| Introduction |
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1 | (2) |
| Part One: The Simplest Geometric Manifolds |
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I. Line-Segment, Area, Volume, as Relative Magnitudes |
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3 | (18) |
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Definition by means of determinants; interpretation of the sign |
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3 | (3) |
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Simplest applications, especially the cross ratio |
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6 | (1) |
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Area of rectilinear polygons |
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7 | (3) |
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10 | (1) |
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Theory of Amsler's polar planimeter |
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11 | (5) |
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Volume of polyhedrons, the law of edges |
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16 | (2) |
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18 | (3) |
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II. The Grassmann Determinant Principle for the Plane |
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21 | (8) |
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22 | (1) |
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Application in statics of rigid systems |
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23 | (1) |
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Classification of geometric magnitudes according to their behavior under transformation of rectangular coordinates |
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24 | (2) |
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Application of the principle of classification to elementary magnitudes |
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26 | (3) |
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III. The Grassmann Principle for Space |
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29 | (10) |
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Line-segment and plane-segment |
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30 | (1) |
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Application to statics of rigid bodies |
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31 | (2) |
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Relation to Möbius' null-system |
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33 | (2) |
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Geometric interpretation of the null-system |
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35 | (2) |
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Connection with the theory of screws |
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37 | (2) |
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IV. Classification of the Elementary Configurations of Space according to their Behavior under Transformation of Rectangular Coordinates |
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39 | (15) |
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Generalities concerning transformations of rectangular space coordinates |
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39 | (3) |
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Transformation formulas for some elementary magnitudes |
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42 | (2) |
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Couple and free plane magnitude as equivalent manifolds |
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44 | (2) |
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Free line-segment and free plane magnitude ("polar" and "axial" vector) |
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46 | (2) |
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Scalars of first and second kind |
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48 | (1) |
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Outlines of a rational vector algebra |
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48 | (3) |
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Lack of a uniform nomenclature in vector calculus |
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51 | (3) |
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54 | (15) |
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Derivatives from points (curves, surfaces, point sets) |
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54 | (1) |
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Difference between analytic and synthetic geometry |
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55 | (1) |
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Projective geometry and the principle of duality |
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56 | (3) |
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Plucker's analytic method and the extension of the principle of duality (line coordinates) |
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59 | (2) |
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Grassmann's Ausdehnungslehre; n-dimensional geometry |
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61 | (2) |
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Scalar and vector fields; rational vector analysis |
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63 | (6) |
| Part Two: Geometric Transformations |
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Transformations and their analytic representation |
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69 | |
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I. Affine Transformations |
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70 | (16) |
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Analytic definition and fundamental properties |
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70 | (6) |
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Application to theory of ellipsoid |
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76 | (2) |
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Parallel projection from one plane upon another |
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78 | (1) |
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Axonometric mapping of space (affine transformation with vanishing determinant) |
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79 | (4) |
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Fundamental theorem of Pohlke |
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83 | (3) |
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II. Projective Transformations |
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86 | (12) |
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Analytic definition; introduction of homogeneous coordinates |
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86 | (2) |
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Geometric definition: Every collineation is a projective transformation |
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88 | (4) |
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Behavior of fundamental manifolds under projective transformation |
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92 | (2) |
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Central projection of space upon a plane (projective transformation with vanishing determinant) |
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94 | (1) |
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95 | (1) |
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Application of projection in deriving properties of conics |
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96 | (2) |
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III. Higher Point Transformations |
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98 | (10) |
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1. The Transformation by Reciprocal Radii |
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98 | (4) |
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Peaucellier's method of drawing a line |
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100 | (1) |
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Stereographic projection of the sphere |
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101 | (1) |
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2. Some More General Map Projections |
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102 | (3) |
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103 | (2) |
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105 | (1) |
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3. The Most General Reversibly Unique Continuous Point Transformations |
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105 | (3) |
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Genus and connectivity of surfaces |
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106 | (2) |
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Euler's theorem on polyhedra |
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108 | (1) |
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IV. Transformations with Change of Space Element |
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108 | (9) |
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1. Dualistic Transformations |
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108 | (3) |
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2. Contact Transformations |
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111 | (2) |
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113 | (5) |
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Forms of algebraic order and class curves |
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113 | (2) |
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Application of contact transformations to theory of cog wheels |
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115 | (2) |
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V. Theory of the Imaginary |
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117 | (13) |
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Imaginary circle-points and imaginary sphere-circle |
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118 | (1) |
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119 | (1) |
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Von Staudt's interpretation of self-conjugate imaginary manifolds by means of real polar systems |
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120 | (3) |
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Von Staudt's complete interpretation of single imaginary elements |
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123 | (4) |
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Space relations of imaginary points and lines |
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127 | (3) |
| Part Three: Systematic Discussion of Geometry and Its Foundations |
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I. The Systematic Discussion |
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130 | (29) |
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1. Survey of the Structure of Geometry |
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130 | (5) |
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Theory of groups as a geometric principle of classification |
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132 | (2) |
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Cayley's fundamental principle: Projective geometry is all geometry |
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134 | (1) |
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2. Digression on the Invariant Theory of Linear Substitutions |
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135 | (9) |
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Systematic discussion of invariant theory |
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136 | (4) |
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140 | (4) |
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3. Application of Invariant Theory to Geometry |
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144 | (4) |
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Interpretation of invariant theory of n variables in affine geometry of Rn with fixed origin |
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144 | (1) |
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Interpretation in projective geometry ofRn1 |
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145 | (3) |
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4. The Systematization of Affine and Metric Geometry Based on Cayley's Principle |
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148 | (14) |
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Fitting the fundamental notions of affine geometry into the projective system |
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149 | (1) |
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Fitting the Grassmann determinant principle into the invariant-theoretic conception of geometry. Concerning tensors |
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150 | (6) |
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Fitting the fundamental notions of metric geometry into the projective system |
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156 | (2) |
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Projective treatment of the geometry of the triangle |
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158 | (1) |
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II. Foundations of Geometry |
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159 | (1) |
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General statement of the question: Attitude to analytic geometry |
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159 | (1) |
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Development of pure projective geometry with subsequent addition of metricgeometry |
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160 | (49) |
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1. Development of Plane Geometry with Emphasis upon Motions |
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162 | (12) |
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Development of affine geometry from translation |
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163 | (4) |
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Addition of rotation to obtain metric geometry |
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167 | (5) |
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Final deduction of expressions for distance and angle |
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172 | (1) |
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Classification of the general notions surface-area and curve-length |
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173 | (1) |
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2. Another Development of Metric Geometry-the Role of the Parallel Axiom |
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174 | (14) |
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Distance, angle, congruence, as fundamental notions |
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175 | (1) |
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Parallel axiom and theory of parallels (non-euclidean geometry) |
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175 | (3) |
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Significance of non-euclidean geometry from standpoint of philosophy |
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178 | (1) |
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Fitting non-euclidean geometry into the projective system |
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179 | (6) |
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Modern geometric theory of axioms |
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185 | (3) |
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188 | (21) |
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Historical place and scientific worth of the Elements |
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188 | (3) |
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Contents of thirteen books of Euclid |
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191 | (3) |
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194 | (1) |
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Beginning of the first book |
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195 | (6) |
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Lack of axiom of betweenness in Euclid; possibility of the sophisms |
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201 | (2) |
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Axiom of Archimedes in Euclid; horn-shaped angles as example of a system of magnitudes excluded by this axiom |
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203 | (6) |
| Index of Names |
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209 | (2) |
| Index of Contents |
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211 | |
Other versions by this Author
