Adequality

Adequality is a mathematical term introduced by Pierre de Fermat, which, roughly speaking, means equating things that are different (the exact meaning is controversial, see below). As a mathematical concept, this term had been used by Fermat and his followers, but fell into disuse from the 18th century onward.

Adequality has been reintroduced by modern historians of mathematics to name Fermat's method of computing maxima, minima and tangents, and to discuss the exact meaning of the word in Fermat's work.

Fermat said he borrowed the term from Diophantus. Diophantus coined the term παρισὀτης to refer to an approximate equality. The term was rendered as adaequalitas in Bachet's Latin translation of Diophantus, and adégalité in French.

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.

Behind the word adequality (Latin: adaequalitas) hides a long story of scholarly interpretations of a technical term used by Pierre de Fermat in his seminal work of extreme values and tangents.

The Latin noun adaequalitas and the accompanying Latin verb adaequare are used by Fermat in his treatises and letters about his method of determining maxima and minima of algebraic terms and of computing tangents to algebraic curves (like conic sections). It is a technical term within this method and does not mean the method itself.

The verb adaequare and the noun adaequalitas are used by Fermat in different idioms like adaequentur, adaequabuntur, comparare per adaequalitatem, debet adaequare, fiat per adaequalitatem, and, in his French correspondence, comparer par adéquation, however, most frequently in the form adaequabitur (third person singular passive future) which means "it will be (made) equal" (indefinite tens).

The verb adaequare is synonymous with the verb aequare (Thesaurus Linguae Latinae, Vol. I, p.562). The latter word was used by Viète nearly exclusively (with very few exceptions) to denote equality in place of the symbol =. This symbol was invented by Robert Recorde in 1557. Viète wrote most frequently aequabitur, and Fermat adopted it from him. However, Fermat used both words, aequabitur and adaequabitur, in place of the equals sign, obviously systematically and in a subtle difference of meanings.

In Fermat’s work about maxima and minima and tangents there is a place where he says that he adopted the word “adaequentur” from Diophantus. In his treatise “’’Methodus ad disquirendam maximam et minimam’’” (Fermat, Œuvres, Vol. I, pp.133-136) Fermat writes: ‘’Adaequentur, ut loquitur Diophantus, duo homogenia maxima aut minima aequalia …’’ (We make, as Diophantus says, two expressions of maximum or minimum in like manner equal.) But there are more questions than answers what that exactly means. This topic is taken up further below.

And this is the cause of a long scholarly debate which still goes on. For the purpose of this article we will denote aequabitur by $\scriptstyle =$ and adaequabitur (and its equivalent idioms) by $\scriptstyle \doteq$.

Fermat’s example Nr. 1. (Œuvres, Vol. I, p.134) To divide the line $\scriptstyle AB$ by the point $\scriptstyle E$ this way that the product of the two segments $\scriptstyle AE$ and $\scriptstyle EB$ becomes a maximum.

Fermat’s solution: Let $\scriptstyle b={\overline {AB}}$. Next let $\scriptstyle x$ be one of the two segments. Then the other one will be $\scriptstyle b-x$. And the product, which shall become a maximum, then equals $\scriptstyle b\,x-x^{2}$. However, if the first segment is $\scriptstyle x+e$ then the second one will be $\scriptstyle b-x-e$. The product of the segments is then

$\scriptstyle (x+e)(x-b-e)\,=\,b\,x-x^{2}-2\,e\,x+b\,e-e^{2}.$.

Next Fermat puts the two terms equal:

$\scriptstyle b\,x-x^{2}\,\doteq \,b\,x-x^{2}-2\,e\,x+b\,e-e^{2}.$.

Subtracting common terms on both sides yields

$\scriptstyle b\,e\,\doteq \,2\,e\,x+e^{2}.$

Now Fermat [tacitly assuming that $\scriptstyle e$ is unequal zero] cancels the common factor $\scriptstyle e$ and gets

$\scriptstyle b\,\doteq \,2\,x+e.$

Finally Fermat deletes $\scriptstyle e$ and yields the solution

$\scriptstyle b\,\,=\,2\,x.$

Fermat then remarks: “The question is answered if we take

$\scriptstyle x\,=\,{\frac {b}{2}}.$

Nec potest generalior dari methodus.” (It is impossible to give a more general method.)

There are several questions one would like to ask. What is Fermat’s idea behind his method? Did he explain how he found it? Has Fermat proven the validity of his proceeding? And, if not, is it possible to do this by means of modern analysis? And, above all: what is the precise meaning of “adaequabitur” ($\scriptstyle \doteq$)?

Let us first hear the voices of the scholarly experts! When Paul Tannery in 1896 published his French translation of Fermat’s Latin treatises on maxima and minima (Fermat, Œuvres, Vol. III, pp. 121-156) he was obviously insecure of the meaning of adaequare and aequalitas and avoided a translation of these words from Latin into modern French. He invented instead the new French verb “adégaler” and adopted Fermat’s “adéquation”. His introduction of the sign $\scriptstyle \backsim$ for “adaequabitur” in mathematical formulas, however, had the most serious consequences.

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. Edwards explains this is because Fermat never described his method with sufficient clarity or completeness to determine precisely what he intended. Fermat never explained whether e was supposed to be taken to be small, infinitesimal, or if he was taking a limit. Depending on how one reads Fermat's work, he either found an algebraic method for computing maxima of polynomials, or he began the field of infinitesimal calculus. Michael Sean Mahoney writes that Fermat's methods were essentially algebraic and not an introduction to limits or infinitesimals. 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. Katz & Katz wrote that Fermat provided the seeds of the solution to the infinitesimal puzzle a century before George Berkeley ever lifted up his pen to write The Analyst. Katz, Schaps and Shnider 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.