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Recently I have started dipping my toes in mathematical waters besides calculus,and with varying success I have started learning bit of something about "everything".

But I have one issue,namely I can not find a satisfactory book regarding geometry. My problem with book choice is two-fold and it will be laid out in few points below.

First issue is that I know almost no geometry. Besides calculating areas of basic shapes,and a few very very basic theorems about chords and circles,I am a blank slate.

Second issue I have is that I can not handle non-axiomatic arguments,which are laid out in many good books.It is not that I find them bad or anything,I just like precision.

So now I will list few points which a book should satisfy,and then I will list a few examples of books which do not satisfy,and why.

Firstly book or book series should contain both plane a 3D geometry(or however it is called).

Exercises should be abundant(not essential)

The more theorems proved in the text,the better.

It should start from scratch.Namely from basic axioms, be it Euclidean or Hilbert or any other axiomatization.Then it should proceed from these axioms,and using strictly them,prove theorems.For reference think of Enderton's Set Theory book,first it lays out the axioms and then proceeds to results, given book should proceed in similar manner.

Any terms used used should be previously or timely defined

It should start from very basic concepts,like lines,circles,angles and etc. and proceed up to higher concepts,whatever they may be.

Now for books which do not satisfy and why. First example is Kiselev's Planimetry and Stereometry. These two books are great, but lack of rigor is very frustrating.

Hartshorne's Geometry book is great according to reviews but my problem with it is that it assumes knowledge which I do not posses. Same goes for Coxeter.

I thank you for your generous help,in advance.

Martin Sleziak
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Vanio Begic
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  • I think it's important for the community to know which kinds of math you've been exposed to and which topics in geometry interest you. I think there are geometric concepts presented in topology, analysis, and linear algebra. If I were to recommend a couple of broad subjects, try would be differential geometry and algebraic geometry. I can speak a bit about former but virtually nothing about the latter. – Mnifldz Feb 08 '15 at 22:23
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    As said,plane and 3D geometry will suffice for start.I need those for my college examination,and they should be the focal point of these books.I have done some calculus(half of Spivak book),linear algebra(very brief so far,only started few days ago),abstract algebra(very brief,definition of groups,orders,isomorphisms),set theory(up to cardinal arithmetic,endertons book). – Vanio Begic Feb 08 '15 at 22:34
  • Although this post is rather old, it's worth noting that Kiselev's books can be made rigorous with just a little effort on behalf of the reader in the early sections. He is a little wishy-washy on things like what 'congruence' is, since he's not clear about things like 'superimposition.' I've managed to fill in the gaps in my reading though. – A. Thomas Yerger Nov 08 '16 at 04:28

4 Answers4

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Yes, Euclid did leave quite a bit to unspoken assumptions. A great more rigorous geometry source that I believe is your level is Foundations of Geometry by Venema.

http://www.amazon.com/gp/aw/d/0136020585/ref=mp_s_a_1_2?qid=1423436640&sr=8-2&keywords=foundation+geometry&pi=AC_SY200_QL40&dpPl=1&dpID=51wVLMhYlTL&ref=plSrch

Paul
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  • I like both of the answers.Yours is more of up to date and modern treatment,which is nice but pure euclidean is what I require.Points given but I will have to accept the other one as final.Thanks for contribution – Vanio Begic Feb 08 '15 at 23:42
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if you want a very axiomatic treatment of elementary geometry, go back to the real source: Euclid. There are many books on this. My favourite is the Byrne book where colour is used instead of names.

http://www.amazon.com/Oliver-Byrne-First-Elements-Euclid/dp/3836544717/

Mark Joshi
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  • Will try good sir.I am kind of skeptical about Euclidean axioms,since many say that they are not good enough.But I will have to see for myself – Vanio Begic Feb 08 '15 at 23:01
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I learned plane & solid Geometry from these old but super nice books, they are what you want. They Start from scratch with axiomatic presentation. You can download them free of charge from archive.org

George Albert Wentworth Plane Geometry

Beman,_David_Eugene_Smith New_Plane Geometry

Wentworth & Smith Solid Geometry

Beman & Smith Solid Geometry

max
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The only book I have found that treats 3d geometry axiomatically (ie, based on set theory) is "Elementary geometry from an advanced standpoint" by Edwin Moise.

Plemath
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  • While Moise' book is wonderful and should be read by everyone interested in classical geometry, the problem with it is he takes Birkoff's metric axioms approach rather then Hilbert's modernization of Euclid's approach. While both are logically equivalent, the poster might not be comfortable with it. – Mathemagician1234 Apr 04 '23 at 22:42