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 (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, as exemplified by Klein’s Erlangen Program, rose to prominence, and looks at Poincaré’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 Hilbert’s Foundations of Geometry; geometry and physics, with a look at some of Einstein’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.
Author(s): Jeremy Gray (auth.)
Series: Springer Undergraduate Mathematics Series
Edition: 1
Publisher: Springer-Verlag London
Year: 2010
Language: English
Pages: 384
Tags: History of Mathematics; Geometry
Front Matter....Pages I-XXV
Mathematics in the French Revolution....Pages 1-10
Poncelet (and Pole and Polar)....Pages 11-24
Theorems in Projective Geometry....Pages 25-41
Poncelet’s Traité ....Pages 43-52
Duality and the Duality Controversy....Pages 53-61
Poncelet, Chasles, and the Early Years of Projective Geometry....Pages 63-78
Euclidean Geometry, the Parallel Postulate, and the Work of Lambert and Legendre....Pages 79-89
Gauss (Schweikart and Taurinus) and Gauss’s Differential Geometry....Pages 91-100
János Bolyai....Pages 101-114
Lobachevskii....Pages 115-127
Publication and Non-Reception up to 1855....Pages 129-135
On Writing the History of Geometry – 1....Pages 137-148
Across the Rhine – Möbius’s Algebraic Version of Projective Geometry....Pages 149-159
Plücker, Hesse, Higher Plane Curves, and the Resolution of the Duality Paradox....Pages 161-171
The Plücker Formulae....Pages 173-178
The Mathematical Theory of Plane Curves....Pages 179-190
Complex Curves....Pages 191-194
Riemann: Geometry and Physics....Pages 195-209
Differential Geometry of Surfaces....Pages 211-225
Beltrami, Klein, and the Acceptance of Non-Euclidean Geometry....Pages 227-240
On Writing the History of Geometry – 2....Pages 241-246
Projective Geometry as the Fundamental Geometry....Pages 247-258
Hilbert and his Grundlagen der Geometrie ....Pages 259-267
The Foundations of Projective Geometry in Italy....Pages 269-279
Henri Poincaré and the Disc Model of non-Euclidean Geometry....Pages 281-297
Is the Geometry of Space Euclidean or Non-Euclidean?....Pages 299-307
Summary: Geometry to 1900....Pages 309-311
What is Geometry? The Formal Side....Pages 313-319
What is Geometry? The Physical Side....Pages 321-331
What is Geometry? Is it True? Why is it Important?....Pages 333-339
On Writing the History of Geometry – 3....Pages 341-344
Back Matter....Pages 345-384
