Memoirs of John Napier of Merchiston

INVENTION OF LOGARITHMS. 457
system, written many years before it came into universal practice, might be transferred
verbatim into a treatise on the subject for the year 1834. It is remarkable that
Sir John Leslie, in connecting Napier with the history of Decimal fractions, had
not referred to the posthumous work rather than to the Rabdologia ; for it was in
the Constructio Logarithmorum, that the ordinary rules of calculation were first dis-
played working with equal facility upon the descending side of the scale. Delambre
(Astronomie Moderne, p. 493, et infra,) was particularly struck with the fact, and I
shall follow so far that illustrious philosopher's profound exposition of the work in ques-
tion. " Napier," says he, " in his definitions, and even in his calculations, makes use of
decimal fractions ; but only gives the notation without any rule of calculation. It is the
earliest example of them I have met with, — it is a first step, and one of the greatest
importance, 1 ' (il est de la plus grande importance.) Delambre then follows Napier through
his method of calculating the terms of his geometrical progression, but takes the aid of
modern algebraic symbols. It would occupy too much space here to give the process,
for which the reader must be referred to Napier's own work, or other recondite sources.
After detailing it, Delambre exclaims, " We here distinctly observe examples of subtrac-
tion in decimal fractions." Passing through some more of the calculations he again ex-
claims, " behold manifestly division in decimal fractions ;" and fnrther on he adds, " I have
already remarked that Napier is the first to afford the idea of the calculation of decimal
fractions, a little more developed afterwards by Briggs."
Such is the hold that Napier has of Decimal fractions, a part of the system, " which"
says Playfair, " completed our arithmetical notation, and formed the second of the three
steps by which in modern times the science of numbers has been so greatly improved."
Of course the Jirst step was Arabic numerals, and the third was the Logarithms ; so when
we take into consideration that decimals only came into active operation with the system of
Logarithms, and that Napier is the first, who affords examples both of the calculation
with decimals, and of their best notation, we may fairly say that his share in the develope-
ment of the great Arabic system is as two to one. The original algorithm, whose his-
tory is lost in distant climes and long past ages, brought as it were the telescope to
numbers. When Napier reversed the notation, and caused it to act in the opposite
direction, he may be said to have added the microscope ; and he did so while creating
the last and greatest revolution in the system, — when to ugd/Mi he added that omnipotent
word, which nor Greeks nor Brahmins knew, >.oya^/j,oi.* How proud a contempla-
* Ayfifjiti't signifies numbers, xaya.^iBfto'i, the ratios of numbers; or, rather, the number of ratios, liyut
d^iS/ui;. Napier compounded the word before his system was known, but subsequent to the date of
his invention. Dr Minto says, " the term Logarithm was first used by Napier after the publication of
the canon in which he uses the term of numerus artificialis." (Buchan and Minto's Life of Napier,
p. 43). This is an extraordinary mistake. In the Constructio Napier used the latter phrase, but a profound
consideration of his own system led him to frame the term Logarithms before he published his canon ;
and the first knowledge of the system that the world obtained was through that nomenclature which
3 M