Adequality is a technique developed by

English translation of Fermat's treatise ''Methodus ad disquirendam maximam et minimam''. (a

Max Miller (1934) wrote:

Jean Itard (1948) wrote:

Joseph Ehrenfried Hofmann (1963) wrote:

Peer Strømholm (1968) wrote:

Claus Jensen (1969) wrote:

Michael Sean Mahoney (1971) wrote:

Charles Henry Edwards, Jr. (1979) wrote:

Herbert Breger (1994) wrote:

John Stillwell (Stillwell 2006 p. 91) wrote:

Enrico Giusti (2009) cites Fermat's letter to Marin Mersenne where Fermat wrote:

Klaus Barner (2011) asserts that Fermat uses two different Latin words (aequabitur and adaequabitur) to replace the nowadays usual equals sign, ''aequabitur'' when the equation concerns a valid identity between two constants, a universally valid (proved) formula, or a conditional equation, ''adaequabitur'', however, when the equation describes a relation between two variables, which are ''not independent'' (and the equation is no valid formula). On page 36, Barner writes: "Why did Fermat continually repeat his inconsistent procedure for all his examples for the method of tangents? Why did he never mention the secant, with which he in fact operated? I do not know." Katz, Schaps, Shnider (2013) 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". They develop a formalisation of Fermat's technique of adequality in modern mathematics as the standard part function which rounds off a finite hyperreal number to its nearest real number.

Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French people, French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In par ...

in his treatise ''Methodus ad disquirendam maximam et minimam''''METHOD FOR THE STUDY OF MAXIMA AND MINIMA''English translation of Fermat's treatise ''Methodus ad disquirendam maximam et minimam''. (a

Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally spoken in the area around Rome, known as Latium. Through the power of the Roman Republic, it became the dominant la ...

treatise circulated in France c. 1636) to calculate maxima and minima
In mathematical analysis, the maxima and minima (the respective plurals of maximum and minimum) of a function, known collectively as extrema (the plural of extremum), are the largest and smallest value of the function, either within a given r ...

of functions, tangent
In geometry
Geometry (from the grc, γεωμετρία; ''wikt:γῆ, geo-'' "earth", ''wikt:μέτρον, -metron'' "measurement") is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space ...

s to curves, area
Area is the quantity that expresses the extent of a two-dimensional region, shape, or planar lamina, in the plane. Surface area is its analog on the two-dimensional surface of a three-dimensional object. Area can be understood as the am ...

, center of mass
In physics, the center of mass of a distribution of mass
Mass is the physical quantity, quantity of ''matter'' in a physical body. It is also a measure (mathematics), measure of the body's ''inertia'', the resistance to acceleration (change ...

, least action
:''This article discusses the history of the principle of least action. For the application, please refer to action (physics)
In physics, action is an attribute of the dynamics of a physical system from which the equations of motion of the sy ...

, and other problems in calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

. According to André Weil
André Weil (; ; 6 May 1906 – 6 August 1998) was a French mathematician, known for his foundational work in number theory and algebraic geometry. He was a founding member and the ''de facto'' early leader of the mathematical Bourbaki group. Th ...

, Fermat "introduces the technical term adaequalitas, adaequare, etc., which he says he has borrowed from Diophantus
Diophantus of Alexandria ( grc, Διόφαντος ὁ Ἀλεξανδρεύς; born probably sometime between AD 200 and 214; died around the age of 84, probably sometime between AD 284 and 298) was an Alexandrian mathematician, who was the autho ...

. 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 TanneryImage:Paul Tannery.jpg, Paul Tannery
Paul Tannery (20 December 1843 – 27 November 1904) was a France, French Mathematics, mathematician and History of Mathematics, historian of mathematics. He was the older brother of mathematician Jules Tannery, t ...

's French translation of Fermat’s Latin treatises on maxima and minima used the words ''adéquation'' and ''adégaler''.
Fermat's method

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 $p(x)$, Fermat equated (or more precisely adequated) $p(x)$ and $p(x+e)$ and after doing algebra he could cancel out a factor of $e,$ and then discard any remaining terms involving $e.$ To illustrate the method by Fermat's own example, consider the problem of finding the maximum of $p(x)=bx-x^2$ (In Fermat's words, it is to divide a line of length $b$ at a point $x$, such that the product of the two resulting parts be a maximum.) Fermat ''adequated'' $bx-x^2$ with $b(x+e)-(x+e)^2=bx-x^2+be-2ex-e^2$. That is (using the notation $\backslash backsim$ to denote adequality, introduced byPaul TanneryImage:Paul Tannery.jpg, Paul Tannery
Paul Tannery (20 December 1843 – 27 November 1904) was a France, French Mathematics, mathematician and History of Mathematics, historian of mathematics. He was the older brother of mathematician Jules Tannery, t ...

):
:$bx-x^2\backslash backsim\; bx-x^2+be-2ex-e^2.$
Canceling terms and dividing by $e$ Fermat arrived at
:$b\backslash backsim\; 2x+e.$
Removing the terms that contained $e$ Fermat arrived at the desired result that the maximum occurred when $x=b/2$.
Fermat also used his principle to give a mathematical derivation of Snell's law
of light at the interface between two media of different refractive index, refractive indices, with n2 > n1. Since the velocity is lower in the second medium (v2 < v_{1}), the angle of refraction θ_{2} is less than the angle of in ...

s of refraction directly from the principle that light takes the quickest path.
Descartes' criticism

Fermat's method was highly criticized by his contemporaries, particularly . Victor Katz suggests this is because Descartes had independently discovered the same new mathematics, known as hismethod of normalsIn calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. The ...

, and Descartes was quite proud of his discovery. Katz also notes that while Fermat's methods were closer to the future developments in calculus, Descartes' methods had a more immediate impact on the development.
Scholarly controversy

Both Newton and Leibniz referred to Fermat's work as an antecedent ofinfinitesimal calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimal
In mathematics, infinitesimals or infinitesimal numbers are quantities that are closer to zero than any standard real number, but are not zero. They do not ex ...

. Nevertheless, there is disagreement amongst modern scholars about the exact meaning of Fermat's adequality. Fermat's ''adequality'' was analyzed in a number of scholarly studies. In 1896, Paul TanneryImage:Paul Tannery.jpg, Paul Tannery
Paul Tannery (20 December 1843 – 27 November 1904) was a France, French Mathematics, mathematician and History of Mathematics, historian of mathematics. He was the older brother of mathematician Jules Tannery, t ...

published a French translation of Fermat’s Latin treatises on maxima and minima (Fermat, Œuvres, Vol. III, pp. 121–156). Tannery translated Fermat's term as “adégaler” and adopted Fermat’s “adéquation”. Tannery also introduced the symbol $\backslash backsim$ for adequality in mathematical formulas.
Heinrich Wieleitner (1929) wrote:Fermat replaces ''A'' with ''A''+''E''. Then he sets the new expression roughly equal (angenähert gleich) to the old one, cancels equal terms on both sides, and divides by the highest possible power of ''E''. He then cancels all terms which contain ''E'' and sets those that remain equal to each other. From that he required''A'' results. That ''E'' should be as small as possible is nowhere said and is at best expressed by the word "adaequalitas".(Wieleitner uses the symbol $\backslash scriptstyle\backslash sim$.)

Max Miller (1934) wrote:

Thereupon one should put the both terms, which express the maximum and the minimum, approximately equal (näherungsweise gleich), as Diophantus says.(Miller uses the symbol $\backslash scriptstyle\; \backslash approx$.)

Jean Itard (1948) wrote:

One knows that the expression "adégaler" is adopted by Fermat from Diophantus, translated by Xylander and by Bachet. It is about an approximate equality (égalité approximative) ".(Itard uses the symbol $\backslash scriptstyle\; \backslash backsim$.)

Joseph Ehrenfried Hofmann (1963) wrote:

Fermat chooses a quantity ''h'', thought as sufficiently small, and puts ''f''(''x'' + ''h'') roughly equal (ungefähr gleich) to ''f''(''x''). His technical term is ''adaequare''.(Hofmann uses the symbol $\backslash scriptstyle\; \backslash approx$.)

Peer Strømholm (1968) 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: $\backslash scriptstyle\; f(A)f(A+E)$. This, I believe, was the real significance of his use of Diophantos' πἀρισον, stressing the ''smallness'' of the variation. The ordinary translation of 'adaequalitas' seems to be "approximate equality", but I much prefer "pseudo-equality" to present Fermat's thought at this point.He further notes that "there was never in M1 (Method 1) any question of the variation ''E'' being put equal to zero. 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) 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 $\backslash scriptstyle\; \backslash approx$.The Latin quotation comes from Tannery's 1891 edition of Fermat, volume 1, page 140.

Michael Sean Mahoney (1971) 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. Borrowing a term from Diophantus, Fermat called this counterfactual equality 'adequality'.(Mahoney uses the symbol $\backslash scriptstyle\backslash approx$.) On p. 164, end of footnote 46, Mahoney notes that one of the meanings of adequality is ''approximate equality'' or ''equality in the limiting case''.

Charles Henry Edwards, Jr. (1979) wrote:

For example, in order to determine how to subdivide a segment of length $\backslash scriptstyle\; b$ into two segments $\backslash scriptstyle\; x$ and $\backslash scriptstyle\; b-x$ whose product $\backslash scriptstyle\; x(b-x)=bx-x^2$ is maximal, that is to find the rectangle with perimeter $\backslash scriptstyle\; 2b$ that has the maximal area, he [Fermat] proceeds as follows. First he substituted $\backslash scriptstyle\; x+e$(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: :$\backslash scriptstyle\; b(x+e)-(x+e)^2=bx+be-x^2-2xe-e^2\backslash ;\; \backslash sim\backslash ;\; bx-x^2.$ After canceling terms, he divided through by ''e'' to obtain $\backslash scriptstyle\; b-2\backslash ,x-e\backslash ;\backslash sim\backslash ;0.$ Finally he discarded the remaining term containing ''e'', transforming the pseudo-equality into the true equality $\backslash scriptstyle\; x=\backslash frac$ that gives the value of ''x'' which makes $\backslash scriptstyle\; bx-x^2$ maximal. 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) wrote:

The two expressions of the maximum or minimum are made ''"adequal"'', which means something like as nearly equal as possible.(Andersen uses the symbol $\backslash scriptstyle\backslash approx$.)

Herbert Breger (1994) wrote:

I want to put forward my hypothesis: ''Fermat used the word "adaequare" in the sense of'' "to put equal" ... 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.(Page 197f.)

John Stillwell (Stillwell 2006 p. 91) wrote:

Fermat introduced the idea of adequality in 1630s but he was ahead of his time. 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) 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.

Klaus Barner (2011) asserts that Fermat uses two different Latin words (aequabitur and adaequabitur) to replace the nowadays usual equals sign, ''aequabitur'' when the equation concerns a valid identity between two constants, a universally valid (proved) formula, or a conditional equation, ''adaequabitur'', however, when the equation describes a relation between two variables, which are ''not independent'' (and the equation is no valid formula). On page 36, Barner writes: "Why did Fermat continually repeat his inconsistent procedure for all his examples for the method of tangents? Why did he never mention the secant, with which he in fact operated? I do not know." Katz, Schaps, Shnider (2013) 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". They develop a formalisation of Fermat's technique of adequality in modern mathematics as the standard part function which rounds off a finite hyperreal number to its nearest real number.

See also

*Fermat's principle *Transcendental law of homogeneityReferences

Bibliography

* Breger, H. (1994) "The mysteries of adaequare: a vindication of Fermat", Archive for History of Exact Sciences 46(3):193–219. * * Enrico Giusti, Giusti, E. (2009) "Les méthodes des maxima et minima de Fermat", Ann. Fac. Sci. Toulouse Math. (6) 18, Fascicule Special, 59–85. * * * Stillwell, J.(2006) ''Yearning for the impossible. The surprising truths of mathematics'', page 91, A K Peters, Ltd., Wellesley, MA. * André Weil, Weil, A., Book Review: The mathematical career of Pierre de Fermat. Bull. Amer. Math. Soc. 79 (1973), no. 6, 1138–1149. {{Infinitesimals Mathematical terminology History of calculus