Adequality

In the history of infinitesimal calculus, adequality is a technique developed by Pierre de Fermat. Fermat said he borrowed the term from Diophantus. Diophantus coined the term παρισὀτης to refer to an approximate equality. Adequality was a technique first used to find maxima for functions and then adapted to find tangent lines to curves. The term adequality has been interpreted by some authors to mean approximate equality (or equality up to an infinitesimal). There is disagreement among scholars as to its precise meaning. To find the maximum of a function $p(x)$, Fermat would 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 use Fermat's own example to illustrate the method, 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 $\approx$ to denote adequality, introduced by Paul Tannery):

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

Canceling terms and dividing by $e$ Fermat arrived at

$b\approx 2x+e.$

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

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

There is disagreement amongst 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. For example, Mahoney's position is that Fermat's methods were essentially algebraic and not an introduction to limits or infinitesimals. On the other hand 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. Such a solution was provided in terms of the standard part function by Abraham Robinson. Breger however holds the view that Fermat used the word adaequare in the sense of "to put equal" and has to be represented by the usual equals sign =. Barner shares Breger's view and shows that Fermat uses "adaequabitur" in place of "aequabitur" when the equation describes a relation between two variables which is not a universally valid formula. The misleading introduction of the symbol $\backsim$ by Paul Tannery had the consequence that many authors tried to give this symbol a proper name: approximately equal, pseudo-equal, counterfactually equal, compared by adégalité, as nearly equal as possible. That's all nonsense. It's equal.

Both Newton and Leibniz referred to Fermat's work as an antecedent of infinitesimal calculus. He used his principle to give a mathematical derivation of Snell's laws of refraction directly from the principle that light takes the quickest path.

See also

• Fermat's principle
• Transcendental Law of Homogeneity

References

• 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

Bibliography

• 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.