Abhandlungen zur Geschichte der Mathematik.

In this paper we shall consider two points in the history of logarithms:
(1) the origin and prevalence, during the seventeenth and eighteenth
centuries, of the error regarding the identity of natural logarithms and those
published by NAPIER in his Mirifici logarit7hmorztm canonis descriptio of 1614;
(2) the earliest publication of a table of natural logarithms.
The theory of natural ("hyperbolic") logarithms apparently first suggested itself to mathematicians engaged in the mensuration of spaces
between the hyperbola and its asymptotes. About a quarter of a century
later, in 1695, EDMUND HALLEY discarded geometrical figures and published
a remarkable artiele containing a purely arithmetioal theory of logarithms.1)
In this original and meritorious investigation he lays great stress upon
what we now call the "modulus". By NAPIER'S logarithms HALLEY understands those which give BRIGGS'S logarithms, when divided by 2.302 585
or when multiplied by 0.43429448. From this statement it appears that
HALLEY considered NAPIER'S logarithms to be identical with natural logarithms, and we must look upon him as one of the first (perhaps the first)
to commit this error. That the two systems are not identical is shown by
the following formula2):
107
logN x = 107 log,, I.
During the eighteenth century this misunderstanding regarding the two
systems does not appear to have been as wide-spread as it was later. On
consulting mathematical dictionaries and other books of the seventeenth
and eighteenth centuries, which are accessible to me, I find that VITALL3)
and OZANAM 4) do not touch the point in question, for the former directs
1) Phil. Trans., 1695, No. 216. His article is known to me only through
the account of it given in HUTTON, Math. Tables, 5th ed., London, 1811, pp. 107-110;
M. CANTOR, Gesch. d. Math., III., pp. 80-82; R. REIFF, Gesch. d. unendlichen
Reihen, Tübingen, 1889, pp. 38-40; E. STONE, New Mathematical Dictionary,
2nd ed., 1743, article "Logarithms".
2) Consult S. GÜNTHER, Vermischte Untersuch., Leipzig, 1876, pp. 271-278.
3) HIERONYMO VITALI, Lexicon Mathematicumz, Parisiis, 1668, article
"Logarithmi".
4) M. OZANAM, Dictionaire mathematique, Paris, 1691, p. 50.
Abh. zur Gesch. d. Mathem, IX 3