Felix Klein (1) (1849–1925)

Indeholder "Preface to the First Edition", "Preface to the Third Edition", "Translators' Preface", "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 show more polyhedrons", " II. The Grassman 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 Grassman Principle for Space", " Line-segment and plane-segment", " Application to statics of rigid bodies", " Relation to Möbius' 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 R_n with fixed origin", " Interpretation in projective geometry of R_n-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: Attitute to analytic geometry", " Development of pure projective geometry with subsequent addition of metric geometry", " 1. Development of Plane Geometry with Emphesis 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", " Historic 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 examples of a system of magnitudes excluded by this axiom", "Index of Names", "Index of Contents".

Et af de simple eksempler i bogen er en koblingskurve, nærmere bestemt Peaucellier's metode til at lave en rotation om til en bevægelse på en ret linie.
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Indeholder "Preface to the First Edition", "Preface to the Third Edition", "Preface to the English Edition", "Introduction", "First Part: Arithmetic", " I. Calculating with Natural Numbers", " 1. Introduction of Numbers in the Schools", " 2. The Fundamental Laws of Reckoning", " 3. The Logical Foundations of Operations with Integers", " 4. Practice in Calculating with Integers", " II. The First Extension of the Notion of Number", " 1. Negative Numbers", " 2. Fractions", " 3. Irrational show more Numbers", " III. Concerning Special Properties of Integers", " IV. Complex Numbers", " 1. Ordinary Complex Numbers", " 2. Higher Complex Numbers, especially Quaternions", " 3. Quaternion Multiplication - Rotation and Expansion", " 4. Complex Numbers in School Instruction", " Concerning the Modern Development and the General Structure of Mathematics", "Second Part: Algebra", " I. Real Equations with Real Unknowns", " 1. Equations with one parameter", " 2. Equations with two parameters", " 3. Equations with three parameters λ, μ, ν", " II. Equations in the field of complex quantities", " A. The fundamental theorem of algebra", " B. Equations with a complex parameter", " 1. The 'pure' equation", " 2. The dihedral equation", " 3. The tetrahedral, the octahedral, and the icosahedral equations", " 4. Continuation: Setting up the Normal Equation", " 5. Concerning the Solution of the Normal Equations", " 6. Uniformization of the Normal Irrationalities by Means of Transcendental Equations", " 7. Solution in Terms of Radicals", " 8. Reduction of General Equations to Normal Equations", "Third Part: Analysis", " I. Logarithmic and Exponential Functions", " 1. Systematic Account of Algebraic Analysis", " 2. The Historical Development of the Theory", " 3. The History of Logarithms in the Schools", " 4. The Standpoint of Function Theory", " II. The Goniometric Functions", " 1. Theory of the Goniometric Functions", " 2. Trigonometric Tables", " A. Purely Trigonometric Tables", " B. Logarithmic - Trigonometric Tables", " 3. Applications of Goniometric Functions", " A. Trigonometry, in particular, spherical trigonometry", " B. Theory of small oscillations, especially those of the pendulum", " C. Representation of periodic functions by means of series of goniometric functions (trigonometric series)", " III. Concerning Infinitesimal Calculus Proper", " 1. General Considerations in Infinitesimal Calculus", " 2. Taylors Theorem", " 3. Historical and Pedagogical Considerations", "Supplement", " I. Transcendence of the Numbers e and π", " II. The Theory of Assemblages", " 1. The Power of an Assemblage", " 2. Arrangements of the Elements of an Assemblage", "Index of Names", "Index of Contents".

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Reconstrucción histórica del extraordinario desarrollo que tuvo lugar durante el siglo XIX en la matemática