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 παρισὀτης (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 ${\displaystyle p(x)}$, Fermat equated (or more precisely adequated) ${\displaystyle p(x)}$ and ${\displaystyle p(x+e)}$ and after doing algebra he could cancel out a factor of ${\displaystyle e,}$ and then discard any remaining terms involving ${\displaystyle e.}$ To illustrate the method by Fermat's own example, consider the problem of finding the maximum of ${\displaystyle p(x)=bx-x^{2}}$ (In Fermat's words, it is to divide a line ${\displaystyle AC}$ at a point ${\displaystyle E}$, such that the product of the two parts ${\displaystyle AE\times EC}$ be a maximum.) Fermat adequated ${\displaystyle bx-x^{2}}$ with ${\displaystyle b(x+e)-(x+e)^{2}=bx-x^{2}+be-2ex-e^{2}}$. That is (using the notation ${\displaystyle \backsim }$ to denote adequality, introduced by Paul Tannery):

...
Wikipedia