Non-Euclidean Geometry (eBook)

Roberto Bonola (Autor)

eBook Download: EPUB

2012
448 Seiten
Dover Publications (Verlag)
978-0-486-15503-6 (ISBN)

Infos

Examines various attempts to prove Euclid's parallel postulate — by the Greeks, Arabs, and Renaissance mathematicians. It considers forerunners and founders such as Saccheri, Lambert, Legendre, W. Bolyai, Gauss, others. Includes 181 diagrams.

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This is an excellent historical and mathematical view by a renowned Italian geometer of the geometries that have risen from a rejection of Euclid's parallel postulate. Students, teachers and mathematicians will find here a ready reference source and guide to a field that has now become overwhelmingly important.Non-Euclidean Geometry first examines the various attempts to prove Euclid's parallel postulate-by the Greeks, Arabs, and mathematicians of the Renaissance. Then, ranging through the 17th, 18th and 19th centuries, it considers the forerunners and founders of non-Euclidean geometry, such as Saccheri, Lambert, Legendre, W. Bolyai, Gauss, Schweikart, Taurinus, J. Bolyai and Lobachevski. In a discussion of later developments, the author treats the work of Riemann, Helmholtz and Lie; the impossibility of proving Euclid's postulate, and similar topics. The complete text of two of the founding monographs is appended to Bonola's study: "e;The Science of Absolute Space"e; by John Bolyai and "e;Geometrical Researches on the Theory of Parallels"e; by Nicholas Lobachevski. "e;Firmly recommended to any scientific reader with some mathematical inclination"e; — Journal of the Royal Naval Scientific Service. "e;Classic on the subject."e; — Scientific American.

Inhalt

Chapter I. The Attempts to prove Euclid's Parallel Postulate.1-5. The Greek Geometers and the Parallel Postulate6. The Arabs and the Parallel Postulate7-10. The Parallel Postulate during the Renaissance and the 17th CenturyChapter II. The Forerunners on Non-Euclidean Geometry.11-17. GEROLAMO SACCHERI (1667-1733)18-22. JOHANN HEINRICH LAMBERT (1728-1777)23-26. The French Geometers towards the End of the 18th Century27-28. ADRIEN MARIE LEGENDRE (1752-1833)29. WOLFGANG BOLYAI (1775-1856)30. FRIEDRICH LUDWIG WACHTER (1792-1817)30. (bis) BERNHARD FRIEDRICH THIBAUT (1776-1832)Chapter III. The Founders of Non-Euclidean Geometry.31-34. KARL FRIEDRICH GAUSS (1777-1855)35. FERDINAND KARL SCHWEIKART (1780-1859)36-38. FRANZ ADOLF TAURINUS (1794-1874)Chapter IV. The Founders of Non-Euclidean Geometry (Cont.).39-45. NICOLAI IVANOVITSCH LOBATSCHEWSKY (1793-1856)46-55. JOHANN BOLYAI (1802-1860)56-58. The Absolute Trigonometry59. Hypotheses equivalent to Euclid's Postulate60-65. The Spread of Non-Euclidean GeometryChapter V. The Later Development of Non-Euclidean Geometry.66. Introduction Differential Geometry and Non-Euclidean Geometry67-69. Geometry upon a Surface70-76. Principles of Plane Geometry on the Ideas of RIEMANN77. Principles of RIEMANN'S Solid Geometry78. The Work of HELMHOLTZ and the Investigations of LIE Projective Geometry and Non-Euclidean Geometry79-83. Subordination of Metrical Geometry to Projective Geometry84-91. Representation of the Geometry of LOBATSCHEWSKY-BOLYAI on the Euclidean Plane92. Representation of RIEMANN'S Elliptic Geometry in Euclidean Space93. Foundation of Geometry upon Descriptive Properties94. The Impossibility of proving Euclid's PostulateAppendix I. The Fundamental Principles of Statistics and Euclid's Postulate.1-3. On the Principle of the Lever4-8. On the Composition of Forces acting at a Point9-10. Non-Euclidean Statics11-12. Deduction of Plane Trigonometry from StaticsAppendix II. CLIFFORD'S Parallels and Surface. Sketch of CLIFFFORD-KLEIN'S Problems.1-4. CLIFFORD'S Parallels5-8. CLIFFORD'S Surface9-11. Sketch of CLIFFORD-KLEIN'S ProblemAppendix III. The Non-Euclidean Parallel Construction and other Allied Constructions.1-3. The Non-Euclidean Parallel Construction4. Construction of the Common Perpendicular to two non-intersecting Straight Lines5. Construction of the Common Parallel to the Straight Lines which bound an Angle6. Construction of the Straight Line which is perpendicular to one of the lines bounding an acute Angle and Parallel to the other7. The Absolute and the Parallel ConstructionAppendix IV. The Independence of Projective Geometry from Euclid's Postulate.1. Statement of the Problem2. Improper Points and the Complete Projective Plane3. The Complete Projective Line4. Combination of Elements5. Improper Lines6. Complete Projective Space7. Indirect Proof of the Independence of Projective Geometry from the Fifth Postulate8. BELTRAMI'S Direct Proof of this IndependenceAppendix V. The Impossibility of proving Euclid's Postulate. An Elementary Demonstration of this Impossibility founded upon the Properties of the System of Circles orthogonal to a Fixed Circle.1. Introduction2-7. The System of Circles passing through a Fixed Point8-12. The System of Circles orthogonal to a Fixed CircleIndex of AuthorsThe Science of Absolute Space and the Theory of Parallels___________________follow

Produkt-Details

• Erscheint lt. Verlag: 15.8.2012
• Reihe/Serie: Dover Books on Mathematics
• Sprache: englisch
• Maße: 140 x 140 mm
• Gewicht: 454 g
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geschichte der Mathematik
• Schlagworte: 17th century mathematics • 18th century math • 19th century math saccheri • advanced mathematics • euclids parallel postulate • GAUSS • greek attempts to prove euclids postulate • Helmholtz • History of math • j bolyai and lobachevski • Lambert • Legendre • Lie • mathematicians of the renaissance historical mathematics • non euclidean geometries • Riemann • schweikart • science and math • taurinus • the impossibility of proving euclids postulate • w bolyai
• ISBN-10: 0-486-15503-X / 048615503X
• ISBN-13: 978-0-486-15503-6 / 9780486155036