# Seeking concise review of Elementary Euclidean Geometry

I'm seeing a presentation of Euclidean geometry that isn't hand-holdy. I've looked at some textbooks used in high schools these days, and it's hard to find the axioms and theorems in the midst of all the condescension. I just want something that states the definitions, axioms and basic theorems.

I know what a Riemann-Christoffel tensor is; and with a bit of scraping off the rust, could derive it. But if you asked me to demonstrate some basic theorem in elementary Euclidean geometry, I would be hard pressed to state the essential definitions and axioms upon which the theorem rests.

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Edwin E. Moise Geometry perhaps?

Why not read Euclid's Elements itself? It's still the best book on the subject. Try to read it with the companion book Harthorne's "Euclid and beyond".

Moise is very good too, but doesn't focus on the approach that Euclid himself followed.

mathwonk and bcrowell
@micromass, would you also recommended the famous Russian book, Kiselev, " Planimetry."? My instructor gifted mea copy and I think its a good book, I could not get farther than chapter 2, due to time constraints.

However, I am not sure if it follows Euclid, i am a geometry noob.

@micromass, would you also recommended the famous Russian book, Kiselev, " Planimetry."? My instructor gifted mea copy and I think its a good book, I could not get farther than chapter 2, due to time constraints.

However, I am not sure if it follows Euclid, i am a geometry noob.
Kiselev is an excellent book, with some flaws. But it is one of the best introductory geometry books out there.

It doesn't follow Euclid however. It doesn't even state any axioms.

What is Euclid's approach? Well, today you have essentially two approaches, and they all deal with how real numbers are treated. One approach takes the real numbers as fundamental. The axioms state will involve a distance function and explicit reference to real numbers. This is the approach Moise takes initially.
Euclid's original approach however did not contain any real numbers what-so-ever. Euclid did say what it means for figure to have equal area or length, but it never gives a number to that area or length. This approach is also dealt with in Moise, but much later in the book.

Kiselev immediately starts of with measuring angles. So it takes the real numbers very clearly for granted. So if he were to state rigorous axioms, he would not follow Euclid.

Kiselev is an excellent book, with some flaws. But it is one of the best introductory geometry books out there.

It doesn't follow Euclid however. It doesn't even state any axioms.

What is Euclid's approach? Well, today you have essentially two approaches, and they all deal with how real numbers are treated. One approach takes the real numbers as fundamental. The axioms state will involve a distance function and explicit reference to real numbers. This is the approach Moise takes initially.
Euclid's original approach however did not contain any real numbers what-so-ever. Euclid did say what it means for figure to have equal area or length, but it never gives a number to that area or length. This approach is also dealt with in Moise, but much later in the book.

Kiselev immediately starts of with measuring angles. So it takes the real numbers very clearly for granted. So if he were to state rigorous axioms, he would not follow Euclid.
Thanks for the very exciting and informative post.

Why not read Euclid's Elements itself? It's still the best book on the subject. Try to read it with the companion book Harthorne's "Euclid and beyond".
I second this. There's no substitute to learning from the masters.