How to best read Euclid's Elements

[jack476]

I checked out a very nice copy of Euclid's Elements from my university library containing unabridged translations of all 13 books. Nice big book, one proof per page, lots of diagrams. It's been interesting so far to flip through and even work a couple of the proofs myself, but now I want to really get into it.

On the other hand, it doesn't seem like a normal textbook where you read through it a few times, take notes, and solve a few problems, and it certainly doesn't seem like an ordinary book one just reads cover to cover.

How do I most productively tackle this thing?

 

[jedishrfu]

The short answer is to get a toga and imagine you're a Greek student studying in Alexandria at the feet of Euclid.

http://en.wikipedia.org/wiki/Euclid

The best you can do now though is to get a modern geometry book and compare and contrast the proofs.

 

[micromass]

Congratulations for wanting to start Euclid. You are going to read a book which literally shaped the mathematical world. We would be far different (and far less advanced) if it weren't for Euclid's book. The book practically invented the theorem-proof-axiom style and it hasn't changed since. It was also the guide into mathematics for many past mathematicians, and it was a standard textbook in schools for centuries.

That said, math has definitely evolved a lot since Euclid. We have found many mistakes in Euclid, many incomplete proofs, and many things which are just easier when we do it the modern way. And then there's the problem of age. Like any thousands year old book, it's outdated. It gave answers to questions that are not asked anymore (likely because it's solved). So let me try to guide you a bit:

First it is important to get some grasp on the historic significance of Euclid. You should try to get a feel for the time in which Euclid lived, for the problems that they were facing, for the many (partial) solutions they had. For this, I recommend the first volume of this comprehensive text: https://www.amazon.com/dp/0195061357/?tag=pfamazon01-20 You only really need to read a few chapters, until you've covered the Greeks. This will help you appreciate Euclid significantly more. You will understand why some sentences in Euclid are so awkward (like the parallel postulate), why some solutions are what they are, what impact certain proofs had and what questions remained unsolved.

And then you're ready to read Euclid. You probably want some additional book which puts things into a more modern perspective. The best book is Hartshorne: https://www.amazon.com/dp/0387986502/?tag=pfamazon01-20 This book is meant to be read together with Euclid. So I suggest to read Euclid a bit, and then read Hartshorne to see a more modern (and in my opinion: better) treatment of the same materials. You will also cover some nice geometry that's not present in Euclid but could be. And you'll see the solution for several flaws in Euclid.

If you're ready and want some more history of geometry, you could always read the sequel of Kline, but there is also this awesome book: https://www.amazon.com/dp/3642291627/?tag=pfamazon01-20

Lastly, while reading Euclid, I recommend highly to use a computer geometry system to experiment. For this, I recommend geogebra: https://www.geogebra.org This is a free, easy to use, but also very powerful geometry program. I recommend making all constructions of Euclid on geogebra. For example if he says "draw a parellel line", then use the program to draw it. So you can really be interactive!

Please do ask if you want some more information!

 

[jbunniii]

@micromass - thanks for the link to geogebra. I wasn't aware of this software. It's very cool and has a lot more features than I expected!