Description
Book SynopsisBased on the premise that in order to write proofs, one needs to read finished proofs as well as study both their logic and grammar, Revolutions in Geometry depicts how to write basic proofs in various fields of geometry.
Trade Review"An excellent supplemental resource or main textbook for an overview of mathematics course for upper-level undergraduate and graduate students." (Choice, October 2010).
Table of ContentsPreface.
Acknowledgments.
PART I FOUNDATIONS.
1 The First Geometers.
1.1 Egypt.
1.2 Babylon.
1.3 China.
2 Thales.
2.1 The Axiomatic System.
2.2 Deductive Logic.
2.3 Proof Writing.
3 Plato and Aristotle.
3.1 Form.
3.2 Categorical Propositions..
3.3 Categorical Syllogisms.
3.4 Figures.
PART II THE GOLDEN AGE.
4 Pythagoras.
4.1 Number Theory.
4.2 The Pythagorean Theorem.
4.3 Archytas.
4.4 The Golden Ratio.
5 Euclid.
5.1 The Elements.
5.2 Constructions.
5.3 Triangles.
5.4 Parallel Lines.
5.5 Circles.
5.6 The Pythagorean Theorem Revisited.
6 Archimedes.
6.1 The Archimedean Library.
6.2 The Method of Exhaustion.
6.3 The Method.
6.4 Preliminaries to the Proof.
6.5 The Volume of a Sphere.
PART III ENLIGHTENMENT.
7 François Viète.
7.1 The Analytic Art.
7.2 Three Problems.
7.3 Conic Sections.
7.4 The Analytic Art in Two Variables.
8 René Descartes.
8.1 Compasses.
8.2 Method.
8.3 Analytic Geometry.
9 Gérard Desargues.
9.1 Projections.
9.2 Points at Infinity.
9.3 Theorems of Desargues and Menelaus.
9.4 Involutions.
PART IV A STRANGE NEW WORLD.
10 Giovanni Saccheri.
10.1 The Question of Parallels.
10.2 The Three Hypotheses.
10.3 Conclusions for Two Hypotheses.
10.4 Properties of Parallel Lines.
10.5 Parallelism Redefined.
11 Johann Lambert.
11.1 The Three Hypotheses Revisited.
11.2 Polygons.
11.3 Omega Triangles.
11.4 Pure Reason.
12 Nicolai Lobachevski and János Bolyai.
12.1 Parallel Fundamentals.
12.2 Horocycles.
12.3 The Surface of a Sphere.
12.4 Horospheres.
12.5 Evaluating the Pi Function.
PART V NEW DIRECTIONS.
13 Bernhard Riemann.
13.1 Metric Spaces.
13.2 Topological Spaces.
13.3 Stereographic Projection.
13.4 Consistency of Non-Euclidean Geometry.
14 Jean-Victor Poncelet.
14.1 The Projective Plane.
14.2 Duality.
14.3 Perspectivity.
14.4 Homogeneous Coordinates.
15 Felix Klein.
15.1 Group Theory.
15.2 Transformation Groups.
15.3 The Principal Group.
15.4 Isometries of the Plane.
15.5 Consistency of Euclidean Geometry.
References.
Index.
