The construction of the wonderful canon of logarithms

Construction of the Canon. 33
Of
column of the Third table is 9900473.57808.
this (by 44) the limits of the logarithm are
100024.9657720 and 100024.9757760. Then the
fourth proportional will be 9999521.661 1850. Of
this the limits of the logarithm, deduced from the
Second table (by 43), are 478.3502290 and
z| 78. 35028 1 2. 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. 3 1 60010, 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
100503.32 10291, 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-
E
pute