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.

The second installation of the Historically Speaking series is from November 1953. In it, editor Phillip Jones describes early versions of a slide rule, Palmer’s Computing Scale and Palmer’s Pocket Scale, from the 1840s:

Phillip S. Jones, “The Oldest American Slide Rule,” Mathematics Teacher, Vol. 46, No. 7 (November 1953), pp. 500–503. Reprinted with permission from Mathematics Teacher, ©1953 by the National Council of Teachers of Mathematics. All rights reserved.

Click on the title to download a pdf file of the article, “The Oldest American Slide Rule.”

The column opens with a suggested bibliography of the history of mathematics consisting of early articles from Scientific American, which our readers may also find interesting. The article about the Palmer slide rules begins on page 501. We additionally provide printable images of Palmer’s Pocket Scale for potential use in the classroom as described below.

Dr. Peggy Aldrich Kidwell, Curator in the Division of Medicine and Science at the National Museum of American History, Smithsonian Institution, offers a present-day response to the article. Dr. Kidwell’s research interests include the history of mathematics and computing, women in science, and the history of mathematical recreations in the United States. She is the author of several publications, including Tools of American Mathematics Teaching, 1800–2000, with Convergence co-editor Amy Ackerberg-Hastings and David Lindsay Roberts.

Phillip S. Jones’s fine article describes the “pocket scale” or “computing scale” copyrighted by Aaron Palmer in the early 1840s and sold until around 1870 by John E. Fuller. The piece illustrates several intriguing aspects of the history of the slide rule in the United States. The first is the transience of the slide rule itself. Jones proclaimed Palmer’s instrument to be “the oldest American slide rule.” It is unquestionably true that it is one of the first—if not the first—American circular slide rules to sell in the United States. One might note, however, that Solomon A. Jones, who was active in Hartford, Connecticut, from 1838 through 1841, sold a linear slide rule designed especially for carpenters—an example survives in the Smithsonian collections. It is perhaps more noteworthy that the slide rule, which became common in the country in the 1890s and was widely known to mathematics students when Jones wrote in 1953, is now primarily of interest to collectors. The handheld electronic calculator displaced it in the 1970s, though of course the underlying mathematics of logarithms are still widely taught and used.

Due to the efforts of diverse historically-minded people—and the ready availability of online sources—we now know a bit more about Palmer himself. According to Carol L. Hannan, writing on a website devoted to Historic Homes of Brockport, New York, he was a native of Canada who moved to western New York, settling in Brockport and then Rochester. Rochester city directories list him as a machinist in the 1860s and early 1870s, and then as an inventor until his death in 1884. His patented inventions focused more on improvements to farm machinery.^[1]

Palmer sold his computing scale with a book of instructions, which can now readily be consulted online. Thus, a curious teacher or student can easily print out a couple of copies of the illustration provided by Jones, cut the round disc out of one of them, hold everything together with a thumbtack or paper fastener, and have a working replica. Consulting the text can give a sense of the kind of arithmetic problems deemed of interest. Different editions of the booklet prepared by Palmer and then by Fuller tell historians more about the device’s sales and the reactions of those who saw it.

To call attention to his ideas, Palmer solicited recommendations from local authorities. The first he cites in his booklet, who wrote to him in 1842, were principals of academies (private secondary educational institutions) in and around Rochester. The following year, his reach extended to Boston, where he gained endorsements from Harvard professor Benjamin Peirce, textbook author Frederick Emerson, and educational reformer William B. Fowle. Palmer also garnered words of praise from businessmen and a lawyer. Curiously, the only recommendation to actually mention logarithms, describe the physical details of the object, and compare it to linear slide rules was one from George Clinton Whitlock, a Middlebury College graduate who was Professor of Mathematics and Natural Science in Genesee Wesleyan Seminary in Lima, New York. By the time of later editions of the instructions, Fuller could include references to exhibitions at world’s fairs and recommendations from such distinguished foreign mathematicians as Augustus De Morgan.^[2]

Finally, in case someone should like to look at an example of Palmer’s computing scale and its various revisions, readers should be forewarned that finding one is not an entirely easy process. The book and the instrument were often sold together, and hence may survive either in library collections or museums. To give only a few examples, Florian Cajori, who published fundamental 1909 articles on Aaron Palmer’s instrument and on its modification by Fuller, consulted examples from his contemporary Artemas Martin.^[3] It seems likely that these survive with other materials collected by Martin now in the library at American University. Jones described an example on display in 1953 at the Henry Ford Museum in Dearborn, Michigan. It seems probable that this is in the special collections of the University of Michigan. As Jones mentions, Harvard University had several examples of Palmer’s instrument. Waywiser, the database of the Harvard Collection of Scientific Instruments, now lists three examples of Palmer’s computing scale (all from 1843) as well as examples from 1845 and 1847 that combine Palmer’s scale with that introduced by Fuller. At the same time, the Harvard University Libraries boast five editions of Palmer’s various instruction manuals, as well as manuals that include Fuller’s changes. How many of these include examples of the actual slide rule is unclear. As these examples—as well as those in other locations—suggest, those seeking further examples of the scale have a daunting task ahead of them! Jones would undoubtedly be delighted.^[4]

Modern readers (and their students) may never have seen, much less used, a slide rule. Those who wish to learn more about them may wish to consult the WikiHow article, “How to Use a Slide Rule,” or the “Additional ISRM Galleries Slide Rule Resources” section of the website of the International Slide Rule Museum. Some thoughts to spur further thinking and discussion:

• The slide rules students were using when this article was published in 1953 were very different from Palmer's "oldest slide rule," just as today's calculators are very different from slide rules of the 1950s. (Images of slide rules typically used in high school and college classrooms can be viewed in the charming 1940 booklet by Don Herold, How to Choose a Slide Rule, or in Eric Marcotte’s gallery of Keuffel & Esser slide rules.) How have those advances in technology affected the way we think about and teach mathematics?
• The mathematical problems addressed by Palmer's slide rule are similar to those covered in arithmetic textbooks of the time period. How do our textbooks reflect current interests and technological capabilities?

Finally, we hope you will follow Dr. Kidwell's advice and experiment with your very own Palmer's Computing Scale. Print these two copies of the scale and attach them to each other as described above. You can find instructions on how to use the scale here. Let us know about your investigations!

This 1954 offering from the Historically Speaking column of the NCTM journal Mathematics Teacher is about Roman numerals—where they came from, how they changed over time, and how teachers might use the story of their development in the classroom:

Phillip S. Jones, “‘Large’ Roman Numerals,” Mathematics Teacher, Vol. 47, No. 3 (March 1954), pp. 194–195. 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, “‘Large’ Roman Numerals.”

Jones begins by describing the problems students might encounter in trying to write their phone numbers in Roman numerals, for instance. He then offers a brief history of a particular “large” numeral, M = 1000. He ends by noting in passing that, although we do owe the Romans something of a debt for their development of a system of weights and measures, they really “contributed little to the development of real mathematics.” Our responder to the column, Professor Victor Katz, examines that claim.

Victor J. Katz, Professor Emeritus of Mathematics at the University of the District of Columbia, is a founding co-editor of Convergence. With Profs. Fred Rickey and Steven Schot, he founded the Institute for the History of Mathematics and Its Use in Teaching, training educators to teach the history of mathematics at their own institutions. He is the author of a widely-used text, A History of Mathematics: An Introduction, and he edited two sourcebooks of mathematics, especially mathematics that came from non-European roots. For these and many other accomplishments, Prof. Katz was awarded the 2023 Yueh-Gin Gung and Charles Y. Hu Award by the MAA.

Phillip Jones begins this article with a discussion of Roman numerals and gives some historical information on how the numerals may have developed. He continues with some mathematical word origins that have come down to us from the Latin. But he then notes that “the Romans contributed little to the development of real mathematics.’’ That is a common enough belief, but when one thinks about it, it does not quite make sense. Remember that by the first century CE, the Roman Empire ruled the entire Mediterranean basin, stretching from Spain in the west to Egypt, Judaea, Mesopotamia, and Babylonia in the east. And although many of the western territories of the empire started to fall away beginning in the third century, as late as the fifth century the eastern domains were still under Roman rule. So shouldn’t we call mathematics developed around that time in that area “Roman mathematics”? It is, of course, true that most of the mathematicians we know of during that time period wrote in Greek. Some of the more famous ones, all living in Egypt in the early centuries CE, were Ptolemy, Diophantus, and Heron, each of whom certainly contributed to “real mathematics”. In common usage, these mathematicians are usually called contributors to Greek mathematics, but why shouldn’t they be included under “Roman” mathematics?

Perhaps a criterion for “Roman” mathematics is mathematics originally written in Latin. Here the situation is different, as there was very little mathematics written in that language during the period of the Roman Empire. The only well-known authors who wrote about mathematics in Latin in that period were Vitruvius (ca 85–ca 20 BCE), most famous for his work on architecture, and Boethius (ca 480–524 CE), who, shortly after the fall of the western Roman Empire, wrote various works that dealt with the basic ideas of the classical quadrivium: arithmetic, geometry, astronomy, and music. On the other hand, it is well to remember that the Romans were excellent engineers, who built roads, aqueducts, ships, and war machines, each of which required mathematical knowledge. Thus, they were certainly users of mathematics, if not creators. Indeed, extant today are some of the Roman writings on the practical uses of mathematics. 

For example, let us look at the work of Lucius Columella (4–70 CE). He was born in what is now Spain, served in the Roman army, and then spent the rest of his life farming his estates in Latium. He is best known for his twelve-volume work on agriculture in the Roman Empire. It dealt with many ideas important for farming, including soils, olive trees, cattle, sheep, chickens, and personnel management. He treated mathematics in a chapter in Book V on the measurement of land in various shapes, for each of which Columella produced an example of the procedure for calculating the area. In doing these calculations, he used three units of measure that were common in the Empire: the jugerum of 28,800 square Roman feet; the scrupulum of 100 square feet, or 1/288 of a jugerum; and the uncia of 2400 square feet, which equals 24 scrupuli or 1/12 of a jugerum.

For example, he calculated the area of an equilateral triangle each of whose sides are 300 feet. He first squared 300, giving 90,000, then took both the third and the tenth of that, namely 30,000 and 9,000. The sum of those, 39,000 square feet, is the area, which he expressed also as 1 jugerum and one-third of a jugerum, and the forty-eighth part of a jugerum. Since we know that the area of an equilateral triangle of side s is \(\frac{\sqrt{3}}{4}s^2\), we see that Columella approximated this result as  \(\left(\frac{1}{3} + \frac{1}{10}\right)s^2\), which is quite a good approximation.

For a semicircle with diameter 140 feet and radius 70 feet, Columella multiplied those two values together to get 9,800, then multiplied further by 11 and divided by 14 to get the area as 7,700 square feet, or three unciae and five scrupuli, i.e., one-fourth part and 5/288 parts of a jugerum. This calculation is a standard one, as he was using the Archimedean approximation of 22/7 for \(\pi\) and therefore found the area as \(\frac{11}{14} rd = \frac{11}{14}\frac{d^2}{2}\).

But this calculation was preliminary to his calculation of the area of a circular segment. Thus, in such a segment where the base is 16 feet and the breadth 4 feet, he added the two together to get 20 feet, then multiplied by 4 to get 80, and took half of that: 40 feet.  Also, he took half of the base, namely 8, and squared it to get 64. Then, a fourteenth of that is 4 and “a little”, which he added to the 40 to get a final answer of 44 and a bit, or half a scrupulum, i.e. 1/576 part of a jugerum, less a twenty-fifth part of a scrupulum.

This calculation might not be familiar to you, but it is one that Heron wrote about in his Metrica: Namely, the area of a segment of a circle where c is the length of the base of the segment and h the breadth is given by \(A = \frac{1}{2} (c + h)h + \frac{1}{14} \left(\frac{c}{2}\right)^2\). In this particular case, Columella’s calculation is quite accurate. I invite you to try to figure out why this formula does give a good approximation to the area of a segment of a circle.

Now Columella was a farmer. There were, of course, Roman surveyors who needed to know some mathematics, and some of their work is preserved in the compilation called the Corpus Agrimensorum Romanorum, put together in the Middle Ages from documents that had survived to that time. One of the authors was Marcus Junius Nipsus, who lived in the second century CE and compiled some information on finding areas of plots of land. What is fascinating is that his methodology was remarkably similar to methods found in some Greek and demotic papyri that have been found in Egypt and also to the Mesopotamian Seleucid tablet BM 34568, which is dated to perhaps the second century BCE.

For example, Nipsus showed how to calculate the legs of a right triangle given that the hypotenuse is 25 and the area is 150. Namely, he squared the hypotenuse, giving 625, then added four times the area, 600, producing 1225. The square root of that is 35. Again, subtract four times the area from the square of the hypotenuse, giving 25, whose square root is 5. Half the sum of 35 and 5 is 20, which is one leg of the triangle, while if we subtract 5 from the 20, we get the other leg: 15. It is easy to draw a diagram to understand Nipsus’s methodology. He also clearly uses diagrams to help produce algorithms to calculate the areas of obtuse and acute triangles.

But rather than give more examples of what seem to be fairly elementary mathematical ideas in Roman work, I will conclude with a paean to mathematics from Vitruvius, who besides giving many reasons as to why architects needed to study mathematics, urged his countrymen to recognize the importance of the subject generally:

What good does it do humanity that Milo of Croton [a well-known athlete from the 6th century BCE] was undefeated, or the others who were champions of this kind, other than that, so long as they were alive, they held distinction among their own fellow citizens? The valuable precepts of Pythagoras, on the other hand, of Democritus, Plato, Aristotle, and the other sages, cultivated by daily industry, not only produce ever fresh and flourishing fruit for their own fellow citizens, but indeed for all the nations. And those who from an early age enjoy an abundance of learning develop the best judgment, and in their cities they have established civilized customs, equal justice, and those laws without which no community can exist safely. Since so many private and public gifts have been prepared for humanity by the wisdom of writers, I conclude that more than palms and garlands should be awarded them—indeed triumphs should be declared for them and to them it ought to be decided to dedicate thrones among the gods [Vitruvius (ca 30–20 BCE) 1999, p. 107].

References

Vitruvius. (ca 30–20 BCE) 1999. Ten Books on Architecture. Translated and edited by Ingrid Rowland and Thomas Howe. Reprint, Cambridge University Press, 1999. For a public domain version of the full work, see the 1914 translation by Morris Hicky Morgan, published by Harvard University Press.

Current scholarship on BM 34568 includes:

Gonçalves, Carlos H. B. 2008. An alternative to the Pythagorean rule? Reevaluating Problem 1 of cuneiform tablet BM 34 568. Historia Mathematica 35: 173–189.

Høyrup, Jens. 2002. Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin. New York: Springer.

Readers who wish to delve deeper into Prof. Jones's history of Roman numerals may consult the books he referenced:

Cajori, Florian. 1928. A History of Mathematical Notations. Chicago: Open Court Press. (The 2011 Dover Publications edition is here.)

Karpinski, L.C. 1925. The History of Arithmetic. Chicago: Rand McNally & Company.

Smith, D.E. 1923. History of Mathematics. 2 vol. Boston: Ginn and Company. (Both volumes are available from the Internet Archive here and here.)

Finally, the delightful small book published by NCTM of which, according to the author, “it is assumed that all students and teachers have [it] available,” is indeed currently available as an ERIC document:

Smith, D.E., and Jekuthiel Ginsburg. 1937. Numbers and Numerals. National Council of Teachers of Mathematics.