Adequality

Adequality is a technique developed by Pierre de Fermat in his treatise Methodus ad disquirendam maximam et minimam (a Latin treatise circulated in France c. 1636 ) to calculate maxima and minima of functions, tangents to curves, area, center of mass, least action, and other problems in calculus. According to André Weil, Fermat "introduces the technical term adaequalitas, adaequare, etc., which he says he has borrowed from Diophantus. As Diophantus V.11 shows, it means an approximate equality, and this is indeed how Fermat explains the word in one of his later writings." (Weil 1973). Diophantus coined the word παρισότης (parisotēs) to refer to an approximate equality. Claude Gaspard Bachet de Méziriac translated Diophantus's Greek word into Latin as adaequalitas. Paul Tannery's French translation of Fermat’s Latin treatises on maxima and minima used the words adéquation and adégaler. Fermat used adequality first to find maxima of functions, and then adapted it to find tangent lines to curves. To find the maximum of a term , Fermat equated (or more precisely adequated) and and after doing algebra he could cancel out a factor of and then discard any remaining terms involving To illustrate the method by Fermat's own example, consider the problem of finding the maximum of (In Fermat's words, it is to divide a line of length at a point , such that the product of the two resulting parts be a maximum.) Fermat adequated with . That is (using the notation to denote adequality, introduced by Paul Tannery): Canceling terms and dividing by Fermat arrived at Removing the terms that contained Fermat arrived at the desired result that the maximum occurred when . Fermat also used his principle to give a mathematical derivation of Snell's laws of refraction directly from the principle that light takes the quickest path. Fermat's method was highly criticized by his contemporaries, particularly Descartes. Victor Katz suggests this is because Descartes had independently discovered the same new mathematics, known as his method of normals, and Descartes was quite proud of his discovery.