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lgli/Hartshorne R. Geometry.. Euclid and beyond (UTM, Springer, 2000)(ISBN 0387986502)(600dpi)(K)(T)(O)(542s)_MD_.djvu
Geometry: Euclid and Beyond (Undergraduate Texts in Mathematics) 🔍
Robin Hartshorne; Robin Hartshorene Springer; Brand: Springer, Undergraduate texts in mathematics, First edition, New York, 2010
英语 [en] · DJVU · 6.9MB · 2010 · 📘 非小说类图书 · 🚀/duxiu/lgli/lgrs · Save
描述
In recent years, I have been teaching a junior-senior-level course on the classi cal geometries. This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa rately. The remainder of the book is an exploration of questions that arise natu rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And in the last chapter we provide what is missing from Euclid's treatment of the five Platonic solids in Book XIII of the Elements. For a one-semester course such as I teach, Chapters 1 and 2 form the core material, which takes six to eight weeks.
Erscheinungsdatum: 08.06.2000
备用文件名
lgrsnf/Hartshorne R. Geometry.. Euclid and beyond (UTM, Springer, 2000)(ISBN 0387986502)(600dpi)(K)(T)(O)(542s)_MD_.djvu
备用出版商
Springer US
备用出版商
Copernicus
备用出版商
Telos
备用版本
Undergraduate texts in mathematics, New York, New York State, 2000
备用版本
Undergraduate texts in mathematics, New York, 2011
备用版本
United States, United States of America
备用版本
Corrected 2000., PS, 2000
备用版本
September 28, 2005
备用版本
Dec 15, 2010
备用版本
1, 2000
元数据中的注释
Includes bibliographical references (p. 495-501) and indexes.
元数据中的注释
Source title: Geometry: Euclid and Beyond (Undergraduate Texts in Mathematics)
备用描述
Undergraduate Texts in Mathematics Series
Title page
Copyright page
Dedication
- letter from Gauss to Schumacher (1846)
Preface
Contents
Introduction
Chapter 1. Euclid's Geometry
1. A First Look at Euclid's Elements
2. Ruler and Compass Constructions
3. Euclid's Axiomatic Method
4. Construction of the Regular Pentagon
5. Some Newer Results
Chapter 2. Hilbert's Axioms
6. Axioms of Incidence
7. Axioms of Betweenness
8. Axioms of Congruence for Line Segments
9. Axioms of Congruence for Angles
10. Hilbert Planes
11. Intersection of Lines and Circles
12. Euclidean Planes
Chapter 3. Geometry over Fields
13. The Real Cartesian Plane
14. Abstract Fields and Incidence
15. Ordered Fields and Betweenness
16. Congruence of Segments and Angles
17. Rigid Motions and SAS
18. Non-Archimedean Geometry
Chapter 4. Segment Arithmetic
19. Addition and Multiplication of Line Segments
20. Similar Triangles
21. Introduction of Coordinates
Chapter 5. Area
22. Area in Euclid's Geometry
23. Measure of Area Functions
24. Dissection
25. Quadrature Circuli
26. Euclid's Theory of Volume
27. Hilbert's Third Problem
Chapter 6. Construction Problems and Field Extensions
28. Three Famous Problems
29. The Regular 17-Sided Polygon
30. Constructions with Compass and Marked Ruler
31. Cubic and Quartic Equations
32. Appendix: Finite Field Extensions
Chapter 7. Non-Euclidean Geometry
33. History of the Parallel Postulate
34. Neutral Geometry
35. Archimedean Neutral Geometry
36. Non-Euclidean Area
37. Circular Inversion
38. Digression: Circles Determined by Three Conditions
39. The Poincare Model
40. Hyperbolic Geometry
41. Hilbert's Arithmetic of Ends
42. Hyperbolic Trigonometry
43. Characterization of Hilbert Planes
Chapter 8. Polyhedra
44. The Five Regular Solids
45. Euler's and Cauchy's Theorems
46. Semiregular and Face-Regular Polyhedra
47. Symmetry Groups of Polyhedra
Appendix: Brief Euclid
Notes
References
List of Axioms
Index of Euclid's Propositions
Index
Undergraduate Texts in Mathematics Series (continued from page II)
备用描述
In recent years, I have been teaching a junior-senior-level course on the classi cal geometries. This book has grown out of that teaching experience. I assume only high-school geometry and some abstract algebra. The course begins in Chapter 1 with a critical examination of Euclid's Elements. Students are expected to read concurrently Books I-IV of Euclid's text, which must be obtained sepa rately. The remainder of the book is an exploration of questions that arise natu rally from this reading, together with their modern answers. To shore up the foundations we use Hilbert's axioms. The Cartesian plane over a field provides an analytic model of the theory, and conversely, we see that one can introduce coordinates into an abstract geometry. The theory of area is analyzed by cutting figures into triangles. The algebra of field extensions provides a method for deciding which geometrical constructions are possible. The investigation of the parallel postulate leads to the various non-Euclidean geometries. And in the last chapter we provide what is missing from Euclid's treatment of the five Platonic solids in Book XIII of the Elements. For a one-semester course such as I teach, Chapters 1 and 2 form the core material, which takes six to eight weeks.
Erscheinungsdatum: 15.12.2010
备用描述
<p><p>this Book Offers A Unique Opportunity To Understand The Essence Of One Of The Great Thinkers Of Western Civilization. A Guided Reading Of Euclid's Elements Leads To A Critical Discussion And Rigorous Modern Treatment Of Euclid's Geometry And Its More Recent Descendants, With Complete Proofs. Topics Include The Introduction Of Coordinates, The Theory Of Area, Geometrical Constructions And Finite Field Extensions, History Of The Parallel Postulate, The Various Non-euclidean Geometries, And The Regular And Semi-regular Polyhedra. The Text Is Intended For Junior- To Senior-level Mathematics Majors. Robin Hartshorne Is A Professor Of Mathematics At The University Of California At Berkeley, And Is The Author Of Foundations Of Projective Geometry (benjamin, 1967) And Algebraic Geometry (springer, 1977).</p>
备用描述
This book offers a unique opportunity to understand the essence of one of the great thinkers of western civilization. A guided reading of Euclid's "Elements" leads to a critical discussion and rigorous modern treatment of Euclid's geometry and its more recent descendants. Also included are complete proofs, introduction of coordinates, the theory of area, geometrical constructions and finite field extensions, history of the parallel postulate, the various non-Euclidean geometries, and the regular and semi-regular polyhedra. The text is intended for junior-senior level mathematics majors. Robin Hartshorne is a professor of mathematics at the University of California, Berkeley, and is the author of "Foundations of Projective Geometry" (Benjamin, 1967) and "Algebraic Geometry" (Springer, 1977)
备用描述
"This book offers a unique opportunity to understand the essence of one of the great thinkers of western civilization. A guided reading of Euclid's Elements leads to a critical discussion and rigorous modern treatment of Euclid's geometry and its more recent descendants, with complete proofs. Topics include the introduction of coordinates, the theory of area, geometrical constructions and finite field extensions, history of the parallel postulate, the various non-Euclidean geometries, and the regular and semi-regular polyhedra. The text is intended for junior- to senior-level mathematics majors."--Book cover
备用描述
Euclid's Elements is regarded as the prime example of the axiomatic method. This book is a fascinating account that brings Euclid to life for the modern reader by presenting the Elements as the starting point for a study of geometry, from a modern mathematical perspective. The reader will become familiar with the geometrical content of Euclid's work, understand the logical structure of his presentation, and consider various mathematical questions that arise naturally from Euclid's presentation.
备用描述
A little after the time of Plato, but before Archimedes, in ancient Greece, a man named Euclid wrote the Elements, gathering and improving the work of his predecessors Pythagoras, Theaetetus, and Eudoxus into one magnificent edifice.
开源日期
2024-02-07
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