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!
