9873

Gauss on Vega's Thesaurus Logarithmorum Completus

In [2] Gauss reviewed Vega's Thesaurus Logarithmorum Completus [5]. The trigonometric parts of these tables reproduce Vlacq's tables from 2 to 45 degrees, giving the logarithms of the trigonometric functions at 10-second intervals. From 0 to 2 degrees, the functions were interpolated for every second by Lieutenant Dorfmund under Vega's direction [4, p. 5; 1, p. 7]. Gauss observed that tabular results in the sine column were always equal to the sum of the corresponding tabular results in the cosine and tangent columns. But this relation is not always true for rounded values. He also compared samples of Vega's values with more accurate new computations and estimated that there were 47,746 inaccurate values in the trigonometric part of the Thesaurus. But he only found errors at the 10th decimal place.
In this Demonstration, Vlacq's part of Vega's trigonometric tables are reconstructed. The differences between rounded and rounded minus rounded are also given. Note that instead of Vega's (and Vlacq's) tables consist of . In tables, "Dif." means the absolute difference between two consecutive values in the preceding column.

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In the introduction to [5], Vega offered one ducat (a coin of 3.5 g of almost pure gold) for the first announcement of each error in his tables. Due to the work of Gauss, this was not a wise idea. Gauss also mentioned that 138 mistakes had been listed in [3, pp. 350–351], when Vega was still alive.
References
[1] G. Faustmann. "Jurij Vega—The Most Internationally Distributed Logarithm Tables." (Oct 13, 2004) www.rechenschieber.org/vega.pdf.
[2] C. F. Gauss, "Einige Bemerkungen zu Vega's Thesaurus Logarithmorum," Astronomische Nachrichten, 32(756), 1851 pp. 181–188.
[3] J. P. Hobert and L. Ideler, Neue Trigonometrische Tafeln für die Decimaleintheilung des Quadranten, Berlin: Realschulbuchhandlung, 1799.
[4] D. Roegel. "A Reconstruction of Adriaan Vlacq's tables in the Trigonometria Artificialis (1633)." (Nov 26, 2011) locomat.loria.fr/vlacq1633/vlacq1633doc.pdf.
[5] G. Vega, Thesaurus Logarithmorum Completus, ex Arithmetica Logarithmica, et ex Trigonometria Artificiali Adriani Vlacci, Leipzig: Weidmann, 1794.
[6] A. Vlacq, Trigonometria Artificialis, Gouda, The Netherlands: Pieter Rammazeyn, 1633.
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