Historical Notes for the Calculus Classroom
Readers of Convergence probably agree that, for many reasons, it is a good idea to include some history of mathematics in a calculus course. The questions that many of us have are what history should be used and how to implement it. As a long-time believer in the necessity of history in teaching mathematics, V. Frederick Rickey has successfully integrated history into the classroom in a wide variety of ways. His list of techniques includes [Rickey 1996, p. 254]:
- History of specific topics
- History of notation
- Etymology of terms
- Pictures of mathematicians
- Quotations by famous mathematicians
- Biography
- Anecdotes
- Title pages from famous books
- Problems from old textbooks
- Historical errors
Fred has further noted that history may organically arise from several typical classroom situations, such as introducing new topics or motivating discussion of advanced, modern topics. Regardless of how one uses history in the classroom, he has always affirmed that “it needs to be tied very closely to the material being discussed in class” [Rickey 1995, p. 124]. This tenet has led him over the years to pursue research into historical details that are missing in most history of mathematics books . . . and to use what he has learned both to enrich his teaching and as a lens for reflection on what we should (and should not) teach to our students.
Beginning in the late 1980s, Fred wrote a number of short pieces on specific calculus topics based on his experiences with historical research and teaching. That collection, Historical Notes for the Calculus Classroom, has thus far been distributed primarily to participants in the Institute on the History of Mathematics and its Use in Teaching (IHMT), which Fred co-directed with Victor Katz and Steven Schot from 1995 to 1999. Even today, IHMT fellows remain passionate about these Notes. On one level, they provided IHMT fellows with a set of interesting problems and historical tidbits to use in their own calculus and analysis classes. The article “Perrault and the Tractrix,” for instance, not only examines the history and solution of one of the earliest inverse tangent problems, but also paints an image of Perrault’s death as a result of a camel dissection that “is too vivid to forget.” On another level, Fred’s Notes gave IHMT fellows a model for how to research the history of a topic on one’s own. Examples of various approaches to using primary sources are found, for example, in Fred’s analysis of a letter written by L’Hospital to Bernoulli on 17 March 1694 in the article “L’Hospital’s Rule,” as well as in his journey through Leibniz’s struggle to discover the correct form of the product rule in manuscripts written during 1675–1677 in the article “The Product Rule.” Many of the articles in Fred’s Notes also identified resources for finding out more about the historical development of a given topic, thus providing instructors with a foundation for further exploration and learning in the area. Perhaps most importantly, Fred’s Notes offered IHMT fellows specific strategies for and thoughtful insights into how and why to bring the results of that historical research into the classroom itself.
V. Frederick Rickey discusses the background of Historical Notes for the Calculus Classroom
in an interview conducted 18 April 2023 via Zoom by Amy Ackerberg-Hastings.
Uploaded to YouTube by Convergence Editors, 1 May 2023. Closed captioning available.
As a collection of independent topics, the articles in the collection Historical Notes for the Calculus Classroom can be perused in any order, making them ideally suited for serial publication. In this series, Convergence is pleased to share a selection of articles from the collection with our readers. While the pieces that will appear in this series have been updated to reflect more recent historical research, every effort has been made to retain the original intentions and distinctive voice of their author. Fred described those intentions in an interview for Convergence, which you can view by clicking on the image above.
Please note that the articles in this series are not intended to represent a comprehensive history of calculus, and some topics from a standard year-long calculus sequence are missing from the collection. The individual articles are also not “plug-and-play” lessons; rather, they provide impressionistic and idiosyncratic encounters with past developments that, as stated above, reflect Fred’s priorities in teaching calculus. Using this series as a guide to including history in a calculus course will thus require an investment of time from readers:
- to decide which articles align with topics they plan to cover in their course;
- to read and understand the historical background;
- to work through and gain firm control over the mathematical content;
- to decide on a format for including a historical episode in the classroom (e.g., a lecture, a class discussion, a homework assignment, a short group project via a handout);
- and, of course, to decide how much time to devote to it (both for preparation and in class).
As Fred has noted elsewhere, however, even “an incomplete historical note can captivate students” [Rickey 1995, p. 133].
In some cases (but not all), Fred has also described how he has personally used the history discussed in a particular article when teaching his own calculus classes. It has long been his hope that sharing his experience using history in the classroom “will inspire others to try [these] ideas in the classroom and also to share their own successes in using history” [Rickey 1995, p. 124]. The editors of this journal invite all those who are so inspired to share their own stories and successes in Convergence!
V. Frederick Rickey and Victor J. Katz at the Institute on the History of Mathematics and its Use in Teaching.
References
Rickey, V. Frederick. 1995. My Favorite Ways of Using History in Teaching Calculus. In Learn from the Masters!, edited by Frank Swetz, John Fauvel, Otto Bekken, Gengt Johansson, and Victor Katz, 123–134. Washington DC: Mathematical Association of America.
Rickey, V. Frederick. 1996. The Necessity of History in Teaching Mathematics. In Vita Mathematica: Historical Research and Integration with Teaching, edited by Ronald Calinger, 251–256. Washington DC: Mathematical Association of America.
Selected Additional Articles by V. Frederick Rickey in MAA Publications
Huber, Michael, and V. Frederick Rickey. 2008, March. What is 0^0? Convergence.
This look at the history and meaning of the expression 0^0 leads to the conclusion that 0^0 = 1, regardless of what is said in some textbooks.
Rickey, V. Frederick, and Amy Shell-Gellasch. 2008, May. Mathematics Education at West Point: The First Hundred Years. Convergence. A survey of the mathematics education of cadets in the first century after the founding of the U.S. Military Academy.
Rickey, V. Frederick, and Philip M. Tuchinsky. 1980, May. An Application of Geography to Mathematics: History of the Integral of the Secant. Mathematics Magazine 53(3):162–166.
Fred’s first publication in the history of mathematics offers a brief history of a calculus topic that Fred and his co-author describe as an “ideal soapbox for discussing the nature of mathematics, the process of mathematical discovery, and the role that mathematics plays in the world” (p. 166).
Historical Notes for the Calculus Classroom: Fermat’s Integration of Powers
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]
Exercise
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].)
References
Boyer, Carl B. 1945. “Fermat's integration of xn.” National Mathematics Magazine 20: 29–32.
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.
Notes
[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].