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.

29.Therefore to find the limits of the logarithm of a given sine.

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.

30.Whence the first proportional of the First table, which is 9999999, has its logarithm between the limits 1.0000001 and 1.0000000.

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.

31.The limits themselves differing insensibly, they or anything between them may be taken as the true logarithm.

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.

32.There being any number of sines decreasing from radius in geometrical proportion, of one of which the logarithm or its limits ts given, to find those of the others.

This