Insights from Euclid's 'Elements'

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)

[-]TurnTrout5y40

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:

The long s (ſ and ſ italicized) was common in older publications and is used throughout the original book. It can be mistaken for the lowercase f but should be read as s whenever seen. The usage of the long s has fallen out of style but in an effort to faithfully reproduce this book, it was used as well.

[-]Adele Lopez5y30

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.

[-]Thomas Kehrenberg5y40

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".

[-]ryan_b5y160

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?

[-]romeostevensit5y30

Inaccessible beauty makes many feel ugly.

[-]TurnTrout5y40

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".

[-]philh5y120

It seems worth noting here that Elements isn'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):

Later, in the fourth construction, he used superposition (moving the triangles on top of each other) to prove that if two sides and their angles are equal, then they are congruent; during these considerations he uses some properties of superposition, but these properties are not described explicitly in the treatise.

Because when we studied Elements at 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.

[-]TurnTrout5y20

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

[-]Rachel Shu5y40

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.

[-]Gordon Seidoh Worley5y40

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 Elements and 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.

[-]TurnTrout5y80

Why is that missing here?

[-]Gordon Seidoh Worley5y40

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.)

[-][anonymous]5y90

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:

(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.)

It's undoubtedly true that I see some difference 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 been super performance 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 not recommend 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.)

[-]TurnTrout5y50

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 and b) 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' Topology and Dummit & Foote's Abstract Algebra?

[-]aphyer5y20

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:

Two triangles are similar when there exists such an affine transformation (i.e., you can scale as well).

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).

[-]TurnTrout5y40

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

As a map , a similarity of ratio $r$ takes the form $f (x) := r A x + t$ , where $A \in O_{n} (R)$ is an $n \times n$ orthogonal matrix and $t \in R^{n}$ is a translation vector.