An Evening with Euclid

Photo: Irina Stelea

On November 16th, students and faculty, led by Michael Weinman, came together for a seminar on Euclid’s Elements which was a supplementary seminar to the Academy Year core course on Plato’s Republic.

The discussion aimed to relate Euclid’s propositions to the concept of the divided line found in Book VI of the Republic and Socrates’ suggested educational schema in Book VII.

In 522c of the Republic, Socrates discusses education as a way to “train” individuals to think dialectically by first learning arithmetic, which is then followed by geometry. The seminar somehow echoed this process presented in the dialogue, particularly since everyone had cursory knowledge of arithmetic and geometric forms and concepts from pre-university education.

The primary task was to follow the statements made by Euclid in Proposition 11 of Book II and Proposition 30 of Book VI of his Elements, which not only meant reading these statements, but also drawing them. If the proposition is drawn successfully, then the proposition is proven. I thought for sure that dealing with two Euclidean propositions need not take ninety minutes, but as it turned out, geometry was not as easy as I remembered it to be.

In 509d of the Republic, Socrates says “take a line cut in two unequal segments […] and go on and cut each segment in the same ratio”[1]. Prof. Weinman remarked that the two propositions demonstrate the thought process behind the divided line.

Indeed, it is not difficult to see the similarity between Socrates’ statement of the divided line and a statement of Euclid’s from proposition 30 in Book VI: “Let AB be the given straight line; thus it is required to cut AB in extreme and mean ratio.” Much confusion ensued when interpreting these statements, among others like: “Thus it is required to cut AB so that the rectangle contained by the whole […]” and, “Let there be applied to AC the parallelogram CD to BC […].”[2]

After completing the drawings of the two propositions, participants found the resulting figures visually inadequate to prove the statements made by Euclid. I did wonder whether that was what Socrates had in mind.

In the Republic, Socrates sees geometry as what enables individuals to conceive a form, and this ability ideally can draw individuals to philosophy. A shape like a sphere is imaginable, though none has actually been perfectly produced in nature or manufactured in laboratories. Such a conundrum can perhaps give rise to philosophical inquiries on the nature of existence or of knowledge.

It is in hindsight that I realized that the way most people, including myself, approach mathematics might be very different from Socrates’ position in the Republic. Geometry, along with other branches of mathematics, is valued for its practical applications.

Its theorems are the principles on which sound architectural plans are based or by which different ways of cutting a sandwich are demonstrated (there is in fact a ham-sandwich theorem which draws upon Euclidean space).

Practical use of mathematics is not to be lamented, for without it mankind would not have produced many of its innovations, but perhaps there is a value to Socrates’ approach that should be instilled in individuals early on in their education.

Should it be enough that children have memorized multiplication tables or that students can parrot algebraic solutions? If anything, an approach like the one stated in the Republic invites curiosity rather than inducing fear or frustration.

In this light, how did the seminar on Euclid hold up to Socrates’ exhortation? Prof. Weinman offered a possible conclusion: that not only does the divided line work suitably with the construction of the divided line, but the propositions also demonstrate a way of following a particular principle or logos such that inquisitive minds can arrive at a starting point for learning and for doing dialectics.

Students seemed to agree, and it is to be hoped that the upcoming winter term elective on logic and the nature of mathematical knowledge will pick up where we left off.

[1] Plato. Republic. Trans. Allan Bloom. USA: Basic Books, 1991. Print.

[2] Euclid. The Thirteen Books of the Elements. Trans. Thomas L. Heath. USA: Dover, 1956. Print.

by April Matias (2nd year BA, Philippines)

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