The construction of the wonderful canon of logarithms

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Notes. 91
10,000,000 he multiplied his numbers and logarithms by that amount,
thereby making them integral to as many places as he intended to print.
In this we follow his
indication of this multiplication.
example, omitting, however, from the formulae the
In sec. 30, Napier shows that the logarithm of 9999999,
the first
proportional after radius in the First table, lies between the limits
i.ooooooiooooooioetc, and 1.000000000000000 etc. And in sec.
31, he proposes to take 1.00000005, the arithmetical mean between
these
limits, as a sufficiently close approximation to the true logarithm
for, the difference of this mean from either limit being .00000005,
it cannot differ from the true logarithm by more than that amount,
which is the twenty millionth part of the logarithm. But there can be
little doubt that Napier was able to • satisfy himself that the difference
would be very much less, and that his published logarithms would be
unaffected.
We proceed to show the precise amount of error thus introduced into
the logarithm of 9999999.
If we employ the formula
substituting 1 0000000 for n, and multiplying the result by 1 0000000, as
before explained, we have
1.000000050000003333333583 etc.
Again, if we take the arithmetical mean of the limits, carried to a similar
number of places, we have
1.000000050000005000000500 etc.
The error introduced is consequently
.000000000000001666666916 etc.
or about a six hundred billionth part in excess of the true logarithm.
It will be observed that besides being very much less, this error is
in the opposite direction from that caused by the mistake in the
Second table.
We have given above the analytical expression for the true logarithm,
namely, ^ + 2V2
+ sSS + 4^ + s^ + ^t