Adequality is a technique developed by Pierre de Fermat in his treatise Methodus ad disquirendam maximam et minimam[1] (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."
[2] Diophantus coined the word παρισότης (parisotēs) to refer to an approximate equality.
[3] Claude Gaspard Bachet de Méziriac translated Diophantus's Greek word into Latin as adaequalitas.
[citation needed] Paul Tannery's French translation of Fermat’s Latin treatises on maxima and minima used the words adéquation and adégaler.
to denote adequality, introduced by Paul Tannery): Canceling terms and dividing by
[5] Both Newton and Leibniz referred to Fermat's work as an antecedent of infinitesimal calculus.
Max Miller (1934)[7] wrote:Thereupon one should put the both terms, which express the maximum and the minimum, approximately equal (näherungsweise gleich), as Diophantus says.
Jean Itard (1948)[8] wrote:One knows that the expression "adégaler" is adopted by Fermat from Diophantus, translated by Xylander and by Bachet.
Joseph Ehrenfried Hofmann (1963)[9] wrote:Fermat chooses a quantity h, thought as sufficiently small, and puts f(x + h) roughly equal (ungefähr gleich) to f(x).
Peer Strømholm (1968)[10] wrote:The basis of Fermat's approach was the comparition of two expressions which, though they had the same form, were not exactly equal.
This part of the process he called "comparare par adaequalitatem" or "comparer per adaequalitatem", and it implied that the otherwise strict identity between the two sides of the "equation" was destroyed by the modification of the variable by a small amount:
The words Fermat used to express the process of suppressing terms containing E was 'elido', 'deleo', and 'expungo', and in French 'i'efface' and 'i'ôte'.
We can hardly believe that a sane man wishing to express his meaning and searching for words, would constantly hit upon such tortuous ways of imparting the simple fact that the terms vanished because E was zero.(p.
51) Claus Jensen (1969)[11] wrote:Moreover, in applying the notion of adégalité – which constitutes the basis of Fermat's general method of constructing tangents, and by which is meant a comparition of two magnitudes as if they were equal, although they are in fact not ("tamquam essent aequalia, licet revera aequalia non sint") – I will employ the nowadays more usual symbol
The Latin quotation comes from Tannery's 1891 edition of Fermat, volume 1, page 140.
Michael Sean Mahoney (1971)[12] wrote:Fermat's Method of maxima and minima, which is clearly applicable to any polynomial P(x), originally rested on purely finitistic algebraic foundations.
It assumed, counterfactually, the inequality of two equal roots in order to determine, by Viete's theory of equations, a relation between those roots and one of the coefficients of the polynomial, a relation that was fully general.
This relation then led to an extreme-value solution when Fermat removed his counterfactual assumption and set the roots equal.
Charles Henry Edwards, Jr. (1979)[13] wrote:For example, in order to determine how to subdivide a segment of length
(he used A, E instead of x, e) for the unknown x, and then wrote down the following "pseudo-equality" to compare the resulting expression with the original one: After canceling terms, he divided through by e to obtain
Finally he discarded the remaining term containing e, transforming the pseudo-equality into the true equality
Unfortunately, Fermat never explained the logical basis for this method with sufficient clarity or completeness to prevent disagreements between historical scholars as to precisely what he meant or intended."
Kirsti Andersen (1980)[14] wrote:The two expressions of the maximum or minimum are made "adequal", which means something like as nearly equal as possible.
In a mathematical context, the only difference between "aequare" and "adaequare" seems to be that the latter gives more stress on the fact that the equality is achieved.
His successors were unwilling to give up the convenience of ordinary equations, preferring to use equality loosely rather than to use adequality accurately.
The idea of adequality was revived only in the twentieth century, in the so-called non-standard analysis.Enrico Giusti (2009)[16] cites Fermat's letter to Marin Mersenne where Fermat wrote:Cette comparaison par adégalité produit deux termes inégaux qui enfin produisent l'égalité (selon ma méthode) qui nous donne la solution de la question" ("This comparison by adequality produces two unequal terms which finally produce the equality (following my method) which gives us the solution of the problem").. Giusti notes in a footnote that this letter seems to have escaped Breger's notice.
On page 36, Barner writes: "Why did Fermat continually repeat his inconsistent procedure for all his examples for the method of tangents?
Katz, Schaps, Shnider (2013)[18] argue that Fermat's application of the technique to transcendental curves such as the cycloid shows that Fermat's technique of adequality goes beyond a purely algebraic algorithm, and that, contrary to Breger's interpretation, the technical terms parisotes as used by Diophantus and adaequalitas as used by Fermat both mean "approximate equality".