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I'm making a historical research on the origins of differential geometry, starting with non-Euclidean geometry introduced by Gauss. Reading different historical accounts, what I don't understand is how far mathematicians were motivated by a merely abstract and theoretical line of research or if the study of non-Euclidean geometry was also motivated by some pragmatic motives. Gauss was busy also doing geodetic surveys, but that was still Euclidean geometry (right...?). As far as I understand it, Gauss did not have in mind any applications of it (like, much later, will happen with Einstein's general relativity). On the other hand, it is clear that he was not just interested in solving the problem of parallel lines, but he (and many others too, like Riemann, Lobachevsky, etc.) were busy in developing an entirely new geometry. Moreover, according to Wikipedia "Gauss also claimed to have discovered the possibility of non-Euclidean geometries but never published it." Today, for us, a manifold in an N-dimensional non-Euclidean space may sound like an established mathematical object with obvious applications in physics, but I feel that at the time of Gauss that must have sounded like an absurdity to some.

So, the question is: What was the real motivation that stood behind the development of DG? Or, rephrasing: At the time of the birth of non-Euclidean differential geometry, what was the general perception about it? Did they perceive it as a fascinating but mere theoretical abstraction, or were they driven by some other theoretical and/or pragmatic necessity?

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    $\begingroup$ Non-Euclidean geometry at the time of inception, and half a century after, was not differential geometry. The first differential geometric model, Beltrami's, appeared in 1868. Gauss's motivations for developing differential geometry of surfaces were quite distinct than his, or Lobachevsky's, or Bolyai's for tackling the puzzle of the parallel postulate that exercised top geometers for centuries. And that was done by axiomatic constructions, not surface models that came out later. $\endgroup$
    – Conifold
    Commented Aug 12, 2022 at 15:06
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    $\begingroup$ On motivations for differential geometry of surfaces see Struik, Outline of a History of Differential Geometry (II):"GAUSS' mathematical investigation of the problem of map projection was written in 1822... it represents a continuation of papers by EULER and LAGRANGE. This work led GAUSS to the general theory of surfaces. A manuscript on this subject dates from 1825... His manuscripts show that he was in possession of the non-Euclidean geometry when writing his paper on surfaces, but he did not hint at it even remotely". $\endgroup$
    – Conifold
    Commented Aug 12, 2022 at 23:50
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    $\begingroup$ @Mark Welcome to HSM SE. Note: You posted a quite similar question on the Math SE site at Non-Euclidean geometry: any practical use at the times of Gauss? about 2 days ago. Although it was suggested there by a couple of members that you post here instead, note that cross-posting is generally frowned upon, especially if it's done fairly quickly (e.g., less than a week or so of the original post). Regardless, to help avoid duplication of efforts, when you do so then please include in each question a link to the other one. $\endgroup$
    – John Omielan
    Commented Aug 13, 2022 at 0:47

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The motivation of discoverers of non-Euclidean geometry (Gauss, Lobachevski and Bolyai) was their attempts to prove the Fifth postulate of Euclid (to deduce it from the other axioms, or to replace by some other "more evident" axiom). The practical concern was the question "what is the true geometry of the physical space".

Since only one geometry (Euclidean) was known at that time, there was a lot of discussion on what it is really based on. People noticed that while other axioms could be convincingly "experimentally verified", this is not so about the Fifth postulate (even Euclid himself seems to understand this).

The discovery of Gauss, Lobachevski and Bolyai was that the Fifth postulate does not follow from the rest (though they had no formal proof of this), and that another geometry, with a negation of this Fifth postulate is possible. And that geometry of the physical space is not a pure logical construction but has to be verified by observation.

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    $\begingroup$ The Columbus of non-Euclidean geometry is Saccheri, who tried to prove the 5th postulate by assuming the postulate did not hold, and thereby derive a contradiction. Columbus died believing that he'd reached India; Saccheri found something that he said was "repugnant to the nature of straight lines", so he died believing that he had proved the 5th postulate. $\endgroup$
    – Simon Crase
    Commented Aug 13, 2022 at 1:18
  • $\begingroup$ @Simon Crase: you are right, Saccheri has to be mentioned. $\endgroup$
    – Alexandre Eremenko
    Commented Aug 13, 2022 at 2:39
  • $\begingroup$ Another point of view is that there was a history of failed attempts to prove the fifth postulate, each of which had the form "since we know THIS does not happen, the fifth postulate is true". Of course THIS, in retrospect, was always some property of hyperbolic geometry. I would guess that Gauss, Lobachevsky, Bolyai, etc., knew what they were looking for from all of the different THIS's. $\endgroup$
    – Lee Mosher
    Commented Nov 12, 2022 at 2:20
  • $\begingroup$ @Lee Mosher: why is this ANOTHER point of view? Is not this the same that I wrote? $\endgroup$
    – Alexandre Eremenko
    Commented Nov 14, 2022 at 4:18

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