Pappus and Pandrosion: the curmudgeon and the professor
There is a paper on Academia that I posted years ago, proudly claiming that Hypatia was the world’s first female mathematician. It’s humbling what years of research will teach you. It so turns out that Hypatia was NOT the world’s first female mathematician. Other women taught mathematics long before Hypatia, including the mathematician Pandrosion. She was one of the first noted female mathematicians even though, for years, many thought she was a man. It figures.
We learn about Pandrosion from the famous Alexandrian mathematician Pappus, who mentioned her in his third book of Mathematical Collection, a body of works that consisted of about 12 encyclopedic volumes of mathematics as written by his predecessors. Noting his mention of her, Pandrosion was probably born and raised in Alexandria, Egypt.
Pappus does not make it apparent that Pandrosion was a woman. The understanding that Pandrosion was a female contemporary comes through a historical analysis of the name Pandrosion, wherein the –ion ending is a form of a nickname, much like the –y ending in Gabby for Gabrielle. As a result, Pandrosion’s full name was Pandrosos, which is a feminine name referenced in the Greek mythologies of Pandrosos, the daughter of Cecrops and Aglauros. The name, Pandrosos, means “all dewey,” which also alludes to a feminine name.[i]
Many people believe that Pappus “dedicated” his third book of the Collection to Pandrosion. However, this was far from the truth. He writes:
“Certain people who claim to have learned mathematics from you, set out the enunciation of the problems in what seemed to us an ignorant manner.”
And so Pappus begins Book III with a repudiation of Pandrosion.
Pappus wrote Mathematical Collection around 320 CE. As a result, Pandrosion might have been an older contemporary of Theon, the famed mathematician, astronomer, and father of Hypatia.[ii]
In Book III, Pappus uses his work as a platform to lambast Pandrosion. However, his anger wasn’t gender-specific and she was not alone. In addition to Pandrosion, in the Collection he also criticizes her students and other scholars as well. [iii] Something tells me that Pappus was a curmudgeon.
In one part of Book III, he directs his rant to Pandrosion, as he explains the difference between a problem and a theorem. He adds that academia must censure any teacher who confuses these terms.[iv] Could you imagine a professor in our current times writing a book to publish for a college curriculum as a platform to insult and attempt to silence other mathematicians? Could you imagine an intelligent professor creating a public drama to insult the aptitude of another? Actually, yes…there was John Wallis and Thomas Hobbes. But, that’s another story for another day. I digress.
Back to Pappus…
In one problem, Pandrosion’s student attempts to explain the harmonic mean using geometry. Pappus describes the student as “someone who seems to be a great geometrician” but appears to be uninformed as Pappus explains that he “set his problems ignorantly.”[v] Pappus uses this student’s work as a way to find the arithmetic, geometric, and harmonic means using a geometrical approach. In his propositions for finding the means, he ridicules the “certain other person” who uses a semi-circle and four lines.[vi] This wouldn’t be so insulting if Pappus were discourteous to Pandrosion. However, he wasn’t. He was mocking one of her students! The equivalent to this in today’s time would be for a renowned mathematician to ridicule a graduate student and her advisor in a math textbook that would be commonly be used by a university. It was a bold and uncouth move on his part to insult her students in such a permanent and public light.
While Pappus might have thought that his insult was a score, this move backfired. What it really did was memorialize Pandrosion’s name in history. We would not have even known who Pandrosion was if it wasn’t for Pappus and his public rant.
Regardless, Pappus does acknowledge that the student did find the arithmetic and geometric means. What follows is a bit of mathematical backstory and some math.
Finding the Arithmetic, Geometric, and Harmonic Means
Finding the arithmetic mean and geometric mean uses Euclid’s first three postulates that state:
- A straight-line segment can be drawn to join any two points.
- A straight-line segment can be extended to any indefinite length.
- Given a straight-line segment, a circle can be drawn such that the straight line is the radius and one endpoint is the center.
Arithmetic Mean
An arithmetic mean is an average of numbers. It is the process of adding up all the numbers and then dividing that sum by the total quantity of numbers. In the case of the semicircle below, if there are two numbers, x and y, that represent the lengths of those segments, the two numbers can be added up and then divided by two to find the mean of the diameter:
\begin{aligned} \frac {x+y}{2} \end{aligned}In the semicircle in Figure 1, OC is the radius. If we let AD = x and let DB = y, then the arithmetic mean of AD and DB is
\begin{aligned} \frac {x+y}{2}=\frac {AB}{2}=OC \end{aligned}Since half the diameter of the semicircle is the radius, we determine that the arithmetic mean of the diameter of this semicircle equals the radius.
Geometric Mean
A geometric mean is useful when a set of values have different units, and it is necessary to eliminate the bias of the range. For example, if you need to find the mean size of a group of cars with different sizes and weights, or if you need to find the mean of two bike races, where each race was a different distance.
A geometric mean is achieved when we multiply numbers together and then take the root of that many numbers. So, if two numbers are multiplied together, we take the square root of the product. If three numbers are multiplied, we take the cube root of the product.
In the case of the semicircle, if two numbers are x and y, then the geometric mean of the following semicircle is:
\begin{aligned} \sqrt {xy} \end{aligned} \begin{aligned} \angle CAD=\angle BCD\: \text{and} \: \angle DCA=\angle DBC\: \text{and} \: \angle ADC=\angle CDB. \\ \end{aligned} \begin{aligned} \text{Also, we see that} \: AD=x\: \text{and} \: BD=y. \end{aligned}Using proportions, we get
\begin{aligned} \frac {CD}{AD}=\frac {BD}{CD} \: \text{therefore} \: \frac {CD}{x}=\frac {y}{CD} \end{aligned}Cross-multiplying this last equality, we then get
\begin{aligned} CD^2=xy \: \text{therefore} \: CD=\sqrt {xy} \end{aligned}And so, we find that the geometric mean of AD and BD is CD.
Harmonic Mean
The harmonic mean is useful for measuring the central tendency, such as averaging rates, or averaging an overall driving speed that includes multiple stops.
A harmonic mean is expressed as the reciprocal of the arithmetic mean of the reciprocals. So, for two numbers x and y, with their respective reciprocals as
\begin{aligned} \frac{1}{x} \: \text{and} \: \frac{1}{y} \end{aligned}the equation would take the reciprocal of
\begin{aligned} \frac {\frac{1}{x}+\frac{1}{y}}{2} \: \text{to get} \end{aligned} \begin{aligned} \frac {2}{\frac{1}{x}+\frac{1}{y}}=\frac {2}{\frac{x}{xy}+\frac{y}{xy}}=\frac {2}{\frac{x+y}{xy}}=\frac{2}{1} \cdotp \frac{xy}{x+y}= \text{H(x,y)}=\frac{2xy}{x+y} \end{aligned}Using a semicircle, the harmonic mean can be represented geometrically as follows:
In Figure 3, triangle COD and CDE are similar, because we have two right triangles that have the same angles as follows:
\begin{aligned} \angle DCE=\angle OCD\: \text{and} \: \angle CDE=\angle COD\: \text{and} \: \angle DEC=\angle ODC. \\ \end{aligned} \begin{aligned} \text{Also, we see that} \: AD=x\: \text{and} \: BD=y. \end{aligned}Using proportions, we then know that
\begin{aligned} \frac {CE}{CD}=\frac {CD}{CO} \end{aligned}When we cross multiply, we see that
\begin{aligned} CD^2=CE \cdotp CO \: \text{and} \:\frac{CD^2}{CO}=CE \end{aligned}In the proof for the Geometric Mean, we found that
\begin{aligned} CD^2=xy \\ \text{as a result} \\ \frac{xy}{CO}=CE \end{aligned}Also, in the proof for the Arithmetic Mean, we find that
\begin{aligned} OC=\frac {x+y}{2}=CO \end{aligned}Using substitution, we then see that
\begin{aligned} CE=\frac{xy}{\frac{x+y}{2}}=\frac{xy}{1} \cdot \frac{2}{x+y}=\frac{2xy}{x+y} \end{aligned}Referring back to our definition of the Harmonic Mean, we then see that CE is the Harmonic Mean of AD and DB.
Back to the story….
In the student’s original presentation, he proved the arithmetic and geometric means on the outset. Regardless, Pappus ridicules the student for finding the arithmetic, geometric, and harmonic means using four lines and a semicircle. Pappus indignant response of the student’s work states that though the student found the arithmetic and geometric mean, the student could not explain how to find the harmonic mean. Pappus writes, “but how BZ is a mean of the harmonic mediety, or of which straight lines, he does not say.” However, the student did not need to say, as the solution was implied in the presentation.
Pandrosion had many male students who, like Hypatia’s students, went on to become renowned in their own right. Pappus even makes a note of this when he mentions one of Pandrosion’s students as “someone who seems to be a great geometrician.”[vii]
For Pappus to deride Pandrosion in a public work indicates that her intellect and reputation might have been a threat to Pappus. Additionally, if Pandrosion did not have a positive reputation or was not a familiar name in the academic circles of Alexandria, she might not have even been on Pappus’ radar, which further validates that Pandrosion was an accomplished mathematician and professor.
There are so many lost stories about historical figures in STEM that have faded away in the chronicles of history, either through age or through destruction. There are also many stories in the history of science that were misconstrued. Between the stories, there are volumes of books written by some of the most brilliant thinkers in history. In these volumes of books are writings that enlighten us into their characteristics, their passions, their frustrations, and their disputes. We get a better understanding of their personalities by carefully reading the texts that functioned as curriculum. What I find particularly fascinating is that more than 2,000 years later we find that the same people we placed upon historical pedestals and showered with admiration were really a lot like some of the characters we encounter in our own life. They were curmudgeons, frustrated teachers, patient educators, nervous students, and diligent professors. What is also enthralling is finding people in these texts that we never knew existed, like the female mathematician Pandrosion!
[i] Winifred Frost, “Pappus and the Pandrosion Puzzlement,” Function 16, no. 3 (June 1992), https://www.qedcat.com/function/16.3.pdf.
[ii] M. Deakin, Mathematician and martyr: A biography of Hypatia of Alexandria (Amherst, NY: Prometheus Books, 2007), 132.
[iii] M. Deakin, Mathematician and martyr: A biography of Hypatia of Alexandria (Amherst, NY: Prometheus Books, 2007), 131.
[iv] Edward J. Watts, Hypatia: The Life and Legend of an Ancient Philosopher (Oxford: University Press, 2017), 94.
[v] Edward J. Watts, Hypatia: The Life and Legend of an Ancient Philosopher (Oxford: University Press, 2017), 95.
[vi] M. Deakin, Mathematician and martyr: A biography of Hypatia of Alexandria (Amherst, NY: Prometheus Books, 2007), 131.
[vii] Edward J. Watts, Hypatia: The Life and Legend of an Ancient Philosopher (Oxford: University Press, 2017), 95.