As is known, Gauss came to similar conclusions as Lobachevskii in the problem of fifth postulate, but I don't know whether Gauss had any impact to the contemporaries in that story. Can anybody enlighten me? Will it be correct to say that Gauss' opinion had any role in the disputes of XIX century about non euclidean geometry? When were his diaries (with the results on non-euclidean geometry) published? And what exacly was contained there?


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There was not much of an influence. Gauss chose to keep his thoughts out of public view to avoid controversy ("uproar of the Boeotians", as he put it poetically) and only shared them with a few correspondents in private letters, Bolyai senior, Gerling, Olbers, Taurinus and Bessel. English translations of relevant excerpts are collected by Burris in Gauss and Non-Euclidean Geometry. Only brief sketches of hyperbolic geometry appear in Gauss's letters, no systematic developments, and they were published only posthumously and long after Beltrami's Euclidean models of the hyperbolic plane (1868) led to its acceptance. Most in volume 8 of his Werke published by Stäckel in 1900, a summary appeared earlier in Die Theorie der Parallellinien von Euklid bis auf Gauss (1895) edited by Engel and Stäckel, see Zormbala, Gauss and the Definition of the Plane Concept in Euclidean Elementary Geometry.

Gauss's Kantian route towards non-Euclidean geometry, and its philosophical relevance, was also independently arrived at by Helmholtz and others. As Gauss wrote to Taurinus, the idea was that physical geometry is not settled a priori, and might require empirical input:

"It seems to me that in spite of the word-mastery of the metaphysicians, we know really too little, or even nothing at all, about the true nature of space to be able to confuse something that seems unnatural with absolutely impossible. If non-Euclidean geometry is the real one and the constant is comparable to the magnitudes that we encounter on earth or in the heavens then it can be determined a posteriori."

One might expect some Gaussian influence filtering through to Bolyai junior, but then his father explicitly and passionately discouraged him from pursuing the matter. When Schweikart sent Gauss his sketch of hyperbolic geometry all Gauss did was ask Gerling to convey to him his "very best". He did write to Schweikart's nephew Taurinus, which apparently encouraged the latter to pursue the parallel lines further. This might have been Gauss's biggest influence. But he did nothing publicly when Bolyai's and Lobachevsky's work appeared in print. Perhaps, Gauss's acceptance played some positive role after his letters were finally published, perhaps he influenced Riemann's outlook, although Riemann credits Gauss's work on surfaces for inspiration in his famous lecture, not his hyperbolic musings. For a discussion of historical context and Gauss's thought process on hyperbolic geometry see Baker, Gauss, Kästner, and Kant.

Gauss explained his attitude in a 1832 letter to Bolyai senior, after his son already published on hyperbolic geometry on his own:

"If I start by saying “I cannot praise it” then you will most likely be taken aback; but I cannot do otherwise; to praise it would be to praise myself; the entire contents of the work, the path that your son has taken and the results to which it leads, are almost perfectly in agreement with my own meditations, some going back 30 – 35 years. In truth I am astonished. My intention was not to release any of my own work in my lifetime. Most people don’t have a true sense of what is involved, and I have found very few who are particularly interested. To appreciate what is going on one must first of all have a real grasp of what is missing, and on this point most are in the dark. On the other hand it was my intention to write everything down so that it didn’t perish with me. So I am truly surprised that I am now spared this effort, and it is the greatest joy for me that precisely the son of my old friend is the one who preceded me in such a remarkable manner."

  • $\begingroup$ Conifold, thank you very much! Could you, please, specify this: " they were published only posthumously"? When did exactly this happen? And I actually need a reference for this. $\endgroup$ Apr 2, 2021 at 4:06
  • $\begingroup$ I see this article in Wikipedia: en.wikipedia.org/wiki/… . Is it correct to say that Gauss' attitude to non Euclidean geometry was not known before 1903? $\endgroup$ Apr 2, 2021 at 6:04
  • $\begingroup$ @SergeiAkbarov There wasn't much on it in his diary, the letters were publicized a little earlier, see edit. $\endgroup$
    – Conifold
    Apr 2, 2021 at 11:17
  • $\begingroup$ Conifold, thank you very much, that is great! $\endgroup$ Apr 2, 2021 at 15:23
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    $\begingroup$ One minor addition: According to Encyclopedia Britannica about Lobachevsky, "In 1842 his work was noticed and highly praised by Gauss, at whose instigation Lobachevsky was elected that year as a corresponding member of the Royal Society of Göttingen." $\endgroup$ Apr 5, 2021 at 21:02

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