The construction of the wonderful canon of logarithms

Construction of the Canon. 9
000502 1 is the same as 9999998xxr
of others.
^^'^ so
6. When the tables are computed, the fractions follotvmg
the period may then be rejected without any sensible error.
For in our large numbers, an error which does not exceed
unity is insensible and as if it were none.
Thus in the completed table, instead of
9987643-8213051, which is 9987643 18000000 . we
may put 9987643 without sensible error.
7. Besides this, there is another rule for accuracy ; that is
to say, when an unknown or incommensurable quantity is
included between numerical limits not differing by many
units.
Thus if the diameter of a circle contain 497
parts, since it is not possible to ascertain precisely
of how many parts the circumference consists, the
more experienced, in accordance with the views of
Archimedes, have enclosed it within limits, namely
1562 and 1561. Again, if the side of a square
contain 1000 parts, the diagonal will be the square
root of the number 2000000. Since this is an incommensurable
number, we seek for its limits by
extraction of the square root, namely 141 5 the
greater limit and 1414 the less limit, or more
accurately I4i4^|ff the greater, and 1414^!^
the less; for as we reduce the difference of the
limits we increase the accuracy.
In place of the unknown quantities themselves, their
limits are to be added, subtracted, multiplied, or divided,
according as there may be need.
8. The two limits of one quantity are added to the two
limits of another, when the less of the one is added to the
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less