Worlds Out of Nothing [electronic resource] : A Course in the History of Geometry in the 19th Century / by Jeremy Gray.

Mathematics in the French Revolution -- Poncelet (and Pole and Polar) -- Theorems in Projective Geometry -- Poncelets TraitȨ -- 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 -- Jnos Bolyai -- Lobachevskii -- Publication and Non-Reception up to 1855 -- On Writing the History of Geometry 1 -- Across the Rhine Mȵbiuss Algebraic Version of Projective Geometry -- Plȭcker, Hesse, Higher Plane Curves, and the Resolution of the Duality Paradox -- The Plȭcker 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 PoincarȨ 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.

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? 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 (Plȭckers 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, as exemplified by Kleins Erlangen Program, rose to prominence, and looks at PoincarȨs 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 Hilberts Foundations of Geometry; geometry and physics, with a look at some of Einsteins 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.

ZDB-2-SMA