Geometry by Its Transformations: Lessons Centered on the History from 1800-1855

Preface
Introduction
Why 1800 to 1855?
Note to Reader
Contents
1 Greek Background
	1 Forgotten Mathematics
	2 Euclid
	3 The Geometry of Euclid's Elements: Book 1
	4 The Geometry of Euclid's Elements: Triangle Similarity and Circles in Books 6 and 3
	5 Ceva's Theorem
	6 Apollonius
	7 Subcontrary Circle in the Conics of Apollonius
	8 Ptolemy
	9 Optional: Circles on the Sphere Are Mapped to Circles on the Tangent Plane
	10 Pappus
	11 Application and Exercise: Angles  Formed  by  Chords,  Secants,  and  Tangents
	12 Exercises—Greek Background
2 The Dilation Transformation
	1 The Dilation Transformation
	2 Dilations in History
	3 Gaspard Monge and Lazare Carnot
	4 Dilations in Poncelet
	5 Application 1 of the Dilation: The Square in a Triangle
	6 Application 2 of the Dilation: Pappus Book 7 Proposition 130
	7 Application 3 of the Dilation: The 9-Point Circle, with Exercises
	8 Application 4 of the Dilation: π and the Area of a Circle and Liu Hui
	9 Exercises—Dilations
3 Institutional Transformation of Geometry: France
4 Affinity and the List of Transformations by Moebius
	1 Transformations as Listed by Moebius
	2 Affinity According to Moebius
	3 Application: The Ellipse as the Image of a Circle Underan Affinity
	4 Exercises—Affinity and Transformations as Listed by Moebius
5 Background for Homology: The Common Secant, the Cross-Ratio, and Harmonic Sets
	1 Common Secant (or Chord) of Two Circles
	2 Harmonic Conjugates, the Cross-Ratio, and Their Invariance
	3 Dual Statements and the Cross-Ratio
	4 Carnot, the Complete Quadrilateral, and Harmonic Conjugates
	5 Conic Sections, Pole and Polar, and Harmonic Sets
	6 Exercises—Background for Homology
6 Plane-to-Plane Projection
	1 Preparing for Plane-to-Plane Projection
	2 Projecting a Line to Infinity: Brianchon, Poncelet, and Desargues' Theorem
	3 Hexagon Theorems of Pascal and of Brianchon
	4 Poncelet's Fourth Principle and His Proof of the Pole-PolarProperty
	5 Optional. Pole and Polar by George Salmon
	6 Application 1. Brianchon's Conic Section Problem
	7 Application 2. Poncelet's Application of a Theorem by Carnot
	8 Exercises—Plane-to-Plane Projection
7 Homology as Developed by La Hire and Poncelet
	1 Poncelet's Homology, and Homogeneous Coordinates
	2 Philippe de la Hire's Plani-conique of 1673
	3 Poncelet's Construction of Images Under a Homology
	4 Application 1 of Homology: Inverse Homologues and Composition of Homologies
	5 Application 2 of Homology: An Involution Inducedby a Homology
	6 Application 3 of Homology: Inscribed and Circumscribed Quadrilaterals to a Conic
	7 Application 4 of Homology: A Homology Whose Center Is a Focus of a Conic
	8 Application 5 of Homology: The Gergonne Point
	9 Exercises—Homology
8 Matrices and Homogeneous Coordinates
	1 Homogeneous Coordinates
	2 Collineation Defined by Matrix Multiplication
	3 Matrices and the Affinity Transformation
	4 Matrices and the Collineation Transformation
	5 Exercises—Homogeneous Coordinates
9 Projective Geometry: Steiner and von Staudt
	1 Steiner, 1832, and von Staudt, 1847, and Projective Geometry
	2 The Projective Geometry of Steiner
	3 Application 1 of Projective Geometry: A Line Conic in Steiner, 1832
	4 Application 2 of Projective Geometry: Construction of Conics
	5 Application 3 of Projective Geometry: The Cross-Joins Theorem
	6 Application 4 of Projective Geometry: A Problem from Poncelet 1822 Art. 495
	7 The Projective Geometry of von Staudt
	8 Foundational Issues for von Staudt
	9 Some Important Theorems in von Staudt, 1847
	10 Application 5 of Projective Geometry: Involution, Notation, and von Staudt
	11 Optional: A Taste of Projective Algebraic Geometry
	12 Exercises—Projective Geometry
10 Transformation in German Universities
11 Geometric Inversion
	1 Geometric Inversion
	2 Introduction: Bellavitis and His Paper of 1836
	3 Inversion of Lines and Circles
	4 Application 1 of Inversion: A Construction Problem from Bellavitis, 1836
	5 Application 2 of Inversion: Steiner's Porism
	6 Application 3. Ten Related Construction Problems
	7 Exercises—Inversion
12 Moebius Transformation
	1 Moebius Transformation
	2 Complex Numbers and Geometry
	3 The Moebius Transform Preserves Angle Measure
	4 Cayley's Matrix Form of the Moebius Transformation
	5 Application 1. Points at Infinity with Moebius Transform
	6 Application 2. of the Moebius Transformation: An Example from Carathéodorie
	7 Exercises—Moebius Transformation
13 Topic After 1855: Beltrami-Klein Model
	1 The Beltrami-Klein Model
	2 Exercise—Beltrami-Klein Model
14 Topic After 1855: Isometries and Dilations in FrenchSchoolbooks
	1 Isometries
	2 Finite Fixed Points of an Isometry
	3 The Context in Which Transformations Appeared in School Mathematics
	4 Similarity and Congruence by Geometric Transformations
	5 Exercises—Isometries
A Additional Exercises
B A Matrix Algebra Primer
C Group Properties: Identity and Inverse Matrices
D Proof, Simplified, of Poncelet's Fourth Principle
E Alternative Solution, Due to Poncelet, 1822, of the Problem of Apollonius
F The 1855 Argument of Moebius for Invariance of the Cross-ratio
G Solutions to Selected Exercises
Bibliography
Index