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| Part One: The Simplest Geometric Manifolds I. Line-Segment, Area, Volume, as Relative Magnitudes Definition by means of determinants; interpretation of the sign Simplest applications, especially the cross ratio Area of rectilinear polygons Curvilinear areas Theory of Amsler's polar planimeter Volume of polyhedrons, the law of edges One-sided polyhedrons II. The Grassmann Determinant Principle for the Plane Line-segment (vectors) Application in statics of rigid systems Classification of geometric magnitudes according to their behavior under trans formation of rectangular coordinates Application of the principle of classification to elementary magnitudes III. The Grassmann Principle for Space Line-segment and plane-segment Application to statics of rigid bodies Relation to MSbius' null-system Geometric interpretation of the null-system. Connection with the theory of screws IV. Classification of the Elementary Configurations of Space according to their Behavior under Transformation of Rectangular Coordinates Generalities concerning transformations of rectangular space coordinates Transformation formulas for some elementary magnitudes Couple and free plane magnitude as equivalent manifolds Free line-segment and free plane magnitude ("polar" and "axial" vector) Scalars of first and second kind Outlines of a rational vector algebra Lack of a uniform nomenclature in vector calculus V. Derivative Manifolds Derivatives from points (carves, surfaces, point sets) Difference between analytic and synthetic geometry Projective geometry and the principle of duality Pliicker's analytic method and the extension of the principle of duality (lin coordinates) Grassmann's Ausdehnungsiehre;n-dimensional geometry Scalar and vector fields; rational vector analysis Part Two: Geometric Transformations Transformations and their analytic representation I. AtBne Transformations Analytic definition and fundamental properties Application to theory of ellipsoid Parallel projection from one plane upon another Axonometric mapping of space (affine transformation with vanishing deter- minant) Fundamental theorem of Poblke II. Projective Transformations Analytic definition; introduction of homogeneous coordinates Geometric definition: Every coUineation is a projective transformation Behavior of fundamental manifolds under projective transformation Central projection of space upon a plane (projective transformation with vanishing determinant) Relief perspective Application of projection in deriving properties of comcs III. Higher Point Transformations 1. The Transformation by Reciprocal Radii Peaucellier's method of drawing a line Stereographic projection of the sphere 2. Some More General Map Projections. Mercator's projection Tissot theorems 3. The Most General Reversibly Unique Continuous Point Transformatins Genus and connectivity of surfaces Euler's theorem on polyhedra IV. Transformations wlth Change of Space Element 1. Dualistic Transformations 2. Contact Transformations 3. Some Examples Forms of algebraic order and class curves Application of contact transformations to theory of cog wheels V. Theory of the Imaginary Imaginary cirde-points and imaginary sphere-circle Imaginary transformation Von Staudt s interpretation of self-conjugate imaginary manifolds by means oJ real polar systems Von Staudt's complete interpretation of single imaginary elements Space relations of imaginary points and lines …… Part Three:Systematic Discussion of Geometry and Its Foundations II Foundations of Geometry Index of Names Index of Contents |
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