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Books > Science & Mathematics > Mathematics > Geometry > Euclidean geometry
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Euclidean Geometry and its Subgeometries (Hardcover, 1st ed. 2015)
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Euclidean Geometry and its Subgeometries (Hardcover, 1st ed. 2015)
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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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