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 ...

Although Gauss may well have been telling the truth about his prior useof the method, he was unsuccessful in whatever attempts he made tocommunicate it before 1805.Gauss’ claim about his early use of least squares is not generallyaccepted, see for example Marsden (1995, p. 185), who nevertheless had notmentioned the opposite opinion of Brendel (1924) and Galle (1924, p. 9) orof Gauss’ contemporaries. Gerardy (1977), drawing on archival sources,discovered that Gauss, in 1802 – 1807, had participated in land surveying(in part, for his own satisfaction) and concluded, on p. 19 (note 16) thatGauss started using the method not later than in 1803. Regrettably, Gerardyconcentrated on describing Gauss’ simple calculations and his statementmentioned just above was not quite definite. Concerning these testimonies,it is not amiss to recall Gauss’ opinion (W-14, pp. 201 – 204) about theapplication of the theory of probability as discussed in a letter of 1841 by W.E. Weber: An approach only based on numbers could be greatly mistaken,the nature of the studied subject also ought to be taken into account.There are many other instances including that mentioned by von Zach(above) in which Gauss could have well applied his invention at least forpreliminary, trial calculations, or short cuts. For him (Gauss 1809, § 185),least squares were not a cut and dry procedure; he allowed himselfapproximate calculations. Then, possible mistakes in calculations andweighing the observations could have made justification impossible.9.2 HelmertHe mainly completed the development of the classical Gaussian theory oferrors and some of his findings were interesting for mathematical statistics.Until the 1930s, Helmert’s treatise (1872) remained the best source forstudying the error theory and the adjustment of triangulation. Whenadjusting a complicated geodetic net, Helmert (1886, pp. 1 and 86)temporarily replaced chains of triangulation by geodetic lines. Hisinnovation had been applied in the Soviet Union. The chains of the nationalprimary triangulation were situated between baselines and astronomicallydetermined azimuths. Before the general adjustment of the entire system,each chain was replaced by the appropriate geodetic line; only they wereadjusted, then the chains were finally dealt with independently one fromanother.Elsewhere Helmert (1868) studied various configurations of geodeticsystems. In accordance with the not yet existing linear programming, heinvestigated how to achieve necessary precision with least possible effort,or, to achieve highest possible precision with a given amount of work. Someequations originating in the adjustment of geodetic networks are not linear,not even algebraic; true, they can be linearized, and perhaps some elementsof linear programming could have emerged then, in 1868, but this had nothappened. Nevertheless, Helmert noted that it was expedient to leave someangles of particular geodetic systems unmeasured, but his remark was purelyacademic: all angles ought to be measured at least for checking the work asa whole.Abbe (1863) derived the chi-square distribution, see also Sheynin (1966)and M. G. Kendall (1971), as the frequency of the sum of the squares of n86

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