What did Gauss read in the Appendix?

Judit Abardia, Agustí Reventós, Carlos J. Rodríguez

Research output: Contribution to journalArticleResearchpeer-review

3 Citations (Scopus)

Abstract

In a clear analogy with spherical geometry, Lambert states that in an " imaginary sphere" the sum of the angles of a triangle would be less than π. In this paper we analyze the role played by this imaginary sphere in the development of non-Euclidean geometry, and how it served Gauss as a guide. More precisely, we analyze Gauss's reading of Bolyai's Appendix in 1832, five years after the publication of Disquisitiones generales circa superficies curvas, on the assumption that his investigations into the foundations of geometry were aimed at finding, among the surfaces in space, Lambert's hypothetical imaginary sphere. We also wish to show that the close relation between differential geometry and non-Euclidean geometry is already present in János Bolyai's Appendix, that is, well before its appearance in Beltrami's Saggio. From this point of view, one is able to answer certain natural questions about the history of non-Euclidean geometry; for instance, why Gauss decided not to write further on the subject after reading the Appendix. © 2012 Elsevier Inc.
Original languageEnglish
Pages (from-to)292-323
JournalHistoria Mathematica
Volume39
Issue number3
DOIs
Publication statusPublished - 1 Aug 2012

Keywords

  • Bolyai
  • Guass
  • Imaginary sphere
  • Lambert
  • Non-Euclidean geometry

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