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Michael O'Leary, "Revolutions of Geometry"
2010 | pages: 607 | ISBN: 0470167556 | PDF | 7,0 mb
Guides readers through the development of geometry and basicproof writing using a historical approach to the topic
In an effort to fully appreciate the logic and structure ofgeometric proofs, Revolutions of Geometry places proofs intothe context of geometry's history, helping readers to understandthat proof writing is crucial to the job of a mathematician.Written for students and educators of mathematics alike, the bookguides readers through the rich history and influential works, fromancient times to the present, behind the development of geometry.As a result, readers are successfully equipped with the necessarylogic to develop a full understanding of geometric theorems.
Following a presentation of the geometry of ancient Egypt,Babylon, and China, the author addresses mathematical philosophyand logic within the context of works by Thales, Plato, andAristotle. Next, the mathematics of the classical Greeks isdiscussed, incorporating the teachings of Pythagoras and hisfollowers along with an overview of lower-level geometry usingEuclid's Elements. Subsequent chapters explore the work ofArchimedes, Viete's revolutionary contributions to algebra,Descartes' merging of algebra and geometry to solve the Pappusproblem, and Desargues' development of projective geometry. Theauthor also supplies an excursion into non-Euclidean geometry,including the three hypotheses of Saccheri and Lambert and the nearsimultaneous discoveries of Lobachevski and Bolyai. Finally, moderngeometry is addressed within the study of manifolds and ellipticgeometry inspired by Riemann's work, Poncelet's return toprojective geometry, and Klein's use of group theory tocharacterize different geometries.
The book promotes the belief that in order to learn how to writeproofs, one needs to read finished proofs, studying both theirlogic and grammar. Each chapter features a concise introduction tothe presented topic, and chapter sections conclude with exercisesthat are designed to reinforce the material and provide readerswith ample practice in writing proofs. In addition, the overallpresentation of topics in the book is in chronological order,helping readers appreciate the relevance of geometry within thehistorical development of mathematics.
Well organized and clearly written, Revolutions ofGeometry is a valuable book for courses on modern geometry andthe history of mathematics at the upper-undergraduate level. It isalso a valuable reference for educators in the field ofmathematics.
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