The Elements of Geometry
Some time ago, I wrote about Alexandria, the most important city in history, briefly discussing the lives of just 17 of the men and women that made it so. Prime to that list, both in sequence and in importance, was Euclid of Alexandria, a personal hero of mine who I consider to be one of the most inspirational and influential people in all of human history. We know next to nothing about Euclid’s life– we do not know where or when he was born, where or when he died, and extremely little about the time between those events. We know that he lived in Alexandria at roughly the same time as Ptolemy I, circa 300 BCE, and we know that he wrote prolifically about mathematics. Yet, even with so very little information as this, I would strongly argue that Euclid contributed far more to the world than did much more well-known figures like the great historian, Herodotus; or the conquering emperor, Julius Caesar; or even the revolutionary preacher, Jesus of Nazareth. What could Euclid have possibly done that outshines these other, great men? Euclid of Alexandria wrote the Elements.
The Elements is a mathematics textbook. Starting with the utter basics– a page of definitions, five postulates, and five axioms– Euclid proceeded to lay out systematic proofs for the entire foundation of mathematics known at his time for the purpose of more easily instructing future mathematicians in the field. This was a first of its kind. There had certainly been educational texts before the Elements, but these tended to be more like workbooks than textbooks. They listed out practical problems for students to solve, but did very little to show how or why those problems could be solved. Other works detailing mathematical proofs had certainly been written, as well, but these were generally specific to a certain problem or class of problems, and were intended for accomplished mathematicians rather than students. Euclid’s Elements was intended to teach a student from the barest of knowledge. They could approach the text knowing nothing at all about mathematics, and use it to learn everything. After its completion, nearly every great mathematician west of India learned their trade from this textbook for over 2000 years, and nearly every development in modern mathematics can be traced back to the rudimentary principles elucidated in the Elements. Euclid’s teaching methodology and his form of rigorous logical proof set the absolute standard for all education that would follow him in math, chemistry, physics, and all of the physical sciences.
I have always loved math, but I have to admit that when I first took geometry, back in High School, I hated it. It was so very different from the arithmetic and algebra with which I had been accustomed. I didn’t understand why I was wasting my time with proofs and postulates. Just give me a formula so I can plug in some numbers! However, the first time I read through Euclid’s Elements, it was like a curtain had been drawn back and I was seeing the sun for the first time. A few scant rays may have slipped through the drapery, before, but now I was viewing Helios in all his glory! As I read, I began to understand so many things that I had simply tried to memorize, before. It was so clear, and so brilliant! Why hadn’t math been taught like this from the very start? It is far better to teach a student to comprehend a subject than to memorize a subject.
The Elements was so incredibly mind-blowing and influential to me, that I got a tattoo of my favorite proposition from the text (Book 1, prop 47; now known as the Pythagorean Theorem). The tattoo is on my chest, over my heart, complete with a diagram and written in the original Greek– a language I began learning largely to be able to read the Elements as it had been penned.
I am a teacher, at heart. In High School, I used to volunteer my time to tutor at-risk youth in math, English, history, and computers. For the past six years, I have spent my evenings teaching Brazilian Jiu-Jitsu to children and adults, rearranging the schedule of my primary occupation as a computer programmer around my classes. Ask me a question about any random topic that I’ve studied, and I’ll chew your ear off with all the details for as long as you are willing to listen. Recently, I came to the realization that I am not happy as a programmer, sitting behind a desk, barely interacting with other people, constantly fixing bugs caused by other bug fixes. I loved the problem-solving, but it felt empty without the ability to really communicate why it had been a problem and how it needed to be solved. I want to improve people, not computers. So, I have switched tack, after nearly a decade in my profession. I’m going back to school, completing a degree in Mathematics, and hoping to find a job teaching algebra and geometry and trigonometry and calculus.
To that end, I have started a little project: an ode to my hero, Euclid. I’m working on creating a Geometry curriculum, aimed at High Schoolers, which will utilize an adaptation of Euclid’s Elements as its primary textbook while adequately covering all of the Common Core standards for the course. In addition to the textbook, I’m planning on creating a digital experience to help illustrate the lessons– perhaps in Wolfram’s CDF format, or utilizing the Geometer’s Sketchpad. I may preview some of that work on Boxing Pythagoras, from time to time; and if I do, I would certainly appreciate any and all feedback that I can get. This is an exciting, new time for me, and I am very happy to share it with all of my friends and readers.