Need the Area of a Triangle? The Pope Can Help! – Gerbert Finds the Area of a Triangle, Part 1: Letter to Adalbold

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.

Roman and Egyptian fractions

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.