The Base of Napier's Logarithms

Abstract

WHAT Napier actually gives in his table is a series of natural sines with a corresponding series of logarithms which diminish as the sines increase. If a Napierian logarithm is considered to be the logarithm of the sine opposite to which it stands, the base is approximately e¹; but we may, if we like, regard the logarithms as logarithms of cosecants, and the base is then approximately e. Or again (as in “Encycl. Brit.,” xvii., 179) we may take Napier's sines as actual integers, and use log Nap n for the logarithm placed opposite n in the table; then we have approximately log Nap n = 10⁷ log_e (10⁷/n).