The construction of the wonderful canon of logarithms

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Construction of the Canon,
ii
and 43.1 produced 166.7 and 166.3 (as in 8), yet
the converse does not follow; for there may be
some quantity between 166.7 and 166.3 frorn
which if you subtract some other which is between
43.2 and 43.1, the remainder may not lie between
123.5 and 123,2, but it is impossible for it not to
lie between the limits 123.6 and 123.1.
1 1
Division of limits is performed by dividing the greater
limit of the dividend by the less of the divisor, and the less
of the dividend by the greater of the divisor.
Thus, in the preceding figure, the rectangle
abed lying between the limits 33.774432 and
33.757500 may be divided by the limits of a c,
which are 3.216 and 3.215, when there will come
out 10.505^4^1 and io.496§|g| for the limits of
a b, and not 10.502 and 10.500, for the same reason
that we stated in the case of subtraction.
1 2, The vulgar fractions of the limits may be removed by
adding unity to the greater limit.
Thus, instead of the preceding limits of a b,
namely, io-505^|ff and io-496|ff|, we may put
10-506 and 10-496,
Thus far concerning accuracy ; what follows concerns
ease in working.
1 3- The construction of every arithmetical progression is
easy ; not so, however, of every geometrical progression.
This is evident, as an arithmetical progression
is very easily formed by addition or subtraction
but a geometrical progression is continued by
multiplications, divisions, or extrac-
very difficult
tions of roots.
Those geometrical progressions alone are carried on
B 2 easily