Read June 5, 1806.
If there already existed as full and extensive logarithmic tables as will ever be wanted, and of whose accuracy we were absolutely certain, and if the evidence for that accuracy could remain unimpaired throughout all ages, then any new method of computing logarithms would be totally superfluous so far as concerns the formation of tables, and could only be valuable indirectly, inasmuch as it might shew some curious and new views of mathematical truth. But this kind of evidence is not in the nature of human affairs. Whatever is recorded is no otherwise believed than on the evidence of testimony; and such evidence weakens by the lapse of time, even while the original record remains; and it weakens on a twofold account, if the record must from time to time be replaced by copies. Nor is this destruction of evidence arising from the uncertainty of the copy's being accurately taken, any where greater than in the case of copied numbers. It is useful then to contrive new and easy methods for computing not only new tables, but even those we already have.
It is useful to contrive methods by which any part of a table may be verified independently of the rest; for by