Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I'm looking for an short and elementary book which does Euclidean geomety with Birkhoff's axioms.

It would be best if it would also include some topics in projective (and/or) hyperbolic geometry.

About the course. The students suppose know some basic calculus, but they did not see real proofs.
Most of the students in my course want to become math teachers. The course description says: "Euclidean and Hyperbolic geometries and their development from postulate systems".

I choose Birkhoff's axioms sinse they use real numbers as a building block. This makes possible to do intro without cheating and without borring details. I know some good books for school students, but I am looking for something bit more advanced.

P.S. I want to thank everyone for comments and answers.

As I stated in the comments, I did not find an appropriate book and wrote the lecture notes my-self. They are available on arXiv, here is the link: Euclidean Plane and its Relatives.

share|improve this question
Out of curiosity: what kind of students will you have? –  Mariano Suárez-Alvarez May 11 '11 at 1:49
Out of curiosity: presumably you made a choice and taught the course by now. What did you do, and how did it go? –  Gerry Myerson May 11 '11 at 6:04
Anton, if your notes are available online I am sure more than a few of us would enjoy seeing them. –  roy smith May 14 '11 at 23:30
Could you say more about who the students were? Were these undergraduate math majors (who want to become teachers)? Did they learn to do proofs? (Teaching students how to do proofs is something I seem to be unable to do at all.) –  Deane Yang May 15 '11 at 4:04
Must it be be CW? –  SNd May 15 '11 at 15:05

6 Answers 6

Have you taught this course before? After teaching it several times from Millman/Parker and other materials using Birkhoff's axioms, I suggest you consider using Euclid himself plus Hartshorne's guide, Geometry: Euclid and beyond, which uses a form of Hilbert's axioms.

The problem for me is that real numbers are much more sophisticated than Euclidean geometry, and the Birkhoff approach is thus a bit backwards except for experts like us who know what real numbers are.

When we covered as much of Millman/Parker as we could manage, the most enjoyable part for the class was the section on neutral geometry, which I learned recently was lifted bodily from Euclid Book I.

If you like assuming that every line in the plane is really the real numbers R, what about going the rest of the way and assuming the plane itself is R^2? Then you can use matrices to define rigid motions and do a lot that connects up to their calculus courses.

Moise is more succinct than the 500 pages suggests as I recall, and is an excellent text from a mathematician's standpoint, but very forbidding probably from a student's. I noticed Moise went from 1.4 to 1.9 pounds from 1st edition to third so maybe the first is also 25% shorter.

The old SMSG books in the 1960's were based on Birkhoff's approach, but are not short. They are also available free on the web.

I just looked at the old SMSG book and found the following circular sort of discussion of real numbers: "if you fill in all those other non rational points on the line, you have the real numbers."

Clint McCrory spent several years developing his own course using Birkhoff's approach at UGA, and made it very successful. Here is a link to his course page. The students loved his class at least in its evolved form after a couple years. they especially appreciated the GSP segment at the start. Apparently many students had little geometric intuition and used that to acquire some. Clint apparently never found an appropriate book to use though.


After teaching this course myself from Greenberg, Millman/Parker, Clemens, supplemented by Moise, and the original works of Saccheri, my own Birkhoff axioms, I finally found Euclid and Hartshorne to be my favorite, by a large margin.

But the beauty of the topic is that there is no perfect choice. You will likely enjoy the search for your favorite too. There is a reason however that Euclid has the longevity it has.

In a nutshell, there are two equivalent concepts, similarity and area, that are treated in opposite order in Euclid and Birkhoff. Euclid's theory of equal content, via equidecomposability, in his Book I, allows him to use area to prove the fundamental principle of similarity in Book VI. Birkhoff assumes similarity as an axiom, and area is relatively easy using similarity, e.g. similarity allows one to show that the formula A = (1/2)BH for area of a triangle is independent of choice of base. Euclid himself uses similarity to deduce a general Pythagorean theorem in Prop. VI.31, whose proof many people prefer as "simpler" than Euclid's own area based proof of Pythagoras in Prop.I.47. The problem is that there has, to my knowledge, never been a civilization in which similarity or proportionality developed before the idea of decomposing and reassembling figures. Briefly, congruence, on which equidecomposability is based, is more fundamental than similarity. Hence, although logically either concept can be used to deduce the other, it seems to me at least that the more primitive concept should be placed first in a course.

share|improve this answer
When I last taught the course from Euclid, we kept a diary of axioms we thought were needed for his proofs and compared them to Hilbert's. The main difference was, since our list added tacit assumptions Euclid had made, we had existence of rigid motions as an axiom, whereas Hilbert had SAS congruence, whose proof used that principle, as an axiom. We were also not as precise on separation properties needed. However Hilbert's own separation axioms were not independent as he claimed. As Robin H. pointed out the several editions of Hilbert mean there is no definitive list of Hilbert's axioms. –  roy smith May 14 '11 at 23:40

I am not sure this qualifies under short and elementary but I recommend:

Geometry: A Metric Approach with Models

Richard Millman and George Parker; Springer-Verlag, NY, 1981

(there may be a more recent edition, I did not check).

This book does treat hypberbolic geometry.

share|improve this answer
I checked this book --- it will definetely scary any student... –  Anton Petrunin Jul 6 '10 at 13:17

If you're willing to use an unpublished manuscript, from the little I've looked at it, this book by Matthew Harvey looks pretty good. However, he uses Hilbert's axioms rather than Birkhoff's. Jack Lee at the University of Washington is writing another book, using a variant on the SMSG postulates, designed for a geometry course for math majors who are considering teaching high school. His book spends several chapters on hyperbolic geometry, but doesn't have any projective geometry. The book is not publicly available, but you could email him and ask him about it.

share|improve this answer

How about Basic Geometry and Basic Geometry - Manual for Teachers by George D. Birkhoff and Ralph Beatley?

In the spring of 1923, Professor Birkhoff was invited to deliver in Boston a series of Lowell Lectures on Relativity. In order to present this subject with as few technicalities as possible he decided to devise the simplest possible system of Euclidean geometry he could think of, and... he hit upon the framework of the system that, with all the details filled in, is now BASIC GEOMETRY.

share|improve this answer
This book does not treat the hyperbolic plane. Another advantage of Millman and Parker, Geometry: A Metric Approach with Models is that it has a much more modern flavor treating things like the taxicab plane, the Moulton plane, etc. –  Joseph Malkevitch Jul 5 '10 at 14:37
Joseph, I haven't read the book you recommend. I do not claim at all that it might be inferior in any sense to the book by Birkhoff and Beatley. –  Andrey Rekalo Jul 5 '10 at 14:43
Thank you. I checked the book, it is written for school students, a lot of motivation, but it does not go far enough for me (even in Euclidean geometry). –  Anton Petrunin Jul 6 '10 at 13:12

Chapter 14 of Prenowitz and Jordan, Basic Concepts of Geometry, begins, "In this chapter a tentative treatment of congruence is given based on a proposal of G. D. Birkhoff (1884-1944) that the real number system should be assumed in the treatment of Euclidean geometry at an elementary level. Birkhoff's development was modified and simplified by the School Mathematics Study Group. Our treatment is an adaptation of theirs and assumes a modification of their Ruler Postulate employed by MacLane."

References are given to Birkhoff and Beatley, to the School Mathematics Study Group textbook Geometry (Yale U. Press, 1961), and to S. MacLane, Metric postulates for plane geometry, American Mathematical Monthly 66 (1959) 543-555.

There is no discussion of projective or hyperbolic geometry in Chapter 14 (but there is an extensive discussion of hyperbolic geometry in earlier chapters).

You might also be interested in Moise, Elementary Geometry From An Advanced Standpoint. In Chapter 8 you find out that the way he has been presenting plane geometry "is not the classical one. It was proposed in the early 1930's by G. D. Birkhoff, and has only recently become popular." In later chapters he does hyperbolic geometry, but I don't know whether he follows the Birkhoff path. I see no mention of projective geometry in this book.

share|improve this answer
Moise,to me,is the definitive work on Euclid from a metric standpoint.Millman and Parker is a good alternative,but not as good as Moise. Frankly,I think thier elementary differential geometry text is MUCH better and it's a tragedy it's out of print. –  The Mathemagician Jul 6 '10 at 7:19
Yes, Moise is quite good. –  Anton Petrunin May 10 '13 at 21:42

Geometry: Plane and Fancy doesn't exactly fit what you describe (it moves on to the fancy stuff a lot sooner than 2/3 of the way through), but might nevertheless be worth a look.

share|improve this answer
Yes, "it moves on to the fancy stuff a lot sooner"... –  Anton Petrunin Jul 6 '10 at 16:03

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.