Introduction 

1  (2) 
Part One: The Simplest Geometric Manifolds 


I. LineSegment, Area, Volume, as Relative Magnitudes 


3  (18) 

Definition by means of determinants; interpretation of the sign 


3  (3) 

Simplest applications, especially the cross ratio 


6  (1) 

Area of rectilinear polygons 


7  (3) 


10  (1) 

Theory of Amsler's polar planimeter 


11  (5) 

Volume of polyhedrons, the law of edges 


16  (2) 


18  (3) 

II. The Grassmann Determinant Principle for the Plane 


21  (8) 


22  (1) 

Application in statics of rigid systems 


23  (1) 

Classification of geometric magnitudes according to their behavior under transformation of rectangular coordinates 


24  (2) 

Application of the principle of classification to elementary magnitudes 


26  (3) 

III. The Grassmann Principle for Space 


29  (10) 

Linesegment and planesegment 


30  (1) 

Application to statics of rigid bodies 


31  (2) 

Relation to MÃ¶bius' nullsystem 


33  (2) 

Geometric interpretation of the nullsystem 


35  (2) 

Connection with the theory of screws 


37  (2) 

IV. Classification of the Elementary Configurations of Space according to their Behavior under Transformation of Rectangular Coordinates 


39  (15) 

Generalities concerning transformations of rectangular space coordinates 


39  (3) 

Transformation formulas for some elementary magnitudes 


42  (2) 

Couple and free plane magnitude as equivalent manifolds 


44  (2) 

Free linesegment and free plane magnitude ("polar" and "axial" vector) 


46  (2) 

Scalars of first and second kind 


48  (1) 

Outlines of a rational vector algebra 


48  (3) 

Lack of a uniform nomenclature in vector calculus 


51  (3) 


54  (15) 

Derivatives from points (curves, surfaces, point sets) 


54  (1) 

Difference between analytic and synthetic geometry 


55  (1) 

Projective geometry and the principle of duality 


56  (3) 

Plucker's analytic method and the extension of the principle of duality (line coordinates) 


59  (2) 

Grassmann's Ausdehnungslehre; ndimensional geometry 


61  (2) 

Scalar and vector fields; rational vector analysis 


63  (6) 
Part Two: Geometric Transformations 


Transformations and their analytic representation 


69  

I. Affine Transformations 


70  (16) 

Analytic definition and fundamental properties 


70  (6) 

Application to theory of ellipsoid 


76  (2) 

Parallel projection from one plane upon another 


78  (1) 

Axonometric mapping of space (affine transformation with vanishing determinant) 


79  (4) 

Fundamental theorem of Pohlke 


83  (3) 

II. Projective Transformations 


86  (12) 

Analytic definition; introduction of homogeneous coordinates 


86  (2) 

Geometric definition: Every collineation is a projective transformation 


88  (4) 

Behavior of fundamental manifolds under projective transformation 


92  (2) 

Central projection of space upon a plane (projective transformation with vanishing determinant) 


94  (1) 


95  (1) 

Application of projection in deriving properties of conics 


96  (2) 

III. Higher Point Transformations 


98  (10) 

1. The Transformation by Reciprocal Radii 


98  (4) 

Peaucellier's method of drawing a line 


100  (1) 

Stereographic projection of the sphere 


101  (1) 

2. Some More General Map Projections 


102  (3) 


103  (2) 


105  (1) 

3. The Most General Reversibly Unique Continuous Point Transformations 


105  (3) 

Genus and connectivity of surfaces 


106  (2) 

Euler's theorem on polyhedra 


108  (1) 

IV. Transformations with Change of Space Element 


108  (9) 

1. Dualistic Transformations 


108  (3) 

2. Contact Transformations 


111  (2) 


113  (5) 

Forms of algebraic order and class curves 


113  (2) 

Application of contact transformations to theory of cog wheels 


115  (2) 

V. Theory of the Imaginary 


117  (13) 

Imaginary circlepoints and imaginary spherecircle 


118  (1) 


119  (1) 

Von Staudt's interpretation of selfconjugate imaginary manifolds by means of real polar systems 


120  (3) 

Von Staudt's complete interpretation of single imaginary elements 


123  (4) 

Space relations of imaginary points and lines 


127  (3) 
Part Three: Systematic Discussion of Geometry and Its Foundations 


I. The Systematic Discussion 


130  (29) 

1. Survey of the Structure of Geometry 


130  (5) 

Theory of groups as a geometric principle of classification 


132  (2) 

Cayley's fundamental principle: Projective geometry is all geometry 


134  (1) 

2. Digression on the Invariant Theory of Linear Substitutions 


135  (9) 

Systematic discussion of invariant theory 


136  (4) 


140  (4) 

3. Application of Invariant Theory to Geometry 


144  (4) 

Interpretation of invariant theory of n variables in affine geometry of Rn with fixed origin 


144  (1) 

Interpretation in projective geometry ofRn1 


145  (3) 

4. The Systematization of Affine and Metric Geometry Based on Cayley's Principle 


148  (14) 

Fitting the fundamental notions of affine geometry into the projective system 


149  (1) 

Fitting the Grassmann determinant principle into the invarianttheoretic conception of geometry. Concerning tensors 


150  (6) 

Fitting the fundamental notions of metric geometry into the projective system 


156  (2) 

Projective treatment of the geometry of the triangle 


158  (1) 

II. Foundations of Geometry 


159  (1) 

General statement of the question: Attitude to analytic geometry 


159  (1) 

Development of pure projective geometry with subsequent addition of metricgeometry 


160  (49) 

1. Development of Plane Geometry with Emphasis upon Motions 


162  (12) 

Development of affine geometry from translation 


163  (4) 

Addition of rotation to obtain metric geometry 


167  (5) 

Final deduction of expressions for distance and angle 


172  (1) 

Classification of the general notions surfacearea and curvelength 


173  (1) 

2. Another Development of Metric Geometrythe Role of the Parallel Axiom 


174  (14) 

Distance, angle, congruence, as fundamental notions 


175  (1) 

Parallel axiom and theory of parallels (noneuclidean geometry) 


175  (3) 

Significance of noneuclidean geometry from standpoint of philosophy 


178  (1) 

Fitting noneuclidean geometry into the projective system 


179  (6) 

Modern geometric theory of axioms 


185  (3) 


188  (21) 

Historical place and scientific worth of the Elements 


188  (3) 

Contents of thirteen books of Euclid 


191  (3) 


194  (1) 

Beginning of the first book 


195  (6) 

Lack of axiom of betweenness in Euclid; possibility of the sophisms 


201  (2) 

Axiom of Archimedes in Euclid; hornshaped angles as example of a system of magnitudes excluded by this axiom 


203  (6) 
Index of Names 

209  (2) 
Index of Contents 

211  