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I am teaching a course in Euclidean geometry at the University of South Carolina, and it seemed highly appropriate and interesting to read Euclid himself. (See here for a wonderful, and completely free, translation and guide.)

We are up through Proposition 17 now, and although this is very instructional for both the students and myself, I sense that before too long the novelty will wear off and it will be good to return to a modern treatment. (I will, though, want to say a little bit about his treatment of parallel lines and elliptic and hyperbolic geometry.)

That said, there are some gems. In Book 2, Euclid constructs square roots, and in Book 4 he describes how to draw a regular pentagon (which is surely not obvious). And, the constructions in the first few books are all very interesting. (Of course there is lots of fascinating number theory too, but my course is on geometry.)

What other particularly fascinating tidbits do the Elements contain, which it may be easy to overlook?

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To me, the basic experience of reading book I of Euclid is that he proves a series of dull and obvious propositions, and then all of a sudden Prop. 47, the Pythagorean theorem, comes along, and suddenly it becomes amazing how much he's accomplished with so little to work from. I don't think I've ever read every single theorem from 1 through 46 -- it's like reading the "begats" in Genesis. What is your audience? Is this course remedial? Penrose's book The Road to Reality has some nice modern interpretation of the postulates as empirically testable statements about physical space. –  Ben Crowell Oct 9 '12 at 23:18
    
The course is certainly not remedial. Math majors of mixed background, most of them future high school teachers. In my opinion it's been interesting so far, in the same way that point set topology is interesting (you have to reconsider what's "obvious" strictly in terms of what you've proved so far)... but it's beginning to wear thin. –  Frank Thorne Oct 10 '12 at 20:52
    
Since you seem to have quite a free reign over what you teach, what about letting your students rein a bit? Surely it would help if they are more interested in some particular area of geometry or even specific problems, and often there would be relationships to other areas that you can point out along the way. And if anything is really important you should be able to convince your students without much effort that it is really interesting. =) –  user21820 May 20 at 7:00
    
@BenCrowell About your last comment's sentence: what pages in the book are you thinking of? (It's a big book.) –  rschwieb Oct 2 at 19:26

2 Answers 2

For me, I like Book 1 Prop 35, whichis the first use of "equal" to mean equi-areal rather than congruent. Book 2 Prop 11, where Pythagoas is used geometrically to prove a construction for the golden ratio (later used for the regular pentagon construction) 3:35 again a geometrical use of Pythagoras to prove the "power of a point" 6:3 showing that the angle bisector cuts the base of a triangle in the same ratio as the other sides. 6:19 showing that "similar triangles are to one another in the duplicate ratio of their corresponding sides 6:31 pythagoras generalized and proved using ratios

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If I were to ever incorporate Euclid's elements in a class, it would be through a critical exposition of Euclid's work, not necessarily directly with his books.

The best source I've gotten into for that is Hartshorne's Euclid and beyond. It really helps you see the importance and impact of Euclid's work, without really having to trudge through it.

As I recall, the gems are strewn along the path of good exposition. The book also introduces Hilbert's axioms, which was the next major revolution of the foundations of geometry that students should really hear about.

It never lingers on Euclid's presentation long enough to become boring. It revisits it throughout the book, of course. Really, there is no reason to linger on Euclid after students have gotten the general feel of the material.

Maybe you can take a look at how Hartshorne presents results, and then devise a plan.

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