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

Geometry: Euclid and Beyond (Undergraduate Texts in Mathematics)

Geometry(Chinese Edition)

几何 英文

Hartshorne, Robin, author; Hartshorene, Robin. Companion to Euclid

HA CI HUO EN Hartshorne.R

Robin Hartshorne著; 哈茨霍恩

(美)哈茨霍恩著

World Publishing Corporation

Springer; Brand: Springer

World Publishing Company

New York: Springer

世界图书出版公司北京公司

Springer US

Undergraduate texts in mathematics, First edition, New York, 2010

Undergraduate texts in mathematics, New York, ©2000

United States, United States of America

China, People's Republic, China

Ying yin ban, Beijing, 2011

Dec 15, 2010

2011-04-01

1997

Source title: Geometry: Euclid and Beyond (Undergraduate Texts in Mathematics)

Bookmarks: p1 (p7): Introduction
p2 (p7): Chapter 1．Euclid's Geometry
p2-1 (p8): 1．A First Look at Euclid's Elements
p2-2 (p18): 2．Ruler and Compass Constructions
p2-3 (p27): 3．Euclid's Axiomatic Method
p2-4 (p45): 4．Construction of the Regular Pentagon
p2-5 (p51): 5．Some Newer Results
p3 (p65): Chapter 2．Hilbert's Axioms
p3-1 (p66): 6．Axioms of Incidence
p3-2 (p73): 7．Axioms of Betweenness
p3-3 (p81): 8．Axioms of Congruence for Line Segments
p3-4 (p90): 9．Axioms of Congruence for Angles
p3-5 (p96): 10．Hilbert Planes
p3-6 (p104): 11．Intersection of Lines and Circles
p3-7 (p112): 12．Euclidean Planes
p4 (p117): Chapter 3．Geometry over Fields
p4-1 (p118): 13．The Real Cartesian Plane
p4-2 (p128): 14．Abstract Fields and Incidence
p4-3 (p135): 15．Ordered Fields and Betweenness
p4-4 (p140): 16．Congruence of Segments and Angles
p4-5 (p148): 17．Rigid Motions and SAS
p4-6 (p158): 18．Non-Archimedean Geometry
p5 (p165): Chapter 4．Segment Arithmetic
p5-1 (p165): 19．Addition and Multiplication of Line Segments
p5-2 (p175): 20．Similar Triangles
p5-3 (p186): 21．Introduction of Coordinates
p6 (p195): Chapter 5．Area
p6-1 (p196): 22．Area in Euclid's Geometry
p6-2 (p205): 23．Measure of Area Functions
p6-3 (p212): 24．Dissection
p6-4 (p221): 25．Quadratura Circuli
p6-5 (p226): 26．Euclid's Theory of Volume
p6-6 (p231): 27．Hilbert's Third Problem
p7 (p241): Chapter 6．Construction Problems and Field Extensions
p7-1 (p242): 28．Three Famous Problems
p7-2 (p250): 29．The Regular 17-Sided Polygon
p7-3 (p259): 30．Constructions with Compass and Marked Ruler
p7-4 (p270): 31．Cubic and Quartic Equations
p7-5 (p280): 32．Appendix:Finite Field Extensions
p8 (p295): Chapter 7．Non-Euclidean Geometry
p8-1 (p296): 33．History of the Parallel Postulate
p8-2 (p304): 34．Neutral Geometry
p8-3 (p319): 35．Archimedean Neutral Geometry
p8-4 (p326): 36．Non-Euclidean Area
p8-5 (p334): 37．Circular Inversion
p8-6 (p346): 38．Digression:Circles Determined by Three Conditions
p8-7 (p355): 39．The Poincaré Model
p8-8 (p373): 40．Hyperbolic Geometry
p8-9 (p388): 41．Hilbert's Arithmetic of Ends
p8-10 (p403): 42．Hyperbolic Trigonometry
p8-11 (p415): 43．Characterization of Hilbert Planes
p9 (p435): Chapter 8．Polyhedra
p9-1 (p436): 44．The Five Regular Solids
p9-2 (p448): 45．Euler's and Cauchy's Theorems
p9-3 (p459): 46．Semiregular and Face-Regular Polyhedra
p9-4 (p469): 47．Symmetry Groups of Polyhedra
p10 (p481): Appendix:Brief Euclid
p11 (p487): Notes
p12 (p495): References
p13 (p503): List of Axioms
p14 (p505): Index of Euclid's Propositions
p15 (p507): Index

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)

xi, 526 pages : 25 cm
"Earlier version of this book appeared as Companion to Euclid : A Course of geometry ... Berkeley Mathematics Lecture Notes, volume 9, American Mathematical Society and Berkeley Center for Pure and Applied Mathematics."--Title page verso
Includes bibliographical references (p. 495-501) and indexes
Euclid's geometry -- Hilbert's axioms -- Geometry over fields -- Segment arithmetic -- Area -- Construction problems and field extensions -- Non-Euclidean geometry -- Polyhedra -- Appendix: Brief Euclid

<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, history of the parallel postulate, the various non-Euclidean geometries, and the regular and semi-regular polyhedra.</p>

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.

2023-06-28