What's so special of Euclid's Elements? Is it even worth paying attention to in the modern world...

What's so special of Euclid's Elements? Is it even worth paying attention to in the modern world? Could a modern person learn from his work, or would it be better to approach mathematics from another area?

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Understanding Euclidean Geometry is very important for mathematicians meaning to master modern geometry. It all comes from there.

Read modern books about it. All of his axioms, theorems and proofs have been "translated" to the modern language of set theory. Reading the original or pre-set theory versions is useless though. All pre-set theory mathematics is outdated and arguably wrong, as it did not base itself on set theory, the only correct foundation.

It is the single greatest book on Math, it is incredibly rigorous, so I am told

>Read modern books about it. All of his axioms, theorems and proofs have been "translated" to the modern language of set theory.
Any particular recommendations?
Preferably ones that fill in the gaps from Euclid, e.g. Pasch's axiom

I own a copy, it is a solid and rigourous work. There is a reasonably cheap copy by green lion print, which includes much of the greek.

It is incredibly useful for personal study of geometry, with many implications for architecture from within the treatises

Sorry, no recommendations. I didn't read a book like this, I took a class about it and the professor had a really old book about it that he taught us from but I never even saw its title.

Just google modern euclidean geometry and first check the first few pages to see if they define points, lines, rays, planes, etc. in terms of set theory. If not then put it in the thrash because geometry without set theory won't allow you to jump into topology.

How does one read it? What sort of process should I be engaged in to get as much out of it?

You may not understand what I'm asking it so let me provide an example: when reading philosophy I constantly take notes and ask myself what the arguments are. This is my process of reading. How would I approach the Elements?

I do it slightly differently, but when I was reading, I would work the proofs through on a pad of paper as I read the work to feel how he had felt it.

Euclid has no concept of algebra, so he often defines algebraic notions within terms of geometry. He had an extraordinary ability to condense geometry into core parts and expand on it as in subsequent chapters.

Much of the work you would have covered in primary school, but there is still plenty of geometric thought.

Okay, thank you very much friend for the responses. I greatly appreciate it.

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Starting reading this in the 9th grade. It was my introduction to proof-based rigorous mathematics. If you're only a beginner at math, I would recommend giving this a read. If you've had training in formal math, don't bother

I'm actually reading it right now and drawing a lot of the proofs with a compass and straight-edge just for fun. It's basically the precursor to all modern math in its attempt to create a rigorous system of axioms and proofs. It's also a good intro to Euclidean geometry and was used as a textbook in schools until the 19th century.

That said, it's still primitive compared to modern geometry which starts from much more basic axioms and is much more rigorous. So I'd say it's useful for historical context, recreation, or to get a normie-level understanding of the math, but modern textbooks are going to be more in-depth.

Math grad here.

The reason why Euclid (by which I simply mean the text, 'The Elements') is important is because mathematical truth has not changed since the time of its writing, nor is it likely to change in the foreseeable future.

Furthermore, the basic style and considerations entailed in mathematical writing have not /ontologically changed/ since Euclid.

What is meant, here, is that Euclid demonstrates how to argue about abstract entities. Euclid shows us, again and again, what a mathematical proof looks like, how to start from premises and then, /making certain other not very controversial assumptions/, yet these sticking points are more closely exaimed in modern mathematics, arrive at a necessary conclusion. Proofs and stamp collecting are not the end-all be-all of mathematics, but they are of course a central activity of mathematics because they, along with some qualified statements about sensory input that I feel necessary to add here but which I won't elaborate just now, are where /knowledge/ comes from, mathematical knowledge anyway.

Furthermore, a cultural thing: the way that Euclid /looks/ on the page is the exact-same god-damned thing that we still do today. Prose, set up some notation, draw the occasional diagram, and develop an elaborate argument of whatever kind. At left is a random sample of Mochizuiki, at right an old Euclid fragment. Just stop thinking about the content of each for a moment, and just /gloss over the pages/: they convey information in similar ways. Writing, odd abstract symbols, and a diagram now and then to give the reader a visual context.

It also sufficed as a textbook of the west until recently, which in this argument is not so much a knock on the slow development of math, as it is a reinforcement of my original point, that mathematical truth does not change with the years. That is exactly what has allowed the text to continue to be used in the capacity of an instruction manual.

ALL OF THIS BEING SAID, Euclid is still the stuff of the /history of math/, so just resign yourself just a bit to doing the meme-project that you actually propose to do If you're reading it properly, yes, what you will be doing will in fact be a proper mathematical activity, just of a low order, in one sense. Reading a Very Very Important classic text of western civilization is never a bad thing, in my view, however.

Ideally, actually draw every prop for yourself, and think the logical/proof process through for yourself until you get to a point where you either understand or at least understand what is being driven at. Take your own notes, come up with your own criticisms, and be willing to "gloss over" fine points in order to finish a proof to its conclusion. Then catalog what you've learned so far and keep going. You may wish to get a little compass set for this.

>reading The Elements is a good intro to Euclidian Geometry

get this man a junior G-man badge, I am over fucking whelmed :^)

What's your point? It's not like many people recommend Principia Mathematica as an introduction Newtonian mechanics.

Meant to attach this image here.

I thought a reasonable push-back might be given along these lines, so I have two points: The Elements is so good as a textbook that people still undertake to read it to this very day, even if partly as a cute historical exercise. The examples still work, the thing still makes sense etc in terms of how math is actually done today. To your point, I cannot recall a single Veeky Forums thread where anyone ever seriously proposed to read the Principia. The Elements still succeeds for us to exposit its subject, where other texts become misty unread memes in time. And this exactly because math doesn't change, as I've insisted.

The other point is that even if we agree that the Principia is "a bad texbook" or similar statement by today's standards, it is still true that from the world historical point of view, the text "introduced" classical mechanics as we understand it. My mockery of your statement is that when a text is synonymous with a subject, regardless of whether people actually read the text, it sounds ridiculous to point out that the text is "a good intro to the subject". You might as well have written that the Bible is a good introduction to Christianity. The point here is not how right or wrong the sentiment is (in terms of a textbook that we'd use), but how silly and ridiculous to point out something seemingly obvious, in natural language.

I loved The Elements. Great intro to geometrical proofs. I went on from there to read a bit of Archimedes works. That is QUITE a step up. Going from Euclid to Archimedes is like going from Elementary school to college.

Wildberger, who has a hardon for geometry, recommended a dover publication in his youtube vid math history 1a i think. Cant recall the author

users.humboldt.edu/flashman/spiralsprop10.htm
This is proposition 20 from On Spirals.

Also, some of Archimedes' works like On the Equilibrium of Planes are engineering works that helped Archimedes build giant death machines to siege castles.

This book could impress the heck out of somebody that barely knows what a circle is ... And, that's definitely not in this millenia. About 2,500 years ago ... In 2013, anybody that reads this in for a big surprise. The fact that, arithmetic notation such as a^2 didn't even exist made my head spin when I was reading the book. The only thing that was cool about this book was the sentence, "things that are equal to each other are equal to one another." And, the only reason this was cool was since I figured out where the movie "Lincoln" got this sentence. This sentence had a good arithmetic meaning 2,500 years ago. In 2013, we say, If A=B and B=C, then A=C. Done.
Don't get me wrong. We owe EVERYTHING to Euclid. I have a Ph.D. and I still can't say that I have 1/1000000th the arithmetic and geometric genius Euclid had. However, this book is simply 2 millenia too old for me. May be, somebody should have a translation of it with the modern arithmetic symbolism added. This, trying to blindly stick to the "original" is reducing the value of the book by about 1000x.

This post reads like a bad, confused, troll-y amazon review, done by a high school student. Odd, oblique "cute" references to things (a recent movie), hyperbole (1/1000000).

The post seems to want to make some sort of a more precise rhetorical point, but it can't decide what that point might be apart from "Euclid is old!"

copied and pasted from Amazon

lel

Because the Elements is actually a genuinely good intro text to both geometry and the concept of mathematical proof, demonstrated in a highly visual form that - with paper, compass, and ruler - can be directly connected to hands-on experience. Plus, it is a good introduction to the the idea that math isn't about numbers - it's about structures, relationships between things. Rather than manipulating numbers and logical rules about how numbers interact, you're manipulating points, lines, and circles and rules about how they interact. Much of abstract algebra can trace its lineage to investigations of exactly what you could and couldn't do with compass and straightedge.

Euclid's Elements was still used in classrooms up into the 20th century, because it's *actually a good introductory textbook *.


But it is an intro text, so if you're already familiar with this it's really only of historical value.