Der Begriff der ,Angleichung' (παρισóτης, adaequatio) bei Diophant und Fermat

Abstract

'Adequality' is an essential step in Fermat's method of finding maxima, minima, tangents, centers of gravity etc. However, it seems to be an enigma what this method really consists of. It is usually understood that a convincing interpretation would require some elements of approximation or (infinitesimal?) smallness, although such elements cannot be found in Fermat's writings. We shall present a reading which is based on Fermat's frequent use of the, less-than'-relation when either a maximal point on a given curve or a point on a tangent outside the curve is considered. In all applications of his method Fermat constructs certain additional polynomials (h) which have the form (h) = hψ(h) and shows that (h) is strictly positive in a certain neighbourhood of 0 for h ≠ 0. This is the core of the fermatian method of, adequality'. It allows one to conclude that (h) has a double root at h = 0 and hence the constant term c of ψ(h) must be zero. In this way Fermat passes from, adequality' to, equality', hoping that the equation c = 0 yields enough information to arrive at a solution of the problem under consideration. It is clear that infinitesimal arguments are not needed in this reading of the fermatian method, and that the only mathematical techniques used herein were available at Fermat's time. In addition the mathematics of Fermat becomes clear and correct. We also carefully analyze the source of Fermat's method of adequality, namely the 'p a r i s ó t ē s ' of Diophantus.