Article ID: CBB001251202

What Did Gauss Read in the Appendix? (2012)

unapi

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 p. 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.

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Authors & Contributors
Bullynck, Maarten
Reichardt, Hans
Braver, Seth Philip
Carter, Jessica
Chen, Huiyong
Christiansen, Andreas
Journals
Archive for History of Exact Sciences
Historia Mathematica
Archives Internationales d'Histoire des Sciences
British Society for the History of Mathematics Bulletin
Historia Scientiarum: International Journal of the History of Science Society of Japan
Isis: International Review Devoted to the History of Science and Its Cultural Influences
Publishers
Teubner
Carocci Editore
Guaraldi
Lang
The MIT Press
University of Montana
Concepts
Mathematics
Non-euclidean geometry
Geometry
Spherical geometry
Solid geometry
Philosophy of mathematics
People
Gauss, Carl Friedrich
Bolyai, Janos (Johann) von
Lambert, Johann Heinrich
Lobachevskii, Nikolai Ivanovich
Menelaos of Alexandria
Poincaré, Jules Henri
Time Periods
19th century
Ancient
18th century
13th century
17th century
20th century
Places
Germany
Greece
Alexandria (Egypt)
Europe
Norway
United States
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