Carl Benjamin Boyer on Jost Bürgi

“[I]n 1575 Western Europe had recovered most of the major mathematical works of antiquity now extant. Arabic algebra had been... mastered and improved... through the solution of the cubic and quartic and through... partial... symbolism; and trigonometry had become an independent discipline. The time was almost ripe for rapid strides... The transition from the Renaissance to the modern world was... made through... intermediate figures, a few of the more important... Galileo Galilei... and Bonaventura Cavalieri... from Italy; several... as Henry Briggs.., Thomas Harriot.., and William Oughtred... were English; two... Simon Stevin... and Albert Girard... were Flemish; others came from varied lands—John Napier... from Scotland, Jobst Bürgi... from Switzerland, and Johann Kepler... from Germany.”

“Napier was... the first... to publish.., but...it is possible that the idea of logarithms had occured to Bürgi as early as 1588... half a dozen years before Napier began work... However, Bürgi printed... in 1620, half a dozen years after Napier published... Descriptio. Bürgi's... book... Arithmetische und geometrische Progress-Tabulen... indicates... the influences were similar... to Napier. Both... proceeded from the properties of arithmetic and geometric sequences, spurred, probably by the method of prosthaphaeresis. The differences... lie chiefly in... terminology and... numerical values..; the fundamental principles were the same. Instead of proceeding from a number a little less than one (...Napier used 1 - 10^{-7}), Bürgi... a little greater than one... 1 + 10^{-4}; and instead of multiplying powers of this number by 10^7, Bürgi multiplied... 10^8. ...[O]ne other minor difference: Bürgi multiplied all... power indices by ten... [I]f N = 10^8(1+10^{-4})^L, Bürgi called 10L the "red"... corresponding to the "black"... N. If... we were to divide all black[s]... by 10^8 and all red[s]... by 10^5, we should have... a system of natural logarithms. ...Bürgi gave for the black...1,000,000,000 the red...230,270.022, which on shifting decimal points, is equivalent to... ln 10 = 2.3027022... not a bad approximation... especially when... (1 - 10^{-4})^{10^4} is not quite the same as lim_{n to infty}(1+ frac{1}{n})^n although... values agree to four significant figures. In publishing... he had... an antilogarithmic table... The essence of the principle is there... Bürgi must be regarded as an independent discoverer who lost credit... because of Napier's priority in publication. In one respect his logarithms come closer to ours than Napier's, for as... black[s] increase, so do the red[s]..; but the two systems share the disadvantage that the logarithm of the product or quotient is not the sum or difference of the logarithms.”