Theory of the Combination of Observations Least Subject to by Carl Friedrich Gauss

By Carl Friedrich Gauss

Within the 1820s Gauss released memoirs on least squares, which comprise his ultimate, definitive remedy of the realm besides a wealth of fabric on chance, records, numerical research, and geodesy. those memoirs, initially released in Latin with German Notices, were inaccessible to the English-speaking group. the following for the 1st time they're amassed in an English translation. For students drawn to comparisons the publication comprises the unique textual content and the English translation on dealing with pages. extra in most cases the booklet might be of curiosity to statisticians, numerical analysts, and different scientists who're drawn to what Gauss did and the way he set approximately doing it. An Afterword through the translator, G. W. Stewart, locations Gauss's contributions in ancient standpoint.

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Extra info for Theory of the Combination of Observations Least Subject to Errors: Part One, Supplement

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Which in turn range over all values corresponding to values of y between 77 and 77'. dy taken over all values of y from y = — oo to y — +00 may be obtained from the integral taken from x = —oo to x = +00, from x' — —oo to x1 = +00, etc. 14. dy taken over all values of y — the mean value of y — is equal to the sum of the terms where the integration extends from x = —oo to x = +00, from x' = — oo to x' = +00, etc. , x"7, etc, have been replaced by their mean values. The truth of this important theorem could easily have been established from other considerations.

Et addendo, obtinemus: quae aequatio manifesto consistere nequit, nisi simul fuerit 0 = 0, & = 0, 9" = 0, etc. Hinc primo colligimus, necessario esse debere K = 0. Dein aequationes (I) docent, functiones v, v', v", etc. ita comparatas esse, ut ipsarum valores non mutentur, si valores quantitatum x, y, z, etc. capiant incrementa vel decrementa ipsis Fj G, H, etc. resp. proportionalia, idemque manifesto de functionibus V, V, V", etc. valebit. Suppositio itaque consistere nequit, nisi in casu tali, ubi vel e valoribus exactis quantitatum V, V, V", etc.

Atque, pro harum valoribus veris error in determinatione valor ipsius T, e valoribus observatis ipsarum V, V, V", etc. , = Ef atque error medius in ista determinatione metuendus = \/KKmm + K'K'm'm' + K"K"m"m" + etc.. Errores E, E' vero manifesto ab invicem iam non erunt independentes, valorque medius producti EE'', secus ac valor medius producti ee', non erit = 0, sed = KXmm + K/A'ra'ra' + K/fX"m"m" + etc. IV. Problema nostrum etiam ad casum eum extendere licet, ubi valores quantitatum V, V, V", etc.

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