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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 **of**least squares became generally accepted, in particular because theobservational errors were (and are) approximately normal and the work **of**Quetelet (§ 10.5) and Maxwell (§ 10.8.5) did much to spread the idea **of**normality whereas his mature contribution (1823b) was extremelyuninviting. However, the pro**of** **of** formula (6a), from which the condition **of**least 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 **of**his 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 **of**extraneous 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 **of**observations 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