The construction of the wonderful canon of logarithms

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24 Construction of the Canon.
these definitions arithmetical increase always the
same corresponds to geometrical decrease similarlyproportioned,
of necessity we conclude that equidifferent
logarithms and their limits correspond to
similarly proportioned sines. As in the above
example from the First table, since there is a like
proportion between 9999999.0000000 the first
proportional after radius, and 9999997.0000003
the third, to that which is between 9999996.0000006
the fourth and 9999994.0000015 the sixth ; therefore
1.00000005 the logarithm of the first differs
from 3.00000015 the logarithm of the third, by the
same difference that 4.00000020 the logarithm of
the fourth, differs from 6.00000030 the logarithm
of the sixth proportional. Also there is the same
ratio of equality between the differences of the
respective limits of the logarithms, namely as the
differences of the less among themselves, so also
of the greater among themselves, of which logarithms
the sines are similarly proportioned.
37- Of three sines continued in geometrical proportion, as
the square of the mean equals the product of the extremes,
so of their logarithms the double of the msan equals the
sum, of the extremes. Whence any two of these logarithms
being given, the third becomes known.
Of the three sines, since the ratio between the
first and the second is that between the second
and the third, therefore (by 36), of their logarithms,
the difference between the first and the second is
that between the second and the third. For
example, let the first logarithm be represented by
the line b c, the second by the line b d, the third
by the line b e, all placed in the one line b c d e,
thus :
and