Worlds Out of Nothing

1. Front Matter

   Pages i-xxiii

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2. Mathematics in the French Revolution

   Pages 1-10

3. Poncelet (and Pole and Polar)

   Pages 11-24

4. Theorems in Projective Geometry

   Pages 25-41

5. Poncelet’s Traité

   Pages 43-52

6. Duality and the Duality Controversy

   Pages 53-61

7. Poncelet, Chasles, and the Early Years of Projective Geometry

   Pages 63-78

8. Euclidean Geometry, the Parallel Postulate, and the Work of Lambert and Legendre

   Pages 79-88

9. Gauss (Schweikart and Taurinus) and Gauss’s Differential Geometry

   Pages 89-98

10. János Bolyai

    Pages 99-111

11. Lobachevskii

    Pages 113-122

12. Publication and Non-Reception up to 1855

    Pages 123-129

13. On Writing the History of Geometry — 1

    Pages 131-142

14. Across the Rhine — Möbius’s Algebraic Version of Projective Geometry

    Pages 143-153

15. Plücker, Hesse, Higher Plane Curves, and the Resolution of the Duality Paradox

    Pages 155-164

16. The Plücker Formulae

    Pages 165-170

17. The Mathematical Theory of Plane Curves

    Pages 171-182

18. Complex Curves

    Pages 183-186

19. Riemann: Geometry and Physics

    Pages 187-201

20. Differential Geometry of Surfaces

    Pages 203-217

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.

From the reviews:

"Gray’s new book will become both a classic reference and a model on how to write a useful course text. With original source material woven in with historical context, this book is a fun read as it examines geometry historically as a connected sequence of diverse ideas … . it includes a useful index and extensive bibliography. If you enjoy mathematics, buy this and read it! Summing Up: Highly recommended. General readers; lower-division undergraduates through faculty." (J. Johnson, CHOICE, Vol. 44 (11), July, 2007)

"Gray (Open Univ., UK) does it again. Complementing his works on Jinos Bolyai or the world of different geometries, Gray's new book will become both a classic reference and a model on how to write a useful course text. With original source material in with historical context, this book is a fun read as it examines geometry historically as a connected sequence of diverse ideas: duality in projective geometry, the problem of parallels in non-Euclidean geometries, the nature of surfaces in differential geometry, and concern for geometry as a true "measure" of space (physical or philosophical). Documenting mathematicians' contributions within this historical sequence, the book becomes a fascinating Who's Who: Poncelet, Desargues, Pascal, Gergonne, Saccheri, Lambert, Legendre, Gauss, Bolyai, Lobachevski, Möbius, Plücker, Beltrami, Klein, Poincaré, Hilbert, and Einstein. Finally, the book's unusual twist is its inclusion of three chapters on writing and critical reading about the history of mathematics; these chapters alone make the book valuable to those helping students improve as writers of mathematics. An offshoot of a University of Warwick course, it includes a useful index and extensive bibliography. If you enjoy mathematics, but this and read it! Summing up: Highly recommended. General readers; lower-division undergraduates through faculty." - J. Johnson, Western WashingtonUniversity

"This is an outstanding text for the well-prepared student who has a basic knowledge of linear algebra, matrix theory, calculus of several variables, and geometry. … an important addition to the mathematical and historical literature. Albeit primarily a textbook for a one-semester upper level or graduate course in the history of mathematics, it will be appreciated and enjoyed by those interested in geometry, history, philosophy, and pedagogy. … The book contains an excellent bibliography. … it provides a useful reference for anyone interested in geometry." (James J. Tattersall, Mathematical Reviews, Issue 2008 b)

"The 19^th century was certainly an exciting time in geometry. To chronicle all of that excitement in one place is a monumental task; to have done so with real clarity and attention to detail, as Jeremy Gray has done, is an impressive achievement. … Gray has succeeded on several levels: as a historical chronicler, as a mathematical scholar, and as an advisor to teachers. Worlds Out Of Nothing is a first-rate addition to the geometry enthusiast’s bookshelf." (Mark Bollman, MathDL, January, 2008)

• The University of Warwick, UK

  Jeremy Gray

• The Centre for the History of the Mathematical Sciences, The Faculty of Mathematics and Computing, The Open University, Buckinghamshire, UK

  Jeremy Gray

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 history of mathematics in the 19th and 20th Centuries and is internationally recognised as an authority on the subject. His book, Ideas of Space, is a standard text on the history of geometry (see competitive literature).

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