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Geometry An Introduction

Ewald, Günther

Verlag: Ishi Press, 2013
ISBN 10: 4871877183 / ISBN 13: 9784871877183
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Titel: Geometry An Introduction

Verlag: Ishi Press

Erscheinungsdatum: 2013

Einband: Soft cover

Zustand: New


Brand New, Unread Copy in Perfect Condition. A+ Customer Service! Summary: Geometry was considered until modern times to be a model science. To be developed more geometrico was a seal of quality for any endeavor, whether mathematical or not. In the 17th century, for example, Spinoza set up his Ethics in a more geometrico manner, to emphasize the perfection, certainty, and clarity of his pronouncements. Geometry achieved this status on the heels of Euclid's Elements, in which, for the first time, a theory was built up in an axiomatic-deductive manner. Euclid started with obvious axioms - he called them "common notions" and "postulates" -, statements whose validity raised no doubts in the reader's mind. His propositions followed deductively from those axioms, so that the truth of the axioms was passed on to the propositions by means of purely logical proofs. In this sense, Euclid's geometry consisted of "eternal truths." Given its prominence, Euclid's Elements was also used as a textbook until the 20th Century. Today geometry has lost the central importance it had during earlier centuries, but it still is an important area of mathematics, and is truly fundamental for mathematics from a variety of points of view. The "Introduction to Geometry" by Ewald tries to address some of these points of view, whose significance will be examined in what follows from a historical perspective. Buchnummer des Verkäufers ABE_book_new_4871877183

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Inhaltsangabe: One of the insights that arose not long after Hilbert's Foundations of Geometry was that it is possible to build geometry without notions of order or continuity. An essential tool in this direction was the calculus of reflections, an idea that owes much to Hjelmslev. Bachmann later deepened the study of reflection geometry in a systematic way and coined the concept of a metric plane, a structure that captures the core of the orthogonality properties common to the Euclidean and the classical non-Euclidean planes. All Hilbert planes, i. e. all models of the plane axioms of Hilbert's axiom system, without the parallel axiom and the continuity axioms, turn out to be metric planes. Metric planes can be embedded in projective-metric planes, and thus can also be described analytically, i. e. in terms of coordinates. Reflection geometry emphasizes the interplay between geometry and group theory. This "Introduction" by Ewald occupies a singular place in the English language literature. Ewald's book treats a central topic of geometry, the theory of metric planes in Bachmann's sense. It makes this theory accessible to readers of English, in a systematic manner, through an axiomatic-deductive approach. Hyperbolic and elliptic geometries are also treated as substructures of a circle geometry, the Mobius geometry. This geometry is also introduced axiomatically by using an axiom system of van der Waerden.

Über den Autor: The author: Professor Ewald (Dr. rer. nat. University of Mainz) is Professor of Mathematics at the Ruhr Universitat. Professor Ewald is the author of a considerable number of research publications and a former Fulbright Scholar. He has taught as a visiting professor at Michigan State University and at the University of Southern California.

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