Almost Equal: the Method of Adequality from Diophantus to Fermat and Beyond

Abstract

We analyze some of the main approaches in the literature to the method of 'adequality' with which Fermat approached the problems of the calculus, as well as its source in the parisâth § of Diophantus, and propose a novel reading thereof. Adequality is a crucial step in Fermat's method of ªnding maxima, minima, tangents, and solving other problems that a modern mathematician would solve using inªnitesimal calculus. The method is presented in a series of short articles in Fermat's collected works (Tannery and Henry 1981, pp. 133-172). We show that at least some of the manifestations of adequality amount to variational techniques exploiting a small, or inªn-itesimal, variation e. Fermat's treatment of geometric and physical applications suggests that an aspect of approximation is inherent in adequality, as well as an aspect of smallness on the part of e. We question the relevance to understanding Fermat of 19th century dictionary deªnitions of parisâth § and adaequare, cited by Breger, and take issue with his interpretation of adequality, including his novel reading of Diophantus, and his hypothesis concerning alleged tampering with Fermat's texts by Carcavy. We argue that Fermat relied on Bachet's reading of Diophantus. Diophantus coined the term parisâth § for mathematical purposes and used it to refer to the way in which 1321/711 is approximately equal to 11/6. Bachet performed a semantic calque in passing from parisoo to adaequo. We note the similar role of, respectively, adequality and the Transcendental Law of Homogeneity in the work of, respectively, Fermat (1896) and Leibniz (1858) on the problem of maxima and minima.We are grateful to Enrico Giusti, Alexander Jones, and David Sherry for helpful comments. The inºuence of Hilton Kramer (1928Kramer ( -2012 is obvious.

Approaches To Analysis With Infinitesimals Following Robinson, Nelson, And Others

“…The view that numbers come in more than one flavor is closely parallel to the dichotomy of assignable vs inassignable quantities (the latter being viewed as useful fictions) in Leibnizian calculus; see Bair et al [8] and Bascelli et al [16] for more details. Even earlier, Fermat's technique of adequality exploited procedures using E in a striking anticipation of later infinitesimal techniques; see [100]. Meanwhile L. Carnot spoke of quantités désignées and quantités auxiliaires in 1797; see Barreau [12, p. 46, 53].…”

Gregory’s Sixth Operation

“…Indeed, Leibniz himself cites an antecedent in Pierre de Fermat's technique (known as the method of adequality; see [Katz et al 2013]), in the following terms:…”

Rubik’s for Cryptographers

“…A recent study of Fermat (Katz, Schaps, and Shnider 2013 [66]) shows how the nature of his contribution to the calculus was distorted in recent Fermat scholarship, similarly due to an "evolutionary dead-end" bias (see the subsection "Fermat, Pierre de").…”

“…However, a detailed analysis (see [66]) of Fermat's treatment of the cycloid reveals that Fermat did rely on issues of "smallness" in his treatment of the cycloid and reveals that Breger's interpretation thereof contains both mathematical errors and errors of textual analysis. Similarly, Fermat's proof of Snell's law, a variational principle, unmistakably relies on ideas of "smallness".…”