Pappus and Pandrosion: the curmudgeon and the professor

Gabriellebirchak/ August 18, 2019/ Ancient History, Classical Antiquity

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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:

1. A straight-line seg­ment can be drawn to join any two points.
2. A straight-line seg­ment can be extend­ed to any indef­i­nite length.
3. 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!

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[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.