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Fermats «adæquare» – und kein Ende?

  • Philosophische und historische Sicht
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Abstract

In his papers on the determination of maxima and minima and on the calculation of tangents Pierre Fermat uses two different Latin verbs, æquare and adæquare, which do not differ semantically but are used by him obviously in different meanings. While æquabitur is used unambiguously in the sense of “is equal” the meaning of adæquabitur is disputed by the experts since Tannery’s French translation (Œuvres complètes 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 adæquare in the sense of “to put equal” and adds: In a mathematical context, the only difference between “æquare” and “adæquare” (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 adæquare 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 spécial, 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 æquare and adæquare. The same distinction is made by Nicolas Bourbaki introducing his «théorie égalitaire». Notwithstanding: both verbs stand for a «relation d’égalité». 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.

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  1. Institut für Mathematik, Universität Kassel, 34109, Kassel, Deutschland

    Klaus Barner

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Barner, K. Fermats «adæquare» – und kein Ende?. Math Semesterber 58, 13–45 (2011). https://doi.org/10.1007/s00591-010-0083-5

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