Dover Publications | 2004 | 224 páginas | rar - epub | 6,2 Mb
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This text begins with the simplest geometric manifolds, the Grassmann determinant principle for the plane and the Grassmann principle for space; and more. Also explores affine and projective transformations; higher point transformations; transformations with change of space element; and the theory of the imaginary. Concludes with a systematic discussion of geometry and its foundations
CONTENTS
Introduction
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 transformation 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 Mobius’ 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 (curves, surfaces, point sets)
Difference between analytic and synthetic geometry
Projective geometry and the principle of duality
Plücker’s analytic method and the extension of the principle of duality (line coordinates)
Grassmann’s Ausdehnungslehre; n-dimensional geometry
Scalar and vector fields; rational vector analysis
Part Two: Geometric Transformations
Transformations and their analytic representation
I. Affine 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 determinant)
Fundamental theorem of Pohlke
II. Projective Transformations
Analytic definition; introduction of homogeneous coordinates
Geometric definition: Every collineation 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 conics
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 Transformations
Genus and connectivity of surfaces
Euler’s theorem on polyhedra
IV. Transformations with 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 circle-points and imaginary sphere-circle
Imaginary transformation
Von Staudt’s interpretation of self-conjugate imaginary manifolds by means of 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
I. The Systematic Discussion
1. Survey of the Structure of Geometry
Theory of groups as a geometric principle of classification
Cayley’s fundamental principle: Projective geometry is all geometry
2. Digression on the Invariant Theory of Linear Substitutions
Systematic discussion of invariant theory
Simple examples
3. Application of Invariant Theory to Geometry
Interpretation of invariant theory of n variables in affine geometry of Rn with fixed origin
Interpretation in projective geometry of Rn−1
4. The Systematization of Affine and Metric Geometry Based on Cayley’s Principle
Fitting the fundamental notions of affine geometry into the projective system
Fitting the Grassmann determinant principle into the invariant-theoretic conception of geometry. Concerning tensors
Fitting the fundamental notions of metric geometry into the projective system
Projective treatment of the geometry of the triangle
II. Foundations of Geometry
General statement of the question: Attitude to analytic geometry
Development of pure projective geometry with subsequent addition of metric geometry
1. Development of Plane Geometry with Emphasis upon Motions
Development of affine geometry from translation
Addition of rotation to obtain metric geometry
Final deduction of expressions for distance and angle
Classification of the general notions surface-area and curve-length
2. Another Development of Metric Geometry—the Role of the Parallel Axiom
Distance, angle, congruence, as fundamental notions
Parallel axiom and theory of parallels (non-euclidean geometry
Significance of non-euclidean geometry from standpoint of philosophy
Fitting non-euclidean geometry into the projective system
Modern geometric theory of axioms
3. Euclid’s Elements
Historical place and scientific worth of the Elements
Contents of thirteen books of Euclid
Foundations
Beginning of the first book
Lack of axiom of betweenness in Euclid; possibility of the sophisms
Axiom of Archimedes in Euclid; horn-shaped angles as example of a system of magnitudes excluded by this axiom
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