22 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 continually in 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 ae will be double of it, the third triple, and so of the others; until the logarithms of all 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 0, and (
by 30) the logarithm of 9999999, the first sine after radius in the First table, lies between the limits 1.0000001 and 10000000; 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 increasing 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
