While we hope that all readers of *Convergence *will enjoy this article, it is meant especially to be accessible and instructive to students of geometry—in a high school class or a university course for pre-service teachers, for instance. It could also be used in a history of mathematics course, especially one for pre- or in-service teachers. Students can read it, either alone or in small groups, working the exercises as they go along.

**Figure 1.** The letter by Gerbert d’Aurillac to Adalbold, who later became Bishop of Utrecht, that is discussed in this article.

12th-century Austrian manuscript owned by the University of Pennsylvania Libraries.

Gerbert d’Aurillac (ca 945–1003) was a 10th-century French churchman and scholar who was named Pope Sylvester II in the year 999. Presumably unlike other popes, he was fascinated by mathematics. He studied it as a young man in the medieval Spanish March; he taught it at the Cathedral School at Rheims; he even wrote a well-known textbook in geometry. He was one of the first people in Western Europe to learn about Hindu-Arabic numerals, and he developed an abacus for working with them.

**Figure 2.** Gerbert d'Aurillac (in blue), as painted by a master of the Reichenau School for the *Evangeliar Ottos III* oder *Evangeliar Heinrichs II* (*The Gospels of Otto III*, also known as *The Gospels of Heinrichs II*), owned by the Bavarian State Library. Public domain, Wikimedia Commons, attributed to The Yorck Project (2002) 10.000 Meisterwerke der Malerei (DVD-ROM), distributed by DIRECTMEDIA Publishing GmbH.

In the spring of 999, at the end of a brief tenure as Archbishop of Ravenna, Gerbert had just been elected Pope and was about to move to Rome to take up his new responsibilities. In the midst of all the associated excitement and busyness, he took the time to write a letter [Lattin 1961, pp. 299-302] to his friend Adalbold (975–1026), a cleric and teacher in central Germany who would later serve as Bishop of Utrecht—the last piece of correspondence we have from Gerbert before he was inaugurated as Pope. And that letter was all about . . . how to find the area of an equilateral triangle.

In this letter, Gerbert described two different methods for finding such an area. The first, the *geometrical rule, *is the rule we usually write as

\(A = \frac{1}{2}bh\)

where \(b\) is the base, or side, of the triangle, and \(h\) is the altitude. The tricky part of using this rule, of course, is that one needs to know not only the side but also the altitude of the triangle, which can be difficult to compute.

Gerbert gave an example: “In these geometrical figures which you have [already] received from us, there was a certain equilateral triangle, whose side was 30 feet, height 26, and according to the product of the side and the height the area is 390” [Lattin 1961, p. 300]. He did not indicate how he arrived at the figure of 26 feet for the altitude, and we should probably be a little suspicious of it. In fact, Gerbert himself was suspicious of it, as we shall see later in the letter.

And so Gerbert introduced a second method, less familiar to us: the *arithmetical rule. *Not having modern algebraic notation at his disposal, he described it in words:

One side is multiplied by the other and the number of one side is added to this multiplication, and from this sum one-half is taken [Lattin 1961, p. 300].

In other words, if \(b\) is the side of the triangle, the area given by the arithmetical rule is

\(A = \frac{b \cdot b + b}{2}\).

In order to help Adalbold visualize this problem, he gave him a simpler example, an equilateral triangle of side 7 feet. And he included a drawing which probably looked something like Figure 3.

**Figure 3. **Building an equilateral triangle [Lattin 1961, p. 301].

He reminded Adalbold that we measure area in square units; if one were to construct an equilateral triangle of side 7 feet out of little \(1 \times 1\) squares, it would undoubtedly look like the one in Figure 3, starting with 7 units on the bottom, then 6, and so forth, until we reach 1 unit at the top. And if we add up the areas of all those little squares, we get

\(A = 1 + 2 + 3 + 4 + 5 + 6 + 7 = \frac{7(7 + 1)}{2}\) sq ft

from a well-known formula for the sum of the first \(b\) positive integers, which is indeed \(\frac{b \cdot b + b}{2}\)*.*

But we can see from the picture that this answer is obviously too big: as Gerbert pointed out, we are including the whole of each square, when often we need only part of it. But because he had drawn the figure very cleverly, he had a solution. According to Gerbert, the altitude of this triangle is clearly 6 feet. And so now we can go back to the geometrical rule and use 6 for the altitude, obtaining

\(A = \frac{1}{2} \cdot 7 \cdot 6 = 21\) sq ft.

This answer was also wrong,^{[1]} but Gerbert blithely continued. The general rule, he said, is to use \(\frac{6}{7} \cdot b\) as the altitude of the triangle and then apply the geometrical rule. For the triangle of side 30 feet, we should use \(\frac{6}{7} \cdot (30) = 25\frac{5}{7}\) for the altitude, giving an area of \(385 \frac{5}{7}\) square feet—a very unusual number, as we will see, for a medieval mathematician—or pope—to use.

In the European Middle Ages, fractions were still written in the Roman* duodecimal* (base twelve) fashion. The Romans’ words for fractions were based on a unit of weight (*as*) which we can think of as a pound. This unit was divided into twelve smaller parts (*unciae*) which we can think of as ounces. And each *uncia* was further divided into twelve parts. So Romans could deal easily with fractions like \(\frac{5}{12}\) or \(\frac{1}{144}\) or even \(\frac{3}{4}\) (because it can be written as \(\frac{9}{12}\)). But \(\frac{5}{7}\) would have presented a problem. Popular fractions had their own Latin names; for instance, the word *quincunx* designated the fraction \(\frac{5}{12}\) [Menninger 1969, pp. 158–162; Olleris 1867, pp. 583–584].

Sometimes medieval European merchants or scholars would return to the Egyptian method of expressing fractions as *unit fractions—*fractions with a numerator of 1 [Lattin 1961, p. 302]. The fraction \(\frac{3}{4}\) had to be written as \(\frac{1}{2} + \frac{1}{4}\); a more complicated fraction like \(\frac{3}{5}\) would be expressed as \(\frac{1}{3} + \frac{1}{6} + \frac{1}{10}\)—just using \(\frac{1}{5}\) three times was not allowed! But expressing \(\frac{5}{7}\) as a sum of unit fractions is certainly not an easy task.

When Gerbert needed to write an unusual fraction like \(\frac{5}{7}\), he wrote it out in words: *quinque septimas, *“five sevenths.” And, despite his reputation as an early adopter of Hindu-Arabic numerals, he wrote whole numbers—in his letters and in his textbook—in Roman numerals. The correct area of an equilateral triangle of side 30, he writes, is* CCCLXXXV et quinque septimas* [Olleris 1867, p. 477], as shown in Figure 4, in lines 7 and 8 of the text.

**Figure 4.** An excerpt of Gerbert's letter to Adalbold [Bubnov 1899, p. 44],

digitized by the University of Michigan. Public domain.

[1] For an interesting, if somewhat cantankerous, discussion of the arithmetical rule and what Gerbert’s drawing might have looked like, see [Miller 1921] and [Cajori 1922].

- The
**arithmetical rule**for finding area, as we have seen, comes from the description for a triangular number*,*usually written \[1 + 2 + \ldots + n = \frac{n(n + 1)}{2},\] where \(n\) is a positive integer. But could the rule also be used for triangles with non-integer sides?

- Use the arithmetical rule \[A = \frac{b(b + 1)}{2}\] to find the (approximate) area of an equilateral triangle of side 2. (We will omit units for the sake of simplicity.)
- Now use it for an equilateral triangle of side 3.
- What result does it give for an equilateral triangle of side \(2\frac{1}{2}\)? Given your previous answers, does it seem reasonable?
- What result does it give for an equilateral triangle of side \(\sqrt{5}\)? Again, does it seem to work?

**Exact and approximate answers**

- Sketch an equilateral triangle of side 7. Draw in its altitude. What is its exact value? How do you know?
- What is the true area of an equilateral triangle of side 7?
- You have access to two tools which Gerbert did not have: the decimal representation of a real number, and a hand-held calculator. Find a decimal approximation for the area of an equilateral triangle of side 7. Round to 3 decimal places.
- What result did Gerbert get, using his \(\frac{6}{7}\) rule? How far off was he?
- What about for the triangle with side 30? Find a decimal representation for the correct area and compare it to Gerbert’s result. How far off was he this time?

*relative error*is defined as \(\frac{error}{true value}\).

- For the triangle of side 7, the error was _____ and the exact area (to 3 decimal places) was _____, so the relative error was the ratio of those two numbers, or __________.
- Find the relative error for the triangle of side 30. What do you notice?
- In general, we might say that Gerbert’s special \(\frac{6}{7}\) rule for finding the area of an equilateral triangle has an error of about _____%. Not bad!

**Why \(\bf{\frac{6}{7}}\)?**

All equilateral triangles are similar, so if we can find the altitude of an equilateral triangle of side 1, we can find the altitude of any such triangle, simply by multiplying the result by the side length \(b\).

- Use the Pythagorean Theorem to find the (exact) measure of the altitude of an equiangular triangle of side 1.

- Use a calculator to find decimal approximations to \(\frac{6}{7}\) and \(\frac{\sqrt{3}}{2}\), say to five decimal places. How close was Gerbert’s approximation?
- Consider all the simple fractions between 0 and 1 of the form \(\frac{n}{n + 1}\): \[\frac{1}{2}, \frac{2}{3}, \ldots, \frac{9}{10}. \]

Which of them is closest to \(\frac{\sqrt{3}}{2}\)? Do you think Gerbert knew that?

In the year 972, when Gerbert was in his late twenties, he was invited to go to the Cathedral School at Rheims (Reims), France, to study and teach mathematics and other subjects. He stayed there for eight years, learning and teaching, revising the curriculum, and acting as the head of the school. Figure 5 shows Rheims in northeast France. Gerbert spent his youth at a monastery in Aurillac (hence his name) in the Auvergne region, also shown on the map.

**Figure 5.** The city of Reims (Rheims) in France. Map courtesy of On the World Map: Free Printable Maps.

Gerbert had a reputation as a brilliant, innovative, open-minded teacher. He constructed spheres for teaching astronomy, had a shield-maker make an abacus for use in the instruction of Hindu-Arabic numerals, used a monochord to teach music, and made a “diagram of rhetoric” out of parchment [Darlington 1947]. He also taught his students how to use an astrolabe [Swetz 2020] to find the height of objects—perhaps even in his geometry classes [Lattin 1961, p. 6]. He left behind several documents which served as textbooks, including one in geometry.

We do not know exactly when Gerbert wrote his geometry textbook—presumably a collection of lessons he had taught his students during his years at Rheims [Olleris 1867, p. 594]. In fact, these writings may have been collected after his death [Høyrup 2014]. At any rate, for much of the Middle Ages, after Boethius’ text on geometry was lost, Gerbert’s textbook was the only Latin geometry text available for the many cathedral and church schools. Figure 6 [Swetz 2019] shows the first page of Gerbert's *Isagoge Geometriae *(*Introduction to Geometry*)*,* from a 12th-century Austrian edition of the text, available in the Mathematical Treasures Collection of this journal. (There are other beautiful images from the book there, along with a copy of a page reprinting Gerbert’s famous letter to Adalbold.)

**Figure 6.** The first page of Gerbert's *Geometria*. University of Pennsylvania Libraries, reference no. LJS 194.

And so, when Gerbert taught his students how to find the area of an equilateral triangle, which method did he teach them? What method is enshrined in his geometry text—the geometrical rule, the arithmetical rule, or the hybrid rule using \(\frac{6}{7} b\) as the altitude?

When Gerbert introduced triangles in his textbook [Olleris 1867, p. 414], he classified them either by the measures of their angles:

*Orthogonius*(right-angled)*Ampligonius*(wide-angled)*Oxygonius*(sharp-angled)

or by the relative lengths of their sides:

*Isopleuros*(same-sided)*Isosceles*(two sides the same)*Scalenos*(no sides the same).

Today we use mostly words derived from Latin, like *equilateral *and *acute, *to describe triangles. But Gerbert, who generally wrote almost exclusively in Latin, used Greek words here: they are exactly the same words Euclid used 1300 years earlier [Fitzpatrick 2008, p. 6]. He used the word *cathetus*, also from the Greek, for the altitude of a triangle.

Prof. Alexandre Olleris, in his edition of Gerbert’s *Geometria *[Olleris 1867], divided the text into very small chapters, each a paragraph or two long, on one topic. Chapter 49 (p. 450) has this heading:

*XLIX. Trigoni isopleuri, cujus sunt singular latera XXX, embadi pedes comprehendere*,

or, roughly,

*To understand how to find the area of an equilateral triangle, each of whose sides is 30 feet.*

We have seen this problem before; this is the problem Gerbert posed to Adalbold in his letter.

Here is an outline of Gerbert’s solution:

- First we must find the altitude:
- Multiply the side by itself: \(30 \cdot 30 = 900\).
- Deduct a quarter [of the result]: \(900 - \frac{900}{4} = 900 - 225 = 675\).
- Which, if you add 1, gives 676.
- And the square root of 676 is 26.
*Ecce cathetum!*Behold the altitude!- Therefore, to find the area, multiply \(\frac{1}{2} \cdot 30 \cdot 26 = 390\).

So now we see where Gerbert got the number 26 for the height of this triangle. But what exactly was he doing? Deduct one-fourth? Add 1? What is this algorithm? Was he completing the square? Or something else?

Actually, up to the point at which Gerbert added 1, he was absolutely correct. He was trying to say, without modern notation, or a complete understanding of irrational numbers, that the altitude of an equilateral triangle of side \(b\) is \(\frac{b}{2} \sqrt{3}\). But then he pulled a fast one on us. He didn’t have a number at his disposal that meant “the square root of 3.” He could not find the square root of 675. So he just found the nearest whole number whose square root he did know and used it. If the side of the triangle had been 29, or 31, this trick of adding 1 would not have worked.

By the time Gerbert wrote to Adalbold, he had presumably figured out that, by using \(\frac{6}{7}\) in place of \(\frac{\sqrt{3}}{2}\) (or, in other words, using \(\frac{12}{7}\) as an approximation for \(\sqrt{3}\) [Folkerts and Hughes 2016, p. 48]) he would be dealing with purely rational numbers and could obtain an answer.

- Show that, for an equilateral triangle of side 30, the square of the altitude is indeed 675.
- Show that, for an equilateral triangle of side \(b\), using Gerbert’s algorithm up through the instruction to “deduct a quarter of it” gives the correct value for the square of the altitude.
- Suppose you are a student in Gerbert’s geometry class, using his textbook, and you have just seen his solution for how to find the area of an equilateral triangle of side 30. Now you are working on your homework assignment.

- You must find the area of an equilateral triangle of side 28. What goes wrong? What number do you need to add to get a perfect square? How far off is the value for the altitude given by this algorithm? How far off is the area? Can you get a closer answer by subtracting instead of adding something?
- What if the measure of the side had been 29? What makes this problem more complicated?

For many in the mathematics community, Gerbert’s 999 letter to Adalbold is their introduction to Gerbert and to his interest in mathematics. And for most of them, the introduction stops there. In this article we have been able to get a glimpse of Gerbert’s growth in his knowledge of computing the area of a triangle, from the rabbit-out-of-the-hat approach he taught his students, to his graphical explanation of the arithmetical rule and its faults, and finally to using insight gleaned from that graphical approach to find a better approximation for the altitude of a triangle. He deserves to be remembered as a creative and deep thinker as well as an accomplished teacher of mathematics.

Other entries about Gerbert and his work in this journal:

Mayfield, B. 2010, August. Gerbert d’Aurillac and the March of Spain: A Convergence of Cultures. *Convergence*.

The importance of Gerbert’s three-year stay in Catalonia to the development of his mathematical knowledge.

Swetz, F.J. 2019, January. Mathematical Treasure: Gerbert's Geometry. *Convergence*.

Images from a 12th-century copy of the *Isagoge Geometriae, *plus part of Gerbert’s letter to Adalbold. Provided courtesy of the University of Pennsylvania Libraries, The Lawrence J. Schoenberg Collection of Late and Early Renaissance Manuscripts, Kislak Center for Special Collections, Rare Books and Manuscripts.

Swetz, F.J. 2020, January. Mathematical Treasure: Mathematics of Gerbert of Aurillac.

Images from a French manuscript of the 12th century on geometry and astronomy, including the use of an astrolabe, and a 13th-century English manuscript illustrating the problem of determining the height of a remote tower using methods based on Gerbert’s geometry. Provided courtesy of The British Library.

Bubnov, N., ed. 1899. *Gerberti Opera Mathematica (972–1003).* Berlin: R. Friedländer und Sohn.

Cajori, F. 1922. Discussions: The Formula 1/2 a(a+1) for the Area of an Equilateral Triangle. *American Mathematical Monthly *29(8): 303–307.

Darlington, O. 1947. Gerbert, the Teacher. *The American Historical Review *52(3): 456–476.

Fitzpatrick, R., trans. and ed. 2008. Euclid’s Elements of Geometry.

Folkerts, M., and B. Hughes. 2016. The Latin Mathematics of Medieval Europe. In V. Katz, ed., *Sourcebook in the Mathematics of Medieval Europe and North Africa*, 4–223. Princeton: Princeton University Press.

Høyrup, J. 2014. Mathematics Education in the European Middle Ages. In G. Schubring and A. Karp, eds., *Handbook on the History of Mathematics Education*, 109–124. New York: Springer.

Lattin, H.P. 1961. *The Letters of Gerbert: With His Papal Privileges as Sylvester II.* New York: Columbia University Press.

Menninger, K. 1969. *Number Words and Number Symbols: A Cultural History of Numbers* (English translation). Cambridge, MA: The MIT Press.

Miller, G.A. 1921. The Formula 1/2 a(a+1) for the Area of an Equilateral Triangle. *American Mathematical Monthly *28(6/7): 256–258.

Olleris, A., ed. 1867. *Oeuvres de Gerbert, Pape sous le nom de Sylvestre II.* Clermont: F. Thibaud.

Betty Mayfield is Professor Emerita of Mathematics at Hood College in Frederick, Maryland, where she loved learning and teaching about the history of mathematics.