Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century (Springer Undergraduate Mathematics Series Book 0)

Worlds Out of Nothing is the first book to provide a course on the history of geometry in the 19 th century. Based on the latest historical research, the book is aimed primarily at undergraduate and graduate students in mathematics but will also appeal to the reader with a general interest in the history of mathematics. Emphasis is placed on understanding the historical significance of the new mathematics: Why was it done? How - if at all - was it appreciated? What new questions did it generate? 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, as exemplified by Klein’s Erlangen Program, rose to prominence, and looks at Poincaré’s ideas about non-Euclidean geometry and their physical and philosophical significance. It then concludes with discussions on geometry and formalism, examining the Italian contribution and Hilbert’s Foundations of Geometry; geometry and physics, with a look at some of Einstein’s ideas; and geometry and truth. 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. Front Matter....Pages I-XXV Mathematics in the French Revolution....Pages 1-10 Poncelet (and Pole and Polar)....Pages 11-24 Theorems in Projective Geometry....Pages 25-41 Poncelet’s Traité ....Pages 43-52 Duality and the Duality Controversy....Pages 53-61 Poncelet, Chasles, and the Early Years of Projective Geometry....Pages 63-78 Euclidean Geometry, the Parallel Postulate, and the Work of Lambert and Legendre....Pages 79-89 Gauss (Schweikart and Taurinus) and Gauss’s Differential Geometry....Pages 91-100 János Bolyai....Pages 101-114 Lobachevskii....Pages 115-127 Publication and Non-Reception up to 1855....Pages 129-135 On Writing the History of Geometry – 1....Pages 137-148 Across the Rhine – Möbius’s Algebraic Version of Projective Geometry....Pages 149-159 Plücker, Hesse, Higher Plane Curves, and the Resolution of the Duality Paradox....Pages 161-171 The Plücker Formulae....Pages 173-178 The Mathematical Theory of Plane Curves....Pages 179-190 Complex Curves....Pages 191-194 Riemann: Geometry and Physics....Pages 195-209 Differential Geometry of Surfaces....Pages 211-225 Beltrami, Klein, and the Acceptance of Non-Euclidean Geometry....Pages 227-240 On Writing the History of Geometry – 2....Pages 241-246 Projective Geometry as the Fundamental Geometry....Pages 247-258 Hilbert and his Grundlagen der Geometrie ....Pages 259-267 The Foundations of Projective Geometry in Italy....Pages 269-279 Henri Poincaré and the Disc Model of non-Euclidean Geometry....Pages 281-297 Is the Geometry of Space Euclidean or Non-Euclidean?....Pages 299-307 Summary: Geometry to 1900....Pages 309-311 What is Geometry? The Formal Side....Pages 313-319 What is Geometry? The Physical Side....Pages 321-331 What is Geometry? Is it True? Why is it Important?....Pages 333-339 On Writing the History of Geometry – 3....Pages 341-344 Back Matter....Pages 345-384 Based 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. Based 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 include projective geometry, especially the concept of duality, non-Euclidean geometry, and more