  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.
