Conclusion
17
Many disputes have been fought in the world of science over the question
"who was first". For example, Napier was certainly the first to publish the concept
of logarithms, but many researchers have tried to make him share more of the
credit with Jost Brgi, or even Longomontanus, Edward Wright and the long list
of those having discovered their own "addition of exponents". We have now seen
the story of Briggs, Vlacq and De Decker in more detail, and we conclude that
the first logarithmic table of numbers from 1 to 100,000 was really completed
by Vlacq and De Decker together, building on the initial computations of Briggs.
In this endeavour, Vlacq's character is not without stains, as he never mentioned
De Decker's contribution in any of his publications, not even his name!
Acuracy of Logarithmic Tables
Early handmade tables contained innumerable errors. The first Vlacq tables
appear to have had hundreds of errors, but many were in the last decimal,
which is not too bad for practical use if the precision covers already 10 decimals.
Then the printer added his own "typo's" in arbitrary decimals, which could be
recognised as standing out in the gradual diminishment of the differences
between successive entries (by the way, most "small" Vlacq tables did not have
differences printed in separate columns, like later tables had). So it is not
surprising that many reprints boasted the inclusion of new corrections, for
example Vlacq's 1742 edition, advertised as "Editio ultima ab innumeris mendis,
quibus postrema scatebat, purgata" ("Ultimate edition, cleansed of the
innumerable errors with wich the last edition was infested"). The only useful
aspect of errors in logarithmic tables has been that - as trace markers - they
allowed literal copies of the tables to be detected in other publications, see [12],
As an example is given the entry in 7-digit tables for the number 9482: it
should have been rounded off to 9769000, but editors of different tables have
copied many times the wrong entry 9768999.
Such lower output precision provided a better balance with the input
precision, because a relative overprecision in the logarithmic values would
become lost in the transformation back to the number domain.
The "differences" between subsequent table entries (indicating the slope
of the logarithmic function) are high in the lower regions and small in the higher
regions, a characteristic of the "anti-exponential" curve. This means that linear
interpolation between two adjacent values, to get an additional decimal of input
precision, gives less precise results in the lower regions. Therefore it is useful to
extend tables with one additional input decimal in a lower region. For this
reason many logarithmic tables have been published with input ranges extended
to - for example - 108,000, or even to 200,000. At least one specimen of the
Arithmetica Logarithmica exists with a range extension to 101,000, see [23].
