Possible Calculation of Logarithms of Cosines in Vlacq's Trigonometria Artificialis

This Demonstration shows that the logarithms of cosines can be calculated using tables of trigonometric functions calculated to 15 decimals (such as the tables of Pitiscus) and tables of logarithms calculated to 10 decimals (such as Vlacq's 1628 tables) with aid of linear interpolation. Eventual errors are only one unit in the last decimal. In the grid, only the fractional part of is given.


  • [Snapshot]
  • [Snapshot]


Errors that Gauss found in the trigonometric part of Vega's Thesaurus were actually from Vlacq's Trigonometria Artificialis.
According to [3, p. 5], Vlacq (1633) first computed the logarithms of the cosines from 0 to 45 degrees. This was done using the sines given by Rheticus in the Opus Palatinum (1596) where they are tabulated to 10 decimal places every 10 seconds, the same step as the one used by Vlacq. There are 16,200 logarithms and they could have been computed using Briggs' radix method.
According to [2, p.16], Vlacq's tables from 1628 were based on Pitiscus' Thesaurus Mathematicus, computing logarithms of known sines using the radix method.
[1] C. F. Gauss, "Einige Bemerkungen zu Vega's Thesaurus Logarithmorum," Astronomische Nachrichten, 32(756), 1851 pp. 181–188.
[2] D. Roegel. "A Reconstruction of De Decker-Vlacq's Tables in the Arithmetica Logarithmica (1628)." (Dec 6, 2010) hal.inria.fr/docs/00/54/39/41/PDF/vlacq1628doc.pdf.
[3] D. Roegel. "A Reconstruction of Adriaan Vlacq's Tables in the Trigonometria Artificialis (1633)." (Nov 26, 2011) locomat.loria.fr/vlacq1633/vlacq1633doc.pdf.
[4] D. Roegel, "A Reconstruction of the Tables of Pitiscus' Thesaurus Mathematicus (1613)." (Dec 6, 2010) hal.inria.fr/docs/00/54/39/33/PDF/pitiscus1613doc.pdf.
[5] D. Roegel, "A Reconstruction of the Tables of Rheticus' Opus Palatinum (1596)." (Dec 6, 2010) hal.inria.fr/docs/00/54/39/32/PDF/rheticus1596doc.pdf.
[6] G. Vega, Thesaurus Logarithmorum Completus, ex Arithmetica Logarithmica, et ex Trigonometria Artificiali Adriani Vlacci, Leipzig: Weidmann, 1794.
[7] A. Vlacq, Trigonometria Artificialis, Gouda, The Netherlands: Pieter Rammazeyn, 1633.
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.

Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-Step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2018 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+