The construction of the wonderful canon of logarithms

32 Construction of the Canon.
iionals in the first column of the Third table may be found
with sufficient exactness^ or may be included between known
limits difiering by an insensible fraction.
For, since (by 43) the logarithm of 9995000,
the first proportional after radius in the first
column of the Third table, is 5001.2485387 with
no sensible error ; therefore! (by 32) the logarithm
of the second proportional, namely 9990002.5000,
will be 10002.4970774; and so of the others, proceeding
up to the last in the column, namely
9900473.57808, the logarithm of which, for a like
reason, will be 100024.9707740, and its limits will
be 100024.9657720 and 100024.9757760.
45. To find the logarithms of natural numbers or sines not
proportionals in the first column of the Third table, but
near or between them ; or to include them, between known
limits differing by an insensible fraction.
Write down the sine in the first column of the
Third table nearest the given sine, whether
greater or less. By 44 seek for the limits of the
logarithm of the table sine.
Then, by one of the
methods described in 43, seek for a fourth proportional,
which shall be to radius as the less of
the given and table sines is to the greater. Having
found the fourth proportional, seek (by 43) for
the limits of its logarithm from the Second table.
When these are found, add them to the limits of
the logarithm of the table sine found above, or
else subtract them (by 8, 10, and 35), and the
limits of the logarithm of the given sine will be
brought out.
Example.
THUS,
let the given sine be 9900000. The
proportional sine nearest it in the first
column