Adequality

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Adequality is a technique developed by
Pierre de Fermat Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
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 a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
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 ran ...
of functions,
tangent In geometry, the tangent line (or simply tangent) to a plane curve at a given point is the straight line that "just touches" the curve at that point. Leibniz defined it as the line through a pair of infinitely close points on the curve. More ...
s to curves,
area Area is the quantity that expresses the extent of a region on the plane or on a curved surface. The area of a plane region or ''plane area'' refers to the area of a shape A shape or figure is a graphics, graphical representation of an obje ...
,
center of mass In physics, the center of mass of a distribution of mass in space (sometimes referred to as the balance point) is the unique point where the weighted relative position of the distributed mass sums to zero. This is the point to which a force may ...
,
least action The stationary-action principle – also known as the principle of least action – is a variational principle that, when applied to the ''action'' of a mechanical system, yields the equations of motion for that system. The principle states that ...
, and other problems in
calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. 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 aut ...
. 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 Claude may refer to: __NOTOC__ People and fictional characters * Claude (given name), a list of people and fictional characters * Claude (surname), a list of people * Claude Lorrain (c. 1600–1682), French landscape painter, draughtsman and etcher ...
translated Diophantus's Greek word into Latin as ''adaequalitas''.
Paul Tannery Paul Tannery (20 December 1843 – 27 November 1904) was a French mathematician and historian of mathematics. He was the older brother of mathematician Jules Tannery, to whose ''Notions Mathématiques'' he contributed an historical chapter. Thou ...
'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 $p\left(x\right)$, Fermat equated (or more precisely adequated) $p\left(x\right)$ and $p\left(x+e\right)$ 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 $p\left(x\right)=bx-x^2$ (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 $b\left(x+e\right)-\left(x+e\right)^2=bx-x^2+be-2ex-e^2$. That is (using the notation $\backsim$ to denote adequality, introduced by
Paul Tannery Paul Tannery (20 December 1843 – 27 November 1904) was a French mathematician and historian of mathematics. He was the older brother of mathematician Jules Tannery, to whose ''Notions Mathématiques'' he contributed an historical chapter. Thou ...
): :$bx-x^2\backsim bx-x^2+be-2ex-e^2.$ Canceling terms and dividing by $e$ Fermat arrived at :$b\backsim 2x+e.$ 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 Snell's law (also known as Snell–Descartes law and ibn-Sahl law and the law of refraction) is a formula used to describe the relationship between the angles of incidence and refraction, when referring to light or other waves passing through ...
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 Descartes. Victor Katz suggests this is because Descartes had independently discovered the same new mathematics, known as his
method of normals In calculus, the method of normals was a technique invented by Descartes for finding normal and tangent lines to curves. It represented one of the earliest methods for constructing tangents to curves. The method hinges on the observation that 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 of
infinitesimal calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...
. 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 Tannery Paul Tannery (20 December 1843 – 27 November 1904) was a French mathematician and historian of mathematics. He was the older brother of mathematician Jules Tannery, to whose ''Notions Mathématiques'' he contributed an historical chapter. Thou ...
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 $\backsim$ 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 $\scriptstyle\sim$.)
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 $\scriptstyle \approx$.)
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 $\scriptstyle \backsim$.)
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 $\scriptstyle \approx$.)
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: $\scriptstyle f\left(A\right)f\left(A+E\right)$. 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 $\scriptstyle \approx$.
The Latin quotation comes from Tannery's 1891 edition of Fermat, volume 1, page 140.
Michael Sean Mahoney Michael Sean Mahoney (June 30, 1939 – July 23, 2008) was a historian of science. Mahoney was born in New York City, and did his undergraduate studies at Harvard University, graduating in 1960. He earned a Ph.D. in history and history of science ...
(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 $\scriptstyle\approx$.) 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 $\scriptstyle b$ into two segments $\scriptstyle x$ and $\scriptstyle b-x$ whose product $\scriptstyle x\left(b-x\right)=bx-x^2$ is maximal, that is to find the rectangle with perimeter $\scriptstyle 2b$ that has the maximal area, he ermatproceeds as follows. First he substituted $\scriptstyle x+e$
(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: :$\scriptstyle b\left(x+e\right)-\left(x+e\right)^2=bx+be-x^2-2xe-e^2\; \sim\; bx-x^2.$ After canceling terms, he divided through by ''e'' to obtain $\scriptstyle b-2\,x-e\;\sim\;0.$ Finally he discarded the remaining term containing ''e'', transforming the pseudo-equality into the true equality $\scriptstyle x=\frac$ that gives the value of ''x'' which makes $\scriptstyle bx-x^2$ 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 Kirsti Andersen (born December 9, 1941, Copenhagen), published under the name Kirsti Pedersen, is a Danish historian of mathematics. She is an Associate Professor of the History of Science at Aarhus University, where she had her Candidate exami ...
(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 $\scriptstyle\approx$.) 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 John Colin Stillwell (born 1942) is an Australian mathematician on the faculties of the University of San Francisco and Monash University. Biography He was born in Melbourne, Australia and lived there until he went to the Massachusetts Instit ...
(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 The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta ...
.
Enrico Giusti Enrico Giusti (born Priverno, 1940), is an Italian mathematician mainly known for his contributions to the fields of calculus of variations, regularity theory of partial differential equations, minimal surfaces and history of mathematics. He ha ...
(2009) cites Fermat's letter to
Marin Mersenne Marin Mersenne, OM (also known as Marinus Mersennus or ''le Père'' Mersenne; ; 8 September 1588 – 1 September 1648) was a French polymath whose works touched a wide variety of fields. He is perhaps best known today among mathematicians for ...
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 In nonstandard analysis, the standard part function is a function from the limited (finite) hyperreal numbers to the real numbers. Briefly, the standard part function "rounds off" a finite hyperreal to the nearest real. It associates to every suc ...
which rounds off a finite
hyperreal number In mathematics, the system of hyperreal numbers is a way of treating infinite and infinitesimal (infinitely small but non-zero) quantities. The hyperreals, or nonstandard reals, *R, are an extension of the real numbers R that contains numbers ...
to its nearest
real number In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
.

# See also

*
Fermat's principle Fermat's principle, also known as the principle of least time, is the link between ray optics and wave optics. In its original "strong" form, Fermat's principle states that the path taken by a ray between two given points is the pat ...
*
Transcendental law of homogeneity In mathematics, the transcendental law of homogeneity (TLH) is a heuristic principle enunciated by Gottfried Wilhelm Leibniz most clearly in a 1710 text entitled ''Symbolismus memorabilis calculi algebraici et infinitesimalis in comparatione potent ...

# Bibliography

* Breger, H. (1994) "The mysteries of adaequare: a vindication of Fermat",
Archive for History of Exact Sciences ''Archive for History of Exact Sciences'' is a peer-reviewed academic journal currently published bimonthly by Springer Science+Business Media, covering the history of mathematics and of astronomy observations and techniques, epistemology of scienc ...
46(3):193–219. * * 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. * 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