Fermats «adæquare» – und kein Ende?

Abstract

In his papers on the determination of maxima and minima and on the calculation of tangents Pierre Fermat uses two different Latin verbs, aequare and adaequare, which do not differ semantically but are used by him obviously in different meanings. While aequabitur is used unambiguously in the sense of “is equal” the meaning of adaequabitur is disputed by the experts since Tannery’s French translation (Œuvres completes de Fermat, Vol. III, 1896). Herbert Breger (Arch. Hist. Exact Sci. 46, 193–219, (1994), p. 197 f), for instance, holds the view that Fermat used the word adaequare in the sense of “to put equal” and adds: In a mathematical context, the only difference between “aequare” and “adaequare” (if there is any) seems to be that the latter gives more stress on the fact that the equality is achieved. In contrast to this Michael Mahoney holds the thesis that adaequare describes a counterfactual equality (Mahoney, M.S.: Fermat, Pierre de. In: Dictionary of Scientific Biography, vol. IV (1971), p. 569) or a pseudo-equality (Mahoney, M.S.: The Mathematical Career of Pierre de Fermat (1601–1665), (1973), p. 164), whatever that may mean. This viewpoint has been taken up again recently by Enrico Giusti (Ann. Fac. Sci. Toulouse, Math. (6), 18 fascicule special, 59–85 (2009)) in order to bring arguments to bear against Breger. In contrast to these (and other) authors, I show that Fermat makes a subtle logical distinction between the words aequare and adaequare. The same distinction is made by Nicolas Bourbaki introducing his «theorie egalitaire». Notwithstanding: both verbs stand for a «relation d’egalite». On this premiss, I describe—using six selected examples—that Fermat’s “method” may be justified right down to the last detail, even from the view of today’s mathematical knowledge.