The mathematical technique in Fermat's deduction of the law of refraction

Brief History of Natural Sciences for Nature-Inspired Computing in Engineering

“…In Europe, the most fundamental natural science, the mechanics of terrestrial and heavenly bodies was developed in the 15 th century. The basic features of many fundamental mathematical methods were developed in the 16 th and 17 th century such as logarithms by John Napier (1550-1617 (Hobson 1914;see Napier 1969), analytical geometry by Rene Descartes (1596Descartes ( -1650 (Forbes 1777; see Descartes 1954), differential and integral calculus by Pierre de Fermat 1 (1601-1665) (Anderson 1983), Sir Isaac Newton (1642-1727) (Cohen 1971;see Newton 1972) and Gottfried Wilhelm Leibnitz (1646Leibnitz ( -1716 (Child 1920;see Leibnitz 1975). The eighteenth century Europe saw a number mathematical prodigies like Leonhard Euler (1707-1783) with significant contribution to the development of mathematics (Grattan-Guinness 1971), Joseph-Louis Lagrange (1736-1813) with tremendous contribution to development of analytical mechanics, calculus of variations and celestial mechanics (Fraser 1983; see Lagrange 1901), Pierre Simon Laplace (1749-1827) contributed to celestial mechanics (see Laplace 1966), Adrien-Marie Legendre (1752-1833) contributed to geometry, differential equation, theories of functions and numbers (Boyer 1991) and Jean Baptiste Joseph Fourier (1768-1830) contributed to mathematical physics (Grattan-Guinness 1972;Jourdian 1912).…”

Das Leben Fermats

“…Jahrhundert an der Universität Or-7 Eine einigermaßen befriedigende Würdigung des wissenschaftlichen Werks Fermats würde den Rahmen dieses Aufsatzes sprengen. Der Leser findet eine detaillierte Diskussion in den Aufsätzen und Monographien [1,8,9,14,22,23,26,27,28,32,35,38,42,46,56,57,58,59]. So kompetent deren Autoren die Mathematik Fermats analysieren, so unzuverlässig sind sie fast alle (Catherine Goldstein ausgenommen), wenn sie Angaben zu Fermats Biographie oder zum zeitgeschichtlichen, sozialen oder religiösen Hintergrund von Fermats Leben machen.…”

Almost Equal: the Method of Adequality from Diophantus to Fermat and Beyond

“…133-172]. The first article, Methodus ad Disquirendam Maximam et Minimam, 2 opens with a summary of an algorithm for finding the maximum or minimum value of an algebraic expression in a variable A. For convenience, we will write such an expression in modern functional notation as f (a).…”

“…See Section 1.2 for a more detailed etymological discussion. 2 In French translation, Méthode pour la Recherche du Maximum et du Minimum. Further quotations will be taken from the French text [19].…”

“…7 The method (leaving aside step (5) for the moment) is immediately understandable to a modern reader as the elementary calculus exercise of finding the extremum by solving the equation f ′ (a) = 0. But the real question is how Fermat understood this algorithm in his own terms, in the mathematical language of his time, prior to the invention of calculus by Barrow, Leibniz, Newton, et al There are two crucial points in trying to understand Fermat's reasoning: first, the meaning of "adequality" in step (2), and second, the justification for suppressing the terms involving positive powers of e in step (6). The two issues are closely related because interpretation of adequality depends on the conditions on e. One condition which Fermat always assumes is that e is positive.…”