Elementary arithmetic, with brief notices of its history... by Robert Potts.

LOGARITHMS.

Napier after his father's death, and it appears that a copy of Napier's,
book on Arithmetic was copied out by Robert Napier for Mr. Henry
Briggs, the Savilian Professor at Oxford. Briggs never made any
claim to be the inventor of logarithms; and the fact of the first
mention of his improvement publicly in his lectures is quite consistent with Napier's statement made at his first interview withBriggs, that he had before been aware of the advantages of the
method Briggs had proposed.
Briggs had also completed before his death, which happened in
1630, a table of the logarithmis of the sines, tangents, and secants.
to fifteen places of figures, and annexed it to his table of the natural
sines, tangents, and secants which he had before calculated. This
work he committed to his friend Henry Gellibrand, then professor of
geometry in Gresham College, who published it in 1633 with the title
of " Trigonometria Britannica." These tables for their accuracy have
been seldom found to differ from the truth by more than a few units
in the fifteenth figure.
No one before Napier ever considered all numbers as expressions
of proportions, which could be included in a series of ratios. This
idea is the basis of his invention, and the effect was, that he devised a
method of finding a series of proportionals, containing all numbers,
and every number having its own exponent, and also a method of
finding the exponent of any given number of the series, or the noumber
of any given exponent.
The merit of Napier consists of having imagined and assigned a
logarithm to any number whatever, by supposing the logarithm of
that number to be one of the terms of a series in arithmetical progression, and the number itself one of the terms of a geometrical
progression whose successive terms differ by very small increments
froml each other.
The natural numbers, 1, 2, 3, 4, 5, &c., form an arithmetical progression, and may be made the logarithms of a series of numbers in
geometrical progression: and the same numbers, 1, 2, 3, &c., may be
also made to represent the logarithms of different geometrical series.
But the natural numbers, of which the logarithms are wanted, form of
themselves an arithmetical series, and can never become a geometrical
series. Here arose the difficulty, how could one series of numbers in
arithmetical progression be made to correspond with the whole series
of the natural numbers, in their order of increase, which are themselves also a series of numbers in arithmetical progression?
The arithmetical progression, 0, 1, 2, 3, 4, 5, &c., corresponds with.
the geometrical progression, 1, 2, 4, 8, 16, 32, &c., and the numbers inl
order of the former can be made the logarithms of the corresponding
numbers of the latter series; thus, 0 may be made the logarithm
of 1, 1 of 2, 2 of 4, 3 of 8, 4 of 16, 5 of 32, &c., and so on; but hero
arose the difficulty, how could the logarithms of the numbers intermediate be represented? the logarithm of 3 lying between 2 and 4;:
of 5, 6, 7, between 4 and 8; of 9, 10, 11, 12, 13, 14, 15, between
8 and 16, &c.; so as to be made to correspond to all numbers in
their natural order of increase, these natural numbers themselves
being in arithmetical progression. Napier had not the aid of the
algebraical notations afterwards devised, and he knew nothing of the
equation x = y, nor had he even conceived the idea of the base of a