Worlds Out of Nothing

A Course in the History of Geometry in the 19th Century

  • Jeremy Gray

Part of the Springer Undergraduate Mathematics Series book series (SUMS)

Table of contents

  1. Front Matter
    Pages I-XXV
  2. Jeremy Gray
    Pages 1-10
  3. Jeremy Gray
    Pages 11-24
  4. Jeremy Gray
    Pages 25-41
  5. Jeremy Gray
    Pages 43-52
  6. Jeremy Gray
    Pages 53-61
  7. Jeremy Gray
    Pages 101-114
  8. Jeremy Gray
    Pages 115-127
  9. Jeremy Gray
    Pages 129-135
  10. Jeremy Gray
    Pages 137-148
  11. Jeremy Gray
    Pages 173-178
  12. Jeremy Gray
    Pages 179-190
  13. Jeremy Gray
    Pages 191-194
  14. Jeremy Gray
    Pages 195-209
  15. Jeremy Gray
    Pages 211-225
  16. Jeremy Gray
    Pages 241-246
  17. Jeremy Gray
    Pages 259-267
  18. Jeremy Gray
    Pages 309-311
  19. Jeremy Gray
    Pages 313-319
  20. Jeremy Gray
    Pages 321-331
  21. Jeremy Gray
    Pages 341-344
  22. Back Matter
    Pages 345-384

About this book


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.


Euclid Geometrie Geometry algebra algebraic curve boundary element method differential geometry duality equation evolution history of mathematics mathematics projective geometry story theorem

Authors and affiliations

  • Jeremy Gray
    • 1
  1. 1.Fac. Mathematics, Walton HallOpen UniversityMilton KeynesUnited Kingdom

Bibliographic information

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