When introducing the integral via the notion of Riemann sums, the problems quickly become too hard to carry out the details in class. Pierre de Fermat’s ingenious integration of \(y={x^n}\) provides an “admirably simple and appropriate example” [Boyer 1945, p. 29] which can easily and profitably be done in class. The only fact that is needed is the sum of the geometric series, a fact the student needs in other situations anyway.

This etching of Fermat (1601–1665) appears in his *Varia opera mathematica *(1679)

and was made by prolific engraver and artist François de Poilly (1623–1693).

*Convergence *Mathematical Treasures.

The 1656 publication of the *Arithmetica infinitorum* (*The **Arithmetic of Infinitesimals*) of John Wallis (1616–1703) prompted Fermat to compose a treatise on the quadrature methods for finding the area under curves that he had been working on for two decades. The result was his posthumous “De aequationum localium transmutatione, & emendatione, ad multimodam curvilineorum inter se, vel cum rectilineis comparationem, cui annectitur proportionis geometricae in quadrandis infinitis parabolis et hyperbolis usus,” or “On the transformation and alteration of local equations for the purpose of variously comparing curvilinear figures among themselves or to rectilinear figures, to which is attached the use of geometric proportions in squaring an infinite number of parabolas and hyperbolas."

More concisely known as the “Treatise on Quadrature,” this work was probably written in 1658, but it was not published (or circulated) until 1679, as part of Fermat’s *Varia opera mathematica*. While it appeared too late to have a profound effect on the historical development of the calculus, the approach to integration found in Fermat’s treatise is closely related to how we introduce integration to students today. Specifically, our interest is in the method he created for dealing with the quadrature of the “higher parabolas,” \(y = k x^\frac{p}{q}\). This is a neat trick that will allow you to use a Riemann sum definition of integrals to evaluate the integrals of powers of *\(x\)* in class—and for an arbitrary integer \(n\).

The clever idea that Fermat had for integrating \(y=x^n\) was not to divide the interval of integration \([0,a]\) into equal subdivisions, but rather to use unequal subdivisions. Now, it is quite clear where this idea came from. He had been finding the area under his generalized hyperbolas \(y = \frac{1}{x^n}\) on the interval \([a,\infty)\), and in this situation it is quite natural to use unequal subdivisions with the width of each rectangle increasing as \(x\) increases.^{[1]} When he considered the generalized parabolas \(y=x^n\) on \([0,1]\) with this new trick in mind, it was natural to invert the ray \([1,\infty)\) into the finite interval \([0,1]\). When this is done, we get unequal partitions of the interval \([0,a]\) with the width of each rectangle now decreasing as \(x\) approaches \(0\). Let’s look at the details of what Fermat did.

Let \(E\) be a positive constant less than \(1\). Use it to divide the closed interval \([0,a]\) into infinitely many subintervals of different lengths at the points \(\ldots aE^3, aE^2, aE, a\). Then construct rectangles on these subintervals so that they circumscribe the curve \(y=x^n\) and add up their areas, which form a geometric progression:

\[\begin{array}{ccl}S_E &= & \displaystyle \sum_{i=0}^{\infty}\left(aE^i\right)^n\left(aE^i-aE^{i+1}\right)\\ &= & a^{n+1}\displaystyle \sum_{i=0}^{\infty}E^{in}E^i\left(1-E\right)\\ &= & a^{n+1}\left(1-E\right)\displaystyle \sum_{i=0}^{\infty}\left(E^{n+1}\right)^i\\ &= & a^{n+1}\left(1-E\right) \frac{1}{1-E^{n+1}}\\ &= & \frac{a^{n+1}}{1+E+E^2+\ldots +E^{n}} \end{array} \]

The last step here follows by elementary algebra. Now as \(E\) approaches 1 we see that \(S_E\) approaches \(\frac{a^{n+1}}{n+1}\). Thus we have \(\int_0^a x^n dx = \frac{a^{n+1}}{n+1}\).

This argument works for all rational \(n\), except for the logarithmic case, \(n = -1\), which Fermat called "the infinite hyperbola of Apollonius."

GeoGebra applet illustrating Fermat's partitioning approach, created by Erik Tou (University of Washington).

The above proof is quite easy for us to understand, but this is primarily because we have translated it into modern notation and nomenclature. For Fermat, the word “quadrature” literally meant finding a rectangular shape with the same area as that of the region under the given curve. He did this using proportions of lines and areas, so his arguments are fairly difficult for modern readers to understand. An error in Dirk Struik’s English translation (line 21 on page 220 should not say “decreasing”) makes Fermat’s argument for the generalized hyperbola case even harder to follow [Struik 1969].^{[2]}

There are very few problems where unequal subdivisions are useful, but here is one. Use the definition of the definite integral to show \[\int_0^a \sqrt{x}dx = \frac{2}{3}a^{2/3}.\] Use the \(n\) partition points \(x_k=\frac{bk^2}{n^2}\) and the right endpoints of these intervals as evaluation points. (Taken from [Simmons 1985, p. 176].)

Boyer, Carl B. 1945. “Fermat's integration of *x ^{n}*.”

Fermat, Pierre de. 1679. “De aequationum localium transmutatione, & emendatione, ad multimodam curvilineorum inter se, vel cum rectilineis comparationem, cui annectitur proportionis geometricae in quadrandis infinitis parabolis et hyperbolis usus.” In *Varia opera mathematica D. Petri de Fermat*, 44–57. Toulouse: Johannes Pech.

Fermat, Pierre de. 1896. “Sur la transformation simplification des équations de lieux, pour la comparaison sous toutes les formes des aires curvilignes, soit entre elles, soit avec les rectilignes, et en meme temps sure l’emploi de la progressions géométrique pour la quadrature des paraboles et hyperboles a l’infini.” In *Oeuvres de Fermat*, 3:216–237. Paris: Gauthier-Villar. French translation by Paul Tannery of [Fermat 1679].

Klyve, Dominic. 2020. “Beyond Riemann Sums: Fermat's Method of Integration.” Digital Commons, TRansforming Instruction in Undergraduate Mathematics via Primary Historical Sources (TRIUMPHS). *Calculus*. 12.

Mahoney, Michael S. 1970. “Fermat, Pierre de.” In *Dictionary of Scientific Biography, *edited by Charles Coulston Gillispie, 4:566–576. New York: Charles Scribner’s Sons.

Mahoney, Michael S. 1973. *The Mathematical Career of Pierre de Fermat, 1601*–*1665*. Princeton: Princeton University Press, 1973. See especially pp. 243–253.

Simmons, George F. 1985. *Calculus with Analytic Geometry*. New York: McGraw Hill.

Stedall, Jacqueline. 2004.*The Arithmetic of Infinitesimals: John Wallis 1656*. New York: Springer Verlag.

Stedall, Jacqueline. 2008. *Mathematics Emerging: A Sourcebook 1540–1900*. Oxford: Oxford University Press.

Struik, Dirk J. 1969. “Fermat. Integration.” In *A Source Book in Mathematics, 1200-1800*, 219–222. Cambridge: Harvard University Press. English translation by Dirk J. Struik of a portion of [Fermat 1896].

Wallis, John. 1656. *Arithmetica infinitorum*. Oxford.

[1] Fermat’s biographer gave a splendid recreation of the likely thought process behind Fermat’s move to unequal subdivisions for generalized hyperbolas in [Mahoney 1973, pp. 245–247].

[2] Struik based his translation on the French translation in [Fermat 1896], in which the same error is committed. Jacqueline Stedall gave an accurate and more authentic English translation of the generalized hyperbola case in [Stedall 2008, pp. 78–84], along with a reproduction of the original Latin source. Another accurate translation of this case that similarly avoids the use of modern notation is found in the student project [Klyve 2020].