The construction of the wonderful canon of logarithms

Construction of the Canon. 21
T d the less limit of the logarithm which b c
represents.
29. Therefore to find the limits ofthe logarithm of a given
sine.
'
By the preceding it is proved that the given
_^
sine being subtracted from radius the less limit
remains, and that radius being multiplied into the
less limit and the product divided by the given
sine, the greater limit is produced, as in the following
example.
30. Whence the first proportional of the First tab14, which is
9999999, has its logarithm between the limits i.ooooooi
and 1.0000000.
For (by 29) subtract 9999999 from radius with
cyphers added, there will remain unity with its own
cyphers for the less limit ; this unity with cyphers
being multiplied into radius, divide by 9999999 and
there will result 1.0000001 for the greater limit, or
3 1
if you require greater accuracy 1.0000001 000000 1.
• The limits themselves differing insensibly, they or anything
between them may be taken as the true logarithm.
Thus in the above example, the logarithm of
the sine 9999999 was found to be either i.0000000
or i.ooooooio, or best of all 1.00000005. For
since the limits themselves, i.0000000 and
I.OOOOOOI, differ from each other by an insensible
fraction like iooooooo > therefore they and whatever
is between them will differ still less from the
true logarithm lying between these limits, and by
a much more insensible error.
3 2 . There being any number of sines decreasingfrom radius
in geometricalproportion, of one of which the logarithm or
its limits is given, to find those of the others.
C 3 This