## Pappus criticises Pandrosion

Pandrosion is mentioned in Book 3 of Pappus's

*Mathematical Collection.*In Edward J Watts' book about Hypatia, namely*Hypatia: The Life and Legend of an Ancient Philosopher*(Oxford University Press, 2017) details are given of the references to Pandrosion in Pappus's*Mathematical Collection*. The reasons why it is believed that Pappus tried to show Pandrosion as an incompetent mathematical teacher are also clearly argued:**Pandrosion of Alexandria**

Hypatia had four significant female contemporaries who were trained as philosophers, taught philosophy or mathematics, or played a public role like the one she assumed. Three of these, Pandrosion of Alexandria, Sosipatra of Pergamum, and the wife of Maximus of Ephesus, are older than Hypatia. The fourth woman, Asclepigenia of Athens, was the daughter of Hypatia's younger rival, the Athenian philosopher Plutarch. Each of these women, however, established a philosophical reputation that was described, recognized, and admired by her male contemporaries. None of her peers were quite as accomplished as Hypatia ...

The activities of the early fourth-century Alexandrian mathematician Pandrosion most closely resembled those of Hypatia. While Pandrosion may have had the highest profile of these four women during her lifetime, she left no significant imprint on the historical record. No narrative source mentions her, none of her writings survive, and nothing written by her students has been preserved. Pandrosion is instead known only from the arguments made against her by Pappus in Book 3 of his

*Collectio.*In this section of the text, Pappus takes issue with a method of finding cube roots that Pandrosion pioneered. His criticism takes the form of an address to Pandrosion that is at times both pedantic and aggressive. It begins with Pappus sarcastically defining for Pandrosion what the basic mathematical terms

*problem*and

*theorem*mean. A problem, he writes, is a test of a proposition that can ultimately be possible or impossible. A theorem, however, must be demonstrated to be correct. Pappus then tells Pandrosion that anyone who claims to know mathematics should be censured if she confuses these things and sets up a mathematical investigation incorrectly.

This matters, Pappus continues, because "some people who claimed to have learned mathematics from you lately gave us an ignorant explanation of problems." Pappus condescendingly offers to explain the proofs of these and related questions to Pandrosion, "for your benefit and that of other lovers of learning. Pappus then proceeds through a series of problems related to him by students of Pandrosion. The first student is described as "someone who seems to be a great geometrician" but who nonetheless "set his problems ignorantly" when he sought to determine cube roots and their squares. When this student brought his work to Pappus and asked the mathematician to critique it, his ignorance moved Pappus to write a thorough repudiation of the method.

Pappus's repudiation is both comprehensive and convincing, but it is also completely unfair. Modern mathematicians have found that the approach attributed to Pandrosion's student works at least as well as that favored by Pappus. It is possible that Pappus may truly have believed that Pandrosion was introducing a terrible mathematical practice that needed explicit correction, but it is more likely that this attack on Pandrosion and her prominent student reflects a genuine rivalry between Pandrosion and Pappus. The second section of Book 3 further suggests that Pappus saw Pandrosion as a genuine rival worth engaging. It focuses on the problem of "exhibiting the arithmetic, geometric, and harmonic means in a semicircle." Pappus mocks "another certain person" who claimed that these could be exhibited simply by drawing four lines within the semicircle. Again, however, Pappus is being somewhat disingenuous. Pappus affirms that this method does discover the geometric and arithmetic means, but he discards it because the speaker does not explicitly say how one of the lines marks the harmonic mean Technically, this may have been true, but it was also self-evident that the method did actually mark the harmonic mean. The missing explanation that Pappus felt doomed Pandrosion's approach would have been essentially a superfluous gloss to an otherwise sound and thorough discussion. Here, too, a modern assessment of the method employed by Pandrosion's student confirms that it works as well as that with which Pappus sought to replace it.

Ultimately, Pappus's critiques provide little direct information about Pandrosion and her activities. He does, however, offer some tantalizing glimpses into the world of this innovative and influential female mathematician. It is clear that Pandrosion was a contemporary of Pappus who was active in Alexandria in the early or middle part of the fourth century. If the methods that Pappus criticizes were truly new, Pandrosion may well have been a younger rival of Pappus and a contemporary of Hypatia's father, Theon. This means that the young Hypatia would certainly have known about Pandrosion's career and may even have known Pandrosion herself.

Pappus's text also shows that Pandrosion's teaching activities resembled those of Hypatia. All of the articles and pronouns that Pappus uses to describe Pandrosion's students are masculine. This suggests that, like Hypatia, Pandrosion taught mathematics to male students, probably in some sort of a public setting. Pappus's comment that one of her students is thought to be a great geometrician also indicates that Pandrosion and the members of her circle had begun to earn a reputation in Alexandria for laying out innovative mathematical approaches.

The blame that Pappus assigns to Pandrosion for the failures of her students also suggests that Pandrosion was somehow involved in the creation, review, or publication of the texts that Pappus had read. This points to a scenario like the one in which Synesius sent some of his philosophical works to Hypatia so that she could comment on them before he distributed them more widely. It is also possible that the mathematical works of Pandrosion's students that Pappus saw may have resulted from an even closer intellectual collaboration between Pandrosion and the men in her circle. If this process resembled that through which students in fifth-century philosophical schools authored their first commentaries under the supervision of their teacher, the work on cube roots that Pappus critiqued may have been the equivalent of a thesis in which Pandrosion and her student worked together to apply her teaching to a new problem. In either case, however, Pappus felt confident that he could convincingly tie to Pandrosion the things that he found to be problematic in the work of her students. This sort of attack indicates that Pandrosion had developed a reputation as a successful mentor of high-profile male students.

The scant in formation that Pappus provides does not allow us to say much more about Pandrosion. We know, however, that Pappus and others in the Alexandrian mathematical establishment appear to have ultimately stifled whatever innovative techniques Pandrosion and her students pioneered. Not only do no texts by Pandrosion currently survive, but none of the great Platonic mathematicians of the later fifth and early sixth centuries like Proclus and Marinus seem to know anything about her, her students, or their ideas. Later Byzantine and Arabic mathematicians also seem not to know about her. This suggests that her scholarly line ended at some point in fourth century, and that, when it did, her work and those of her students was eventually forgotten. Pappus prevailed over Pandrosion. This victory had the unfortunate effect of almost completely obscuring the legacy of the Alexandrian female intellectual whose career likely served as a model for the young Hypatia.