Worlds Out of Nothing: A Course in the History of Geometry in the 19th Century [Paperback]

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 (Pl?ckers equations) and their role in resolving a paradox in the theory of duality; to Riemanns work on differential geometry; and to Beltramis 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 Poincar?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.

Mathematics in the French Revolution.- Poncelet (and Pole and Polar).- Theorems in Projective Geometry.- Poncelets Trait?.- Duality and the Duality Controversy.- Poncelet, Chasles, and the Early Years of Projective Geometry.- Euclidean Geometry, the Parallel Postulate, and the Work of Lambert and Legendre.- Gauss (Schweikart and Taurinus) and Gausss Differential Geometry.- J?nos Bolyai.- Lobachevskii.- Publication and Non-Reception up to 1855.- On Writing the History of Geometry  1.- Across the Rhine  M?biuss Algebraic Version of Projective Geometry.- Pl?cker, Hesse, Higher Plane Curves, and the Resolution of the Duality Paradox.- The Pl?cker Formulae.- The Mathematical Theory of Plane Curves.- Complex Curves.- Riemann: Geometry and Physics.- Differential Geometry of Surfaces.- Beltrami, Klein, and the Acceptance of Non-Euclidean Geometry.- On Writing the History of Geometry  2.- Projective Geometry as the Fundamental GeometrlÃw