The construction of the wonderful canon of logarithms

1
Trigonometrical Propositions. 71
Now add the logarithm of the half sum, namely
1819061,
either
or
to the logarithm 12 18382, to the logarithm of the
and the sum will be complement of the half
3037443
; from this subtract
the logarithm 1 90022 and the sum will be
difference, namely 11 307,
and there will remain 1830368; from this subtract
1 137222.
693147 and there
will remain 1137221.
Halve the latter and you have the logarithm
56861 1, to which corresponds the arc 34° 30', and
double this arc is the base required, namely 69°.
Conversely, given the three sides, to find any angle. The
solution of this problem is given in my work on Logarithms,
Book II. chap. vi. sect. 8, but partly by logarithms
and partly by prosthaphcsresis of arcs.
It is to be observed that in the preceding and following
problems there is no need to
discriminate between the different
cases, since the form and magnitude of the several
parts appear in the course of the calculation.
Another direct converse of the preceding problemfollows.—
[Given the sides and the base, to find the vertical angle.]
HALVE
the given base, namely 69°, and you have
34° 30', the logarithm of which is 568611.
Double the latter and you have 11 37222 ;
corresponding
to this is the arc 18° 42', which note as the
second found.
As before, take for the first found the arc 0° 44',
corresponding to the logarithm 4357140.
The complements of the two arcs are 89° 16' and
71° 18'; their half sum is 80° 17', and its logarithm
I 4 14449;