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

LOG RITHMS.

7

which the logarithm of 1 is 0.1 I have also published some lucubrations upon the new species of logarithms, by that most excellent
mathematician, Henry Briggs, public professor in London, who undertook most willingly the very severe labour of calculating this canon,
in consequence of the singular affection that existed between him and
my father of illustrious memory." Robert Napier in this volume
makes no allusion to the claim of Briggs, as his father, in the " Rabdologia," laid claim to that improvement, and stated that he had committed the execution of it to Briggs. From an expression in Briggs's
preface to his " Chilias Prima Logarithmorum," it would appear he
expected a recognition of his claim in the posthumous work of Napier.
But as it had been passed over in silence, Briggs, in the preface to his
"Arithmetica Logarithmica," clearly declared the part he had taken,
and that he had first suggested the improvement in his lectures.
In the year 1624 Briggs published his great work " Arithmetica
Logarithmica," of which a translation in English appeared in 1631.
In the address to his readers he gives the following account of the
part he took in the improvement of logarithms, and his great
labour in the calculation of the improved tables. "Be not surprised that these logarithms are different from those which that
illustrious man, the Baron of Marchiston, published in his ' Canon
Mirificus.' For when explaining publicly the doctrine of them to my
auditors at Gresham College, in London, I remarked that it would be
much more convenient that 0 should stand for the logarithm of the
whole sine [or radius] as in the 'Canon Mirificus,' but that the
logarithm of the tenth part of the same whole sine, namely, of 5~ 44' 21",
should be 10,000,000,000. And concerning this matter I immediately
wrote to the author himself; and as soon as the season of the year,
and my public teaching would permit, I went to Edinburgh, where
being most kindly received by him, I staid a whole month. But
when we began to converse about this change in the system, he
said that for some time [dudum] he had been sensible of the same
thing, and had desired to accomplish it, but however he had published those that he had already prepared, until he could make others
more convenient if his duties and feeble health would permit. But
1 In this appendix he shows how the logarithms of all composite numbers can
be found from the logarithms of prime numbers, and thus describes his method.
"In order to find the logarithms of all numbers, it is necessary that the logarithms
of some two natural numbers be given, or at least assumed, as in the former first
construction 0, or cipher, was assumed for the logarithm of the natural number
1, and 10,000,000,000 for the logarithm of the natural number 10. These, therefore, being given, the logarithm of the natural number 5 (which is a prime number) is sought in this manner. Between 10 and 1 is sought the geometric mean,
which is 31o6o22o70606' So between 10,000,000,000 and 0 is sought the arithmetic mean, which, is 5,000,000,000. Next between 10 and,6207o is
taken the geometric mean, which is -5623413251^ And similarly between
10000000ooooo,'
5,000,000,000 and 10 is taken the arithmetic mean, which is 75,000,000,000."
It will be seen by this process that the successive arithmetic means are the
logarithms of the corresponding geometric means. But as these geometric means
are not the natural prime numbers, if the process be continued it will be found,
after twenty-five operations, that the geometric mean (taking seven places of
figures) will be very nearly equal to 9, being defective only by the five millionth
part of an unit; and the corresponding arithmetic mean may be taken without
sensible error as the logarithm of 9. Thus the logarithm of 9 being known, the
logarithm of the prime number 3 is also known.