The suggestion of a classicist colleague a few years ago that I read Sophocles marked the start of my fascination with ancient Greek culture. I read Sophocles, the other tragedians, Homer, Herodotus, Thucydides, Plato, Aristophanes, and pretty much any ancient Greek literature I could find. I loved it all. It was through Plato, and the co-direction of a Union College student’s senior thesis, that I discovered the beauty and excitement of ancient Greek mathematics. This student, who was a joint major in classics and mathematics, wrote her senior thesis on Plato’s Theaetetus, a dialogue that deals with the theory of knowledge. The particular part of the dialogue on which she focused concerned the existence of what the Greeks called “incommensurable magnitudes,” which are equivalent to what we would call “irrational numbers.” I found it to be a fascinating project that involved also learning the Pythagorean and then the Eudoxian approaches to proportionality.
After this first project, I supervised a number of additional theses on various subjects in ancient Greek mathematics, such as the Platonic Solids and the Mechanical Method of Archimedes. I was enjoying learning this material and working on student projects immensely (always warning my students in advance that I’m a beginner, but an enthusiastic beginner). However, something was missing, and that was a sense of place, a sense of orientation. Let’s see: Thales was from Miletus, Pythagoras from Samos, Plato founded his academy in Athens but traveled to Sicily, Euclid received his training from Plato’s students and then founded an academy in Alexandria… Then I heard about the first MAA study tour to Greece, and I knew it was for me.
I had high expectations for the trip, and it exceeded them all. Let me start with my favorite, Delphi. Visiting Delphi, with its Temple of Apollo on towering Mount Parnassus, I thought about the problem of doubling the cube (i.e., for a given cube, construct a cube having double its volume). According to legend, this problem originated when the people of the Greek island of Delos, suffering from a terrible plague, asked the oracle at Delphi what they should do, and the oracle, who was believed to be speaking for the god Apollo, replied that they should build a cubical altar twice the size of their present cubical altar. (Plato’s reaction to this was to argue that Apollo did not really want an altar built, but rather wanted to shame the Greeks for their neglect of geometry.)
Then there was our brief visit to the west coast of Turkey, an area that the ancient Greeks had colonized. As we walked past the ancient building, I imagined Thales walking in exactly the same place. Thales started it all. He is considered by most historians to have begun the entire ancient Greek scientific enterprise.
Near Thales’ home of Miletus is the Greek island of Samos, where we stayed for a few days in the town of Pythagoria. Not too hard to guess this town’s claim to fame. Pythagoras, perhaps as much a guru as a philosopher/mathematician, started a school and attracted many followers. It is fair to say that this school began the study of number theory and also music theory. And here we all were, drinking ouzo and eating baklava in Pythagoria. (My guess is that Pythagoras did neither.)
Also on the island of Samos, we toured the tunnel of Eupalinus, site of an ancient tunnel and a modern mystery. (How did the ancient builders of this tunnel manage to make the two ends of the tunnel meet in the middle of a mountain?)
In Athens, we visited the site of Plato’s academy, probably the very first academic institution and toured the Agora where Socrates used to hang out and insist that people clearly define their terms and justify their beliefs.
Throughout the trip, we also heard wonderful lectures by Greek historians of mathematics on such topics as: the contrast between the mathematical philosophies of Plato and Aristotle, the planetary theory of Aristarchus (who described the orbits of the Earth and moon about the sun long before Copernicus), and the burning mirrors of Archimedes (used in the defense of Sicily against the Romans).
Of course, we also saw many wonderful sites that did not directly connect with mathematics, such as Olympia (site of the original Olympic games), the Temple of Hera on Samos, and the ancient palace at Mycenae (the possible home of Agamemnon, one of the central characters in Homer’s Iliad).
I am in the process of developing a course on ancient Greek mathematics for non-math majors. For this reason, Union College helped provide support (through the Hewlett Foundation) for my participation in the MAA study tour. I’m confident that this was money well spent. When I teach this course for the first time next year, I will be able to reinforce the mathematics of ancient Greece with pictures and stories from our trip and with a deeper sense of history and place.
Julius Barbanel is a professor of mathematics at Union College in Schenectady, New York.