Worlds Out of Nothing is the first book to provide a course on the history of geometry in the 19th 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?

Mathematics in the French Revolution.- Poncelet (and Pole and Polar).- Theorems in Projective Geometry.- Poncelet's Traité.- 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 Gauss's Differential Geometry.- János Bolyai.- Lobachevskii.- Publication and Non-Reception up to 1855.- On Writing the History of Geometry - 1.- Across the Rhine - Möbius's Algebraic Version of Projective Geometry.- Plücker, Hesse, Higher Plane Curves, and the Resolution of the Duality Paradox.- The Plücker 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 Geometry.- Hilbert and his Grundlagen der Geometrie.- The Foundations of Projective Geometry in Italy.- Henri Poincaré and the Disc Model of non-Euclidean Geometry.- Is the Geometry of Space Euclidean or Non-Euclidean?.- Summary: Geometry to 1900.- What is Geometry? The Formal Side.- What is Geometry? The Physical Side.- What is Geometry? Is it True? Why is it Important?.- On Writing the History of Geometry - 3.