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Geometry by Its History / by Alexander Ostermann, Gerhard Wanner
Contributor(s): Resource type: Ressourcentyp: Buch (Online)Book (Online)Language: English Series: Undergraduate Texts in Mathematics | SpringerLink BücherPublisher: Berlin, Heidelberg : Springer Berlin Heidelberg, 2012Description: Online-Ressource (XII, 437p. 452 illus., 61 illus. in color, digital)ISBN:- 9783642291630
- 516.009
- 516
- QA440-699
Contents:
Summary: Preface -- Part I: Classical Geometry -- Thales and Pythagoras -- The Elements of Euclid -- Conic Sections -- Further Results on Euclidean Geometry -- Trigonometry -- Part II: Analytic Geometry -- Descartes' Geometry -- Cartesian Coordinates -- To be Constructible, or not to be -- Spatial Geometry and Vector Algebra -- Matrices and Linear Mappings -- Projective Geometry -- Solutions to Exercises -- References -- Figure Source and Copyright -- Index.Summary: In this textbook the authors present first-year geometry roughly in the order in which it was discovered. The first five chapters show how the ancient Greeks established geometry, together with its numerous practical applications, while more recent findings on Euclidian geometry are discussed as well. The following three chapters explain the revolution in geometry due to the progress made in the field of algebra by Descartes, Euler and Gauss. Spatial geometry, vector algebra and matrices are treated in chapters 9 and 10. The last chapter offers an introduction to projective geometry, which emerged in the 19th century. Complemented by numerous examples, exercises, figures and pictures, the book offers both motivation and insightful explanations, and provides stimulating and enjoyable reading for students and teachers alike.PPN: PPN: 1651473293Package identifier: Produktsigel: ZDB-2-SEB | ZDB-2-SMA | ZDB-2-SXMS
""Geometry by Its History""; ""Preface""; ""Contents""; ""Part I Classical Geometry""; ""1 Thales and Pythagoras""; ""1.1 Thales� Theorem""; ""1.2 Similar Figures""; ""1.3 Properties of Angles""; ""1.4 The Regular Pentagon""; ""1.5 The Computation of Areas""; ""1.6 A Remarkable Babylonian Document""; ""1.7 The Pythagorean Theorem""; ""1.8 Three Famous Problems of Greek Geometry""; ""1.9 Exercises""; ""2 The Elements of Euclid""; ""2.1 Book I""; ""2.2 Book III. Properties of Circles and Angles""; ""2.3 Books V and VI. Real Numbers and Thales�""; ""Theorem""
""2.4 Books VII and IX. Number Theory""""2.5 Book XI. Spatial Geometry and Solids""; ""2.6 Book XII. Areas and Volumes of Circles, Pyramids,""; ""Cones and Spheres""; ""2.7 Epilogue""; ""2.8 Exercises""; ""3 Conic Sections""; ""3.1 The Parabola""; ""3.2 The Ellipse""; ""3.3 The Hyperbola""; ""3.4 The Area of a Parabola""; ""3.5 Exercises""; ""4 Further Results in Euclidean Geometry""; ""4.1 The Conchoid of Nicomedes, the Trisection of an""; ""Angle""; ""4.2 The Archimedean Spiral""; ""4.3 The Four Classical Centres of the Triangle""; ""4.4 The Theorems of Menelaus and Ceva""
""4.5 The Theorems of Apollonius�Pappus�Stewart""""4.6 The Euler Line and the Nine-Point Circle""; ""4.7 Excircles and the Nagel Point""; ""4.8 Miquel�s Theorems""; ""4.9 Steiner�s Circle Theorems""; ""4.10 Morley�s Theorem""; ""4.11 Exercises""; ""5 Trigonometry""; ""5.1 Ptolemy and the Chord Function""; ""5.2 Regiomontanus and Euler�s Trigonometric Functions""; ""5.3 Arbitrary Triangles""; ""5.4 Trigonometric Solution of Malfatti�s Problem""; ""5.5 The Stereographic Projection""; ""5.6 The Spherical Trigonometry of Right-Angled""; ""Triangles""
""5.7 The Spherical Trigonometry of General Triangles""""5.8 The Area of a Spherical Triangle""; ""5.9 Trigonometric Formulas for the Conics""; ""5.10 The Great Discoveries of Kepler and Newton""; ""5.11 Exercises""; ""Part II Analytic Geometry""; ""6 Descartes� Geometry""; ""6.1 The Principles of Descartes� Geometry""; ""6.2 The Regular Heptagon and Enneagon""; ""6.3 The Trisection of an Angle and Cubic Equations""; ""6.4 Regular Polygons in the Unit Circle""; ""6.5 Van Roomen�s Famous Challenge""; ""6.6 A Geometric Theorem of Fermat""
""6.7 Heronâ€?s Formula for the Area of a Triangle""""6.8 The Eulerâ€?Brahmagupta Formula for a Cyclic""; ""Quadrilateral""; ""6.9 The Cramerâ€?Castillon Problem""; ""6.10 Exercises""; ""7 Cartesian Coordinates""; ""7.1 Equations of Lines and Circles""; ""7.2 The Problem of Pappus""; ""7.3 Conic Sections: Poles, Polars and Tangents""; ""7.4 Problems of Minimum and Maximum""; ""7.5 Some Famous Curves and Their Tangents""; ""7.6 Curvature""; ""7.7 The Euler Line by Euler""; ""7.8 The Simson Line and Sturmâ€?s Circles""; ""7.9 The ErdoË?sâ€?Mordell Inequality and the""
""Steiner�Lehmus Theorem""
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