Logarithms: The Early History of a Familiar Function - The Challenges of Parallel Insights in the History of Mathematics

When French mathematician Pierre-Simon Laplace noted that logarithms “by shortening the labors, doubled the life of an astronomer,”** he hardly exaggerated. Where previously practitioners had been slowed down by long and tedious calculations required in their disciplines, now that logarithms had been introduced, they literally had to spend only half as much time on these necessary computations. Therefore, right from its inception, there has never been any doubt as to the importance of the logarithmic relation. Furthermore, its usefulness has persisted in different areas of mathematics right to the present day.

As a result, the logarithm has always formed an important part of historical scholarship, with many historians reflecting on various aspects of its development. In particular, the introduction of the logarithm by (at least) two individuals seemingly simultaneously has led historians of mathematics to wonder to whom to attribute the “discovery.” This issue of priority is a compelling one. When a mathematical idea or concept is introduced independently at roughly the same time by different scholars, historians are faced with an interesting dilemma. They can identify a single individual as ultimately responsible for the idea. Or, they can resist this inclination to focus on a single person as preeminent over all others, and instead consider on equal footing the multiple proposals given by the various scholars and what these proposals reveal more broadly about the mathematical scene in which these ideas flourished. The former approach often has the consequence of selecting that individual whose proposal most resembles how we understand the concept today, which can often be considered anachronistic and even ahistorical. The latter approach tends to result in a broader contextual historical account and for this reason, many historians would argue, is preferable.

One particular historian of mathematics, Florian Cajori, concluded at the very beginning of his work on the history of logarithms, “Few inventors have a clearer title to priority than has Napier to the invention of logarithms” (Cajori, 1915, p. 93; for other historians who hold a similar position, see Amy Shell-Gellasch (“Napier’s e,” 2008) and David E. Smith (1959, p. 149)). From the outset then, Cajori's focus was on Napier's claim to preeminence with respect to the logarithm concept, and his evaluation consisted of a systematic examination of Napier's contemporaries to reject any claims they might have had to this priority.

Indeed, Cajori brings to mind the challenges related to the retrospective appraisal of ideas in history. To find traces of an idea in the work of mathematicians past is too easily done from a future vantage point, and it risks anachronism or ignoring the context in which these ideas were understood. Accordingly, to speak of Napier’s “clear title to priority” to logarithms obscures the fact that his proposal remained always within a particular context for Napier. His logarithm concept was not expansive enough to include the general notion of a function under which we understand logarithms today.

Another approach for historians is rather to examine more roundly the various proposals and to refrain from attaching a single historical figure to the emergence of the logarithm. Focusing on the question of priority in mathematics causes scholars to look at the similarities between different manifestations of the same concept. This is very valuable for the history of mathematics, but emphasizing these similarities can distract from the more insightful issue of how mathematical inventors experimented with the concept, both imperfectly and successfully, and articulated it. Parallel insights reveal an abundant legacy of the motivations of these inventors and of challenges they faced and the ways in which they responded. They reveal the various individual and creative ways in which mathematicians reacted to the intellectual environment around then, and the unique methods or products they developed as a result of these stimuli. Parallel insights are in fact immensely rich for the historian; that legacy is too significant to be obscured by the determination to single out one individual as ultimately preeminent over the others. The parallel insights of Napier and Bürgi epitomize this. As we have seen, both presentations were richly and meaningfully different in many respects. Their conceptions were characterized by distinctive motivations, general designs, presentations, theoretical backdrops, and were also different in many of the details.

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**Note: This quote is frequently invoked in modern scholarship, but its origin remains a mystery to the authors of this paper, who have been unable to track it down in primary source material. Whether or not Laplace actually made the statement remains an outstanding question.