C. F. Gauss and geodetic observations
The author gives an investigation on the precision and the dependence of observations. At the time of Gauss, numerical analysis was not yet developed, Gauss did not publish his ideas on the propagation of errors in his geodetic calculations. There are two main questions concerning the theory of errors: 1. How do errors influence the final result? 2. Which observation is the best among several ones? Tobias Mayer developed first ideas in regarding precision. Daniel Bernoulli stated his law without giving a proof. In 1809 Gauss published his ideas on the measurements of precision by means of the weight of the arithmetic mean: \[ \overline x={e^ 2_ 1x^ 2_ 1+e^ 2_ 2x^ 2_ 2+\cdots+e^ 2_ nx^ 2_ n\over e^ 2_ 1+e^ 2_ 2+\cdots+e^ 2_ n}, \] \(e_ i=\) weight of observations. He also determined the relative measure of precision of a system of linear equations in calculating the square root of weights. The probable error was first discussed by Lambert, D. Bernoulli and Delambre. Gauss contributed various methods of calculation in the case of normally distributed observations. In 1823, however, Gauss also developed the mean square error of indirect observations: \(m=\sqrt{{[vv]\over n-\rho}}\), \(\rho=\) number of the unknowns, in the case of direct observations \(\rho=1\). In 1814 Gauss had still calculated with \(\rho=0\). In 1975 Lyapunov was able to give a proof of this law on the basis of modern terminology. The influence of several observations was also discussed by Struve (1824), Cournot (1843) and Schreiber (1879) as well as by some later authors. The estimation of precision is connected with the rejection of outlying observations. Gauss mentioned this possibility, but did not want to make use of it generally. Gauss also mentioned the independence of observations and discussed cases where independence is not given. Though correlation theory was well known in the 19th century in biology and statistics, it was introduced in astronomy only in 1912 by Kapteyn. The method of least squares was first published by Gauss in 1809. Gauss also assumed that the arithmetic mean of observations was the most probable value, he thought the principle of least squares to be an axiom. There were lots of controversies on this statement.
scientific article; zbMATH DE number 519640
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