CONTENTS
Page
Introduction 1
Part One: The Simplest Geometric Manifolds
I. Line-Segment, Area, Volume, as Relative Magnitudes 3
Definition by means of determinants; interpretation of the sign 3
Simplest applications, especially the cross ratio 6
Area of rectilinear polygons 7
Curvilinear areas 10
Theory of Amsler's polar planimeter 11
Volume of polyhedrons, the law of edges 16
One-sided polyhedrons 18
II. The Grassmann Determinant Principle for the Plane 21
Line-segment (vectors) 22
Application in statics of rigid systems 23
Classification of geometric magnitudes according to their behavior under trans-
formation of rectangular coordinates 24
Application of the principle of classification to elementary magnitudes ... 26
III. The Grassmann Principle for Space 29
Line-segment and plane-segment 30
Application to statics of rigid bodies 31
Relation to M6bius' null-system 33
Geometric interpretation of the null-system 35
Connection with the theory of screws 37
IV. Classification of the Elementary Configurations of Space according to their
Behavior under Transformation of Rectangular Coordinates 39
Generalities concerning transformations of rectangular space coordinates . . 39
Transformation formulas for some elementary magnitudes 42
Couple and free plane magnitude as equivalent manifolds 44
Free line-segment and free plane magnitude ("polar" and "axial" vector). . 46
Scalars of first and second kind . 48
Outlines of a rational vector algebra 48
Lack of a uniform nomenclature in vector calculus 51
V. Derivative Manifolds 54
Derivatives from points (curves, surfaces, point sets) 54
Difference between analytic and synthetic geometry 55
Projective geometry and the principle of duality 56
Plttcker's analytic method and the extension of the principle of duality (line
coordinates) 59
Grassmann's Ausdehnungslehre; »-dimensional geometry 61
Scalar and vector fields; rational vector analysis 63
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iii Contents
Part Two: Geometric Transformations
"age
Transformations and their analytic representation 69
I. Affine Transformations 70
Analytic definition and fundamental properties 70
Application to theory of ellipsoid 76
Parallel projection from one plane upon another 78
Axonometric mapping of space (affine transformation with vanishing deter-
minant) , . 79
Fundamental theorem of Pohlke 83
II. Protective Transformations 86
Analytic definition; introduction of homogeneous coordinates 86
Geometric definition: Every collineation is a projective transformation . . . 88
Behavior of fundamental manifolds under projective transformation ... 92
Central projection of space upon a plane (projective transformation with
vanishing determinant) 94
Relief perspective 95
Application of projection in deriving properties of conies 96
III. Higher Point Transformations 98
1. The Transformation by Reciprocal Radii 98
Peaucellier's method of drawing a line 100
Stereographic projection of the sphere 101
2. Some More General Map Projections 102
Mercator's projection 103
Tissot theorems 105
3. The Most General Reversibly Unique Continuous Point Transformations . 105
Genus and connectivity of surfaces 106
Euler's theorem on polyhedra 108
IV. Transformations with Change of Space Element 108
1. Dualistic Transformations 108
2. Contact Transformations Ill
3. Some Examples 113
Forms of algebraic order and class curves 113
Application of contact transformations to theory of cog wheels .... 115
V. Theory of the Imaginary 117
Imaginary circle-points and imaginary sphere-circle 118
Imaginary transformation 119
Von Staudt's interpretation of self-conjugate imaginary manifolds by means of
real polar systems 120
Von Staudt's complete interpretation of single imaginary elements .... 123
Space relations of imaginary points and lines 127
Contents ix
Part Three: Systematic Discussion of Geometry and Its Foundations
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I. The Systematic Discussion 130
1. Survey of the Structure of Geometry 130
Theory of groups as a geometric principle of classification 132
Cayley's fundamental principle: Projective geometry is all geometry . . . 134
2. Digression on the Invariant Theory of Linear Substitutions 135
Systematic discussion of invariant theory 136
Simple examples 140
3. Application of Invariant Theory to Geometry 144
Interpretation of invariant theory of n variables in affine geometry of Rn with
fixed origin 144
Interpretation in projective geometry of Rn-i 145
4. The Systematization of Affine and Metric Geometry Based on Cayley's
Principle 148
Fitting the fundamental notions of affine geometry into the projective system 149
Fitting the Grassmann determinant principle into the invariant-theoretic con-
ception of geometry. Concerning tensors 150
Fitting the fundamental notions of metric geometry into the projective
system 156
Projective treatment of the geometry of the triangle 158
II. Foundations of Geometry 159
General statement of the question: Attitude to analytic geometry 159
Development of pure projective geometry with subsequent addition of metric
geometry 160
1. Development of Plane Geometry with Emphasis upon Motions .... 162
Development of affine geometry from translation 163
Addition of rotation to obtain metric geometry 167
Final deduction of expressions for distance and angle 172
Classification of the general notions surface-area and curve-length . . . 173
2. Another Development of Metric Geometry—the Role of the Parallel Axiom 174
Distance, angle, congruence, as fundamental notions 175
Parallel axiom and theory of parallels (non-euclidean geometry) .... 175
Significance of non-euclideangeometryfrom standpoint of philosophy . . . 178
Fitting non-euclidean geometry into the projective system 179
Modern geometric theory of axioms 185
3. Euclid's Elements 188
Historical place and scientific worth of the Elements 1S8
Contents of thirteen books of Euclid 191
Foundations 194
Beginning of the first book 195
Lack of axiom of betweenness in Euclid; possibility of the sophisms . . . ■ . 201
Axiom of Archimedes in Euclid; horn-shaped angles as example of a system of
magnitudes excluded by this axiom 203
Index of Names 209
Index of Contents 211