Worlds Out of Nothing: A Course in the History of Geometry in the 19th CenturyBased on the latest historical research, Worlds Out of Nothing is the first book to provide a course on the history of geometry in the 19th century. Topics covered in the first part of the book are projective geometry, especially the concept of duality, and non-Euclidean geometry. The book then moves on to the study of the singular points of algebraic curves (Plücker’s equations) and their role in resolving a paradox in the theory of duality; to Riemann’s work on differential geometry; and to Beltrami’s role in successfully establishing non-Euclidean geometry as a rigorous mathematical subject. The final part of the book considers how projective geometry rose to prominence, and looks at Poincaré’s ideas about non-Euclidean geometry and their physical and philosophical significance. Three chapters are devoted to writing and assessing work in the history of mathematics, with examples of sample questions in the subject, advice on how to write essays, and comments on what instructors should be looking for. |
Contents
Mathematics in the French Revolution | 1 |
Poncelet and Pole and Polar | 11 |
Theorems in Projective Geometry | 25 |
Poncelets Traité | 43 |
Duality and the Duality Controversy | 53 |
Poncelet Chasles and the Early Years of Projective Geometry | 62 |
Euclidean Geometry the Parallel Postulate and the Work of Lambert and Legendre | 79 |
Gauss Schweikart and Taurinus and Gausss Differential Geometry | 91 |
Differential Geometry of Surfaces | 210 |
Beltrami Klein and the Acceptance of NonEuclidean Geometry | 227 |
On Writing the History of Geometry 2 | 241 |
Projective Geometry as the Fundamental Geometry | 247 |
Hilbert and his Grundlagen der Geometrie | 259 |
The Foundations of Projective Geometry in Italy | 268 |
Henri Poincaré and the Disc Model of nonEuclidean Geometry | 281 |
Is the Geometry of Space Euclidean or NonEuclidean? | 298 |
János Bolyai | 101 |
Lobachevskii | 115 |
Publication and NonReception up to 1855 | 128 |
On Writing the History of Geometry 1 | 137 |
Across the Rhine Möbiuss Algebraic Version of Projective Geometry | 149 |
Plücker Hesse Higher Plane Curves and the Resolution of the Duality Paradox | 160 |
The Plücker Formulae | 173 |
The Mathematical Theory of Plane Curves | 179 |
Complex Curves | 191 |
Geometry and Physics | 195 |
Geometry to 1900 | 309 |
What is Geometry? The Formal Side | 312 |
What is Geometry? The Physical Side | 321 |
What is Geometry? Is it True? Why is it Important? | 332 |
On Writing the History of Geometry 3 | 341 |
Von Staudt and his Influence | 345 |
Bibliography | 359 |
| 377 | |
| 379 | |
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Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century Jeremy Gray No preview available - 2010 |