Historically Speaking: 1. The Quadrature of the Parabola

The editors’ choice for the first Historically Speaking reprint is one of the earliest, from January 1954. In it, mathematical historian Carl Boyer provides a new proof, using determinants, of Archimedes’ formula for finding the area enclosed by a portion of a parabola:

Carl Boyer and Phillip S. Jones, “The Quadrature of the Parabola: An Ancient Theorem in Modern Form,” Mathematics Teacher, Vol. 47, No. 1 (January 1954), pp. 36–37. Reprinted with permission from Mathematics Teacher, ©1954 by the National Council of Teachers of Mathematics. All rights reserved.

Click on the title to download a pdf file of the article, “The Quadrature of the Parabola: An Ancient Theorem in Modern Form.”

Professor William Dunham offers a present-day response to the article. Dunham, the author of four books, including Journey through Genius: The Great Theorems of Mathematics, has won the MAA’s equivalent of an EGOT: the Pòlya, Trevor Evans, Beckenbach, Allendoerfer, Halmos-Ford, and Chauvenet Prizes for his writing about the history of mathematics.

In this short article for “Historically Speaking,” Carl Boyer revisited a theorem of Archimedes. Boyer was a distinguished math historian from the mid-20th century, and Archimedes, of course, was an extremely distinguished mathematician from 23 centuries before. The theorem in question addressed the quadrature of the parabola, which to us means finding (in some sense) parabolic area. Boyer called this “one of the best-known of the classics of the history of mathematics” and said its “familiar” proof proceeded “in the usual Archimedean manner.”

As these quotations suggest, Boyer imagined his readers to be familiar with ideas that today are a bit obscure. In order to follow his argument, it helps to reside in that corner of mathematics where history, linear algebra, and analytic geometry intersect. Those who reside elsewhere might need some review. With this in mind, let me offer three caveats.

First, Boyer presented his complicated Figure 1 as a fait accompli. This obscured its multi-step, chronological development. Here’s where the diagram came from: begin with the parabola \(y^2=2px\); draw any chord \(P_1P_5\), creating the parabolic segment whose area Archimedes sought; bisect \(P_1P_5\) at \(M_3\); construct \(M_3P_3\) parallel to the axis of the parabola; draw \(P_1P_3\) and \(P_5P_3\), forming the crucial \(\Delta P_1P_3P_5\); bisect this triangle’s sides at \(M_2\) and \(M_4\); and construct \(M_2P_2\) and \(M_4P_4\) parallel to the parabola’s axis. Whew!

A second caveat is that readers might want to dust off some ideas from analytic geometry and linear algebra. For instance, Boyer breezily employed the “point-of-division formula” from analytic geometry. That sent me scurrying to an old textbook. His primary mathematical argument rested upon

\(\Delta P_1P_2P_3 = \frac{1}{2} \begin{vmatrix} x_1 & y_1 & 1 \\ x_2 & y_2 & 1 \\ x_3 & y_3 & 1 \end{vmatrix}\)

which connects determinants and triangular area. I had to guess—correctly, it turned out—that the subscripted terms in the determinant were the coordinates of the triangle’s vertices (Boyer never mentioned this). And the area formula actually requires the absolute value of the determinant, a point that Boyer did not state but repeatedly used.

Finally, I have a caveat about Boyer’s assertion that

\(1 + \frac{1}{4} + \frac{1}{16} + \ldots + \frac{1}{4}n + \ldots = \frac{4}{3}.\)

OK, this is not so much a caveat as a cringe-worthy typo.

If this sounds like the prologue to a negative review, it is not. In fact, once I absorbed the telegraphic nature of Boyer’s argument, I found it delightful. Determinants were everywhere, the parabola’s analytic representation was essential, the aforementioned point-of-division formula took center stage, and I loved Boyer’s verbal recipe for matrix row operations:

One subtracts from the elements of the second row twelve times the corresponding elements of the last row . . . and six times the corresponding elements of the first row. . . .

Implausible as it might seem, these are precisely the operations necessary to derive the proof’s most critical formula: \(\Delta P_1P_2P_3 = \frac{1}{8} \Delta P_1P_3P_5.\)

In the end, I found the linkages between analytic geometry and linear algebra to be as fascinating as they were unanticipated. Professor Boyer surely did what his title promised, providing a short, clever proof of “an ancient theorem in modern form.”

As Dunham’s commentary suggests, Archimedes’ own proof of his theorem looked quite different from Boyer’s modern proof. Readers who are interested in reading Archimedes’ purely geometrical approach can find his proof in Propositions 18–24 of Quadrature of the Parabola [Heath 1897, pp. 233–252]. Archimedes prefaced his geometric proof with a description of how he “first investigated [the proof] by means of mechanics” [Heath 1897, pp. 231]. He also described his discovery of the theorem by means of mechanics (i.e., the fulcrum principle) in Proposition 1 of his “lost work on The Method” [Heath 1897, p. 15],^[1] a text hidden in the famous Archimedes’ Palimpsest that was discovered by Johan Ludvig Heiberg in 1906. Bonsangue and Shultz [2016] recreate Archimedes’ discovery approach with the aid of contemporary algebraic language and suggest how instructors might present that derivation (or, at least, its result) in a calculus course. Other teaching materials based on Archimedes’ original geometric approach to the quadrature of a parabola include “Quadrature of the Parabola” in [Ebert et al. 2004, pp. 30–58] and “Archimedes’ Quadrature of the Parabola” in [Laubenbacher and Pengelley 1999, pp. 118–122].

Boyer’s approach to the quadrature of the parabola is itself an intriguing example of how ancient mathematics that we think of as contributing to one field (e.g., geometry) can stimulate a teaching moment in a field that didn’t really exist when that mathematics was originally created (e.g., linear algebra). Although determinants are no longer typically taught in an analytic geometry course, they do appear in linear algebra courses. The use of 3x3 determinants to calculate the area of a triangle given the coordinates of its vertices is a standard linear algebra topic. Typically, students are given only mundane numerical examples to work out, rather than the more complicated algebra that Boyer's argument involves. That argument, however, suggests a nice extension exercise that would teach students something useful about matrix reduction (and its connection to the computation of determinants), as well as give them insight into problem solving more generally. In addition to being an interesting and surprising mathematical argument, Boyer's clever use of determinants to give a modern proof of a key lemma behind Archimedes' actual theorem thus has the potential to deepen student learning of a topic from today’s curriculum.

References

Bonsangue, Martin V., and Harris S. Shultz. 2016. In Search of Archimedes: Quadrature of the Parabola. The Mathematics Teacher 109(9): 712–716.

Ebert, Jim, Rebecca Kessler, Gail Kaplan, and Ed Sandifer. 2004. Archimedes Module. In Historical Modules for the Teaching and Learning of Mathematics, edited by V. Katz and K. D. Michalowicz. Mathematical Association of America.

Heath, T. L. 1897. The Works of Archimedes. Cambridge University Press.

Heath, T. L. 1912. The Method of Archimedes, recently discovered by Heiberg; A supplement to the Works of Archimedes. Cambridge University Press.

Laubenbacher, Reinhard, and David Pengelley. 1999. Mathematical Expeditions: Chronicles by the Explorers. Springer.

Mendell, Henry. n.d.-a. Archimedes Mechanical Method with Indivisibles, The Method, Prop. 1. Vignettes of Ancient Mathematics. Annotated online translation with illustrative diagrams.

Mendell, Henry. n.d.-b. Archimedes, Quadrature of the Parabola. Vignettes of Ancient Mathematics. Annotated online translation with illustrative diagrams.

Osler, T. J. 2006. Archimedes’ Quadrature of the Parabola: A Mechanical View. The College Mathematics Journal 37(1): 24–28.