The construction of the wonderful canon of logarithms

30 Construction of the Canon.
limit, and 100.0004950 for the less limit, between
which the required logarithm of the given sine is
included.
42. Hence it follows that the logarithms of all the proportionals
in the Second table may be found with sufficient
exactness, or may be included between known limits differing
by an insensiblefraction.
Thus since the logarithm of the sine 9999900,
the first proportional of the Second table, was
shown in the preceding example to lie between
the limits 100.0005050 and 100.0004950; necessarily
(by 32) the logarithm of the second proportional
will lie between the limits 200.0010100 and
200.0009900 ; and the logarithm of the third proportional
between the limits 300.0015150 and
300.0014850, &c. And finally, the logarithm of
the last sine of the Second table, namely 9995001.
222927, is included between the limits 5000.
0252500 and 5000.0247500. Now, having all
these limits, you will be able (by 31) to find the
actual logarithms.
43. To find the logarithms of sines or natural numbers not
proportionals in the Second table, but near or between
them ; or to include them between known limits differing
by an insensible fraction.
Write down the sine in the Second table nearest
the given sine, whether greater or less. By
42 find the limits of the logarithm of the table
sine. Then by the rule of proportion seek for a
fourth proportional, which shall be to radius as
the less of the given and table sines is to the
greater. This may be done in one way by multiplying
the less sine into radius and dividing the
product by the greater. Or, in an easier way, by
. multiplying