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.
Through 35 centuries of the history of geometry, discover the
birth and follow the evolution of those innovative ideas that
allowed humankind to develop so many aspects of contemporary
mathematics. Understand the various levels of rigor which
successively established themselves through the centuries. Be
amazed, as mathematicians of the 19th century were, when observing
that both an axiom and its contradiction can be chosen as a valid
basis for developing a mathematical theory. Pass through the door
of this incredible world of axiomatic mathematical theories
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