Advances in Mathematics
His work, Mirifici Logarithmorum Canonis Descriptio (1614) contained fifty-seven pages of explanatory matter and ninety pages of tables of numbers related to natural logarithms. The book also has an excellent discussion of theorems in spherical trigonometry, usually known as Napier's Rules of Circular Parts. Modern English translations of both Napier's books on logarithms, and their description can be found on the web, as well as a discussion of Napier's Bones (see below) and Promptuary (another early calculating device). His invention of logarithms was quickly taken up at Gresham College, and prominent English mathematician Henry Briggs visited Napier in 1615. Among the matters they discussed was a re-scaling of Napier's logarithms, in which the presence of the mathematical constant e (more accurately, e times a large power of 10 rounded to an integer) was a practical difficulty. Napier delegated to Briggs the computation of a revised table. The computational advance available via logarithms, the converse of powered numbers or exponential notation, was such that it made calculations by hand much quicker. The way was opened to later scientific advances, in astronomy, dynamics, physics; and also in astrology.
Napier made further contributions. He improved Simon Stevin's decimal notation. Arab lattice multiplication, used by Fibonacci, was made more convenient by his introduction of Napier's bones, a multiplication tool using a set of numbered rods.
Napier may have worked largely in isolation, but he had contact with Tycho Brahe who corresponded with his friend John Craig. Craig certainly announced the discovery of logarithms to Brahe in the 1590s (the name itself came later); there is a story from Anthony à Wood, perhaps not well substantiated, that Napier had a hint from Craig that Longomontanus, a follower of Brahe, was working in a similar direction. It has been shown that Craig had notes on a method of Paul Wittich that used trigonometric identities to reduce a multiplication formula for the sine function to additions.
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