Page:The Construction of the Wonderful Canon of Logarithms.djvu/47

CONSTRUCTION OF THE CANON. 23

way, if you please, continue the logarithms themselves in an exactly similar progression with little and insensible error; in which case the logarithm of radius will be 0, the logarithm of the first sine after radius (by 31) will be 1.00000005, of the second 2.00000010, of the third 3.00000015, and so of the rest.

34.The difference of the logarithms of radius and a given sine is the logarithm of the given sine itself.

This is evident, for (by 27) the logarithm of radius is nothing, and when nothing is subtracted from the logarithm of a given sine, the logarithm of the given sine necessarily remains entire.

35.The difference of the logarithms of two sines must be added to the logarithm of the greater that you may have the logarithm of the less, and subtracted from the logarithm of the less that you may have the logarithm of the greater.

Necessarily this is so, since the logarithms increase as the sines decrease, and the less logarithm is the logarithm of the greater sine, and the greater logarithm of the less sine. And therefore it is right to add the difference to the less logarithm, that you may have the greater logarithm though corresponding to the less sine, and on the other hand to subtract the difference from the greater logarithm that you may have the less logarithm though corresponding to the greater sine.

36.The logarithms of similarly proportioned sines are equi-different.

This necessarily follows from the definitions of a logarithm and of the two motions, For since by

these