Worlds out of nothing: a course in the history of geometry in the 19th Century.
Springer Undergraduate Mathematics Series.
London, UK: Springer.
Worlds out of Nothing is the first book to provide a course on the history of geometry in the 19th Century, and it is based on the latest historical research. Emphasis is placed on understanding the historical significance of the new mathematics: why it was done, how, if at all, it was appreciated, what new questions did it generate? Topics covered in the first part of the course are projective geometry, especially the concept of duality (in the work of Gergonne, Poncelet and Chasles), and non-Euclidean geometry (the work of Gauss, Bolyai and Lobachevskii). The course 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 course considers how projective geometry rose to a central position in geometry (exemplified by Klein’s Erlangen Program) and then looks at Poincaré’s ideas about non-Euclidean geometry and its possible physical and philosophical significance. It ends with a series of discussions about geometry: geometry and formalism (Italian work and Hilbert’s Foundations of Geometry), geometry and physics (a look at some ideas of Einstein’s), and geometry and truth. An Appendix describes how von Staudt gave an independent foundation for projective geometry and how his work it was received.
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