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My question is based on Dunnington's biography of Gauss: Gauss: titan of science. In it Dunnington mentions that Gauss, in his response to Janos Bolyai's paper, sent him a synthetic proof of the angular deficit theorem in hyperbolic geometry, and in addition recommended him to find the hyperbolic volume of a tetrahedron in this new geometry.

Also in Dunnington biography, he mentions a note from 1841 by Gauss, again on the volume of the tetrahedron, found among the pages of Lobachevsky's paper.

So what is the importance of the problem, and where are the relevant notes in Gauss's Nachlass?

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  • $\begingroup$ At first is this know a famous story between Bolyai and Gauss? Then by the story, it is assumed that Gauss is narcissist and powerful mathematician, he recommended category which wasn't discover well yet, due to avoid same Gauss' research. $\endgroup$ – Takahiro Waki May 12 '17 at 12:44
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On the importance of the volume of a tetrahedron in hyperbolic geometry, see the survey article

John Milnor, Hyperbolic geometry: the first 150 years. Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 1, 9–24.

freely available online. He discusses the volume of the tetrahedron in detail, which shows that it is important. How Gauss was aware of its importance? Well, this is an established (empirical) fact that Gauss somehow anticipated (or maybe influenced, determined?) the development of mathematics for the next 200 years:-)

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  • $\begingroup$ Can you tell where are the relevant notes in Gauss's nachlass? Is there any english translation of those notes? And what is the result that Gauss obtained? $\endgroup$ – user2554 May 6 '17 at 12:39
  • $\begingroup$ No. I don't read German or Latin. I am sure there is no English translation. $\endgroup$ – Alexandre Eremenko May 6 '17 at 12:41
  • $\begingroup$ I think the relevant note in Gauss's nachlass is in page 228 of volume 8 of his work: gdz.sub.uni-goettingen.de/dms/load/img/…. it's a little not note entitled "CUBIRUNG DER TETRAEDER ". Can you tell something about it? $\endgroup$ – user2554 Jun 20 '17 at 12:27

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