Geometry: Euclid and Beyond by Hartshorne

[micromass]

• Author: Robin Hartshorne
• Title: Geometry: Euclid and Beyond
• Amazon Link: https://www.amazon.com/dp/0387986502/?tag=pfamazon01-20
• Prerequisities: Proofs, intro to abstract algebra
• Level: Undergrad

Table of Contents:

[LIST]
[*] Euclid's Geometry
[LIST]
[*] A First Look at Euclid's Elements
[*] Ruler and Compass Constructions
[*] Euclid's Axiomatic Method
[*] Construction of the Regular Pentagon
[/LIST]
[*] Hilbert's Axioms
[LIST]
[*] Axioms of Incidence
[*] Axioms of Betweenness
[*] Axioms of Congruence for Line Segments
[*] Axioms of Congruence for Angles
[*] Hilber Planes
[*] Intersection of Lines and Circles
[*] Euclidean Planes
[/LIST]
[*] Geometry over Fields
[LIST]
[*] The Real Cartesian Plane
[*] Abstract Fields and Incidence
[*] Ordered Field and Betweenness
[*] Congruence of Segments and Angles
[*] Rigid Motions and SAS
[*] Non-Archimedean Geometry
[/LIST]
[*] Segment Arithmetic
[LIST]
[*] Addition and Multiplication of Line Segments
[*] Similar Triangles
[*] Introduction of Coordinates
[/LIST]
[*] Area
[LIST]
[*] Area in Euclid's Geometry
[*] Measure of Area Functions
[*] Dissection
[*] Quadratura Circuli
[*] Euclid's Theory of Volume
[*] Hilbert's Third Problem
[/LIST]
[*] Construction Problems and Field Extensions
[LIST]
[*] Three Famous Problems
[*] The Regular 17-Sided Polygon
[*] Constructions with Compass and Marked Ruler
[*] Cubic and Quartic Equation
[*] Appendix: Finite Field Extensions
[/LIST]
[*] Non-Euclidean Geometry
[LIST]
[*] History of the Parallel Postulate
[*] Neutral Geometry
[*] Archimedean Neutral Geometry
[*] Non-Euclidean Area
[*] Circular Inversion
[*] Digression: Circles Determined by Three Conditions
[*] The Poincaré Geometry
[*] Hilbert's Arithmetic of Ends
[*] Hyperbolic Trigonometry
[*] Characterization of Hilbert Planes
[/LIST]
[*] Polyhedra
[LIST]
[*] The Five Regular Solids
[*] Euler's and Cauchy's Theorems
[*] Semiregular and Face-Regular Polyhedra
[*] Symmetry Groups of Polyhedra
[/LIST]
[*] Appendix: Brief Euclid
[*] Notes
[*] References
[*] List of Axioms
[*] Index of Euclid's Propositions
[*] Index
[/LIST]

 

[mathwonk]

Outstanding guide to Euclid. This opens up Euclid and makes one see why that is such a great book. And goes much further. Enough beautiful material for years of self study.