In Elements of Geometry, students study Euclid and learn to question assumptions.
 

For more than 2200 years, there was only one geometry textbook in the Western world: Euclid's Elements. It was second only to the Bible in the number of translations and editions published. From the Middle Ages until the early 20th century, the Elements was required reading for every educated person and was considered a model of logical reasoning and argument by mathematicians and non-mathematicians alike. For example, here is Abraham Lincoln's account of his student years:


In the course of my law reading, I constantly came upon the word ‘demonstrate.’ I thought, at first, that I understood its meaning, but soon became satisfied that I did not. ... I consulted all the dictionaries and books of reference I could find, but with no better results... At last I said,—Lincoln, you never can make a lawyer if you do not understand what ‘demonstrate’ means; and I left my situation in Springfield, went home to my father’s house, and stayed there till I could give any proposition in the six books of Euclid at sight. I then found out what ‘demonstrate’ means, and went back to my law studies.
— Abraham Lincoln

In our class, we did not get quite as far as Lincoln: we only made it three quarters of the way through Book 1 of the Elements, after which we switched to a more modern treatment. (At some point, the fact that Euclid did not have the modern concept of "number" becomes too much of a distraction from the geometric content of the book.) Still, it was thrilling to guide the students in an exploration of this ancient work that seems remarkably relevant and radical, even today. 

In American high schools, geometry is traditionally the course in which students learn to write proofs. This is Euclid's legacy: his great contribution in the Elements is the axiomatic method, which underlies all of mathematics as we know it today.  Starting from just five geometric "postulates" (such as "there exists a unique line segment through any two points") and some general "common notions" (such as "two quantities equal to the same quantity are equal to each other"), Euclid builds up the entire edifice of geometry, brick by logical brick, completely from scratch. It is an awe-inspiring demonstration of the power of reason to create something from virtually nothing. No wonder the book has been a bestseller for two millennia.

Our first few weeks of class were spent getting used to the axiomatic method, which meant learning to question our intuitions at every step. No matter how obvious something seemed, if it was not rigorously provable from the postulates, then we could not take it as true. While to a non-mathematician it may seem silly to keep insisting on proving the obvious, this is exactly the kind of mental discipline that ultimately gives mathematics its power. Abraham Lincoln understood its appeal, and after a couple of weeks, so did our Proofniks. Occasionally they outdid Euclid himself in noticing places where he takes things for granted that he should not. (Alas, even Euclid is not infallible; I had to defend him to the kids by reminding them of the many generations of giants on whose shoulders we are standing.)

Our study of Euclid culminated in Week 3 of the block with an in-depth look at his famous Fifth Postulate (also known as the Parallel Postulate). The fate of Euclid's Fifth Postulate is probably the most vivid demonstration of the power of the axiomatic method in the history of mathematics, and I cannot resist the opportunity to tell it here. Please bear with me: I promise that it does all get back to the students in the end!

Euclid's first four postulates are very straightforward statements that seem like obvious candidates for foundational geometric truths. The Fifth is different. At first reading, it is almost inscrutable. What it actually asserts is that under certain conditions, two lines are guaranteed to cross -- a statement that (once you figure out what it's saying) seems obviously true, but too complicated to be a postulate. It feels more like a theorem to be proved than like a foundational truth.  Euclid himself clearly felt uncomfortable with having to take it as given: he goes out of his way not to rely on the Fifth Postulate in his proofs for as long as he can. 

For the next two thousand years, generations of mathematicians devoted hundreds of pages to "proving" that the Fifth Postulate follows logically from the other four and therefore does not need to be stated as a postulate. None of these arguments were valid. It was not until the 19th century that mathematicans finally realized that the Fifth Postulate can never be proved, because, despite being "obvious", it is not necessarily true. The search for its proof had led several people to discover alternative (non-Euclidean) forms of geometry that are perfectly self-consistent, yet in which the Fifth Postulate is false. The first of these geometries to be described was hyperbolic geometry, a bizarre world in which everything is completely at odds with our intuition: straight lines can get arbitrarily close to each other without intersecting, the sum of the angles of a triangle can be as small as you want, rectangles do not exist, and the Pythagorean Theorem is false. Yet it was the discovery of hyperbolic geometry that ultimately paved the way for the mathematical framework underlying Einstein's general theory of relativity.  

Without the practice of suspending one's intuitive thinking as demanded by Euclid's axiomatic method, it would have been impossible for anyone to stumble upon this "strange new universe" (as one of the discoverers of hyperbolic geometry described it). We, in the twenty-first century, have gotten used to the fact that science can outstrip our intuition in describing reality; we throw around words like "quantum effect" and "black hole" without a second thought. Yet few of us get to experience first-hand the invigorating discomfort of having our fundamental intuitions and assumptions shattered by reason.

This was the experience I wanted the students to have at the end of our Euclid unit, when we spent a week teasing out the many "obvious" things that, in a world without the Fifth Postulate, are simply false. What I hope the students got out of this exercise is the sense that in mathematics, one's unexamined intuitions can be limiting and misleading. On the other hand, when one takes the time to think things through carefully, one's intuition can adapt and expand, to encompass a larger, more flexible, more interesting version of the universe.

(If all of this seems like a metaphor for social or moral progress, it is perhaps not a coincidence. This may be a good time to mention that Euclid's influence on Abraham Lincoln was not limited to legal arguments. Lincoln explicitly appeals to Euclid as a model in several anti-slavery debates, where he enjoins the audience to think through the logical implications of arguments and to question assumptions.)

But back to geometry. In the second half of the block, we returned to the familiar world of Euclidean geometry, where the Fifth Postulate is safely in place and one can more or less believe the pictures that one draws. Here, we discovered another payoff to the axiomatic method: increasingly, we found that it had the power to prove not only things that seemed intuitively obvious, but also things that seemed at first glance quite mysterious:

  • Why is it that, when you connect the midpoints of the adjacent sides of any quadrilateral, the result is always a parallelogram? 

  • Why is it that, when you draw lines connecting the three vertices of a triangle to the midpoints of the opposite sides, these three lines always intersect at a single point? 

  • Why is it that, if you choose a pair of opposite points on a circle and draw lines connecting them to a third point on the same circle, the two lines will always meet at a right angle? 

By the end of the block, we could answer all these questions, and others, with nothing more at our disposal than what we got from Euclid: five postulates, a few common notions, rigorous logical thinking, and an eagerness to question the obvious.

--Mira Bernstein