Almost equal: Method of adequality from Diophantus to Fermat and beyond

TL;DR

This paper explores Fermat's method of adequality, tracing its historical roots from Diophantus to Fermat, and offers a new interpretation emphasizing its variational and approximation aspects in calculus.

Contribution

It provides a novel reading of adequality, clarifies its connection to variational techniques, and challenges previous interpretations based on 19th-century dictionary definitions.

Findings

Adequality involves variational techniques with infinitesimal variations.

Fermat's approach incorporates inherent approximation and smallness.

Historical analysis links adequality to Diophantus and Bachet's interpretations.

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 parisotes of Diophantus, and propose a novel reading thereof. 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. 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. 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 parisoo 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.