The construction of the wonderful canon of logarithms

90 Notes.
Notwithstanding the simplicity and elegance of the last of these, it was
long after Napier's time—in fact, not till the eighteenth century—that it
came into general use.
The subject is referred to by Mark Napier in the Memoirs,' ' pp. 451-
455, and by Mr Glaisher in the Report of the 1873 Meeting of the
British Association, Transactions of the Sections, p. 16.
On the Occurrence of a Mistake in the Computation
of the Second Table ; with an Enquiry into
the Accuracy of Napier's Method of Computing his
Logarithms.
It is evident that a mistake must somewhere have occurred in the
computation of the Second table, since the last proportional therein
is given (sec. 17) as 9995001.222927, whereas on trial it will be found
to be 9995001.224804.
This mistake introduced an error into the logarithms of the Radical
table, as the logarithm of the first proportional in that table is deduced
from the logarithm of the last proportional in the Second table by finding
the limits of their difierence. But these limits are obtained from the
proportionals themselves, and, as shown above, one of these proportionals
was incorrect : the limits therefore are incorrect, and consequently
the logarithm of the first proportional in the Radical table.
We see the effect of this in the logarithm of the last proportional in
the Radical table, which is given (sec. 47) as 6934250.8, whereas it
should be 6934253.4, the given logarithm thus being less than the true
logarithm by 2.6, or rather more than a three millionth part.
The logarithms as published in the original Canon are affected by the
above mistake, and also, as mentioned in sec. 60, by the imperfection
of the table of sines.
It seems desirable, therefore, to enquire whether
in addition any error might have been introduced by the method of
computation employed.
Before entering on this enquiry, we should premise that in comparing
Napier's logarithms with those to the base e~^ (which is the base required
by his
reasoning, though the conception of a base was not formally
known to him), it must be kept in view that in making radius
10,000,000