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Can someone recommend a good basic book on Geometry? Let me be more specific on what I am looking for. I'd like a book that starts with Euclid's definitions and postulates and goes on from there to prove thereoms about triangles, circles and other plane shapes. I'm not interested (at this time) in a book that ties geometry in with linear algebra or explains non-euclidean geometry. All the books I've seen seem to be of two types 1) Let us use linear algebra and other techniques to put geometry on a firm footing. Let us assume you already know Eucliean geometry and move on to more interesting stuff. These are probably too advanced for me. 2) Let us skip all that complicated postulates/proofs and make geometry fun. Too easy.

When I was in 9th grade Honors Geometry way back in 1981/82 we had a textbook that, as I recall through my foggy memory, was very good and did systematically build up geometry from the propositions (5 of them?). And it certainly didn't attempt to address Hilbert's contributions to geometry or explain non-Euclidean geometry. Unfortunately, I don't recall the title or author. I suspect the book was used nationally though and for quite a number of years, so perhaps someone with a better memory than me can provide a title/author. I do recall it had all the postulates and most major thereoms listed at the end of the book.

My motivation is two fold... 1. Review the material myself in preparation for my advanced texts. 2. Help my children in a few years when they take geometry.

Books I could get for the iPad/Kindle or free books (Google?) would be most appreciatied, but I'm not opposed to killing a tree either :) Thanks for all the help. Dave

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The classic Elements (Euclid, c. $-300$) comes to mind ... –  Henning Makholm Feb 10 '12 at 19:14
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possible duplicate of Book recommendation on plain Euclidean geometry –  lhf Feb 10 '12 at 19:20
    
Maybe too simple, but geometry book by Harold Jacobs is nicely written. –  André Nicolas Feb 10 '12 at 20:09
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@HenningMakholm: I would recomment to the average person reading the first 47 propositions of Euclid. Reading all of Euclid is kind of silly unless you're a historian of mathematics. –  Ben Crowell Feb 11 '12 at 0:00
    
I would go further than @BenCrowell and advise against reading the original Euclid at all. The text's definitions and axiomatics don't meet a modern standard of coherence, and the corresponding murkiness in some proofs is present as soon as proposition I.4 (the "superposition"). Of course, most modern texts at the high school level are no better, but at least they're written in modern language that makes logical gaps easier to detect. Students reading Euclid often assume the original text to be faultless, and attribute anything they don't quite understand to the translator's choice of words. –  leslie townes Feb 11 '12 at 1:03

5 Answers 5

  • Introduction to Geometry by Coxeter.
  • Elementary Geometry From An Advanced Viewpoint by Moise.
  • Geometry: Euclid and Beyond by Hartshorne.
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Coxeter's Introduction to Geometry is a very nice book, great exposition, some very hard problems. However, it is not at all the kind of book that the OP describes wanting. –  André Nicolas Feb 10 '12 at 19:57
    
I second the recommendation of the Hartshorne text--- the first few chapters are a great companion to any exploration of Euclidean geometry (indeed, the title of earlier editions of the book was something like "Companion to Euclid"). It would work well as an "adult" companion to a less rigorous textbook (e.g. any high school axiomatic geometry book). [The later chapters get quite algebraic, and also concern stuff related to non-Euclidean geometry, but those first few chapters are very much what the OP wants.] –  leslie townes Feb 11 '12 at 1:06

Searching for Geometry textbooks between about 1970 & 1982 in one possible database (probably not very complete), I found:

Could one of these or the "related" books be the one you used?

A free possibility, alas without figures, might be Geometry Unbound (Kedlaya 1996).

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You might want to look at "The Harpur Euclid". It's a free Google book which covers basic Euclidean Geometry based on Euclid's five postulates, has a large number of exercises, and some enrichment material on what was called "Modern Geometry" in 1894, topics like Simson's line, the radical axis, poles and polars, inversion, etc. I'm using it to tutor some home-school kids in Geometry. You can find it here:

http://books.google.com/books?id=PkHP78BglQMC&printsec=frontcover&dq=harpur+euclid&hl=en&sa=X&ei=BM02T52CPe-P0QGGwLGkAg&ved=0CDIQ6AEwAA#v=onepage&q=harpur%20euclid&f=false

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My recommendations from the earlier thread : Book recommendation on plane Euclidean geometry - are still to me as good a set of recommendations as there are.

I'll also throw my support behind Robin Hartshorne's Geometry: Euclid and Beyond as a great book for showing the connections between abstract algebra and classical geometry,but it's considerably more difficult then the other books on my list. You really need to be very comfortable with basic group and ring theory to be able to read it without a struggle.

Elmer Rees' classic Notes On Geometry is still the classic textbook on plane geometry from an advanced standpoint and should be in every mathematician's library. Again,to be able to read it,one needs to be very comfortable with both linear algebra and group theory and a certain comfort with basic real and complex analysis will help. But it's well worth the effort as there's probably no better single source on this material at this level.

Those 2 lists should get you started. Good luck!

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And this was worth a point off,why,exactly?I wish they'd do a better job controlling the trolls on here. –  Mathemagician1234 Feb 12 '12 at 0:40
    
+1. Hope you feel better. –  Tim Feb 12 '12 at 1:20
    
Thanks and appreciate the kind words. –  Mathemagician1234 Feb 12 '12 at 1:53
    
And my fan club's at it again,how nice. –  Mathemagician1234 Oct 1 '12 at 1:19

"Geometry for Enjoyment and Challenge" by Rhoad, Milauskas, and Whipple (published by McDougal-Littell) is a fairly common proof-based high school honors geometry textbook, at least in the greater Chicago metropolitan area.

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