I apologize for posting and deleting a long post after misreading your request as for a book using Hilbert's axioms.

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

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

Have you looked at Moise, Elementary geometry from an advanced standpoint? Igave my copy away but maybe that uses Birkhoff's axioms.But I really recommend Euclid + Hartshorne.

This also shows why I question the point of Birkhoff's approach, as it is neither fish nor fowl about using numbers as coordinates but only half of each.

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 noticed Moise went from 1.4 to 1.9 pounds from 1st edition to third so maybe the first is also 25% shorter.

I.e. real numbers are usually studied after lines and motivated in terms of them, so to me Birkhoff's approach is logically inadequate unless the students have had a careful real analysis course, which they seldom have.

Nonetheless,

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. He 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.

I apologize for posting and deleting a long post after misreading your request as for a book using Hilbert's axioms.

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

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 only really enjoyable part for the class was the section on neutral geometry, which I learned recently was lifted bodily from Euclid Book I.

Have you looked at Moise, Elementary geometry from an advanced standpoint? I gave my copy away but maybe that uses Birkhoff's axioms. But I really recommend Euclid + Hartshorne.

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.

This also shows why I question the point of Birkhoff's approach, as it is neither fish nor fowl about using numbers as coordinates but only half of each.

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.

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 noticed Moise went from 1.4 to 1.9 pounds from 1st edition to third so maybe the first is also 25% shorter.

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."

I.e. real numbers are usually studied after lines and motivated in terms of them, so to me Birkhoff's approach is logically inadequate unless the students have had a careful real analysis course, which they seldom have.

Imagine teaching the exponential function to undergrads as

Nonetheless, Clint McCrory spent several years developing his own course using Birkhoff's approach at UGA, and made it very successful. Here is a special case of the exponential map of lie groupslink to his course page. He apparently never found an appropriate book to use though.

http://www.math.uga.edu/~clint/2008/geomF08/home.htm

I apologize for posting and deleting a long post after misreading your request as for a book using Hilbert's axioms.

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

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 only really enjoyable part for the class was the section on neutral geometry, which I learned recently was lifted bodily from Euclid Book I.

Have you looked at Moise, Elementary geometry from an advanced standpoint? I gave my copy away but maybe that uses Birkhoff's axioms. But I really recommend Euclid + Hartshorne.

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.

This also shows why I question the point of Birkhoff's approach, as it is neither fish nor fowl about using numbers as coordinates but only half of each.

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.

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 noticed Moise went from 1.4 to 1.9 pounds from 1st edition to third so maybe the first is also 25% shorter.

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."

I.e. real numbers are usually studied after lines and motivated in terms of them, so Birkhoff's approach is logically inadequate unless the students have had a careful real analysis course, which they seldom have.

Imagine teaching the exponential function to undergrads as a special case of the exponential map of lie groups.

I apologize for posting and deleting a long post after misreading your request as for a book using Hilbert's axioms.

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

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 only really enjoyable part for the class was the section on neutral geometry, which I learned recently was lifted bodily from Euclid Book I.

Have you looked at Moise, Elementary geometry from an advanced standpoint? I gave my copy away but maybe that uses Birkhoff's axioms. But I really recommend Euclid + Hartshorne.

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.

This also shows why I question the point of Birkhoff's approach, as it is neither fish nor fowl about using numbers as coordinates but only half of each.

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.

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 noticed Moise went from 1.4 to 1.9 pounds from 1st edition to third so maybe the first is also 25% shorter.

I apologize for posting and deleting a long post after misreading your request as for a book using Hilbert's axioms.

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

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 only really enjoyable part for the class was the section on neutral geometry, which I learned recently was lifted bodily (and without credit), from Euclid Book I.

Have you looked at Moise, Elementary geometry from an advanced standpoint? I gave my copy away but maybe that uses Birkhoff's axioms. But I really recommend Euclid + Hartshorne.

This does not quite fit your conditions, but after many years of teaching geometry it is my absolute best recommendation.

I believe the best geometry book in the world is Euclid, apologize for posting and there is deleting a beautiful edition from Green Lion Press. There is now long post after misreading your request as for a fantastic guide to Euclid based on book using Hilbert's axiomscalled: .

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

I feel there

The problem for me is very little flaw in the original Euclid. He did omit explicit separation axioms insuring certain lines and circles do meet, which are implicit in his proofs, but aside from that real numbers are much more sophisticated than Euclidean geometry, about and the only thing missing Birkhoff approach is thus a failure to assume there exist rigid motions, needed bit backwards except for his proof of SAS congruence. (Oh and at some point one uses Archimedes' axiom, maybe in treating similarity.)

Thus Hilbert assumes instead SAS as an axiom, and makes several separation axioms about lines and planes, but Hilbert himself makes an erroneous statement that his own axioms experts like us who know what real numbers areindependent, failing to note that his separation axiom for planes implies the one for lines. This is corrected in Hartshorne's version of Hilbert's axioms.

One may note however that Hartshorne's axioms still differ slightly from Hilbert's.

When I asked Robin about this, he pointed out there are many editions of Hilbert and hence no definitive list of Hilbert's axioms.

I myself prefer to assume existence of rigid motions, since Euclid used them, and Hartshorne also includes a proof that this is permissible in place of Hilbert's SAS.

To comply more closely with your "shortness" and "Hilbert" request, I would direct you to Hilbert's own very short book maybe "Foundations we covered as much of Geometry", flaws and all, from Open Court publishing. Mine is from about 1905 Millman/Parker as I recall, and cost 95 cents. Hartshorne is helpful however in understanding his proofs.

I have taught from Millman and Parkerwe could manage, and although I liked the models and clever proofs, I found that it only became really enjoyable to part for the class in was the section on neutral geometry, some 150 pages in. which I learned later that part recently was lifted bodily from Euclid, Book I, I think.

In general I regret the overblown focus on the few tiny almost invisible axioms missed by Euclid, to the exclusion of the beautiful geometry he does do. Students as well, usually know so little Euclid that to teach as if they know it cold and only need a course on the consequences of not assuming Playfair's postulate resulted for me in a much less fun and interesting course.

It took me 50 years to find this out. I thought it was cheating to teach a graduate course from Euclid until I realized I myself did not know that stuff as well as I thought. E.g. Euclid makes it clear he understood perfectly that a tangent line to a circle is a limit of secant lines, whereas some of us thought he treated it only as the perpendicular to the radius.

I believe Hartshorne also includes hyperbolic, but maybe not projective geometry.

Birkhoff's approach of course assumes the theory of the real numbers before doing geometry, which I find sets a sophisticated prerequisite for learning a more simple subject, hence is pedagogically backwards from my own viewpoint. It is hard for me to see how anyone who has not studied geometry first, can understand real numbers. The other practical problem was that my students did not know their trig, hence could not follow the development in Millman and Parker, which I admit I had assumed would be a great approach before trying it. Euclid himself has essentially no prerequisites, a big plus.

I agree that a teacher who enjoys any approach can make it work well, Birkhoff, Hilbert, or any other. I have tried all these approaches, (and I liked them all in advance myselfwithout credit), but they did not all work at my university. I have taught from Millman/Parker, Greenberg, Clemens, and all taught me something. The most successful one though by far, for me and my students, was Euclid with guide by Hartshorne.

Archimedes is a nice complement as well to Euclid , for areas and volumes. In particular, at the age of 67, Book Ilearned that he knew what we call the Cavalieri principle quite well, and used it very effectively, e.g. to compute the volume of not only a sphere but also a "bicylinder", much more easily than we do in calculus class. I.e. a bycylinder, by Cavalieri and Pythagoras, is complementary to an octahedron inscribed in a cube, hence has volume 2/3 of the circumscribed cube. This point is entirely missed in calculus books which give the volume formula in terms of the radius of the cylinders forming the bicylinder.

This does not quite fit your conditions, but after many years of teaching geometry it is my absolute best recommendation. The I believe the best geometry book in the world is Euclid, and there is a beautiful edition from Green Lion Press. There is now a fantastic guide to Euclid based on Hilbert's axioms called: Geometry: Euclid and beyond, by Robin Hartshorne.

My point is

I feel there is very little flaw in the original Euclid. He did omit to assume explicitly to state explicit separation axioms insuring certain lines and circles do meet, which are implicit in his proofs, but aside from that, about the only thing missing is a failure to assume there exist rigid motions, needed for his proof of SAS congruence. (Oh and maybe at some point one uses Archimedes' axiom, maybe in treating similarity.)

Thus Hilbert assumes instead SAS as an axiom, and makes several separation axioms about lines and planes. , but Hilbert himself makes an erroneous statement that his own axioms are independent, failing to note that his separation axiom for planes implies the one for lines. This is corrected in Hartshorne's version of Hilbert's axioms.

One may note however that Hartshorne's axioms still differ slightly from Hilbert's. When I asked Robin about thisis , he pointed out there are many editions of Hilbert and hence no definitive list of Hilbert's axioms.

I have taught from Millman and Parker, and although I liked the models and clever proofs, I found that it only became enjoyable to the class in the section on neutral geometry, some 150 pages in. I learned later that part was lifted bodily from Euclid, Book I, I think.

In general I eschew regret the overblown focus on those the few tiny almost invisible axioms missed by Euclid, to the exclusion of the beautiful geometry he does do. Students as well, usually know so little Euclid that to teach as if they know it cold and only need a course on the consequences of not assuming Playfair's postulate resulted for me in a much less fun and interesting course.

It took me 50 years to find this out. I thought it was cheating to teach a graduate course from Euclid until i I realized I myself did not know that stuff as well as I thought. E.g. Euclid makes it clear he understood perfectly that a tangent line to a circle is a limit of secant lines, whereas many some of us thought he treated it only as the perpendicular to the radius.

Birkhoff's approach of course assumes the theory of the real numbers before doing geometry, which I find sets a far more sophisticated prerequisite for learning a more simple subject,hence it subject, hence is pedagogically backwards from my own viewpoint. I.e. it It is hard for me to see how anyone who has not studied geometry first, can understand real numbers. The other practical problem was that my students did not know their trig, hence could not follow the development in Millman and Parker, which I admit I had assumed would be a great approach before trying it. Euclid himself has essentially no prerequisites, a big plus.

I agree that a teacher who enjoys any approach can make it work well, Birkhoff, Hilbert, or any other. I have tried all these approaches, and I liked them all in advance myself, but they did not all work at my university. I have taught from Millman/Parker, Greenberg, Clemens, and all taught me something. The most successful one though by far, for me and my students, was Euclid with guide by Hartshorne.

Archimedes is a nice complement as well to Euclid, for areas and volumes. In particular, at the age of 67, I learned that he knew what we call the Cavalieri principle quite well, and used it very effectively, e.g. to compute the volume of not only a sphere but also a "bicylinder", much more easily than we do in calculus class. I.e. a bycylinder, by Cavalieri and Pythagoras, is complementary to an octahedron inscribed in a cube, hence has volume 2/3 of the circumscribed cube. This point is entirely missed in calculus books which give the volume formula in terms of the radius of the cylinders forming the bicylinder.

This does not quite fit your conditions, but after many years of teaching geometry it is my absolute best recommendation. The best geometry book in the world is Euclid, and there is beautiful edition from Green Lion Press. There is now a fantastic guide to Euclid based on Hilbert's axioms called: Geometry: Euclid and beyond, by Robin Hartshorne.

My point is there is very little flaw in the original Euclid. He did omit to assume explicitly to state separation axioms insuring certain lines and circles do meet, which are implicit in his proofs, but aside from that, about the only thing missing is a failure to assume there exist rigid motions, needed for his proof of SAS congruence. (Oh and maybe at some point one uses Archimedes' axiom, maybe in treating similarity.)

Thus Hilbert assumes instead SAS as an axiom, and makes several separation axioms about lines and planes. Hilbert himself makes an erroneous statement that his own axioms are independent, failing to note that his separation axiom for planes implies the one for lines. This is corrected in Hartshorne's version of Hilbert's axioms.

One may note however that Hartshorne's axioms still differ slightly from Hilbert's. When I asked Robin about this is he pointed out there are many editions of Hilbert and hence no definitive list of Hilbert's axioms.

I myself prefer to assume existence of rigid motions, since Euclid used them, and Hartshorne also includes a proof that this is permissible in place of Hilbert's SAS.

To comply more closely with your "shortness" and "Hilbert" request, I would direct you to Hilbert's own very short book maybe "Foundations of Geometry", flaws and all, from Open Court publishing. Mine is from about 1905 as I recall, and cost 95 cents. Hartshorne is helpful however in understanding his proofs.

I have taught from Millman and Parker, and found that it only became enjoyable in the section on neutral geometry, some 150 pages in. I learned later that part was lifted bodily from Euclid.

In general I eschew the overblown focus on those few tiny almost invisible axioms missed by Euclid, to the exclusion of the beautiful geometry he does do. Students as well, usually know so little Euclid that to teach as if they know it cold and only need a course on the consequences of not assuming Playfair's postulate resulted for me in a much less fun and interesting course.

It took me 50 years to find this out. I thought it was cheating to teach from Euclid until i realized I myself did not know that stuff as well as I thought. E.g. Euclid makes it clear he understood perfectly that a tangent line to a circle is a limit of secant lines, whereas many of us thought he treated it only as the perpendicular to the radius.

I believe Hartshorne also includes hyperbolic, but maybe not projective geometry.

Birkhoff's approach of course assumes the theory of the real numbers before doing geometry, which sets a far more sophisticated prerequisite for learning a more simple subject,hence it is pedagogically backwards from my own viewpoint. I.e. it is hard for me to see how anyone who has not studied geometry first, can understand real numbers. The other practical problem was that my students did not know their trig, hence could not follow the development in Millman and Parker, which I admit I had assumed would be a great approach before trying it. Euclid himself has essentially no prerequisites, a big plus.

This does not quite fit your conditions, but after many years of teaching geometry it is my absolute best recommendation. The best geometry book in the world is Euclid, and there is beautiful edition from Green Lion Press. There is now a fantastic guide to Euclid based on Hilbert's axioms called: Geometry: Euclid and beyond, by Robin Hartshorne.

My point is there is very little flaw in the original Euclid. He did omit to assume explicitly to state separation axioms insuring certain lines and circles do meet, which are implicit in his proofs, but aside from that, about the only thing missing is a failure to assume there exist rigid motions, needed for his proof of SAS congruence. (Oh and maybe at some point one uses Archimedes' axiom, maybe in treating similarity.)

Thus Hilbert assumes instead SAS as an axiom, and makes several separation axioms about lines and planes. Hilbert himself makes an erroneous statement that his own axioms are independent, failing to note that his separation axiom for planes implies the one for lines. This is corrected in Hartshorne's version of Hilbert's axioms.

One may note however that Hartshorne's axioms still differ slightly from Hilbert's. When I asked Robin about this is he pointed out there are many editions of Hilbert and hence no definitive list of Hilbert's axioms.

I myself prefer to assume existence of rigid motions, since Euclid used them, and Hartshorne also includes a proof that this is permissible in place of Hilbert's SAS.

To comply more closely with your "shortness" and "Hilbert" request, I would direct you to Hilbert's own very short book maybe "Foundations of Geometry", flaws and all, from Open Court publishing. Mine is from about 1905 as I recall, and cost 95 cents. Hartshorne is helpful however in understanding his proofs.

I have taught from Millman and Parker, and found that it only became enjoyable in the section on neutral geometry, some 150 pages in. I learned later that part was lifted bodily from Euclid.

In general I eschew the overblown focus on those few tiny almost invisible axioms missed by Euclid, to the exclusion of the beautiful geometry he does do. Students as well, usually know so little Euclid that to teach as if they know it cold and only need a course on the consequences of not assuming Playfair's postulate resulted for me in a much less fun and interesting course.

It took me 50 years to find this out. I thought it was cheating to teach from Euclid until i realized I myself did not know that stuff as well as I thought. E.g. Euclid makes it clear he understood perfectly that a tangent line to a circle is a limit of secant lines, whereas many of us thought he treated it only as the perpendicular to the radius.

I believe Hartshorne also includes hyperbolic, but maybe not projective geometry.