An axiomatic approach to geometry: geometric trilogy I Borceux, Francis

Table of contents:

1. Pre-Hellenic Antiquity
2. Some Pioneers of Greek Geometry
3. Euclid's Elements
4. Some Masters of Greek Geometry
5. Post-Hellenic Euclidean Geometry
6. Projective Geometry
7. Non-Euclidean Geometry
8. Hilbert's Axiomatization of the Plane

Focusing methodologically on those historical aspects that are relevant to supporting intuition in axiomatic approaches to geometry, the book develops systematic and modern approaches to the three core aspects of axiomatic geometry: Euclidean, non-Euclidean and projective. Historically, axiomatic geometry marks the origin of formalized mathematical activity. It is in this discipline that most historically famous problems can be found, the solutions of which have led to various presently very active domains of research, especially in algebra. The recognition of the coherence of two-by-two contradictory axiomatic systems for geometry (like one single parallel, no parallel at all, several parallels) has led to the emergence of mathematical theories based on an arbitrary system of axioms, an essential feature of contemporary mathematics.
This is a fascinating book for all those who teach or study axiomatic geometry, and who are interested in the history of geometry or who want to see a complete proof of one of the famous problems encountered, but not solved, during their studies: circle squaring, duplication of the cube, trisection of the angle, construction of regular polygons, construction of models of non-Euclidean geometries, etc. It also provides hundreds of figures that support intuition.

(http://www.springer.com/gp/book/9783319017297#aboutBook)