In the first few weeks of the pandemic, I was tasked with tutoring my younger brother in geometry to help punch up his grades. This was no trifle for someone who had spent so much time in the humanities, but his personality did not mesh well with the teacher’s and remote learning only put further strain on his education. I took a week to catch up to where he was on top of my other obligations before I began. With no hesitation I can say that I understand geometry better now, three years out from that affair, than I did fresh out of high school. My secret weapon? Euclid’s Elements.
The one “modernization” of common editions of this book that has taken place over the last 2,000 years was Byrne’s “coloured” edition, which replaces the lettering used to define angles and sides of shapes with differing tints on the drawings and painted swatches in the text. A Github user by the name of Jemmy Button1 has made use of online hypertext to link each of the relevant propositions to the postulates and vice versa, which clears up any remaining potential for confusion. This is the best edition for my money (in that it’s free), but Byrne only transcribed the first 6 books with this method. A publishing house Kickstarted a complete version in 2019, but its current retail price is far beyond the budget for this blog.2 I used this website to supplement the last 7 books. Some of the content therein, like prime numbers, can be found in other areas of mathematics, but books 11-13 cover solid geometry and cannot be skipped. Instead of presenting information in lengthy paragraphs that often go unread, Euclid sums up all information into pithy sentences that are categorized as axioms, postulates, definitions, common notions, and propositions. Little space, and therefore little time, is wasted in these books.
Outside of a few additional proofs and axioms (that tend to appear in the footnotes anyway), the factual matters of what Euclid presents differ little from modern day geometry. There are more exciting and interesting ways to teach the Pythagorean theorem now, sure, but any one proof is just as valid as the next. Of course, you’d know all this if you’ve ever taken a geometry course. In fact you’d know every part of this if you’ve ever taken a geometry course. Why, then, do publishers charge hundreds of dollars for high school and college textbooks? The simple answer is that greed has corrupted the industry, but they will tell you that generating new problems to avoid cheating justifies these steep costs. On the other hand, technology has made it easier than ever for teachers to come up with their own examples in the classroom. It may be taboo to ask people in that overworked and underpaid occupation to do extra work, but I would argue that the time saved in not bothering with the ephemeral activities that they ask you to download apps for makes it a net positive.
There are some reasonable arguments against this method. The impetus behind the widespread abandonment of Euclid’s Elements was a change in the philosophy of education. Rather than expecting students to learn hard and fast rules by rote, the modern system wants to guide students all the way through the process of solving a problem. Both of these outlooks have advantages and disadvantages: while the more recent changes aid in situations where people are left to their own devices to solve math problems, the advent of the smartphone has leveled the playing field. This is not the first mathematical discipline to see technology change the way it is taught either. Trigonometry classes used to teach proofs for sine, cosine, and tangent. That complex process has been simplified to the press of a button on a calculator. Chances are if you put a slide rule in the hands of anyone under 40, they would not be able to multiply with one, much less solve a trigonometric function.3 The Elements tends to have a different problem with students; a dense text requires active engagement with the content. Teaching vast swathes of teenagers who don’t want to be in school and who only slept 6 hours the night before is hard enough as it is.
Classes in the modern system revolve around active participation as a way to mitigate some of these problems, but even if the structure of schools could accommodate telling students to read a book cover-to-cover, they would hardly be prepared for such an assignment. That’s why I feel it’s important to assert that this method should only be applied remedially.4 Chances are if you’re reading this, you have struggled with geometry in the past. Neither this article in particular nor this blog in general is written with a STEM audience in mind. One of the louder complaints about the current education system is that our culture attempts to push square pegs into round holes. Theorists have long described how diversely human minds will react to teaching styles, but the STEM fields have proven to be most resistant to these developments. They used to cater exclusively to one style when Euclid was the only book used, and now they cater to something different. If we recommended that this book be given to struggling students either as an alternative to regular coursework or instead of attending summer school. This avoids incurring the wrath of teachers who want to stick to their current curriculum, while also benefiting those most in need of a change.
Or Slyusarev Sergey. The reason I’ve linked it like this is that it’s open source and easy to translate. Those who wish for conventional designations could easily swap out the pictures as well, meaning that both tints and letters could be used in the same classroom on the same page of the same textbook.
If I knew how to get in touch with both parties, I would beg the publishing house to permit open source use of their diagrams in digital editions while maintaining their copyright on the published book, because they could really serve a higher purpose. If I had the money I’d buy them out and put them in the public domain myself; that’s how much I believe in Euclid’s efficacy.
This statement, dear reader, includes myself. Truth be told I had to look up what people used to do trigonometry before the invention of the calculator for this part of the essay.
"Remedial math" tends to be the punchline of a joke about euphemisms, but if we want to raise the intelligence of the average American, we need to treat those who fall through the cracks of a traditional education path with more respect.