The construction of the wonderful canon of logarithms

42 Construction of the Canon.
45 degrees, to e i, now radius. Consequently (by
37) double the logarithm of the sine of 45 degrees
is equal to the logarithms of the extremes, namely
radius and its half. But the sum of the logarithms
of both these is the logarithm of half radius only,
because (by 27) the logarithm of radius is nothing.
Necessarily, therefore, the double of the logarithm
of an arc of 45 degrees is the logarithm of half
radius.
5 7. The sum of the logarithms ofhalf radius and any given
arc is equal to the sum, of the logarithms of half the arc
and the complement of the half arc. Whence the logarithm
of the half arc may be found if the logarithms of
the other three be given.
Since (by 55) half radius is to the sine of half
the given arc as the sine of the complement of
that half arc is to the sine of the whole arc, therefore
(by 38) the sum of the logarithms of the two
extremes, namely half radius and the whole arc,
will be equal to the sum of the logarithms of the
means, namely the half arc and the complement
of the half arc. Whence, also (by 38), if you add
the logarithm of half radius, found by 51 or 56, to
the given logarithm of the whole arc, and subtract
the given logarithm of the complement of the half
arc, there will remain the required logarithm of
the half arc.
Example.
LET
there be given the logarithm of half
radius (by 51) 6931469; also the arc 69
degrees 20 minutes, and its logarithm 665143,
The half arc is 34 degrees 40 minutes, whose
logarithm