, S.M. (1986): The History of Statistics, Belknap, Harvard. Hald, A. (1998): A History of Mathematical
Statistics, Wiley-lnterscience.
4 Mayer, T. (1750): Abhandlung ueber die Umwalzung des Monds urn seine Axe und die scheinbare Bewegung
der Mondsflecten. Kosmogr. Nachr. Samml. auf das Jahr 1748,1, 52-183.
5 Forbes, E.G. (1974): The Birth of Scientific Navigation. Maritime Monographs and Reports, No. 10, National
Maritime Museum, Greenwich, London.
6 Sobel, D. (1995): Longitude. Walker, New York. Dutch translation by E. van Altena (1997): Dava Sobel:
Lengtegraad. Ambo.
7 Newton, I. (1687): Philosophiae Naturalis Principia Mathematica.
8 Boscovich, R.J., Maire, C. (1770): Voyage astronomique et géographique dans l'etat de Tégiise. French
translation of original 1755 publication. For Boscovich' contributions, see also Sheynin, O.B. (1973): Arch.
History Exact Sci., 9: 306-324.
9 In 1795 the French government invited other European governments to delegate scientists to Paris for
completing and checking the computations for the standardization of weights and measures. The Dutch
delegates were professor Jan Hendrik van Swinden and the navy officer Henricus Aeneae. The European
scientists met in Paris in 1798 and reported on their findings in 1799. The report on the meter was given by
van Swinden. The standard meter, a bar of platinum with rectangular section of 254 millimeters, was placed in
the French State Archives, Mètre et Kilogramme des Archives. In 1983 the standard meter was defined as the
length traveled by light in vacuum in 1/299.792.458 seconds.
10 Legendre, A.M. (1805): Nouvelles méthodes pour la détermination des orbites des comètes. (Appendix: Sur la
méthode des moindre carrés).
11 When Carl Friedrich Gauss published his first probabilistic version of the method of least-squares in 1809, he
claimed that he had been using the method ('our principle') already since 1795. This claim resulted in a priority
dispute between Legendre and Gauss, for a discussion see e.g. Placket, R.L. (1972): The discovery of the
method of least-squares. Biometrica 59:239-251.
