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Geometric Trilogy - Axiomatic, Algebraic and Differential Approaches to Geometry (Hardcover, 2013 ed.)
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Geometric Trilogy - Axiomatic, Algebraic and Differential Approaches to Geometry (Hardcover, 2013 ed.)
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The Trilogy intends to introduce the reader to the multiple
complementary aspects of geometry, paying attention to the
historical birth and growth of the ideas and results, and
concluding with a contemporary presentation of the various topics
considered. Three essentially independent volumes approach geometry
via the axiomatic, the algebraic and the differential points of
view. The "ruler and compass" approach to geometry, developed by
the Greek mathematicians of the Antiquity, remained the only
reference in Geometry - and even in Mathematics -- for more than
two millenniums. The fruitless efforts for solving the so-called
"classical problems" of Greek geometry lead eventually to a deeper
reflection on the axiomatic bases of geometry, and in particular to
the discovery of projective geometry and non-Euclidean geometries.
During the Renaissance, mathematicians start liberating themselves
from the "ruler and compass" dogma and use algebraic techniques to
investigate geometric situations. The nineteenth century, with the
birth of linear algebra and the theory of polynomials, opens new
doors and in particular, the fascinating world of algebraic curves.
The introduction of differential calculus during the eighteenth
century allows widening considerably the range of curves and
surfaces considered. The notion of curvature -under multiple forms
-- imposes itself as an essential tool for studying the properties
of curves and surfaces. And a keen study of some geometrical
properties of surfaces gives rise to the theory of algebraic
topology. This trilogy is of interest to all those who have to
teach or study geometry and need to have a good global overview of
the numerous facets of this fascinating topic. It provides both the
intuitive and the technical ingredients needed to find one's way
through Euclidean, non-Euclidean, projective, algebraic or
differential geometry at a high level.
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