The construction of the wonderful canon of logarithms

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38 Construction of the Canon.
51. All sines in theproportion of two to one have 6931 469. 2 2
for the difference of their logarithms.
For since the ratio of every sine to its half is
the same as that of radius to 5000000, therefore
(by 36) the difference of the logarithms of any
sine and of its half is the same as the difference of
the logarithms of radius and of its half 5000000.
But (by 34) the difference of the logarithms of
radius and of the sine 5000000 is the same as the
logarithm itself of the sine 5000000, and this logarithm
(by 50) will be 6931469.22. Therefore,
also, 6931469.22 will be the difference of all logarithms
whose sines are in the proportion of two to
one. Consequently the double of it, namely
13862938.44, will be the difference of all logarithms
whose sines are in the ratio of four to one
and the triple of it, namely 20794407.66, will be
the difference of all logarithms whose sines are in
the ratio of eight to one.
52. All sines in theproportion often to one have 23025842.34
for the difference of their logarithms.
For (by 50) the sine 8000000 will have for its
logarithm 2231434.68; and (by 51) the difference
between the logarithms of the sine 8000000 and
of its eighth part 1 000000, will be 20794407.66 ;
whence by addition will be produced 23025842.34
for the logarithm of the sine 1 000000. And since
radius is ten times this, all sines in the ratio of ten
to one will have the same difference, 23025842.34,
between their logarithms, for the reason and cause
already stated (in 51) in reference to the proportion
of two to one. And consequently the double
of this logarithm, namely 46051684.68, will, as regards
the difference of the logarithms, correspond
to