The construction of the wonderful canon of logarithms

2 2 Construction of the Canon.
This necessarily follows from the definitions of
arithmetical increase, of geometrical decrease, and
of a logarithm. For by these definitions, as the
sines decrease continuallyJn geometrical proportion,
so at the same time their logarithms increase
by equal additions in continuous arithmetical progression.
Wherefore to any sine in the decreasing
geometrical progression there corresponds a logarithm
in the increasing arithmetical progression,
namely the first to the first, and the second to the
second, and so on.
So that, if the first logarithm corresponding to
the first sine after radius be given, the second
logarithm will be double of it, the third triple, and
so of the others ; until the logarithms of all the
sines be known, as the following example will
show.
33- Hence the logarithms of all the proportional sines of the
First table may be included between near limits, and consequently
given with sufficient exactness.
Thus since (by 27) the logarithm of radius is o,
and (by 30) the logarithm of 9999999, the first
sine after radius in the First table, lies between the
limits 1.000000 1 and 1 0000000; necessarily the
logarithm of 9999998.0000001, the second sine
after radius, will be contained between the double
of these limits, namely between 2.0000002 and
2.0000000 ; and the logarithm of 9999997.0000003,
the third will be between the triple of the same,
namely between 3.0000003 and 3.0000000. And
so with the others, always by equally incfeasing
the limits by the limits of the first, until you have
completed the limits of the logarithms of all the
proportionals of the First table. You may in this
way