Adequality

Adequality is a term introduced by historians of mathematics to discuss the mathematical meaning of the Latin word adaequalitas, as it was used by Pierre de Fermat in his work on finding maxima, minima and tangents. André Weil wrote: "[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 term παρισὀτης to refer to an approximate equality. The term was rendered as adaequalitas in Claude Gaspard Bachet de Méziriac's Latin translation of Diophantus, and adéquation and adégaler in Paul Tannery's French translation of Fermat’s Latin treatises on maxima and minima.

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 did equate (or more precisely adequate) $p(x)$ and $p(x+e)$ and after doing algebra he could divide by 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 $p(x)=bx-x^{2}$. Fermat adequated $bx-x^{2}$ with $b(x+e)-(x+e)^{2}=bx-x^{2}+be-2ex-e^{2}$. That is (using the notation $\backsim$ to denote adequality, introduced by Paul Tannery):

$bx-x^{2}\backsim bx-x^{2}+be-2ex-e^{2}.$

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 laws 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. V. 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. He 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. 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 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 $\scriptstyle \backsim$ 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 [the 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:

$\scriptstyle f(A){\sim }f(A+E)$.

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."

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 - I will employ the nowdays mor usual symbol $\scriptstyle \approx$."

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 $\scriptstyle \approx$.) 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 $\scriptstyle x(b-x)=bx-x^{2}$ is maximal, that is to find the rectangle with perimeter $\scriptstyle 2b$ that has the maximal area, he [Fermat] proceeds 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:

$\scriptstyle b(x+e)-(x+e)^{2}=bx+be-x^{2}-2xe-e^{2}\;\sim \;bx-x^{2}.$

After canceling terms, he divided through by e to obtain $\scriptstyle 2\,x+b\;\sim \;b.$ Finally he discarded the remaining term containing e, transforming the pseudo-equality into the true equality $\scriptstyle x={\frac {b}{2}}$ 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 (1980) wrote: "The two expressions of the maximum or minimum are made "adequal", which means something like as nearly equal as possible." (Anderson 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 (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."

Klaus Barner (2011) champions Breger and claims 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). Barner emphasizes that his claim is not based on any theory but only on a simple observation, which may be checked by every mathematically inclined person who can read Latin and French.

Katz, Schaps, Shnider (2013) argue that Fermat's application of the technique to transcendental curves such as the cycloid shows that adequality goes beyond a purely algebraic algorithm, and that, contrary to Breger's interpretation, the terms parisotes and adaequalitas mean "approximate equality". They develop a formalisation of Fermat's technique of adequality in modern mathematics as the standard part function sending a finite hyperreal number to the real number infinitely close to it.

See also

• Fermat's principle
• Transcendental Law of Homogeneity

Bibliography

• Edwards, C. H. Jr. (1994), The Historical Development of the Calculus, Springer
• Grabiner, Judith V. (1983), "The Changing Concept of Change: The Derivative from Fermat to Weierstrass", Mathematics Magazine, 56 (4): 195–206 {{citation}}: Unknown parameter |month= ignored (help)
• Katz, V. (2008), A History of Mathematics: An Introduction, Addison Wesley

• Barner, K. (2011) "Fermats <<adaequare>> - und kein Ende?" Mathematische Semesterberichte (58), pp. 13-45
• Breger, H. (1994) "The mysteries of adaequare: a vindication of Fermat", Archive for History of Exact Sciences 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.