The construction of the wonderful canon of logarithms

Construction of the Canon, 29
sine and the table sine (by 40), either both
Hmits or one or other of them, since they are
almost equal, as is evident from the above example.
Now these, or either of them, being found,
add to them the limits above noted down, or else
subtract (by 8, 10, and 35), according as the given
sine is less or greater than the table sine. The
numbers thence produced will be near limits between
which is included the logarithm of the
given sine.
Example,
Let the given sine be 9999975.5000000, to
which the nearest sine in the table is 9999975.
0000300, less than the given sine. By 33 the
limits of the logarithm of the latter are 25.0000025
and 25.0000000. Again (by 40), the difference of
the logarithms of the given sine and the table sine
is .4999712. By 35, subtract this from the above
limits, which are the limits of the less sine, and
there will come out 24.5000313 and 24.5000288,
the required limits of the logarithm of the given
sine 9999975.5000000, Accordingly the actual
logarithm of the sine may be placed without
sensible error in either of the limits, or best of all
(by 31) in 24.5000300,
Another Example.
LET the given sine be 9999900.0000000, the
table sine nearest it 9999900.0004950. By
33 the limits of the logarithm of the latter are
1 00.0000 1 00 and 100.0000000. Then (by 40) the
difference of the logarithms of the sines will be
.0004950. Add this (by 35) to the above limits
and they become 100.0005050 for the greater
D 3 limit,