## Info

Published in Challis (1830).

English translation in Gauss (1963). From the Preface: 'Several astronomers wished me to publish the methods employed in these calculations immediately after the second discovery of Ceres; but many things ... prevented my complying at the time with these friendly solicitations. I .. .have no cause to regret the delay. For, the methods first employed have undergone so many and such great changes, that scarcely any resemblance remains between the method in which the orbit of Ceres was first computed, and the form given in this work.' Gauss did send a brief manuscript containing his early ideas to Olbers in 1802 (Dunnington (1955)).

practice to try and make J2 i \ei I as small as possible. In other words, one should minimize n s =

1 Xi + ... + Q6X6 - bi)2, which canbe achieved by setting all the derivatives ds/dXj, j = 1,... , 6 equal to zero. This leads to a 6 x 6 linear system of equations which can be solved by standard methods.

Legendre pointed out that if one has a set of observations corresponding to just one unknown, then the method of least squares suggests that the arithmetic mean of the observations is the most appropriate choice, exactly as one would like, and in contrast to the conclusions from Laplace's theory of errors (see pp. 343-4). Gauss took this property of the arithmetic mean as his starting point:

It has been customary certainly to regard as an axiom the hypothesis that if any quantity has been determined by several direct observations, made under the same circumstances and with equal care, the arithmetical mean of the observed values affords the most probable value, if not rigorously, yet very nearly at least, so that it is always safe to adhere to it.

Working from this assumption, he calculated that the error distribution should have density

¿(x ) = A e-h2 x 2, n in which the parameter h can be considered as a measure of the precision of the data, since the probability that x e (-a, a) for a distribution with parameter h is the same as the probability that x e ( - Q , Q )fora distribution with parameter ch (c > 0).32

Gauss then went on to examine the consequences of his form for \$. In particular, if one has a set of observations ti, then the probability of the unknown true value lying between r and r + dr is proportional to exp[-h2 Y,i (ti - r)2] and so the most probable value of r is found by minimizing Y,i (ti - r )2. Thus, Gauss provided a probabilistic justification for the method of least squares. He also showed how a set of observations, each of different accuracy, should

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