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.

Mathematics in the French Revolution.- Poncelet (and Pole and Polar).- Theorems in Projective Geometry.- Poncelet's Traité.- Duality and the Duality Controversy.- Poncelet and Chasles.- Lambert and Legendre.- Gauss.- Janos Bolyai.- Lobachevskii.- To 1855.- Writing.- Möbius.- The Duality Paradox.- The Plücker Formulae.- Higher Plane Curves.- Complex Curves.- Riemann.- Differential Geometry of Surfaces.- Non-Euclidean Geometry Accepted.- Writing.- Fundamental Geometry.- Hilbert.- Italian Foundations.- The Disc Model.- The Geometry of Space.- Summary: Geometry to 1900.- The Formal Side.- The Physical Side.- Is Geometry True?- Writing.- Appendix: Von Staudt and his Influence.- Bibliography.- Index.