Page:The Construction of the Wonderful Canon of Logarithms.djvu/45

CONSTRUCTION OF THE CANON. 21 T d the less limit of the logarithm which b c represents.

By the preceding it is proved that the given sine being subtracted from radius the less limit remains, and that radius being multiplied into the less limit and the product divided by the given sine, the greater limit is produced, as in the following example.

For (by 29) subtract 9999999 from radius with cyphers added, there will remain unity with its own cyphers for the less limit; this unity with cyphers being multiplied into radius, divide by 9999999 and there will result 1.0000001 for the greater limit, or if you require greater accuracy 1.00000010000001.

Thus in the above example, the logarithm of the sine 9999999 was found to be either 1.0000000 or 1.00000010, or best of all 1.00000005. For since the limits themselves, 1.0000000 and 1.0000001, differ from each other by an insensible fraction like $$\scriptstyle \frac{1}{10000000}$$ therefore they and whatever is between them will differ still less from the true logarithm lying between these limits, and by a much more insensible error.

This