# Historically Speaking

Author(s):
Betty Mayfield (Hood College)

In October 1953, a new column appeared in the pages of Mathematics Teacher (MT), a practitioner journal of the National Council of Teachers of Mathematics (NCTM) published between 1908 and 2019 that was dedicated to improving mathematics instruction for grades 8–14 and supporting teacher education programs. “Historically Speaking” was edited first by Phillip S. Jones and later by Howard Eves, two giants in the field of the history of mathematics and its use in teaching. The columns appeared in every issue for the next sixteen years, covering topics from the oldest American slide rule to the beginnings of set theory. Many of the columns were written by the editors themselves; others were submitted by other scholars or readers. Sometimes contributors responded to others’ submissions, setting up a conversation among the journal’s readers over the course of several issues.

In this series, Convergence features reprints of selections from the “Historically Speaking” column,[1] each complemented by a guest introduction written by a contemporary expert on the history of mathematics. These columns, or notes, are not traditional articles of the type Convergence reprinted from MT in its previous series. Rather, they are (mostly) short gems that offer a glimpse into topics which caught the fancies of the editors. Installments in the series thus represent two layers of history: the historical topic examined in the column and the column as a historical artifact in its own right. What do we know about the authors of the columns and their approaches to history or its use in teaching? What would a bibliography for the historical topic of a particular column look like today? Within the mathematics curriculum, what changes have occurred with respect to the popularity of certain topics, or advances in explaining them, or approaches taken to using history to teach them? Join us and our guest commentators in reflecting on changes that have taken place (or not) over the past 70 years, as we stroll through the pages of “Historically Speaking”!

The two editors of “Historically Speaking” were almost exact contemporaries. Between them, they helped to define what it meant to study and teach the history of mathematics in the U.S. in the 20th century, from children to university students.

 Philip S. Jones, HPM Newsletter, March 2007. Phillip Sanford Jones (1912–2002) spent his career at the University of Michigan, his alma mater, where he held a joint appointment in the Department of Mathematics and the School of Education. During a distinguished career, he served as President of NCTM, Governor of the Michigan Section of the Mathematical Association of America (MAA), and founding member of the International Study Group on Relations between the History and Pedagogy of Mathematics (HPM). During his NCTM presidency, he encouraged the organization to devote a yearbook to the history of mathematics, which they did in 1969, dedicating the volume to him and inviting him to write the first chapter. The next year, Jones himself edited the yearbook, A History of Mathematics Education in the United States and Canada. He gave an invited address on The History of Mathematics Education at the 50th anniversary meeting of the MAA. Howard Eves, Convergence Portrait Gallery. Howard Whitley Eves (1911–2004) held degrees in mathematics from the University of Virginia, Harvard University, and Oregon State University. In a long career at the University of Maine, he published widely in both geometry and the history of mathematics. He was also an engaging and sought-after speaker, equally popular with high school students and mathematics professors. He was a close friend of Albert Einstein at Princeton. His textbook, An Introduction to the History of Mathematics, first published in 1953, is one of the most popular texts of its type; it is still being published and is now available in Kindle form. He was also the author of six volumes of In Mathematical Circles, collections of anecdotes about mathematics and mathematicians, and two volumes of Great Moments in Mathematics (Before 1650 and After 1650). He was a devoted member of the Mathematical Association of America and was a founder of its Northeast Section.

The National Council of Teachers of Mathematics (NCTM) is the public voice of mathematics education, supporting teachers to ensure equitable mathematics learning of the highest quality for all students through vision, leadership, professional development, and research. In addition to its current practitioner journal Mathematics Teacher: Learning & Teaching PK–12 (MTLT), the council publishes a mathematics education research journal, as well as an online journal for teacher educators (jointly with the Association of Mathematics Teacher Educators). With 80,000 members and more than 200 Affiliates, NCTM is the world’s largest organization dedicated to improving mathematics education in prekindergarten through grade 12. For more information on NCTM membership, visit http://www.nctm.org/membership.

[1] The reprints themselves are posted in pdf format, thanks to an arrangement made possible by the efforts of Convergence founding co-editor Frank Swetz, through which NCTM has allowed Convergence to republish up to two articles from Mathematics Teacher annually since 2015.

# Historically Speaking: 1. The Quadrature of the Parabola

Author(s):
Carl Boyer (Brooklyn College) and William Dunham (Bryn Mawr College)

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): 712716.

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 ViewThe College Mathematics Journal 37(1): 24–28.

[1] As an extension of the digitizations of Heath’s translations of these two works by Archimedes listed in the References, readers may find it useful to consult the annotated online translations produced by Henry Mendell [n.d.-a, n.d.-b]. Note that permission to use the translations, diagrams, or other texts produced by Mendell is only granted for personal use and for use in a classroom using links on the Vignettes of Ancient Mathematics website.