Euclid and The Mathematician’s Process

Last year, Dr. Seeley of the Boethius Institute connected me with Rosary College, a start-up junior college looking for an instructor to teach Euclidian geometry. Unsure how to teach Euclid’s Elements in a way that was authentic to both the classical liberal arts curriculum and my experience as a mathematician, I looked through the library in the math department at Colorado State University, my alma mater. To my surprise, I found a text titled Geometry: Euclid and Beyond authored by Robin Hartshorne. This text provided me with both insight into how fellow mathematicians think about Euclid in the modern day, and sympathy for how intimidating Euclid’s Elements might be to students reading them for the first time.

Robin Hartshorne is a professor emeritus at the University of California, Berkley, and a highly influential mathematician. His 1977 text Algebraic Geometry was one of the first sources that made algebraic geometry – a highly technical and evolving field in modern mathematics born in the 20th century out of the works of Diophantine – “accessible” to graduate students. Yet it was still notoriously difficult, and especially intimidating to me as a first year graduate student. At the time, it seemed just a series of complicated nonsense: an intricate definition, a proposition, another definition, several more definitions, and a long list of proofs I could not follow until I really understood the definitions. Finally, in my third year, I took an algebraic geometry course taught by my advisor. His carefully selected exercises allowed me and my classmates to interact with the content in a meaningful way. Most importantly, these examples were fun to work on together. In a playful spirit, we made progress reading and writing arguments.
That same year, I started to make progress in my research for the first time. Research in mathematics – that is, proving novel theorems – requires a deep understanding of the definitions at hand. Most of the time, building this foundation is a tedious rhythm of reading and taking notes. However, when I went back to basics and played with lower-level examples similar to those my advisor selected for his course, I finally learned the definitions that were previously too complicated to understand.
Such play is not only an essential aspect of modern mathematics, but also of Euclid’s development of the Elements. Starting with five fundamental axioms, Euclid would have played with points, lines, and plane figures until he recognized certain provable truths. Then, using only the axioms and previously proven results, he could construct arguments demonstrating such truth beyond refute. Through the act of reading these arguments, a modern audience might gain an appreciation for Euclid’s use of logic and the axiomatic method but miss the playful discovery of such truths. Euclid’s long list of definitions and propositions may even seem like a series of complicated nonsense like Algebraic Geometry was to me as a graduate student. As I built my syllabus for MATH 101: Euclidean Geometry, I wanted to make the material come to life just as my advisor had done for me.
In the late 1980s, Hartshorne found himself in a similar position – teaching classical geometries as a modern mathematician. In Geometry: Euclid and Beyond, Hartshorne “takes Euclid’s Elements as the starting point for a study of geometry from a modern mathematical perspective,” by walking the reader through Euclid’s most significant propositions and connecting them to newer problems (Hartshorne, 2000, p.1). Although this text ultimately covers material suited for a senior level mathematics major, the first chapter, meant to be read concurrently with the Elements, is filled with accessible discovery-based exercises designed to put any reader in Euclid’s shoes.
Early in the course, I had my students write conjectures based on their explorations in Geogebra, an online calculator that allows for compass and ruler constructions. For example, on a discussion board problem in the first few weeks, I had students answer true and false questions about polygons circumscribed by circles, and write their own conjectures. They had not yet read Books III and IV of Elements which deal with these relationships, but through collaboration and play, they found their own reasoning for why all triangles can be circumscribed by a circle, and used results from Books I and II to justify their reasoning. They also correctly conjectured relationships between angles in a quadrilateral circumscribed by a circle (namely that opposite angles are supplementary). These results were significantly more meaningful when we later discussed their proofs because my students knew them first hand.
Along these lines, one of Hartshorne’s classical geometry students argued that in order to have a truly authentic experience, it would be better to have no accompanying textbook at all, not even the Elements (Hartshorne, 2000 p.13). Students would live the axiomatic method as Euclid did and discover geometry from scratch. Of course, one pitfall of this idea is that a semester is only 15 weeks long. Exploration takes time, but constructing an argument after the play is done takes even more time. Under an exploration-only model, it would be impossible to cover all of the material in the Elements. Additionally, the act of reading written arguments is an important part of the learning process. How could one write a proof having never read one?
With this in mind, I had my students write their own proofs to problems not found in Euclid’s Elements after reading several similar arguments. For example, the third week, I assigned the following homework problem.

As to not spoil the problem solving process for the interested reader, I refrain from including a solution here. However, using known results, and other strategies, my students constructed excellent arguments.
It is worth mentioning that play is sometimes messy. My students did not always find clear-cut solutions like they did for the above exercise. For example, halfway through the semester, I assigned the exercise below.

While students creatively worked toward a proof, the process for this problem was significantly more open-ended and involved; I could not expect routine answers. For certain problems, all of my students observed and communicated the same ideas. Other times, I received as many different answers as I had students because each individual chose to explore a different angle. In class, having open-ended conversations sometimes meant that I stumbled through the material too. Yes, this process was sometimes frustrating, but it was also exactly the point, and my students rose to the challenge. Because this course included rote activities directly from Euclid’s Elements, structured written homework assignments, and a playful activities adapted from both Geometry: Euclid and Beyond and my own undergraduate geometry curriculum (written by Nathaniel Miller1), my students got to live the entirety of the mathematician’s process. From reading the definitions and proofs to playing with lower-level examples, they persevered through the messiness. Guided by the structure of Euclid’s Elements, my students ultimately communicated their arguments and found success writing proofs. Teaching this course reminded me that people are inherently good at recognizing truth when given the opportunity to reckon with it.
References
Hartshorne, R. (2000). Geometry: Euclid and Beyond. Undergraduate Texts in Mathematics. Springer, New York, NY. https://doi.org/10.1007/978-0-387-22676-7
- Nathaniel Miller is a professor at the University of Northern Colorado and has several papers about teaching proof-writing to pre-service secondary teachers. In his dissertation, A DIAGRAMMATIC FORMAL SYSTEM FOR EUCLIDEAN GEOMETRY, he formalizes Euclid’s diagram proofs using computer programming. ↩︎
















I fell in love with liberal education during the pandemic. I was teaching first grade at a poor school that had only recently decided to renew its curriculum and embrace the liberal arts. But through all of the training sessions, retreats, and curriculum writing, I continually encountered the same frustration: All of this would be so useful if only my students could read!
repeated lessons comes to mind. Each year, my own five- and six-year-old students spent our much-anticipated Dinosaur Day studying fossils, biological adaptation, deductive reasoning, and earning “doctorates” in paleontology. The crowning moment of the event was when, donning their handmade T-Rex hats, they “became” dinosaurs. With elbows tucked to their sides and secured with soft, oversized yarn, the young T-Rexes were simply asked to extend two fingers from their closed fists and then go about the rest of the day. It wasn’t a particularly interesting itinerary for human students – eating a snack, putting on a backpack, opening the door, drawing a picture, free play with friends – but the dinosaurs alternated between laughter, frustration, and exhaustion as they discovered the evolutionary disadvantages of a T-Rex’s short arms and few, non-opposable digits. Some students resorted to holding pencils in their mouths. A pair of boys playing Tic-Tac-Toe with sidewalk chalk repeatedly lost balance as their truncated arms failed to reach the ground, even from a kneeling position. Catching a fall was hard, getting up was even worse. Duck-Duck-Goose had to be adapted. Cretaceous chaos reigned.
education is the ideal time to bring ancient and modern understandings of leisure together by making learning truly delightful. Examples are truly endless. Preschoolers may encounter evaporation firsthand as they “paint” with water on a hot sidewalk and watches their art disappear before their very eyes; kindergarteners may dissolve into fits of giggles as they learn to manipulate words by changing the first letter of “cat” to an “f”; first graders may be confronted with the difficulty of making a teepee stand on its own as they explore the difficult implications of a nomadic lifestyle; second graders alternating between laughter, frustration, and gratitude for the human form as they go about their day with their elbows tucked to their sides and only two fingers extended from their closed fists, emulating the evolutionary disadvantages of a T-Rex’s short arms and few, non-opposable digits.
As we mentioned in our last bulletin, a group of classical educators and scholars has launched Principia, a peer-reviewed journal dedicated to advancing scholarship on classical education. As Brian Williams, General Editor, reports in his article introducing the journal, forty years of education renewal has spawned a growing body of scholarly research and writing. Principia provides a “venue for robust and vigorous dialogue and debate about classical education” that “will make substantive and positive impacts on the practical implementation of classical education in schools and homes around the world.” Williams' own summative expression and description of classical education provides a strong beginning.
To understand the minds of any one of those authors takes a great deal of time and effort. The general mastery of words and quantities given by the traditional liberal arts makes understanding the authors possible but not easy. It takes docility and receptiveness, which are dangerously given to the sophistical and brilliant. Plato’s dialogue,
What do the liberal arts produce in those who become proficient? In answer, Schlect is guided by the claim of
I have spent my twenty-five years as a teacher further and further refining my purpose, to its present obsessive focus. My animating ambition, the one I live, sleep, and breathe, is to help people learn to love – to love – great literature.
My own eyes were first opened when I read Victor Hugo’s novel Ninety-Three aloud to my little group of homeschooled students so many years ago. We were wholly absorbed and focused. We were riveted by the plot. We gasped in chorus at the sudden twists and sighed over sentimental passages. We discussed our reactions as we read, and we worked to decipher Hugo’s message. The experience was as much a life-altering one for me as it was for them.