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 παρισότης (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 of length ${\displaystyle b}$at a point ${\displaystyle x}$, such that the product of the two resulting parts 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):

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

Canceling terms and dividing by ${\displaystyle e}$Fermat arrived at

${\displaystyle b\backsim 2x+e.}$

Removing the terms that contained ${\displaystyle e}$Fermat arrived at the desired result that the maximum occurred when ${\displaystyle x=b/2}$.

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.

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, 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.

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 ${\displaystyle \backsim }$for adequality in mathematical formulas.

Heinrich Wieleitner (1929) wrote:

(Wieleitner uses the symbol ${\displaystyle \scriptstyle \sim }$.)

Max Miller (1934) wrote:

(Miller uses the symbol ${\displaystyle \scriptstyle \approx }$.)

Jean Itard (1948) wrote:

(Itard uses the symbol ${\displaystyle \scriptstyle \backsim }$.)

Joseph Ehrenfried Hofmann (1963) wrote:

(Hofmann uses the symbol ${\displaystyle \scriptstyle \approx }$.)

Peer Strømholm (1968) wrote:

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:

The Latin quotation comes from Tannery's 1891 edition of Fermat, volume 1, page 140. Michael Sean Mahoney (1971) wrote:

(Mahoney uses the symbol ${\displaystyle \scriptstyle \approx }$.) 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:

(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:

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

After canceling terms, he divided through by e to obtain ${\displaystyle \scriptstyle b-2\,x-e\;\sim \;0.}$Finally he discarded the remaining term containing e, transforming the pseudo-equality into the true equality ${\displaystyle \scriptstyle x={\frac {b}{2}}}$that gives the value of x which makes ${\displaystyle \scriptstyle bx-x^{2}}$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:

(Andersen uses the symbol ${\displaystyle \scriptstyle \approx }$.) Herbert Breger (1994) wrote:

(Page 197f.) John Stillwell (Stillwell 2006 p. 91) wrote:

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

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.

• Fermat's principle
• Transcendental law of homogeneity

• Breger, Herbert (1994). "The mysteries of adaequare: A vindication of fermat". Archive for History of Exact Sciences. 46 (3): 193–219. doi:10.1007/BF01686277. S2CID 119440472.
• Edwards, C. H. (1979). The Historical Development of the Calculus. doi:10.1007/978-1-4612-6230-5. ISBN 978-0-387-94313-8.
• Giusti, E. (2009) "Les méthodes des maxima et minima de Fermat", Ann. Fac. Sci. Toulouse Math. (6) 18, Fascicule Special, 59–85.
• Grabiner, Judith V. (Sep 1983), "The Changing Concept of Change: The Derivative from Fermat to Weierstrass", Mathematics Magazine, 56 (4): 195–206, doi:10.2307/2689807, JSTOR 2689807
• Katz, V. (2008), A History of Mathematics: An Introduction, Addison Wesley
• 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.