1 Oscar Sheynin History of Statistics Berlin, 2012 ISBN 978-3 ...

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1 Oscar Sheynin History of Statistics Berlin, 2012 ISBN 978-3 ...

evealing the influence of systematic errors as much as possible. Inparticular, during observations at a given station, formula (6b) should not berelied upon; indeed, Gauss observed each angle at each triangulation stationuntil being satisfied that further work was useless. The rejection of outliersremains a most delicate problem; at best, statistical criteria are onlymarginally helpful.Gauss’ opinion notwithstanding, his first justification of the principle ofleast squares became generally accepted, in particular because theobservational errors were (and are) approximately normal and the work ofQuetelet (§ 10.5) and Maxwell (§ 10.8.5) did much to spread the idea ofnormality whereas his mature contribution (1823b) was extremelyuninviting. However, the proof of formula (6a), from which the condition ofleast squares immediately follows, is not difficult; Gauss himself providedit; it demands linearity and independence of the initial equations (2) andunbiassedness of the sought estimators of the unknowns. Therefore, it ismethodically possible to disregard Gauss’ main extremely difficultjustification of that condition, to rest content with the actual secondsubstantiation. Then, it will not be practically necessary to restrict thedescription of least squares by his initial reasoning of 1809.Why then did not Gauss himself change his description accordingly? Atleast he could have additionally mentioned the alternative. May (1972/1977,p. 309) provided a general comment which likely answers my question: Inparticular, by careful and conscious removal from his writings of all trace ofhis heuristic methods [Gauss] maintained an advantage that materiallycontributed to his reputation. Much earlier, Kronecker (1901, p. 42) voicedthe first part of May’s pronouncement.Examples of deviation from the normal law were accumulating both inastronomy and in other branches of natural sciences as well as in statistics,see the same § 10.5 (and the missed opportunity mentioned in § 9.3), whichsupported the rejection of the first substantiation of the principle of leastsquares.Tsinger (1862, p. 1) wrongly compared the importance of the Gaussianand the Laplacean approaches:Laplace provided a rigorous [?] and impartial investigation […]; it canbe seen from his analysis that the results of the method of least squaresreceive a more or less significant probability only on the condition of alarge number of observations; […] Gauss endeavoured, on the basis ofextraneous considerations, to attach to this method an absolute significance[…]. With a restricted number of observations we have no possibility at allto expect a mutual cancellation of errors and […] any combination ofobservations can […] equally lead to an increase of errors as to theirdiminution.Tsinger lumped together both justifications of the principle of leastsquares due to Gauss. Then, practice demanded the treatment of a finite (andsometimes a small) number of observations rather than limit theorems.Tsinger’s high-handed attitude towards Gauss (and his blind respect forLaplace) was not an isolated occurrence, see § 12.2-5. Even a recent author(Eisenhart 1964, p. 24) noted that the existence of the second Gaussian83

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