# Insights from Euclid’s ‘Elements’

Presumably, I was taught geometry as a child. I do not remember.

Recently, I’d made my way halfway through a complex analysis textbook, only to find another which seemed more suitable and engaging. Unfortunately, the exposition was geometric. I knew something was wrong – I knew something had to change – when, asked to prove the similarity of two triangles, I got stuck on *page 7*.

I’d been reluctant to tackle geometry, and when authors reasoned geometrically, I’d find another way to understand. Can you blame me, when most geometric proofs look like *this*?

Distasteful. In a graph with vertices, you’d need to commit things to memory (e.g. triangles, angles) in order to read the proof without continually glancing at the illustration. In a normal equation with variables, it’s .

Sometimes, we just need a little beauty to fall in love.

Welcome to Oliver Byrne’s rendition of Euclid’s *Elements*, digitized and freely available online.

# Elements

Propoſitions are placed before a ſtudent, who though having a ſufficient underſtanding, is told juſt as much about them on entering at the very threſhold of the ſcience, as gives him a prepoſſeſſion moſt unfavourable to his future ſtudy of this delightful ſubject; or “the formalities and paraphernalia of rigour are ſo oſtentatiouſly put forward, as almoſt to hide the reality. Endleſs and perplexing repetitions, which do not confer greater exactitude on the reaſoning, render the demonſtrations involved and obſcure, and conceal from the view of the ſtudent the conſecution of evidence.”

Thus an averſion is created in the mind of the pupil, and a ſubject fo calculated to improve the reaſoning powers, and give the habit of cloſe thinking, is degraded by a dry and rigid courſe of inſtruction into an unintereſting exerciſe of the memory.

## Equality and Similarity

Old mathematical writing lacks modern precision. Euclid says that two triangles are “equal”, without specifying what that means. It means that one triangle can be turned into another via an isometric transformation. That is, if you rotate, translate, and/or reflect triangle , it turns into triangle .

Similarity is a bit more lenient, because you can rescale as well:

My favorite characterization of similarities is:

As a map , a similarity of ratio takes the form , where is an orthogonal matrix and is a translation vector.

The only difference compared to congruence is that congruence requires .

## Synthetic/analytic

I find it strange that Euclid got so far by axiomatizing informal notions without any grounding in formal set theory (e.g. ZFC). I mean, you’d get *absolutely blown away* if you tried to pull these shenanigans in topology. But apparently, Euclidean geometry is sufficiently well-behaved that it basically matches our intuitions without much qualification?

## Area invariance

This says: suppose you draw two parallel lines, and then make a dash of length 2 on each line. Then, make another dash of length 2 on the upper line. The two parallelograms so defined have equal area. This is clarified in the next theorem.

If you take one of the dashes and slide it around on the upper parallel line, the resultant parallelograms all have the same area. I thought this was cool.

## Notes

There aren’t any exercises; instead, I tried to first prove the theorems myself.

Book III treats circles, with wonderful results on arcs and their relation to angles. I search for a snappy example, a gem of an insight to share, but my words fail me. It’s just good.

I read books I, III, IV, and skimmed II. Not all books of the Elements are about plane geometry; some are archaic introductions to number theory, for example. Those looking to learn number theory would do much better with the gorgeous *Illustrated Theory of Numbers*.

# Forward

*Elements* is a *tour de force*. Theorem, theorem, problem, theorem, all laid out in confident succession. It was not always known that from simple rules you could rigorously deduce beautiful facts. It was *not always known* that you could start with so little, and end with so much.

Before I found this resource, I’d checked out several geometry books, all of which seemed bad. To salt the wound, many books were explicitly aimed at middle-schoolers. This… was a bit of a blow.

However, it doesn’t matter when something is normally presented. If you don’t know something, you don’t know it, and there’s nothing wrong with learning it. Even if you feel late. Even if you feel sheepish.

## Against completionism

I’m glad I didn’t read all of the books, even though they’re beautiful. I’d picked up a bad “completionist” habit – if I don’t read the whole book, obviously I haven’t completed it, and obviously I’m not allowed to make a post about it. Of course.

But I’m trying to pick up useful skills, to expand the types of qualitative reasoning available to me, to get the most benefit per unit of reading. I stopped because I have what I need for my complex analysis book.

## Read around

Reading relevant Wikipedia pages / other textbooks helps me cross-examine my knowledge. It also helps connect the new knowledge to existing knowledge. For example, I now have a wonderfully enriched understanding of the geometric mean.

Over time, as you expand and read more books, you’ll find yourself reading faster and faster, understanding more and more subsections. I don’t recommend learning new areas via Wikipedia, but it’s good reinforcement.

## Re-deriving dependencies as a habit

Ever since I learned real analysis, I reflexively reprove all new elementary mathematics whenever I use it. For real analysis, that meant *continually reproving* e.g. whenever I used that property in a proof. Did it feel silly and tedious? A bit, yes.

But with (this) tedium comes power. I can now regenerate a formal foundation for the real numbers from the Peano axioms, proving the necessary properties about the natural numbers, then the integers, then the rationals, and then the reals, and then complex numbers too. (But please, no quaternions!)

With this habit, you continually ask yourself, “how do I know this?”. I think this is a useful subskill of Actually Thinking.

## Commemoration

In college, I taught myself a bit of Japanese. Through a combination of spaced repetition software and memory palaces, and over the course of three months, I learned to read the 2,136 standard use characters. After those three months, I proudly displayed this poster on my wall:

I look forward to another beautiful poster.

As the ſenſes of ſight and hearing can be ſo forcibly and inſtantaneously addreſſed alike with one thouſand as with one,

the millionmight be taught geometry and other branches of mathematics with great eaſe, this would advance the purpoſe of education more than any thing thatmightbe named, for it would teach the people how to think, and not what to think; it is in this particular the great error of education originates.

Curated. I found a lot to be interested in here.

First, I’m just grateful for being introduced to Byrne’s Elements. I think “how to use visuals to improve pedagogy” is a practically important question. I haven’t yet worked through it myself to have a clear sense of “does the improved pedagogy

work(for me)?”, but even at a glance, it looks like a treasure trove of artistry that is worth exploring and learning from.I found reading through Turntrout’s learning process also helpful, to give me some insight into a cohesive worldview that includes “how to learn, how to be rigorous about it, and how to be finding beauty in the world along the way.”

I… do sure find it annoying that the letter S is for reason a weird ſ, which doesn’t seem like the sort of thing it was that important to preserve at the expense of clarity on the new site, but that part isn’t Turntrout’s fault (I’d be interested if there’s a more compelling reason than “that’s just how Byrne did it at the time and we’re faithfully recreating it)

Nope, that’s the reason. Nicholas Rougeaux explains:

I also find the long S super annoying, but it at least should be pretty easy to make a browser plugin or something to replace ‘ſ’ with ‘s’ everywhere.

I found “Word Replacer II” for Chrome works perfectly. You can limit it to only be active on specific sites. And then just specify that you want to replace “ſ” by “s”.

This is glorious. On the flip side of the coin, I struggle with outrage that we had copped to the problem of presenting information and basically had it licked in the middle of the 19th century, and then apparently systematically purged such knowledge during the 20th. For example, there’s this interesting piece about Emma Willard, who drew gorgeous visuals providing perspective to history. She began in ~1837. Good use of images seems only now to be undergoing a renaissance, and that owing to the availability of computer graphics more than anything else.

What the devil happened erstwhile?

Inaccessible beauty makes many feel ugly.

I don’t see why that would explain these deficiencies, even if true. I imagine the answer’s more along the lines of “lack of incentives for textbook writers and publishers, as determined by the scholastic purchasing committees”.

It seems worth noting here that

Elementsisn’t entirely rigorous. I don’t remember many details about that, but https://en.wikipedia.org/wiki/Euclid’s_Elements#Criticism has some. I do remember this bit (or at least something very similar):Because when we studied

Elementsat math camp when I was ~16 I remember this standing out to me. I think we were going through it as a group, and the instructor asked if anyone could prove each theorem in turn before giving us the answer if we couldn’t. Unsurprisingly, no one could prove this one. When he showed us how it was done I felt a bit… cheated? because no one had told us we could do that. But I didn’t do anything with this feeling, I think I just assumed that everything was fine, I should have been able to work out that we could do that.Later I learned that no, it was in fact cheating and we could not do that.

Yeah, and sometimes his case analysis was a little less than exhaustive. I think Byrne fixed that, though.

On the bus from NYC to Boston for EAGxBoston 2019 I chanced to sit next to a topology professor. I don’t have any higher math background, but mentioning that I’d recently read the first few books of the Elements opened the door to a long and interesting conversation. I was amazed that something written two thousand years prior compared so favorably with my own 8th grade geometry experience, which despite having a cool teacher managed to teach me only the rudiments of geometry, and nothing substantial about proofs or theorems.) Minus the annoying long s, I’d gift Byrne’s illustrated Elements to any smart kid in a heartbeat—it’s surprisingly cheap on Amazon.

While this is great, I wonder if something is lost. Specifically I’m remembering when I learned geometry and the class was simply to work through

Elementsand prove each theorem. This happened when I was in 8th grade (US), and it was a frustrating and similarly beautiful and powerful experience. At the time nothing had quite honed my skills for reasoning about abstractions, loading models into my head, and working with those models like geometry did. Without having spent a semester fighting to earn the right to say “QED”, I don’t know if I would have made as good of progress as I did on becoming a programmer and a mathematician by virtue of having had that earlier experience where I learned the basic methods those callings require.Why is that missing here?

I guess my concern is that this makes it too easy in a way that rips out part of the difficulty that encourages learning. Learning geometry the way I did was helpful specifically because I had to go through the process of taking dense and difficult to reason about words and build up my models to understand them. The lack of assistive pedagogy like the kind here forced me to work out something like it for myself inside my head.

This is not to make a general argument against assistive devices; often they are helpful if what matters is getting something done. But I didn’t work through Elements to solve geometry problems where the solution had a positive impact on my life, but to learn a thinking process using geometry.

I also don’t mean to make an argument that no one should get to have some pretty pictures that help them learn. I’m sure the use of pictures like these helps many folks learn geometry who otherwise wouldn’t or wouldn’t learn it as well. I only mean to say that I think we give up something of value by making geometry easier to learn.

(FWIW I’ve made the same argument in the context of training programmers, preferring that they have to learn to work with assembly, FORTRAN, and C because the difficulty forced me to understand a lot of useful details that help me even when working in higher level languages that can’t be fully appreciated if you are, for example, trying to simulate the experience of managing memory or creating loops with JUMPIF in a language where it’s not necessary. Not exactly the same as what’s going on here but of the same type.)

FWIW as someone who learned Python first, was exposed to C but didn’t really understand it, and then only really learned C later (by playing around with / hacking on the OpenBSD operating system and also working on a project that used C++ with mainly only features from C), I’ve always found the following argument quite suspect with respect to programming:

It’s undoubtedly true that I see

somedifference before & after “grokking” low-level programming in terms of being able to better debug issues with low-level networking code and maybe having a better intuition for performance. Now in fairness, most of my programming work hasn’t beensuperperformance focused. But, at the same time, I found learning lower level programming much easier after having already internalized decent programming practices (like writing tests and structuring my code) which allowed me to focus on the unique difficulties of C and assembly. Furthermore, I was much more motivated to understand C & assembly because I felt like I had a reason to do so rather than just doing it because (no snark intended) old-school programmers had to do so when they were learning.For these reasons, I definitely would

notrecommend someone who wants to learn programming start with C & assembly unless they have a goal that requires it. This just seems to me like going to hard mode directly primarily because that’s what people used to have to do. As I said above, I’m fairly convinced that the lessons you learn from doing so are things you can pick up later and not so necessary that you’ll be handicapped without them.(Of course, all of this is predicated on the assumption that I have the skills you claim one learns from learning these languages, which I admit you have no reason to believe purely based on my comments / posts.)

The main difference is that the original is harder to follow because of shortcomings of the human short-term memory system. You’re still thinking about exactly the same abstract concepts. The potential danger is the lack of exercises, I suspect – that’s where

a)first proving things yourself andb)the rederivation habit, come in handy.I also suspect math students have ample opportunities to crunch through dense thickets of words… why oh why do I suddenly find myself thinking of Munkres’

Topologyand Dummit & Foote’sAbstract Algebra?A bit pedantic but still worth pointing out—unless I’m very mistaken or Euclid uses the term very differently from modern mathematicians, this bit:

is not the correct definition of ‘affine transformation’. If two triangles are similar, there will exist an affine transform between them, but the reverse is not necessarily true. The category of affine transformations is broader than you think—an affine transformation allows you to skew things as well (e.g. convert a square into a parallelogram), or to scale things differently along different axes (e.g. convert a square into a rectangle). The ratio between two distances in an affine transform must remain the same only if those distances are parallel. In point of fact, I believe affine transformations can convert any triangle into any other triangle.

(Checks online: Wolfram says yes).

Yes, this is correct; this phrasing was misleading. IMO, the most succinct formally correct characterization of similarities is:

The only difference compared to congruence is that congruence requires r=1.