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 from 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.
11.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 a b c d 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 and 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.
12.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, and , we may put 10.506 and 10.496.
12.Thus far concerning accuracy; what follows concerns ease in working.
13.The construction of every arithmetical progression ts 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 very difficult multiplications, divisions, or extractions of roots.
13.Those geometrical progressions alone are carried on
easily
