By Carl Friedrich Gauss
Within the 1820s Gauss released memoirs on least squares, which comprise his ultimate, definitive remedy of the world besides a wealth of fabric on chance, information, numerical research, and geodesy. those memoirs, initially released in Latin with German Notices, were inaccessible to the English-speaking neighborhood. the following for the 1st time they're amassed in an English translation. For students drawn to comparisons the booklet comprises the unique textual content and the English translation on dealing with pages. extra mostly the ebook might be of curiosity to statisticians, numerical analysts, and different scientists who're attracted to what Gauss did and the way he set approximately doing it. An Afterword via the translator, G. W. Stewart, areas Gauss's contributions in old standpoint.
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Additional resources for Theory of the Combination of Observations Least Subject to Errors: Part One, Supplement
Sample text
Since we have taken the observation errors to be quantities of the first order and have neglected quantities of higher orders, we may use the values of the differential quotients ^, etc. that come from the observed quantities V, V, V", etc. to evaluate our formula instead of A, A', A", etc. Obviously this substitution makes no difference at all when U is a linear function. II. Let p, p', p", etc. be the weights of the observation errors with respect to an arbitrary unit, and let P be the weight of the estimate of U derived from the observed quantities V, V, V", etc.
According to the principles of Art. 6, this error is the square root of the mean value of the function To compute this mean value, it is sufficient to observe that the mean value of a term like ^ is ^ (here n has the same meaning as in Art. 11) and the mean value of a term like 2xxx/x> is —. , we are quite safe in using the formula to obtain an approximate value of m. The mean error to be feared in this estimate (with respect to the square mm) is This last formula contains the quantity n. If we merely wish to form a rough idea of the precision of the estimate, it suffices to make some hypothesis about the function (p.
At pro multitudine observationum modica, res intacta mansit, ita ut si lex nostra hypothetica respuature, methodus quadratorum minimorum eo tantum nomine prae aliis commendabilis habenda sit, quod calculorum concinnitati maxime est adaptata. Geometris itaque gratum fore speramus, si in hac nova argumenti tractatione docuerimus, methodum quadratorum. minimorum exhibere combinationem ex omnibus optimam, non quidem proxime, sed absolute, quaecunque fuerit lex probabilitatis errorum, quaecunque observationum multitude, si modo notionem erroris medii non ad menterm ill.