# Proper axiomatization of Euclidean Geometry that Euclid would approve of

It is relatively common knowledge that Euclid's axiomatization is not sufficient to prove all the things that Euclid wants to, and that there are other axiomatizations out there that strengthen Euclid's in order to make Euclidean geometry fully rigorous.

So now, imagine Euclid comes back from the dead and looks around at what the world has done with his geometry. There are many examples of texts that treat Euclidean geometry, but in some sense they are all different from what Euclid's Elements were - a theorem, a construction and a proof. It seems that the overwhelming majority of Euclidean geometry texts are Euclidean in the sense of the parallel postulate, but not Euclidean in the manner of exposition. In a lot of ways, analytic geometry has replaced the old straight-edge & compass method, and not without good reason, but I can't help but feel that something is lost by ditching this method, if nothing but the hands-on approach.

So the question is this: Is there a text in Euclidean geometry which has a sufficient axiomatization to prove (and shows to the reader) all the propositions in The Elements, and is written in the same style?

For clarity, I am seeking a text which captures the two things about Euclid that make him stand out - the axiomatic approach as well as the constructive straightedge/compass method. I am not sure if a contemporary text has been written that uses solely these tools, but if one existed, that would be the most ideal.

• Of course, the book he is looking for is Hartshorne's Geometry. – Marius Kempe Apr 12 '15 at 20:59
• This book is a step in the right direction for sure. I'm going to keep looking but I appreciate your answer. – Alfred Yerger Apr 12 '15 at 23:55
• You're welcome! – Marius Kempe Apr 13 '15 at 0:06

## 2 Answers

I am not sure if you would agree that this is in the same "style" as Euclid's Elements -- style is something of an ill-defined term -- but I would nominate Hilbert's Foundations of Geometry (or, for purists, Grundlagen der Geometrie). It is definitely "synthetic" (as opposed to "analytic") and unequivocally rigorous. On the other hand, while it definitely is sufficiently powerful to prove all of the theorems in Euclid, it does not attempt to do so.

• This is also very good. It definitely captures the axiomatic component of Euclid - it's no surprise it's famous. – Alfred Yerger Apr 17 '15 at 0:34
• I'm going to award the bounty to you for this problem, as it's definitely the best answer so far, and I'd like that to be recognized, but going to leave the question open because I want others to see the question and continue to contribute. – Alfred Yerger Apr 19 '15 at 5:29

Though it is much more inclusive than Euclid, I have a fondness for Coxeter's "Introduction to Geometry."

It is definitely not elementary.

• The approach is synthetic rather than analytic? – user89 Apr 19 '15 at 3:46
• As I recall, it does both, depending on the subject matter. – marty cohen Apr 19 '15 at 5:32