10
Vlacq after 1628
Vlacq lived in Gouda until 1633, working probably on the construction of the new
tables in the Trigonometria Artificia/is of 1633. He published in 1632 the
Ephemerides Motuum Coe/estium [6d], with tables for the positions of stars and
planets for the years 1633 to 1636. The tables were taken from Tabu/ae Motuum
Coe/estium by the Flemish astronomer Philip van Lansberg and from Kepler's
Rudoiphine Tables, but Vlacq edited the introduction and the instructions for use.
Trigonometric Tables
The massive amount of calculations needed in astronomy, was the primary motive for
inventing the logarithm. That is the reason why Napier started with logarithms of sines.
But many equations in trigonometry require the logarithms of both
numbers and sines, tangents etc. Briggs started with calculating logarithms of
numbers for his Arithmetica Logarithmica, and only in 1633 his work on
trigonometric logarithms was published [7].
It was actually Edmund Gunter who published the first combined
logarithmic table of sines, tangents and numbers in his Canon Trianguiorum
(1620, see [2]). He copied the logarithms of numbers from the Chiiias Prima of his
friend and colleague Briggs, but the logarithms of the sines and tangents must
have been calculated by himself, possibly with some help of Briggs.
It should be noted - to his credit - that Vlacq in his 1628 Arithmetica
Logarithmica, Part II, had added trigonometric tables, and with a higher precision
than Gunter had published in 1620 (10 decimals in stead of Gunter's 7 decimals).
This means that Vlacq must have recalculated (by himself, or with others) the
logarithms of sines and tangents, because his own logarithms of numbers had not
enough precision to be used for table look-up. This was a computational
accomplishment which has been largely neglected by most researchers.
Another large effort must have been spent by Vlacq to construct the
trigonometric tables (per 10 seconds of arc) in his 1633 Trigonometria Artificia/is.
This work was very close - and even competitive - to Briggs' last table, the
Trigonometria Britannica of that same year, which gave trigonometric logarithms
to a higher precision of 14 decimals for sines (10 decimals for tangents), but by
"centesimal" intervals of 1/100 degree. Briggs (and also Gunter, although his
Canon Trianguiorum had been sexagesimal) were early advocates of the
centesimal system for angle units, but the world was not yet ready for that. It
appears that Vlacq's sexagesimal-based table was more popular with the public.
Some researchers have turned this issue upside-down, claiming that Vlacq,
by publishing his sexagesimal tables, has blocked acceptance of the centesimal
units at that time (sic!). Suspicions that Vlacq copied in his own book parts from
Briggs' tables, are unfounded as the sexagesimal and the centesimal ranges do
not coincide except in some whole minutes. It should be noted that Gellibrand's
extensive introductory chapters in Trigonometria Britannica have been reused in
Vlacq's Trigonometria Artificia/is (with or without Gellibrand's agreement).
