Historically Speaking: 3. “Large” Roman Numerals

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.