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Logarithms: The Early History of a Familiar Function - Parallel Insights and the Reception of Mathematical Ideas

Kathleen M. Clark (The Florida State University) and Clemency Montelle (University of Canterbury)

All of this leads us to the question: what causes some mathematical ideas to get a better reception than others? History shows us that it is not simply a question of mathematical merit, nor the precise instant at which they officially entered the public domain, but that many other factors can be influential. Napier was keen to spread his ideas. We can see this clearly in his endorsement of the translation into English of his work by his contemporary Edward Wright (1616). Napier expressed his delight in “this secret invention ... so much the better as it shall be the more common.” Evidently, Napier was concerned with disseminating his ideas as widely as possible.

In contrast, even Bürgi's contemporaries noted that Bürgi made little effort in promoting his work. Johann Kepler, who was most likely aware of Bürgi's endeavors while the two served in the court of Duke Wilhelm II, the Langrave of Hesse-Kassel (d. 1592), portrayed Bürgi as being neglectful of his ideas and failing to promote them more widely. Kepler noted:

These logistic indices ... showed Iustus Byrgi many years before the edition of Napier the way to these very same logarithms. Even so, this man, a procrastinator and guardian of his secrets, abandoned his baby in childbirth and did not nurture it for public use. (As cited in Kepler, Tabulae Rudolphinae, 1627, p. 48 (Gronau, 1996, p. 6))

In addition, Napier had two works on logarithms published. Bürgi had only one published, though another was promised. One can only guess at the difference in reception had this second, more “theoretical” work made its way to public consumption. Napier's first work was published in 1614 and was typeset and printed. Bürgi's was published in 1620, and although his tables were printed, his original manuscript remains to the present day only in handwritten form. Napier was mindful of his audience, tailoring his system to those who used arithmetical operations in a trigonometric context. His table entries, as we have seen, were essentially logarithms of sines of uniformly increasing angles. Bürgi's tables were not compiled in this way and needed (at least) an additional step to be useful in a trigonometric context.

Napier was part of the busier academic community, and had his work translated into the vernacular by Wright almost immediately. Bürgi wrote in German, which was arguably less accessible to the wider community. Furthermore, Napier seems to have responded more quickly to the reaction of his contemporaries and to suggestions they had for improving the system. Napier updated his system and made various accommodations so that it would be easier to use. His contemporaries, chief among them Briggs, helped figure out and initiate these changes, and published the results (Jagger, 2003). A primary improvement of Napier's logarithms was the modification that (log 1 = 0), usually credited to Henry Briggs. Most likely this important change was not solely Briggs' idea. However, the very quick meeting and collaboration between the two men enabled “the idea [to come] about in discussions between Napier and Briggs” (O'Connor & Robertson, 1999). Napier was an established scholar; Bürgi was an employee of the court with (presumably) limited resources and connections, despite the presence of Kepler.

Kathleen M. Clark (The Florida State University) and Clemency Montelle (University of Canterbury), "Logarithms: The Early History of a Familiar Function - Parallel Insights and the Reception of Mathematical Ideas," Convergence (January 2011), DOI:10.4169/loci003495