The construction of the wonderful canon of logarithms

Notes. 93
adapted to meet any requirements of accuracy. In sec. 60, Napier, in
suggesting the construction of a table of logarithms to a greater number
of places, proposes to take 1 00000000 as radius. The effect of this
would be to throw still further back the error involved in taking the
arithmetical mean of the limits for the true logarithm.
Thus, using the
formula given, substituting 1 00000000 for n, and multiplying the result
by that amount as already explained, we should have for the true
logarithm of 99999999, the first proportional after radius in the new
First
table,
1.000000005000000033333 etc.
If we take the arithmetical mean of the limits, we have
1.000000005000000050000 etc.
This brings out a difference of
.000000000000000016666 etc.,
or a sixty thousand billionth part of the logarithm. We see that the
logarithms only begin to differ in the i8th place, and that thus to however
many places the radius is taken, the logarithms of proportionals
deduced from it will be given with absolute accuracy to a very much
greater number of places.
To ensure accuracy in the figures given above, the three
tables were recomputed strictly according to
preparatory
the methods described in
the Constructio, fourth proportionals being found in all the preceding
tables, and both limits of their logarithms being calculated, the work
being carried to the 2 7th place after the decimal point.
As logarithms to base e~' are now quite superseded, it is not worth
while printing these preparatory tables. The following values (pp. 94-95),
however, may be of service for comparison, and as a check to any one
who may desire to work out for himself the tables and examples in the
Constructio. The values given are the first proportional after radius, and
the last proportional in each of the three tables, and also in the Third
table, the last proportional in col. i, and the first proportionals in col.
2 and 69. Opposite these are given their logarithms to base e~^, computed,
first, according to Napier's method, and second, by the present
method of series which gives the value true to the last place, which is
increased by unit when the next figure is 5 or more. The proportionals
and logarithms are each multiplied by 1 0000000, as explained
above.
Though the logarithms in the Canon of 1614 were affected by the
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