Worlds Out Of Nothing: A Course In The History Of Geometry In The 19th Century

Synopsis

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 (Pl'ckera's equations) and their role in resolving a paradox in the theory of duality; to Riemanna's work on differential geometry; and to Beltramia'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 Kleina's Erlangen Program, rose to prominence, and looks at Poincar'a'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 Hilberta's Foundations of Geometry; geometry and physics, with a look at some of Einsteina'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. Jeremy Gray is Professor of the History of Mathematics and Director of the Centre for the History of the Mathematical Sciences at the Open University in England, and is an Honorary Professor in the Mathematics Department at the University of Warwick. He is the author, co-author, or editor of 14 books on the h