CONSTRUCTION OF THE CANON. 33
column of the Third table is 9900473.57808. Of this (by 44) the limits of the logarithm are 100024.9657720 and 100024.9757760. Then the fourth proportional will be 9999521.6611850. Of this the limits the logarithm, deduced from the Second table (by 43), are 478.3502290 and 478.3502812. These limits (by 8 and 35) being added to the above limits of the logarithm of the table sine, there will come out the limits 100503. 3260572 and 100503.3160010, between which necessarily falls the logarithm sought for. Whence the number midway between them, which is 100503.3210291, may be put without sensible error for the true logarithm of the given sine 9900000.
46.Hence it follows that the logarithms of all the proportionals of the Third table may be given with sufficient exactness.
For, as (by 45) 100503.3210291 is the logarithm of the first sine in the second column, namely 9900000; and since the other first sines of the remaining columns progress in the same proportion, necessarily (by 32 and 36) the logarithms of these increase always by the same difference 100503.3210291, which is added to the logarithm last found, that the following may be made. Therefore, the first logarithms of all the columns being obtained in this way, and all the logarithms of the first column being obtained by 44, you may choose whether you prefer to build up, at one time, all the logarithms in the same column, by continuously adding 5001.2485387, the difference of the logarithms, to the last found logarithm in the column, that the next lower logarithm in the same column be made; or whether you prefer to com-
pute
