Page:The Construction of the Wonderful Canon of Logarithms.djvu/116

92 Notes.

obtained by Napier’s method of computation is identical with that to the base $\scriptstyle e^{-1}$. If, however, he had used the base $\scriptstyle \left(1-{\frac {1}{n}}\right)^{n}$, where n= 10000000, then the logarithm of 9999999, multiplied by 10000000, as in the other two cases, would necessarily have been unity, or 1.000000000 etc., which would have agreed with the true logarithm to the 8th place only, and would not have left his published logarithms unaffected.

The small error found above in Napier’s logarithm of 9999999 is successively multiplied on its way through the tables: thus, in the First table it is multiplied by 100, in the Second by 50, and in the Third by 20 and again by 69, or in all by 6900000; so that, multiplying the error in the first proportional by that amount, we should have for the error in the logarithm of the last proportional of the Radical table about .0000000115. The error, however, although continually increasing, yet retains always the same ratio to the logarithm, except for a very small disturbing element to be afterwards referred to, so that the true logarithm will always be very nearly equal to the logarithm found by Napier’s method of computation less a six hundred billionth part.

Let us take, for example, the logarithm of 5000000 or half radius. ’ When computed according to Napier’s method, we find it comes out

6931471.80559946464604 etc.

The true logarithm to the base $\scriptstyle e^{-1}$ is

6931471.80559945309422 etc.

So that the difference between the two is

6931471‍.00000001155181 etc.

The six hundred billionth part of the logarithm is

6931471‍.00000001155245 etc.

The latter agrees very closely with the difference found above, and would have agreed to the last place given except for the small disturbing element referred to above, which is introduced in passing from the logarithms of one table to those of the next, or in finding the logarithm of any number not given exactly in the tables as in this case of half radius, but this element is seen to have little effect in modifying the proportionate amount of the original error.

From the above example we see that the error in the logarithm found by Napier’s method amounts only to unity in the 15th place, so that his method of computation clearly gives accurate results far in excess of his requirements, But it is easy to show that Napier’s method may be

adapted