EAN: 9783319237749

EUCLIDEAN GEOMETRY AND ITS SUBGEOMETRIES - EDWA...

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.

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