Euclidean Geometry and its Subgeometries

In this monograph, the authors present a modern development of Euclidean geometry from independent axioms, using up-to-date language and providing detailed proofs. The axioms for incidence, betweenness, and plane separation are close to those of Hilbert. This is the only axiomatic treatment of Euclidean geometry that uses axioms not involving metric notions and that explores congruence and isometries by means of reflection mappings. The authors present thirteen axioms in sequence, proving as many theorems as possible at each stage and, in the process, building up subgeometries, most notably the Pasch and neutral geometries. Standard topics such as the congruence theorems for triangles, embedding the real numbers in a line, and coordinatization of the plane are included, as well as theorems of Pythagoras, Desargues, Pappas, Menelaus, and Ceva. The final chapter covers consistency and independence of axioms, as well as independence of definition properties. There are over 300 exercises; solutions to many of these, including all that are needed for this development, are available online at the homepage for the book at www.springer.com. Supplementary material is available online covering construction of complex numbers, arc length, the circular functions, angle measure, and the polygonal form of the Jordan Curve theorem. Euclidean Geometry and Its Subgeometries is intended for advanced students and mature mathematicians, but the proofs are thoroughly worked out to make it accessible to undergraduate students as well. It can be regarded as a completion, updating, and expansion of Hilbert's work, filling a gap in the existing literature.

Table of Contents:
Preface.- ​Preliminaries and Incidence Geometry (I).- Affine Geometry: Incidence with Parallelism (IP).- Collineations of an Affine Plane (CAP).- Incidence and Betweenness (IB).- Pasch Geometry (PSH).- Ordering a Line in the Pasch Plane (ORD).- Collineations Preserving Betweenness (COBE).- Neutral Geometry (NEUT).- Free Segments of a Neutral Plane (FSEG).- Rotations about a Point of a Neutral Plane (ROT).- Euclidean Geometry Basics (EUC).- Isometries of a Euclidean Plane (ISM).- Dilations of a Euclidean Plane (DLN).- Every Line in a Euclidean Plane is an Ordered Field (OF).- Similarity on a Euclidean Plane (SIM).- Axial Affinities of a Euclidean Plane (AX).- Rational Points on a Line (QX).- A Line as Real Numbers (REAL); Coordinatization of a Plane (RR).- Belineations on a Euclidean/LUB Plane (AA).- Ratios of Sensed Segments (RS).- Consistency and Independence of Axioms; Other Matters Involving Models.- References.- Index.

Review :
“This is the most detailed undergraduate textbook on the axiomatic foundation of Euclidean geometry ever written.” (Victor V. Pambuccian, Mathematical Reviews, July, 2016) “The authors do a commendable job of writing out proofs in detail and attempting to make the text accessible to undergraduates. … It makes a very useful reference source, and … there aren’t very many current textbooks that discuss geometry from this particular point of view. I commend this book to the attention of instructors with an interest in the foundations of geometry, and to university librarians.” (Mark Hunacek, MAA Reviews, maa.org, March, 2016)