Pierre de Fermat

The method of adequality was the method with which Fermat approached the problems of the calculus. We analyze its source in the parisotes of Diophantus. Adequality is a crucial step in Fermat's method of finding maxima, minima, tangents, and solving other problems that a modern mathematician would solve using infinitesimal calculus. The method is presented in a series of short articles in Fermat's collected works (Tannery).

We show that at least some of the manifestations of adequality amount to variational techniques exploiting a small, or infinitesimal, 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, as in the familiar difference f(A+E)-f(A). We question the relevance to understanding Fermat of 19th century dictionary definitions of parisotes 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 parisotes 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 parisoō to adaequo. We note the similar role of, respectively, adequality and the Transcendental Law of Homogeneity in the work of, respectively, Fermat and Leibniz on the problem of maxima and minima.

The first half of the 17th century was a time of intellectual ferment when wars of natural philosophy were echoes of religious wars, as we illustrate by a case study of an apparently innocuous mathematical technique called adequality pioneered by the honorable judge Pierre de Fermat, its relation to indivisibles, as well as to other hocus-pocus. André Weil noted that simple applications of adequality involving polynomials can be treated purely algebraically but more general problems like the cycloid curve cannot be so treated and require additional tools - leading the mathematician Fermat potentially into troubled waters. Breger attacks Tannery for tampering with Fermat's manuscript but it is Breger who tampers with Fermat's procedure by moving all terms to the left-hand side so as to accord better with Breger's own interpretation emphasizing the double root idea. We provide modern proxies for Fermat's procedures in terms of relations of infinite proximity as well as the standard part principle. For an explanation of the technical terms see Introduction to infinitesimal analysis without the axiom of choice

See also
Fermat's opponents (MO)
Lalouvère's activities as censor (MO)
More on infinitesimals
Stevin
Leibniz
Euler
Cauchy
Riemann
Cantor
Skolem
Infinitesimal topics
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