I no longer have this book having unwisely given it away when I moved out of my office. However I only gave away books I did not enjoy or felt little further need for. So I am guessing this is a book based on the Birkhoff approach to geometry, presuming familiarity with real numbers first, not the way I think geometry should be done.
I also seem to remember that the book is oppressively rigorous, in a non pedagogical way, over concerned with the sort of rigid precision dear to a mathematician but not so much to a student hoping to learn to understand something. So I recall this as a precise highly expert account that somehow removed much of the beauty of my favorite subject. There are several rave reviews on amazon for this book though, and it certainly has mathematical virtues. I still recommend Hartshorne's book above all others, including this one.
But if you want to see how elementary geometry can be derived rigorously from more sophisticated notions, namely the real number system, then this may be for you. At least the title is accurate in that sense.
However, Hilbert's approach espoused by Hartshorne, is much more general and reveals infinitely more Euclidean geometries that are related to this one as the infinitely many other Euclidean fields are related to the special case of the real numbers. I.e. there is no good geometrical reason to prefer the real numbers for geometry. The field theoretic properties that are needed however are only made clear by taking a more geometric approach first.
Still I wish I had not given it away, since it is a rigorous and expert treatment, just not one I enjoy. I hope I am not greatly in error about this book, but there is way to search it on amazon. This is a fine book, well written with high mathematical standards, as is Rudin, but it's also not my style.